Moduli space and deformations of special Lagrangian submanifolds with edge singularities

Abstract

Special Lagrangian submanifolds are submanifolds of a Calabi–Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

1 Introduction

The study of moduli spaces of deformations of a special Lagrangian submanifold in a Calabi–Yau manifold started with the work of McLean [26], where he studied the deformation of compact special Lagrangian submanifolds (without boundary). He proved that the moduli space is a finite dimensional, smooth manifold with dimension equal to the dimension of its space of harmonic 1-forms. The role of special Lagrangian fibrations of Calabi–Yau manifolds in mirror symmetry and especially the presence of singular fibers, motivated the study of special Lagrangian submanifolds with conical singularities/ends [13, 24, 30, 31]. Moreover in the simplest example of a Calabi–Yau manifold, ${\mathbb {C}}^{n}$, the fact that special Lagrangian submanifolds are minimal implies the nonexistence of compact special Lagrangian submanifolds. Hence the search for special Lagrangian submanifolds in ${\mathbb {C}}^{n}$ must be done in the category of non-compact or singular spaces.

Broadly speaking, the study of moduli spaces of special Lagrangian deformations is performed by identifying nearby special Lagrangian submanifolds with elements in the zero set of a non-linear elliptic partial differential operator that governs the deformations. By means of the Implicit Function theorem for Banach spaces, this is reduced to the analysis of the linearised equation. Hence the study of moduli spaces of deformations of a special Lagrangian submanifold requires a good understanding of elliptic equations on the base space.

The theory of linear elliptic partial differential equations on smooth, compact manifolds without boundary is well-developed: the construction of parametrices of inverse order, a complete calculus of elliptic $\Psi DOs$, elliptic regularity, the equivalence between ellipticity and the existence of an a priori estimate, the equivalence of ellipticity and Fredholmness and finally the celebrated Atiyah-Singer index formula. All these elements are already classical tools when studying elliptic equations on compact manifolds.

In contrast, in non-compact or singular spaces there is no canonical approach or methods to study elliptic equations. Even the concept of ellipticity on a non-compact or singular manifold is not canonical as it is in the compact case. The basic model of singularity in the theory of PDEs on singular manifolds is the conical singularity. Near a vertex, a manifold with conical singularities looks like $\frac{\overline{{\mathbb {R}}}^{+}\times {\mathcal {X}}}{ \{0\}\times {\mathcal {X}}}$, where ${\mathcal {X}}$ is a compact manifold without boundary. The usual approach in this case is to blow-up the vertices to obtain a compact manifold with boundary, the stretched manifold, with collar neighborhood $\overline{{\mathbb {R}}}^{+}\times {\mathcal {X}}$.

The most common type of degenerate differential operator studied on the collar neighborhood is the Fuchs type operator

$$\begin{aligned} \text {P}=r^{-m}\sum \limits _{j\le m}a_{j}(r)\left( -r\partial _{r}\right) ^{j} \end{aligned}$$

with coefficients $a_{j }\in C^{\infty } (\overline{{\mathbb {R}}}^{+}, \text {Diff}^{m- j }\left( {\mathcal {X}}\right) )$, where $\text {Diff}^{m- j }\left( {\mathcal {X}}\right)$ is the set of classical differential operators of order $m-j$ on the compact manifold ${\mathcal {X}}$.

Many authors have studied this type of equation with different approaches: Lockhart and McOwen [20, 21], Melrose  [27], Schulze [9, 32, 33], Kozlov, Mazya and Rossmann [16], among possibly others. The b-calculus of Melrose [27] and the cone algebra of Schulze [32] are robust and systematic approaches to Fuchs operators with the goal of constructing a calculus or an algebra of $\Psi DOs$ that contains parametrices of Fusch type operators (see [22] for a comparison of both approaches).

The edgification of a manifold with conical singularities produces a manifold with edge singularities, where locally near the singularity it looks like the ${\mathbb {R}}^{n}\times \left( \frac{\overline{{\mathbb {R}}}^{+}\times {\mathcal {X}}}{ \{0\}\times {\mathcal {X}}}\right)$. Here we also have a class of edge-degenerate differential operators. A typical example is the Laplace–Beltrami operator associated to an edge metric $r^{2}g_{_{{\mathcal {X}}}}+dr^{2}+g_{_{{\mathcal {E}}}}$ where $g_{_{{\mathcal {E}}}}$ is a Riemannian metric on a smooth manifold ${\mathcal {E}}$ (the edge) without boundary (see Sect. 3.1).

This paper is concerned with deformations of special Lagrangian submanifolds with edge singularity (see Sect. 3.1 for the precise definition of a manifold with edge singularity). The motivations to study special Lagrangian submanifolds with edge singularities are the following: on one hand, it is a natural next step in the category of singularities where there is a well-developed elliptic theory [9, 32, 33], hence the analysis of the linearised equation that governs the deformations is accessible. An alternative approach to study edge-degenerate operators developed by R. Mazzeo and B. Vertman can be found in [25, 28].

On the other hand we are interested in the deformation of calibrated vector bundles, specially, special Lagrangian submanifolds obtained as a conormal bundle ${\mathcal {N}}^{*}(M)\subset T^{*}({\mathbb {R}}^{n})\cong {\mathbb {C}}^{n}$ of an austere submanifold M in ${\mathbb {R}}^{n}$. In this direction, Karigiannis and Leung [15] obtained special Lagrangian deformations of ${\mathcal {N}}^{*}(M)$ by affinely translating the fibers. One of the main examples of austere submanifolds in ${\mathbb {R}}^{n}$ is the class of austere cones [6]. These cones are of the form ${\mathbb {R}}^{+}\times {\mathcal {X}}\subset {\mathbb {R}}^{n}$ where ${\mathcal {X}}\subset S^{n-1}$ is an austere submanifold of the sphere. If we assume that the conormal bundle is a trivial bundle near the vertex of the cone (for example if ${\mathbb {R}}^{+}\times {\mathcal {X}}$ is an orientable hypersurface in ${\mathbb {R}}^{n}$ then the conormal bundle is trivial) then it is diffeomorphic to ${\mathbb {R}}^{+}\times {\mathcal {X}}\times {\mathbb {R}}^{q}$ where q is the codimension of ${\mathbb {R}}^{+}\times {\mathcal {X}}$ in ${\mathbb {R}}^{n}$. This implies that we can consider ${\mathcal {N}}^{*}(M)$ as a manifold with an edge singularity. More general, we can consider special Lagrangian submanifolds M with edge singularity ${\mathcal {E}}$ in a Calabi–Yau manifold ${\mathfrak {X}}$. For example if ${\mathfrak {X}}$ is a Calabi–Yau manifold, ${\mathcal {E}}$ a compact special Lagrangian submanifold in ${\mathfrak {X}}$ and M a special Lagrangian submanifold with conical singularity in ${\mathbb {C}}^{n}$ then $M\times {\mathcal {E}}$ is a special Lagrangian submanifold in ${\mathbb {C}}^{n}\times {\mathfrak {X}}$ singular along ${\mathcal {E}}$.

In [9, 32, 33], Schulze and coauthors have developed a comprehensive elliptic theory of edge-degenerate differential operators:

$$\begin{aligned} \text {P}=r^{-m}\sum \limits _{j+\left| \alpha \right| \le m}a_{j\alpha }(r,y)\left( -r\partial _{r}\right) ^{j}\left( rD_{s}\right) ^{\alpha }. \end{aligned}$$

with coefficients $a_{j \alpha }\in C^{\infty } (\overline{{\mathbb {R}}}^{+}\times \Omega , , \text {Diff}^{m- \left( j+\left| \alpha \right| \right) }\left( {\mathcal {X}}\right) ).$

In this paper we use Schulze’s approach to study deformation of special Lagrangian submanifolds with edge singularities. We use Schulze’s approach to analyze the Hodge–Laplace $\Delta$ and Hodge–deRham $d+d^{*}$ operators acting on sections of differential forms induced by edge-degenerate vector fields on M, see Sect. 5.

In previous works of Joyce, McLean, Marshall and Pacini, see [13, 24, 26, 30, 31], finite dimension of the moduli space follows from the Fredholmness of the Hodge–Laplace operator acting on (weighted) Sobolev spaces. As it was mentioned above, there is no canonical notion of ellipticity in non-compact or singular manifolds, however in most approaches, once suitable Banach spaces have been defined, the ellipticity of an operator is defined in such a way that it implies the Fredholm property of the operator acting between those Banach spaces. In manifolds with conical singularities with local model ${\mathbb {R}}^{+}\times {\mathcal {X}}$, the concept of ellipticity is based on the symbolic structure of the Fuchs operator. This is given by two symbols $(\sigma ^{m}_{b}(\text {P}),\sigma ^{m}_{M}(\text {P})(z))$. The first symbol $\sigma ^{m}_{b}(\text {P})$ is the homogeneous boundary principal symbol and $\sigma ^{m}_{M}(\text {P})(z)$ is the Mellin conormal symbol. The symbol $\sigma ^{m}_{M}(\text {P})(z)$ is an operator-valued symbol given by a holomorphic family of continuous operators parametrized by $z\in {\mathbb {C}}$ and acting on the base of the cone ${\mathcal {X}}$, $\sigma ^{m}_{M}(\text {P})(z):H^{s}({\mathcal {X}})\rightarrow H^{s-m}({\mathcal {X}})$. The ellipticity of $\text {P}$ on ${\mathcal {X}}^{\wedge }:={\mathbb {R}}^{+}\times {\mathcal {X}}$ implies that $\sigma ^{m}_{M}(\text {P})(z)$ is a family of isomorphisms for all $z\in \Gamma _{\frac{n+1}{2}-\gamma }=\lbrace z\in {\mathbb {C}}: \text {Re}(z)=\frac{n+1}{2}-\gamma \rbrace$ for some weight $\gamma \in {\mathbb {R}}$. In the approaches of Melrose and Lockhart–McOwen similar symbolic structures are used to define ellipticity. See Sect. 5.1 for a complete discussion of the symbolic structure.

In the analysis of edge-degenerate operators, the symbolic structure for an adequate notion of ellipticity involves the edge symbol $\sigma ^{m}_{\wedge }(u,\eta )$. This is an operator-valued symbol given by a family of continuous operators acting on cone-Sobolev spaces ${\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ (see Definition 3.3) and parametrized by the cotangent bundle of the edge. For $(u,\eta )\in T^{*}{\mathcal {E}}{\setminus }\{0\}$ we have a continuous operator

$$\begin{aligned} \sigma ^{m}_{\wedge }(u,\eta ):{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }) \longrightarrow {\mathcal {K}}^{s-m,\gamma -m}({\mathcal {X}}^{\wedge }). \end{aligned}$$

Analogous to the conical case, a necessary condition for the ellipticity of $\text {P}$ is that $\sigma ^{m}_{\wedge }(u,\eta )$ is an isomorphism for every $(u,\eta )\in T^{*}{\mathcal {E}}{\setminus }\lbrace 0 \rbrace$. However, this is rarely the case (for example, in general, the Laplace–Beltrami operator induced by an edge metric does not satisfy this condition). It is more natural to expect the family $\sigma ^{m}_{\wedge }(u,\eta )$ to be only Fredholm for every $(u,\eta )\in T^{*}{\mathcal {E}}{\setminus }\{ 0\}$.

In this case, in order to have a family of isomorphisms, we need to complete the edge symbol with boundary and coboundary conditions.The need for completing the symbols means that $\text {P}$ is not Fredholm unless we impose complementary edge boundary conditions. Moreover boundary and coboundary conditions are an essential part of the regularity of solutions of elliptic edge-degenerate equations (see Sect. 5.2). Therefore, if we are interested in studying moduli spaces of deformations of a special Lagrangian submanifold with edge singularities, we need to consider deformations with boundary conditions in the edge in order to obtain regular enough deformations that allow the existence of a smooth, finite dimensional moduli space of deformations. These boundary conditions are given by the trace pseudo-differential operator that appears in the completion of the symbol. Moreover, solutions of elliptic equations near singularities have a well-known conormal asymptotic expansion. Our case is not an exception and our deformations have conormal asymptotic expansions near the edge, see Sect. 7.

Once the symbol is completed (this is possible because, in our case, the topological obstruction vanishes, see 5.3) we obtain a Fredholm operator in the edge algebra with a parametrix (with asymptotics) of inverse order. At this point we want to use the Implicit Function theorem for Banach spaces to obtain finite dimensionality and smoothness of the moduli space of deformations. However, the possible non-surjectivity of the linearised deformation map produces an obstruction space. The presence of an obstruction space is not unexpected because even in the case of a compact manifold with isolated conical singularities each of the singular cones contributes to the obstruction space. This was studied in detail by Joyce [13].

Given a special Lagrangian submanifold in ${\mathbb {C}}^{n}$ with edge singularity, $\Phi :M\longrightarrow {\mathbb {C}}^{n}$, our moduli space has as parameters an admissible weight $\gamma >\frac{\text {dim}{{\mathcal {X}}+3}}{2}$ and a trace pseudo-differential operator, ${\mathcal {T}}$, such that it belongs to a set of boundary condition for an elliptic edge boundary value problem for the Hodge–deRham operator on M.

Our main result, Theorem 8.3 in Sect. 8, is a theorem describing the local structure of the moduli space ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ considering the possible obstructions (see Sect. 8 for the precise definition of the moduli space and further details).

Theorem 1.1

Locally near M the moduli space ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ is homeomorphic to the zero set of a smooth map ${\mathfrak {G}}$ between smooth manifolds ${\mathcal {M}}_1$, ${\mathcal {M}}_2$ given as neighborhoods of zero in finite dimensional Banach spaces. The map ${\mathfrak {G}}: {\mathcal {M}}_1 \longrightarrow {\mathcal {M}}_2$ satisfies ${\mathfrak {G}}(0)=0$ and ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ near M is a smooth manifold of finite dimension when ${\mathfrak {G}}$ is the zero map.

2 Special Lagrangian submanifolds and deformations

In this section we introduce the basics of deformation of special Lagrangian submanifolds in ${\mathbb {C}}^{n}$. We will use the definitions and notation of this section throughout this paper. For further details on special Lagrangian geometry the reader is referred to [10, 14].

2.1 Special Lagrangian submanifolds in ${\mathbb {C}}^n$

Let ${\mathbb {C}}^{n}:=\lbrace (z_1,\ldots ,z_n):z_k\in {\mathbb {C}}\text { for all } 1\le k\le n \rbrace$ be the complex n-dimensional space. We identify ${\mathbb {C}}^{n}$ with ${\mathbb {R}}^{2n}={\mathbb {R}}^{n}_x\oplus {\mathbb {R}}^{n}_y$ in the following, specific way

$$\begin{aligned} (x_1,\ldots ,x_n,y_1,\ldots y_n)\longrightarrow (x_1+\sqrt{-1}y_1,\ldots ,x_n+\sqrt{-1}y_n), \end{aligned}$$

(2.1)

hence with this identification $z_k=x_k+\sqrt{-1}y_k$. Now let’s consider the automorphism $J:{\mathbb {C}}^{n}\longrightarrow {\mathbb {C}}^{n}$ given by $J(z)=\sqrt{-1}z$. Then, under the identification ${\mathbb {C}}^{n}\cong {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}$ we have

$$\begin{aligned} J=\begin{bmatrix} 0&\quad -\text {Id}_{{\mathbb {R}}^{n}} \\ \text {Id}_{{\mathbb {R}}^{n}}&\quad 0 \end{bmatrix} :{\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}\longrightarrow {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}. \end{aligned}$$

(2.2)

Definition 2.1

Let $\xi =\xi _1\wedge \cdots \wedge \xi _n$ be an oriented, real n-plane in ${\mathbb {C}}^{n}$, where $\xi _1,\ldots ,\xi _n$ is an oriented, orthonormal basis of $\xi$. We say that $\xi$ is a Lagrangian n-plane if $J(\xi )=\xi ^{\bot }$ where

$$\begin{aligned} \xi ^{\bot }=\lbrace \eta \in {\mathbb {C}}^{n}: \langle \eta ,v \rangle _{g_{{\mathbb {R}}^{2n}}}=0 \text { for all } v\in \xi \rbrace . \end{aligned}$$

(2.3)

Definition 2.2

An oriented n-submanifold $\psi : M\longrightarrow {\mathbb {C}}^{n}$ is a Lagrangian submanifold of ${\mathbb {C}}^{n}$ if each tangent plane $\psi _{*}(T_{p}M)\subset T{\mathbb {C}}^{n}\cong {\mathbb {C}}^{n}$ is a Lagrangian n-plane in ${\mathbb {C}}^{n}$ for every $p\in M$, where $\psi _{*}$ denotes the push-forward.

Definition 2.3

The complex n-form $\Omega =dz_1\wedge \cdots \wedge dz_n$ is called the holomorphic volume form of ${\mathbb {C}}^{n}$.

Definition 2.4

An oriented Langrangian submanifold M in ${\mathbb {C}}^{n}$ is called a special Lagrangian submanifold with phase $\theta _0$ if the following equations are satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Re}(e^{-\sqrt{-1}\theta _{0}}\Omega )\big |_{M}=dV_{M} \\ \text {Im}(e^{-\sqrt{-1}\theta _{0}}\Omega )\big |_{M}=0 \end{array}\right. }. \end{aligned}$$

(2.4)

Observe that given a special Lagrangian submanifold $\Phi : M\longrightarrow {\mathbb {C}}^{n}$ with phase $\theta$ i.e. a Lagrangian submanifold calibrated by $\text {Re}(e^{-\sqrt{-1}\theta _{}}\Omega )$, the submanifold given by

$$\begin{aligned} e^{-\sqrt{-1}\frac{\theta }{n}}\Phi :M\longrightarrow {\mathbb {C}}^{n} \end{aligned}$$

is a special Lagrangian submanifold with phase $\theta =0$.

Therefore by rotating a special Lagrangian submanifold with phase $\theta$ we transform it into a special Lagrangian submanifold with phase zero. Henceforth, when we consider special Lagrangian submanifolds in ${\mathbb {C}}^n$, we shall focus and discuss only the case with phase zero.

Let $({\mathfrak {X}}, \omega , J, g_{_{{\mathfrak {X}}}})$ be a Kähler manifold of complex dimension n with Kähler form $\omega$, complex structure J and Kähler metric $g_{_{{\mathfrak {X}}}}$. Recall that ${\mathfrak {X}}$ is called a Calabi–Yau manifold if the holonomy group of $g_{_{{\mathfrak {X}}}}$ is a subgroup of $\text {SU}(n)$, i.e.

$$\begin{aligned} \text {Hol}(g_{_{{\mathfrak {X}}}})\subseteq \text {SU}(n). \end{aligned}$$

Note that ${\mathbb {C}}^{n}$ is a Calabi–Yau manifold with the structure

$$\begin{aligned} ({\mathbb {C}}^{n},g_{_{{\mathbb {C}}^{n}}},\omega _{_{{\mathbb {C}}^{n}}}, \text {}\Omega ) \end{aligned}$$

where $g_{_{{\mathbb {C}}^n}}= |{dz_1}|^2+\cdots +|{dz_N}|^2$, $\omega _{{\mathbb {C}}^n}=\frac{\sqrt{-1}}{2}\sum \nolimits _{i=1}^{n}dz_i \wedge d\bar{z} _{i}$ and $\Omega =dz_{1}\wedge \cdots \wedge dz_{N}$.

Harvey and Lawson [10] characterized special Lagrangian submanifolds in a way that has been extremely useful to study the deformation theory.

Proposition 2.5

Let $({\mathfrak {X}}, \omega , J, g_{_{{\mathfrak {X}}}},\Omega _{{\mathfrak {X}}})$ be a Calabi–Yau manifold and M a n-dimensional real submanifold. Then M admits an orientation making it into a special Lagrangian submanifold if and only if

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega \big |_{M} \equiv 0\\ \text {Im}\Omega _{{\mathfrak {X}}} \big |_{M} \equiv 0 \end{array}\right. }. \end{aligned}$$

(2.5)

3 Manifolds with singularities

We are interested in the deformation of special Lagrangian submanifolds with singularities. In this section we provide the definitions and concepts related to singular manifolds that we use throughout this paper. We refer the reader to chapter 1 and 2 of [29] for further details.

We denote by

$$\begin{aligned} \text {Diff}_{}(M):=\bigcup \limits _{l\ge 0}\text {Diff}^{l}(M) \end{aligned}$$

the algebra of classical differential operators on M.

Definition 3.1

A singular manifold is a pair $(M, {\mathfrak {D}})$ where M is a smooth manifold possibly non-compact and ${\mathfrak {D}}\subset \text {Diff}_{}(M)$ is a subalgebra of differential operators such that its restriction ${\mathfrak {D}}\big |_{U}$ at every open subset U with compact closure $\overline{U}\subset M$ is equal to the restriction of the algebra of all differential operators $\text {Diff}_{}(M)\big |_{U}.$

The algebra ${\mathfrak {D}}$ is generated by a function space ${\mathcal {F}}$ such that $C^{\infty }_{0}(M)\subset {\mathcal {F}}\subset C^{\infty }(M)$ and a space of vector fields $\mathbf {V}$ on M such that $C^{\infty }_{0}(M,TM)\subset \mathbf {V}\subset C^{\infty }(M, TM).$ We obtain the function space ${\mathcal {F}}$ by embedding M into a compact manifold with boundary $\mathbb {M}$ and defining ${\mathcal {F}}$ as the restriction of the space of smooth functions on $\mathbb {M}$ i.e. ${\mathcal {F}}:=C^{\infty }(\mathbb {M})\big |_{M}.$

3.1 Examples

• (i) Let’s consider a smooth manifold $\mathbb {M}$ with smooth boundary $\partial \mathbb {M}={\mathcal {X}}$, $\text {dim}{\mathcal {X}}=m$ and define $M:=\mathbb {M}{\setminus }\partial \mathbb {M}$. Let $g_{_{M}}$ be a conical metric on M i.e. $g_{_{M}}$ is a Riemannian metric on M such that on a collar neighborhood of $\partial \mathbb {M}$ given by $(0,1)\times {\mathcal {X}}\subset M$ we have

  $$\begin{aligned} g_{_{M}}=r^{2}g_{_{{\mathcal {X}}}}+dr^{2} \end{aligned}$$

  where $g_{_{{\mathcal {X}}}}$ is a Riemannian metric on ${\mathcal {X}}$. Then $g_{_{M}}$ extends to a smooth, symmetric 2-tensor on $\mathbb {M}$ that degenerates in each tangent direction to ${\mathcal {X}}$.

  Now, let ${\mathcal {V}}\in C^{\infty }(\mathbb {M},T\mathbb {M})$ be a vector field with length of the order of unity with respect to $g_{_{M}}$, i.e.

  $$\begin{aligned} \left| {\mathcal {V}}(p)\right| _{g_{_{M}}}\le C \end{aligned}$$

  for any $p\in \mathbb {M}$ and $C>0$ independent of p.

  On a neighborhood $[0,1)\times {\mathcal {U}}\subset [0,1)\times {\mathcal {X}}$ it is easy to see that

  $$\begin{aligned} {\mathcal {V}}={\mathcal {A}}\partial _{r}+\sum \limits ^{m}_{k=1}{\mathcal {B}}_{k}\frac{1}{r}\partial _{k} \end{aligned}$$

  (3.1)

  where $\partial _k$ are the local coordinate vector fields on ${\mathcal {U}}\subset {\mathcal {X}}$ and ${\mathcal {A}},{\mathcal {B}}_k\in C^{\infty }([0,1)\times {\mathcal {U}})$.

  The algebra of degenerate operators ${\mathfrak {D}}$ is generated by functions on M smooth up to $r=0$ i.e. $C^{\infty }(\mathbb {M})\big |_{M}$ and vector fields ${\mathcal {V}}$ such that on the collar neighborhood $[0,1)\times {\mathcal {X}}$ are given by

  $$\begin{aligned} {\mathcal {V}}={\mathcal {A}}\partial _{r}+\Theta \end{aligned}$$

  where ${\mathcal {A}}\in C^{\infty }([0,1)\times {\mathcal {X}})$ and $r\Theta \in C^{\infty }([0,1), T{\mathcal {X}}).$

  This algebra is called the algebra of cone-degenerate operators $\text {Diff}_{\text {cone}}(M).$ From the local expressions above we have that every cone-degenerate operator $\text {P}$ of order l can be written in the collar neighborhood as

  $$\begin{aligned} \text {P}=r^{-l}\sum \limits _{i\le l}a_{i}(r)(-r\partial _{r})^{i} \end{aligned}$$

  (3.2)

  where $a_{i}\in C^{\infty }\left( [0,1),\text {Diff}^{l-i}({\mathcal {X}})\right) ,$ with $\text {Diff}^{l-i}({\mathcal {X}})$ denoting the space of classical differential operators of order $l-i$ on ${\mathcal {X}}$. Cone-degenerate operators are also called Fuchs-type operators.

  We say that $(M,\text {Diff}_{\text {cone}}(M))$ is a manifold with conical singularity.

• (ii) Let $\mathbb {M}$ be a smooth compact manifold with boundary $\partial \mathbb {M}$. Let ${\mathcal {X}}$ and ${\mathcal {E}}$ be smooth, compact manifolds without boundary such that $\partial \mathbb {M}$ is the total space of a smooth ${\mathcal {X}}$-fibration over ${\mathcal {E}}$

  $$\begin{aligned} \pi :\partial \mathbb {M}\longrightarrow {\mathcal {E}}. \end{aligned}$$

  Observe that any collar neighborhood of the boundary $[0,1)\times \partial \mathbb {M}$ has the structure of a ${\mathcal {X}}$-fibration over ${\mathcal {E}}\times [0,1)$. By fixing a collar neighborhood, we use the bundle coordinates on $[0,1)\times \partial \mathbb {M}$ as admissible coordinates i.e. coordinates of the form $(r,\sigma _k,u_l)$ where $(u_l,r)$ are coordinates on ${\mathcal {E}}\times [0,1)$ and $(\sigma _k)$ local coordinates on the fiber ${\mathcal {X}}.$

  Now, we apply the scheme to construct singular manifolds. Let $M=\mathbb {M}{\setminus }\partial \mathbb {M}$ and equip M with an edge metric

  $$\begin{aligned} g_{_{M}}=r^{2}g_{_{{\mathcal {X}}}}+dr^2+g_{_{{\mathcal {E}}}} \end{aligned}$$

  where $g_{_{{\mathcal {E}}}}$ is a smooth Riemannian metric on ${\mathcal {E}}$.

  Observe that the edge metric $g_{_{M}}$ extends to a smooth symmetric 2-tensor on $\mathbb {M}$ that degenerates on each ${\mathcal {X}}$-fiber over $\partial \mathbb {M}.$ In order to define the algebra of degenerate differential operators on M we set ${\mathcal {F}}=C^{\infty }(\mathbb {M})\big |_{M}.$ In admissible coordinates on the collar neighborhood $[0,1)\times {\mathcal {U}}\times \Omega \subset [0,1)\times \partial \mathbb {M}$, the set of vector fields with length of the order of unity with respect to the edge metric $g_{_{M}}$ is given by

  $$\begin{aligned} {V}={\mathcal {A}}\partial _{r}+\sum \limits ^{m}_{k=1} {\mathcal {B}}_k\frac{1}{r}\partial _{k}+\sum \limits ^{q}_{l=1}{\mathcal {C}}_l\partial _{u_l} \end{aligned}$$

  (3.3)

  where ${\mathcal {A}},{\mathcal {B}}_k,{\mathcal {C}}_l \in C^{\infty }([0,1)\times {\mathcal {U}}\times \Omega )$, $\partial _k$ are local coordinate vector fields on ${\mathcal {U}}\subset {\mathcal {X}}$ and $\partial _{u_{l}}$ are local coordinate vector fields on $\Omega \subset {\mathcal {E}}$.

  The algebra generated by ${\mathcal {F}}$ and $\mathbf {V}$ is called the algebra of edge-degenerate operators $\text {Diff}_{\text {edge}}(M).$ In admissible coordinates, every edge-degenerate operator of order l is given by

  $$\begin{aligned} \text {P}=r^{-l}\sum \limits _{i\le l}a_{i,\alpha }(r,u)(-r\partial _{r})^{i}(rD_u)^{\alpha } \end{aligned}$$

  (3.4)

  where $a_{i,\alpha }\in C^{\infty }\left( [0,1)\times \Omega ,\text {Diff}^{l-i-\left| \alpha \right| }({\mathcal {X}})\right)$ and $D_{u_l}=-\sqrt{-1}\partial _{u_l}$.

  We say that $(M,\text {Diff}_{\text {edge}}(M))$ is a manifold with edge singularity.

3.2 Analysis on manifolds with edges

In this section we describe the necessary elements to study partial differential equations on manifolds with conical or edge singularities. In particular we introduce the relevant concepts and definitions needed for the analysis of deformations of singular special Lagrangian submanifolds carried out in Sect. 4. We refer the reader interested in full details and explanations to [33, chapters 2 and 3].

3.3 Sobolev spaces on singular manifolds

We start by introducing suitable Banach spaces on which cone-degenerate operators act. Recall that the Mellin transformation ${\mathcal {M}}$ is a continuous operator ${\mathcal {M}}:C^{\infty }_{0}({\mathbb {R}}^{+})\longrightarrow {\mathcal {A}}({\mathbb {C}})$ given by the integral formula

$$\begin{aligned} ({\mathcal {M}}f)(z)=\displaystyle \int _{0}^{\infty }r^{z-1}f(r)dr, \end{aligned}$$

where ${\mathcal {A}}({\mathbb {C}})$ is the space of holomorphic functions on ${\mathbb {C}}$.

Very often we need the restriction of the holomorphic function ${\mathcal {M}}f$ to subsets isomorphic to ${\mathbb {R}}$ given by

$$\begin{aligned} \Gamma _{\beta }=\lbrace z\in {\mathbb {C}}: Re(z)=\beta \rbrace . \end{aligned}$$

(3.5)

The role of the Mellin transformation in cone-degenerate operators is given by the following basic fact: $(-r\partial _{r})f={\mathcal {M}}^{-1}z{\mathcal {M}}f$ for any $f(r,\sigma )\in C^{\infty }_{0}({\mathbb {R}}^{+}_{r}\times {\mathbb {R}}^{m}_{\sigma })$. Therefore, any cone-degenerate differential operator $\text {P}=r^{-l}\sum \nolimits _{i\le l}a_{i}(r)(-r\partial _{r})^{i}$ is given in terms of the Mellin transformation as follows

$$\begin{aligned} \text {P}=r^{-l}{\mathcal {M}}^{-1}h(r,z){\mathcal {M}}, \end{aligned}$$

where $h(r,z)=\sum \nolimits _{i\le l}a_{i}(r)z^{i}.$

Definition 3.2

The local cone-Sobolev space of order s and weight $\gamma$, denoted by ${\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})$, with $s,\gamma \in {\mathbb {R}}$, is defined as the closure of $C^{\infty }_{0}({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})$ with respect to the norm

$$\begin{aligned}&\left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}\\&\quad :=\left( \frac{1}{2\pi i}\displaystyle \int \limits _{\Gamma _{\frac{m+1}{2}-\gamma }}\displaystyle \int \limits _{{\mathbb {R}}^m}(1+\left| z\right| ^{2}+\left| \xi \right| ^{2})^{s}\left| {\mathcal {M}}_{\gamma -\frac{m}{2},r\rightarrow z}{\mathcal {F}}_{\sigma \rightarrow \xi }f \right| ^{2}dzd\xi \right) ^{\frac{1}{2}}, \end{aligned}$$

where ${\mathcal {F}}_{\sigma \rightarrow \xi }$ denotes the Fourier transformation in the variable $\sigma \in {\mathbb {R}}^{m}$ and the symbol ${\mathcal {M}}_{\gamma -\frac{m}{2},r\rightarrow z}$ denotes the restricted Mellin transformation acting on the variable $r\in {\mathbb {R}}^{+}$.

The local cone-Sobolev spaces are Hilbert spaces with inner product given by

$$\begin{aligned}&\frac{1}{2\pi i}\left\langle (1+\left| z\right| ^{2}+\left| \xi \right| ^{2})^{\frac{s}{2}}{\mathcal {M}}_{\gamma -\frac{m}{2}, r\rightarrow z}{\mathcal {F}}_{\sigma \rightarrow \xi }f,\right. \\&\quad (1+\left| z\right| ^{2}+\left| \xi \right| ^{2})^{\frac{s}{2}}\left. {\mathcal {M}}_{\gamma -\frac{m}{2}, r\rightarrow z}{\mathcal {F}}_{\sigma \rightarrow \xi }g \right\rangle _{L^{2}({\mathbb {R}}\times {\mathbb {R}}^{m})}. \end{aligned}$$

The relation between the local-cone spaces ${\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})$ and the standard Sobolev spaces $H^{s}({\mathbb {R}}^{m+1})$ is given by the following transformation. First consider the transformation

$$\begin{aligned} S_{\gamma -\frac{m}{2}}:C^{\infty }_{0}({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})\longrightarrow C^{\infty }_{0}({\mathbb {R}}_{t}\times {\mathbb {R}}^{m}_x) \end{aligned}$$

(3.6)

such that $S_{\gamma -\frac{m}{2}}(f)(t,x):=e^{-(\frac{1}{2}-(\gamma -\frac{m}{2}))t}f(e^{-t},x).$ This transformation extends to a Banach space isomorphism between ${\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})$ and the standard Sobolev spaces $H^{s}({\mathbb {R}}^{m+1}).$ Therefore the norms $\left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}$ and $\left\| S_{\gamma -\frac{m}{2}}(f)\right\| _{H^{s}({\mathbb {R}}^{m+1})}$ are equivalent.

In order to define the global cone-Sobolev space on a manifold with conical singularities M, we choose a finite open covering $\lbrace {\mathcal {U}}_{_{\lambda }},\chi _{_{\lambda }}\rbrace$ of ${\mathcal {X}}$ given by coordinate neighborhoods such that $\chi _{_{\lambda }}:{\mathcal {U}}_{_{\lambda }}\longrightarrow {\mathbb {R}}^{m}$ and $I\times \chi _{_{\lambda }}:{\mathbb {R}}^{+}\times {\mathcal {U}}_{_{\lambda }}\longrightarrow {\mathbb {R}}^{+}\times {\mathbb {R}}^{m}$ with $(I\times \chi _{_{\lambda }})(r,p)=(r,\chi _{_{\lambda }}(p))$ are diffeomorphisms for every $\lambda$. Let $\lbrace \varphi _{_{\lambda }} \rbrace$ be a partition of unity subordinate to $\lbrace {\mathcal {U}}_{_{\lambda }}\rbrace$. The global cone-Sobolev space near the conical singularities is modelled on the space ${\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ defined on the open cone ${\mathcal {X}}^{\wedge }:={\mathbb {R}}^{+}\times {\mathcal {X}}$ as the closure of $C^{\infty }_{0}({\mathcal {X}}^{\wedge })$ with respect to the norm

$$\begin{aligned} \left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}:=\left( \sum _{\lambda }\left\| (I\times \chi _{_{\lambda }}^{*})^{-1}\varphi _{_{\lambda }}f\right\| ^{2}_{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})} \right) ^{\frac{1}{2}} \end{aligned}$$

(3.7)

where $m=\text {dim}{\mathcal {X}}.$

In order to glue together the space ${\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ with the classical Sobolev space away from the edge we use a cut-off function $\omega (r)\in C^{\infty }_{0}(\overline{{\mathbb {R}}}^{+})$ such that $\omega (r)=1$ for $0\le r <\varepsilon _1$ and $\omega (r)=0$ for $r\ge \varepsilon _2$ for some $0<\varepsilon _1<\varepsilon _2$.

Definition 3.3

Given a compact manifold M with conical singularity, the cone-Sobolev space of order s and weight $\gamma$ is defined as follows

$$\begin{aligned} {\mathcal {H}}^{s,\gamma }(M):=[\omega ] {\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })+[1-\omega ]H^{s}(2\mathbb {M}). \end{aligned}$$

The cone-Sobolev space on the open cone ${\mathcal {X}}^{\wedge }$ is defined in an analogous manner

$$\begin{aligned} {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }):=[\omega ]{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }) +[1-\omega ]H^{s}_{\text {cone}}({\mathcal {X}}^{\wedge }). \end{aligned}$$

Both of these spaces are endowed with the topology of the non-direct sum.

Proposition 3.4

Let $\text {P}\in \text {Diff}^{l}_{\text {cone}}(M)$, then

$$\begin{aligned} \text {P}:{\mathcal {H}}^{s,\gamma }(M)\longrightarrow {\mathcal {H}}^{s-l,\gamma -l}(M) \end{aligned}$$

is a continuous operator for any $s,\gamma$.

See [9, page 153].

The ${\mathbb {R}}^{+}$-action on the cone-Sobolev space is given by $(\kappa _{\lambda }f)(r,\sigma ):=\lambda ^{\frac{m+1}{2}}f(\lambda r,\sigma )$. This defines a continuous one-parameter group of invertible operators with the strong operator topology.

Definition 3.5

Edge-Sobolev spaces.

• (i) We define the edge-Sobolev space on the open edge ${\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q}$ as the the completion of the Schwartz space ${\mathcal {S}}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))$ with respect to the norm

  $$\begin{aligned} \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}= \left( \displaystyle \int [\eta ]^{2s}\left\| \kappa ^{-1}_{[\eta ]}({\mathcal {F}}_{u\rightarrow \eta } f(\eta ))\right\| ^{2}_{{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })} d \eta \right) _{.}^{\frac{1}{2}} \end{aligned}$$

  (3.8)

• (ii) Given a compact manifold M with edge singularity ${\mathcal {E}}$ the edge-Sobolev space ${\mathcal {W}}^{s,\gamma }(M)$ is defined as the closure of $C^{\infty }_{0}(M)$ with respect to the norm

  $$\begin{aligned} \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}=\left( \sum _{j} \left\| \omega \phi _{j}f\right\| ^{2}_{{\mathcal {W}}^{s,\gamma } ({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}+\left\| (1-\omega )f \right\| ^{2}_{H^{s}(2\mathbb {M})}\right) _{.}^{\frac{1}{2}} \end{aligned}$$

  (3.9)

where $\phi _{j}$ is a partition of unity associated to a finite open cover $\lbrace \Omega _{j}\rbrace$ of ${\mathcal {E}}$ and $\omega$ is the cut-off function supported near the edge.

Similarly we can define ${\mathcal {W}}^{s,\gamma }(M, E)$ with E an admissible vector bundle over M (see Definition 4.3).

Proposition 3.6

Let $\text {P}\in \text {Diff}^{l}_{\text {edge}}(M)$, then

$$\begin{aligned} \text {P}:{\mathcal {W}}^{s,\gamma }(M)\longrightarrow {\mathcal {W}}^{s-l,\gamma -l}(M) \end{aligned}$$

is a continuous operator for any $s,\gamma$.

See [32, section 3.1, proposition 5].

4 Deformation of special Lagrangian submanfolds with edges

In this section we consider the problem of deforming special Lagrangian submanfolds with edges. We study and analyze the non-linear differential operator that governs the special Lagrangian deformations and its linearisation.

4.1 Preliminaries

Given a Calabi–Yau manifold $({\mathfrak {X}}, \omega , J, g_{_{{\mathfrak {X}}}},\Omega _{{\mathfrak {X}}})$ and a special Lagrangian submanifold

$$\begin{aligned} \Phi :M\longrightarrow {\mathfrak {X}}, \end{aligned}$$

we are interested in deformations of M, as a submanifold of ${\mathfrak {X}}$, such that the deformed submanifold is special Lagrangian. More precisely we are looking for submanifolds $\Psi :M\longrightarrow {\mathfrak {X}}$ such that $\Phi$ is isotopic to $\Psi$ and $\Psi (M):=M_{\Psi }$ is special Lagrangian. If we are able to find special Lagrangian deformations of M then we want to investigate the structure of the space containing those special Lagrangian deformations i.e. the moduli space of special Lagrangian deformations ${\mathfrak {M}}(M,\Phi )$. In general the moduli space will be the space of special Lagrangian embeddings $\Psi :M\longrightarrow {\mathfrak {X}}$ (equivalent up to diffeomorphism) isotopic to our original $\Phi$. If we require the isotopy through special Lagrangian submanifolds then we are considering only the connected component in the moduli space containing M. If we do not require the intermediate submanifolds to be special Lagrangian then we are considering all the connected components of the moduli space.

If we consider nearby enough submanifolds then it is possible to obtained deformations of M by moving it in a normal direction ${\mathcal {V}}$ given by a section of the normal bundle ${\mathcal {V}}\in C^{\infty }(M,{\mathcal {N}}(M))$. This is possible thanks to the tubular neighborhood theorem (see [18, Ch 4, theorem 5.1]). Observe that the submanifold M is not required to be closed, see [19, theorem 10.19].

Theorem 4.1

Let $({\mathfrak {X}},g_{_{{\mathfrak {X}}}})$ be a Riemannian manifold and M an embedded submanifold. Then there exists an open neighborhood ${\mathfrak {A}}$ of the zero section in ${\mathcal {N}}(M)$ and an open neighborhood U of M in ${\mathfrak {X}}$ such that the exponential map $\text {exp}_{g_{_{{\mathfrak {X}}}}}:{\mathfrak {A}}\subset {\mathcal {N}}(M)\longrightarrow U\subset {\mathfrak {X}}$ is a diffeomorphism.

Therefore any normal section ${\mathcal {V}}$ lying in ${\mathfrak {A}}$ will produce an embedded submanifold given by

$$\begin{aligned} \text {exp}_{g_{_{{\mathfrak {X}}}}}({\mathcal {V}})\circ \Phi :M\longrightarrow {\mathfrak {X}} \end{aligned}$$

such that $(\text {exp}_{g_{_{{\mathfrak {X}}}}}({\mathcal {V}}) \circ \Phi )(M):=M_{{\mathcal {V}}}\subset U$. Once we have the deformed submanifold $M_{{\mathcal {V}}}$ we want to investigate if it is special Lagrangian. Equations (2.5) imply that $M_{{\mathcal {V}}}$ is special Lagrangian if and only if

$$\begin{aligned} {\left\{ \begin{array}{ll} (\text {exp}_{g_{_{{\mathfrak {X}}}}}({\mathcal {V}})\circ \Phi )^{*}(\omega ) \equiv 0 \\ (\text {exp}_{g_{_{{\mathfrak {X}}}}}({\mathcal {V}})\circ \Phi )^{*}( \text {Im}\Omega _{{\mathfrak {X}}}) \equiv 0 \end{array}\right. }. \end{aligned}$$

(4.1)

This provides us with two explicit equations that ${\mathcal {V}}$ must satisfy in order to produce a special Lagrangian deformation of M. Taking advantage of the fact that M is a Lagrangian submanifold we can use the bundle isomorphisms

$$\begin{aligned} T^{*}M \overset{g_{_{{\mathfrak {X}}}}^{-1}}{\cong } TM \overset{J}{\cong } {\mathcal {N}}(M) \end{aligned}$$

so that we obtain that each differential form $\Xi \in {\mathcal {C}}^{\infty }(M,T^{*}M)$ defines a unique section of the normal bundle

$$\begin{aligned} {\mathcal {V}}_{\Xi }=J(g_{_{{\mathfrak {X}}}}^{-1}(\Xi ))\in {\mathcal {C}}^{\infty }(M,{\mathcal {N}}(M)). \end{aligned}$$

Therefore, we can express the deformation problem as a non-linear operator $\text {P}$ acting on differential forms on M:

$$\begin{aligned} \text {P}: {\mathcal {C}}^{\infty }_{}(M,T^{*}M)\big |_{{\mathfrak {A}}}\longrightarrow {\mathcal {C}}^{\infty }\left( M, \bigwedge \nolimits ^{2}T^{*}M \right) \oplus {\mathcal {C}}^{\infty }\left( M, \bigwedge \nolimits ^{n}T^{*}M \right) \end{aligned}$$

(4.2)

given by

$$\begin{aligned} \text {P}(\Xi )=\left( (\text {exp}_{\text {g}_{_{{\mathfrak {X}}}}}({\mathcal {V}}_{\Xi })\circ \Phi )^{*}(\omega ), (\text {exp}_{\text {g}_{_{{\mathfrak {X}}}}}({\mathcal {V}}_{\Xi })\circ \Phi )^{*}(\text {Im}\Omega )\right) , \end{aligned}$$

(4.3)

where ${\mathcal {C}}^{\infty }(M,T^{*}M)\big |_{{\mathfrak {A}}}$ is the space of differential forms $\Xi$ such that their image under the bundle map $J\circ g^{-1}_{{\mathfrak {X}}}$ belongs to ${\mathfrak {A}}$.

The zero set of this operator contains those special Lagrangian deformations of M lying in U, that is,

$$\begin{aligned}&\text {P}^{-1}(0)\\&\quad =\left\{ \Xi \in C^{\infty }(M,T^{*}M)|_{{\mathfrak {A}}}: \text {exp}_{g_{_{{\mathfrak {X}}}}}({\mathcal {V}}_{\Xi })\circ \Phi :M\longrightarrow {\mathfrak {X}} \quad \text {is a S.L. embedding}\right\} . \end{aligned}$$

Hence, the local structure of the moduli space ${\mathfrak {M}}(M,\Phi )$ near M is given by $\text {P}^{-1}(0)$, the zero set of a non-linear operator. A classical result in non-linear functional analysis that has been used to describe the zero set of non-linear operators in deformation of calibrated submanifolds is the Implicit Function Theorem for Banach spaces (see, for example, [17, chapter 14, theorem 2.1]).

Theorem 4.2

Let X and Y be Banach spaces, ${\mathfrak {A}}\subset X$ an open neighborhood of zero and ${\text {P}}:{\mathfrak {A}}\subset X\longrightarrow Y$ a ${\mathcal {C}}^{k}$-map such that

• (i) ${\text {P}}(0)=0$

• (ii) $\text {DP}[0]: X\longrightarrow Y$ is surjective

• (iii) $\text {DP}[0]$ splits X i.e. $X={\text {Ker DP}}[0]\oplus Z$ for some closed subspace Z

then there exist open subsets $W_1\subset {\text {Ker DP}}[0]$, $W_2\subset Z$ and a unique ${\mathcal {C}}^{k}$-map $G:W_1\subset {\text {Ker DP}}[0] \longrightarrow W_2\subset Z$ such that

• (i) $0\in W_1\cap W_2$

• (ii) $W_1\oplus W_2 \subset {\mathcal {U}}$

• (iii) $\text {P}^{-1}(0)\cap (W_1\oplus W_2)=\lbrace (x,G(x)) : x\in W_1\rbrace .$

Ideally, in order to apply Theorem 4.2, we expect to define Banach spaces of differential forms X and Y such that the deformation operator $\text {P}$ acts smoothly, adapt the tubular neighborhood given by Theorem 4.1 such that an open neighborhood of zero ${\mathfrak {A}}\subset X$ fits into it. Moreover we would like that with this choice of Banach spaces the linearisation of the deformation operator at zero $\text {DP}[0]$ is a Fredholm operator. Thus its kernel is a finite dimensional space and it splits X. Moreover if the cokernel of $\text {DP}[0]$ vanishes, then Theorem 4.2 applies immediately and gives us that the moduli space ${\mathfrak {M}}(M,\Phi )$, locally around M, is a finite dimensional, smooth manifold with dimension equal to $\text {dim} \text {Ker}DP[0]$. Moreover any infinitesimal deformation, i.e. $x \in W_1\subset {\text {Ker DP}}[0]$, can be lifted to an authentic deformation given by (x, G(x)) with $\text {P}(x+G(x))=0$ i.e. there are no obstructions.

This ideal situation turned out to be true in the compact case. In [26], McLean studied the deformation of a compact special Lagrangian submanifold inside a Calabi–Yau manifold ${\mathfrak {X}}$. McLean used the classical and well-developed elliptic theory on compact manifolds to analyze the deformation operation and its linearisation.

In order to set up a deformation framework on a singular manifold M with edge singularity, we shall use edge-degenerate differential forms. These forms are dual to the edge-degenerate vector fields in (3.3) Sect. 3.1 with respect to the edge metric $g_{_{M}}=r^{2}g_{_{{\mathcal {X}}}}+dr^2+g_{_{{\mathcal {E}}}}.$ More precisely, let’s consider the following space of differential forms $\gamma$ on the stretched manifold $\mathbb {M}$ such that they vanish on all tangent directions to the fibers on $\partial \mathbb {M}$:

$$\begin{aligned} C^{\infty }(T^{*}_{\wedge } \mathbb {M}):=\lbrace \gamma \in C^{\infty }(T^{*}\mathbb {M}): \gamma \vert _{T {\mathcal {X}}_y}=0\; \forall y\in {\mathcal {E}}\rbrace . \end{aligned}$$

The space $C^{\infty }(T^{*}_{\wedge } \mathbb {M})$ is a locally free $C^{\infty }(\mathbb {M})$-module. By the Swan theorem [35], this is the space of sections of a vector bundle $T^{*}_{\wedge } \mathbb {M}$ over $\mathbb {M}$. The vector bundle $T^{*}_{\wedge } \mathbb {M}$ is called the stretched cotangent bundle of the manifold with edges M (see [29, section 1.3.1]). In local coordinates $(r,\sigma _{k},u_{l})$ we have

$$\begin{aligned} \gamma ={\mathcal {A}}dr+\sum \limits _{k=1}^{m}{\mathcal {B}}_{k}rd\sigma _{k} +{\mathcal {C}}_{l}du_{l} \end{aligned}$$

with ${\mathcal {A}},{\mathcal {B}}_{k},{\mathcal {C}}_{l}$ in $C^{\infty }(\mathbb {M})$. Observe that these are differential forms that degenerate at each direction tangent to ${\mathcal {X}}$. We will denote by $T^{*}_{\wedge }M$ the restriction of the stretched cotangent bundle to M.

At this point we have to make an assumption on the vector bundles we consider on a manifold with edge (or conical) singularity.

Definition 4.3

Let M be a manifold with edge singularity. We say that a vector bundle E over M is admissible if on a collar neighborhood $(0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$ the restriction $E|_{(0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}}$ is the pull-back of a vector bundle $E_{{\mathcal {X}}}$ over ${\mathcal {X}}$.

Now let’s consider the stretched cotangent bundle $T^{*}_{\wedge }M$ as an admissible vector bundle. In order to do that let’s define

$$\begin{aligned} E_{{\mathcal {X}}}:=\wedge ^{0}{\mathcal {X}}\oplus T^{*}{\mathcal {X}}\oplus \underbrace{\wedge ^{0}{\mathcal {X}}\oplus \wedge ^{0}{\mathcal {X}}\oplus ... \oplus \wedge ^{0}{\mathcal {X}}}_\text {q-times} \end{aligned}$$

where $q=\text {dim}{\mathcal {E}}$.

We shall assume that on the collar neighborhood

$$\begin{aligned}{}[0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}\end{aligned}$$

the stretched cotangent bundle $T^{*}_{\wedge }M$ is isomorphic to the pull-back vector bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}E_{{\mathcal {X}}}$ where $\pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}$ is the projection

$$\begin{aligned} \pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}:{\mathcal {X}}^{\wedge }\times {\mathcal {E}}\longrightarrow {\mathcal {X}}\end{aligned}$$

(4.4)

and ${\mathcal {X}}^{\wedge }={\mathbb {R}}^{+}\times {\mathcal {X}}$. We shall also define the bundle $E_{{\mathcal {X}}^{\wedge }}:=\pi ^{*}_{{\mathbb {R}}^{+}}E_{{\mathcal {X}}}$ as the pull-back of the bundle $E_{{\mathcal {X}}}$ by the projection

$$\begin{aligned} \pi ^{}_{{\mathbb {R}}^{+}}:{\mathcal {X}}^{\wedge }\longrightarrow {\mathcal {X}}. \end{aligned}$$

(4.5)

In particular if the edge ${\mathcal {E}}$ is a parallelizable manifold then the stretched cotangent bundle $T^{*}_{\wedge }M$ is an admissible bundle.

Throughout this section we consider ${\mathbb {C}}^{n}$ with its standard Calabi–Yau structure

$$\begin{aligned} ({\mathbb {C}}^{n},g_{_{{\mathbb {C}}^{n}}},\omega _{_{{\mathbb {C}}^{n}}}, \text {}\Omega ) \end{aligned}$$

where $g_{_{{\mathbb {C}}^n}}= |{dz_1}|^2+\cdots +|{dz_n}|^2$, $\omega _{{\mathbb {C}}^n}=\frac{\sqrt{-1}}{2}\sum \nolimits _{i=1}^{n}dz_i \wedge d\bar{z} _{i}$ and $\Omega =dz_{1}\wedge \cdots \wedge dz_{n}$. We consider ${\mathbb {C}}^{n}$ with a fictitious edge structure as follows:

$$\begin{aligned} {\mathbb {C}}^{n}={\mathbb {R}}^{n}\oplus {\mathbb {R}}^{n}\cong \frac{\overline{{\mathbb {R}}}^{+}\times S^{n-1}}{\lbrace 0 \rbrace \times S^{n-1}}\times {\mathbb {R}}^{n}. \end{aligned}$$

Associated with this edge structure we have the stretched space

$$\begin{aligned} {\mathbb {C}}^{n}_{\text {Str}}:=\left( \overline{{\mathbb {R}}}^{+} \times S^{n-1}\right) \times {\mathbb {R}}^{n} \end{aligned}$$

such that

$$\begin{aligned} {\mathbb {C}}^{n}_{\text {Str}}{\setminus }\partial {\mathbb {C}}^{n}_{\text {Str}}=\left( {\mathbb {R}}^{n}{\setminus }\lbrace 0 \rbrace \right) \times {\mathbb {R}}^{n}\cong {\mathbb {C}}^{n}{\setminus }\left( \lbrace 0 \rbrace \times {\mathbb {R}}^{n}\right) . \end{aligned}$$

4.2 Submanifolds with edge singularities in ${\mathbb {C}}^{n}$

Let M be a compact manifold with edge singularity ${\mathcal {E}}$ (see Sect. 3.1). Then the boundary of the stretched manifold $\mathbb {M}$ has a ${\mathcal {X}}$-fibration structure over ${\mathcal {E}}$, $\pi :\partial \mathbb {M}\rightarrow {\mathcal {E}}$, where ${\mathcal {X}}$ and ${\mathcal {E}}$ are compact smooth manifolds (without boundary) with $q=\text {dim}{\mathcal {E}}$ and $m=\text {dim}{\mathcal {X}}$. We assume that ${\mathcal {X}}$ is diffeomorphic to an embedded submanifold of the sphere $S^{n-1}$ with diffeomorphism given by $\theta :{\mathcal {X}}\longrightarrow S^{n-1}\subset {\mathbb {R}}^{n}$. Consider the cone ${\mathcal {X}}^{\wedge }$ with cross section ${\mathcal {X}}$ i.e. ${\mathcal {X}}^{\wedge }={\mathcal {X}}\times {\mathbb {R}}^{+}$ and let’s define a diffeomorphism of ${\mathcal {X}}^{\wedge }$ with a cone ${\mathcal {C}}\subset {\mathbb {R}}^{n}$ by $\psi :{\mathcal {X}}^{\wedge }\longrightarrow {\mathcal {C}}\subset {\mathbb {R}}^{n}$ where $\psi (r,p):=(r\theta _{1}(p),\dots ,r \theta _{n}(p))\in {\mathbb {R}}^{n}.$ We shall also assume that ${\mathcal {E}}$ is embedded in ${\mathbb {R}}^{n}$ by $\tau :{\mathcal {E}}\longrightarrow {\mathbb {R}}^{n}$.

Definition 4.4

Let M be a compact manifold with edge singularity ${\mathcal {E}}$.

• (i) A smooth embedding $\Phi :M\longrightarrow {\mathbb {C}}^{n}$ is called an edge embedding if on a collar neighborhood $(0,\varepsilon )\times \partial \mathbb {M}$, which has the structure of a ${\mathcal {X}}^{\wedge }$-bundle over ${\mathcal {E}}$, the embedding $\Phi$ splits as $\Phi (r,p,v)=(\psi (r,p),\tau (v))$ with respect to the identification ${\mathbb {C}}^{n}\cong {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}$.

• (ii) If $\Phi :M\longrightarrow {\mathbb {C}}^{n}$ is an edge embedding such that $\Phi (M)$ is a special Lagrangian submanifold of ${\mathbb {C}}^{n}$, we say that $(M,\Phi )$ is a special Lagrangian submanifold with edge singularity.

4.3 The deformation operator

Let $\Phi :M\longrightarrow {\mathbb {C}}^{n}$ be a compact special Lagrangian submanifold with edge singularity, see Definition 4.4 in Sect. 4.2. In order to study the moduli space of special Lagrangian deformations of M as a manifold with edge singularities, we have to study small deformations of M inside ${\mathbb {C}}^{n}$. These deformations are produced by sections of the normal bundle $\varphi \in {\mathcal {N}}(M)$ via the exponential map $\text {exp}_{g_{_{{\mathbb {C}}^{n}}}}$. The equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega _{_{{\mathbb {C}}^n}} \big |_{M} \equiv 0 \\ \text {Im}\Omega \big |_{M} \equiv 0 \end{array}\right. } \end{aligned}$$

(4.6)

define a first order non-linear partial differential operator $\text {P}$ such that $\varphi$ must satisfy the equation $\text {P}(\varphi )=0$ in order to produce a special Lagrangian deformation (see (4.3)). Because we are interested in small deformations we can use the Implicit Function Theorem for Banach spaces (if applicable) to describe small solutions of the equation $\text {P}(\varphi )=0$ in terms of solutions of the linearised equation at zero i.e. $\text {DP}[0](\varphi )=0$. In particular, on a collar neighborhood $(0,\varepsilon )\times \partial \mathbb {M}$, equipped with the edge metric $g_{_{M}}=r^{2}g_{_{{\mathcal {X}}}}+dr^{2}+g_{_{{\mathcal {E}}}}$, we want to solve the equation $\text {DP}[0](\varphi )=0$. This is a problem of analysis of PDEs on singular spaces and this observation suggests the approach to follow. First, we expect the operators $\text {P}$ and $\text {DP}[0]$ to be edge-degenerate on $(0,\varepsilon )\times \partial \mathbb {M}$. This is achieved by using sections of the stretched cotangent bundle $T^{*}_{\wedge }M$ to produce small deformations. This is natural as differential forms in $T^{*}_{\wedge }M$ have a degenerate behavior compatible with the edge singularity of M in the sense that their degenerations are induced by the pairing of the edge metric $g_{_{M}}$ with edge-degenerate vector fields. Then, in order to invoke the Implicit Function Theorem for Banach spaces we need that $\text {DP}[0]$ is an elliptic operator in the edge calculus (hence a Fredholm operator by Theorem 5.8 in Sect. 5.2). This is achieved by completing the edge symbol $\sigma _{\wedge }^{1}(\text {DP}[0])$ with boundary and coboundary conditions as the Atiyah-Bott obstruction vanishes, see Sect. 5 below.

4.4 The non-linear deformation operator

Given a compact special Lagrangian submanifold with edge singularity $\Phi : M\longrightarrow {\mathbb {C}}^{n}$, let ${\mathcal {N}}(M)\subset T({\mathbb {C}}^{n})$ be the normal bundle. By using the identification ${\mathbb {C}}^{n}\cong {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}$ we have $T^{*}({\mathbb {C}}^{n})\cong T^{*}({\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n})$. Now, the complex structure J induces an isomorphism of vector bundles $J:T(M)\longrightarrow {\mathcal {N}}(M)$, hence we have a bundle isomorphism $J\circ \Phi _{*}\circ g^{*}_{_{M}}:T^{*}(M)\longrightarrow {\mathcal {N}}(M)\subset T^{}({\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n})|_{M}$ where $g^{*}_{_{M}}$ is the dual metric on the cotangent bundle $T^{*}M$ inducing a bundle map $g_{_{M}}^{*}:T^{*}M\longrightarrow TM$.

Lemma 4.5

Let $\Xi \in C^{\infty }(M,T^{*}M)$ with local expression in an edge neighborhood $(0,\varepsilon )\times {\mathcal {U}}\times \Omega \subset {\mathcal {X}}^{\wedge }\times {\mathcal {E}}$, in local coordinates $(r,\sigma ,u)$, be given by

$$\begin{aligned} \Xi (r,\sigma ,u)={\mathcal {A}}(r,\sigma ,u)dr+\sum \limits _{k=1}^{m}{\mathcal {B}}_k(r,\sigma ,u)d\sigma _k+\sum \limits _{l=1}^{q}{\mathcal {C}}_l(r,\sigma ,u)du_{l}, \end{aligned}$$

then, its image under the map $J\circ \Phi _{*}\circ g^{*}_{M}$ is given by the following expression in the restriction of the tangent bundle $T({\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n})\big |_{(0,\varepsilon )\times {\mathcal {U}}\times \Omega }$

$$\begin{aligned} {\mathcal {V}}_{\Xi }:=J(\Phi _{*}(g^{*}_{M}(\Xi )))=\sum \limits _{i=1}^{n}-\tilde{{\mathcal {C}}}_{i}(r,\sigma ,u)\partial _{x_i}+({\mathcal {A}}(r,\sigma ,u)\theta ^{\circ }_{i}+\frac{1}{r_{}}\tilde{{\mathcal {B}}}_{i}(r,\sigma ,u))\partial _{y_i} \end{aligned}$$

where $\tilde{{\mathcal {B}}}_i$ and $\tilde{{\mathcal {C}}}_l$ are the components of the corresponding pushforwards

$$\begin{aligned}&\theta _{*}\left( g^{*}_{_{{\mathcal {X}}}}\left( \sum \limits _{k=1}^{m}{\mathcal {B}}_k(r,\sigma ,u)d\sigma _k \right) \right) ,\\&\tau _{*}\left( g^{*}_{_{{\mathcal {E}}}}\left( \sum \limits _{l=1}^{q}{\mathcal {C}}_l(r,\sigma ,u)du_l\right) \right) \end{aligned}$$

and $(\theta ^{\circ }_1,\dots \,\theta ^{\circ }_{n})=\theta (\sigma _1,\dots \sigma _{m})$.

Proof

It follows from the expression of the dual edge metric $g^{*}_{_{M}}=\frac{1}{r^2}g^{*}_{_{{\mathcal {X}}}}+\partial _r \otimes \partial _r+g^{*}_{_{{\mathcal {E}}}}$ that

$$\begin{aligned} g^{*}_{_{M}}(\Xi )={\mathcal {A}}\partial _r+\frac{1}{r^2}\sum \limits _{k=1}^{m}{\hat{{\mathcal {B}}_k}} \partial _k+\sum \limits _{l=1}^{q}{\hat{{\mathcal {C}}_l}} \partial _{u_l} \end{aligned}$$

where $\sum \nolimits _{k=1}^{m}{\hat{{\mathcal {B}}_k}} \partial _k=g^{*}_{_{{\mathcal {X}}}}(\sum \nolimits _{k=1}^{m}{\mathcal {B}}_kd\sigma _k)$ and $\sum \nolimits _{l=1}^{q}{\hat{{\mathcal {C}}_l}} \partial _{u_l}=g^{*}_{_{{\mathcal {E}}}}(\sum \nolimits _{l=1}^{q}{\mathcal {C}}_ld u_l).$ Let $p\in (0,\varepsilon )\times {\mathcal {U}}\times \Omega$ and take a curve ${\mathcal {J}}:I\subset {\mathbb {R}}\longrightarrow M$ given by ${\mathcal {J}}(t)=(r(t),\sigma _{}(t),u_{}(t))$, such that ${\mathcal {J}}(0)=p$ and ${\mathcal {J}}'(0)=g^{*}_{_{M}}(\Xi (p))$. Then

$$\begin{aligned} \Phi \circ {\mathcal {J}}:I\subset {\mathbb {R}}\longrightarrow {\mathbb {C}}^{n}\cong {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}\end{aligned}$$

defines a curve given by

$$\begin{aligned} (\Phi \circ {\mathcal {J}})(t)&=\Phi (r(t),\sigma _{}(t),u_{}(t))\\&=(r(t)\theta _{1}(\sigma _{}(t)),\dots ,r(t)\theta _{n}(\sigma _{}(t)), \tau _{1}(u_{}(t)),\dots ,\tau _{n}(u_{}(t))), \end{aligned}$$

therefore

$$\begin{aligned}&(\Phi \circ {\mathcal {J}})'(0)\\&\quad =(r'(0)\theta _{1}(\sigma _{k}(0))+r(0)\theta '_{1}(\sigma _{k}(0)),\dots ,r'(0)\theta _{n}(\sigma _{k}(0))+r(0)\theta '_{n}(\sigma _{k}(0)),\\&\qquad \,\, \tau '_{1}(u_{l}(0)),\dots ,\tau '_{n}(u_{l}(0)))\\&\quad =\left( {\mathcal {A}}\theta ^{\circ }_{1}+\frac{1}{r}\tilde{{\mathcal {B}}}_{1},\dots ,{\mathcal {A}}\theta ^{\circ }_{n}+\frac{1}{r}\tilde{{\mathcal {B}}}_{n},\tilde{{\mathcal {C}}}_{1},\dots ,\tilde{{\mathcal {C}}}_{n}\right) \in {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}\cong T_{p}({\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}). \end{aligned}$$

By applying the standard complex structure J on ${\mathbb {C}}^{n}$ under the identification (2.2) we obtain the result. $\square$

Proposition 4.6

If $\Xi \in {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)$ then ${\mathcal {V}}_{\Xi }$ belongs to ${\mathcal {W}}^{s,\gamma }(M,{\mathcal {N}}(M))$ where ${\mathcal {N}}(M)$ is endowed with the restriction of the standard flat metric $g_{_{{\mathbb {C}}^{n}}}=g_{_{{\mathbb {R}}^{2n}}}$.

Proof

We have to prove that $\left\| {\mathcal {V}}_{\Xi }\right\| _{{\mathcal {W}}^{s,\gamma }(M,{\mathcal {N}}(M))}<\infty$. By (3.9) we have to estimate near the edge with the edge-Sobolev norm

$$\begin{aligned} \left\| \omega {\mathcal {V}}_{\Xi }\right\| _{{\mathcal {W}}^{s,\gamma }(M,{\mathcal {N}}(M))} \end{aligned}$$

and away from the edge with the classical Sobolev norm

$$\begin{aligned} \left\| (1-\omega ){\mathcal {V}}_{\Xi }\right\| _{H^{s}(2\mathbb {M},{\mathcal {N}}(M))}. \end{aligned}$$

Let $\lbrace \Omega _{j},\beta _{j}\rbrace$ and $\lbrace {\mathcal {U}}_{\lambda },\chi _{_{\lambda }} \rbrace$ be finite coverings of ${\mathcal {E}}$ and ${\mathcal {X}}$ respectively, given by coordinate neighborhoods such that

$$\begin{aligned} \beta _{j}:\Omega _{j}\rightarrow {\mathbb {R}}^{q} \end{aligned}$$

and

$$\begin{aligned} I\times \chi _{_{\lambda }}:{\mathbb {R}}^{+}\times {\mathcal {U}}_{\lambda } \longrightarrow {\mathbb {R}}^{+}\times {\mathbb {R}}^{m} \end{aligned}$$

are diffeomorphism and let $\lbrace \phi _{j}\rbrace$ and $\lbrace \varphi _{\lambda } \rbrace$ be corresponding subordinate partitions of unity. Let $\omega (r)$ be the cut-off function defining the edge-Sobolev space (see Definition 3.3). In an edge neighborhood ${\mathbb {R}}^{+}\times {\mathcal {U}}_{\lambda }\times \Omega _j$ we have

$$\begin{aligned} \Xi (r,\sigma ,u)={\mathcal {A}}(r,\sigma ,u)dr+\sum \limits _{k=1}^{m}{\mathcal {B}}_k(r,\sigma ,u)rd\sigma _k+\sum \limits _{l=1}^{q}{\mathcal {C}}_l(r,\sigma ,u)du_l, \end{aligned}$$

and the fact that $\Xi \in {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)$ implies that $\omega \phi _{j}\varphi _{\lambda }{\mathcal {A}}$, $\omega \phi _{j}\varphi _{\lambda }{\mathcal {B}}_k$ and $\omega \phi _{j}\varphi _{\lambda }{\mathcal {C}}_l$ belong to ${\mathcal {W}}^{s,\gamma }(M)$. This follows from (3.7) and (3.9).

By Lemma 4.5 we have

$$\begin{aligned} {\mathcal {V}}_{\Xi }(r,\sigma ,u)=\sum \limits _{i=1}^{n} -\tilde{{\mathcal {C}}}_{i}(r,\sigma ,u)\partial _{x_i}+({\mathcal {A}}(r,\sigma ,u)\theta ^{\circ }_{i}+\tilde{{\mathcal {B}}}_{i}(r,\sigma ,u))\partial _{y_i}. \end{aligned}$$

(4.7)

By 3.8, near the edge we want to estimate the terms

$$\begin{aligned} \left\| \omega \phi _{j}\varphi _{\lambda } \tilde{{\mathcal {B}}_{k}}\right\| ^{2}_{{\mathcal {W}}^{s,\gamma } ({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}=\int \limits _{{\mathbb {R}}^{q}_{\eta }}[\eta ]^{2s} \left\| \kappa ^{-1}_{[\eta ]}{\mathcal {F}}_{u\rightarrow \eta } (\omega \phi _{j}\varphi _{\lambda }\tilde{{\mathcal {B}}_{k}})\right\| ^{2} _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}d\eta \end{aligned}$$

and

$$\begin{aligned} \left\| \omega \phi _{j}\varphi _{\lambda } \tilde{{\mathcal {C}}_{l}}\right\| ^{2}_{{\mathcal {W}}^{s,\gamma } ({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}=\int \limits _{{\mathbb {R}}^{q}_{\eta }}[\eta ]^{2s} \left\| \kappa ^{-1}_{[\eta ]}{\mathcal {F}}_{u\rightarrow \eta }(\omega \phi _{j} \varphi _{\lambda }\tilde{{\mathcal {C}}_{l}})\right\| ^{2}_{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}d\eta . \end{aligned}$$

First, we observe that

$$\begin{aligned}&\left\| \kappa ^{-1}_{[\eta ]}{\mathcal {F}}_{u\rightarrow \eta }(\omega \phi _{j} \varphi _{\lambda }\tilde{{\mathcal {B}}_{k}})\right\| ^{2}_{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })} \end{aligned}$$

(4.8)

$$\begin{aligned}&\quad \approx \sum _{\lambda }\left\| ((I\times \chi _{\lambda })^{*})^{-1}\kappa ^{-1}_{[\eta ]})\varphi _{\lambda } \omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j} \tilde{{\mathcal {B}}}_{k})(\eta )\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}^{2} \end{aligned}$$

(4.9)

by (3.7). By Lemma 4.5 $\tilde{{\mathcal {B}}}_k$ is obtained by applying $\theta _{*}g^{*}_{_{{\mathcal {X}}}}$. This pull-back and push-forward acts locally on the components ${\mathcal {B}}_{k}$ by multiplying by $g^{ij}_{_{{\mathcal {X}}}}$ and partial derivatives of the component functions of $\theta :{\mathcal {X}}\rightarrow S^{n-1}\subset {\mathbb {R}}^{n}$. We claim that both of these operations preserve the membership in ${\mathcal {W}}^{s,\gamma }(M)$. Indeed, in local coordinates ${\mathcal {U}}_{\lambda }$ we have $g^{*}_{_{{\mathcal {X}}}}=\sum \nolimits _{i,j=1}^{m}g^{ij}_{_{{\mathcal {X}}}}\partial _{i}\otimes \partial _{j}$ and

$$\begin{aligned} g^{*}_{_{{\mathcal {X}}}}\left( \sum \limits _{k=1}^{m}{\mathcal {B}}_{k}(r,\sigma ,u)d\sigma _{k}\right) =\sum \limits _{j=1}^{m}\left( \sum \limits _{k=1}^{m} g^{kj}_{_{{\mathcal {X}}}}{\mathcal {B}}_{k}(r,\sigma ,u)\right) \partial _{j}. \end{aligned}$$

The norm

$$\begin{aligned} \left\| ((I\times \chi _{\lambda })^{*})^{-1}\kappa ^{-1}_{[\eta ]} \varphi _{\lambda }\omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j} g^{kj}_{_{{\mathcal {X}}}}{\mathcal {B}}_{k})(\eta )\right\| _{{\mathcal {H}}^{s,\gamma } ({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})} \end{aligned}$$

is equivalent to

$$\begin{aligned} \left\| \varphi _{\lambda } g^{kj}_{_{{\mathcal {X}}}} S_{{\gamma }-\frac{m}{2}} \left( ((I\times \chi _{\lambda })^{*})^{-1} \kappa ^{-1}_{[\eta ]}\omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j} {\mathcal {B}}_{k})(\eta )\right) \right\| _{H^{s}({\mathbb {R}}^{m+1})} \end{aligned}$$

by (3.6). The functions $g^{kj}_{_{{\mathcal {X}}}}$ are bounded on the support of $\varphi _{\lambda }$, hence they are bounded functions on ${\mathbb {R}}^{m+1}$ under the coordinate map $I\times \chi _{\lambda }$. By the general theory of multipliers on Sobolev spaces $H^{s}({\mathbb {R}}^{m+1})$, ([2, theorem 1.9.1 and 1.9.2]), multiplication by any bounded function defines a bounded operator, therefore there exists a constant $C_{kj}$ depending only on $g^{kj}_{{\mathcal {X}}}$ such that

$$\begin{aligned}&\left\| ((I\times \chi _{\lambda })^{*})^{-1}\kappa ^{-1}_{[\eta ]}\varphi _{\lambda }\omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j}g^{kj}_{{\mathcal {X}}}{\mathcal {B}}_{k})(\eta )\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})} \\&\quad \le C_{kj} \left\| ((I\times \chi _{\lambda })^{*})^{-1}\kappa ^{-1}_{[\eta ]}\varphi _{\lambda }\omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j}{\mathcal {B}}_{k})(\eta )\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}. \end{aligned}$$

By hypothesis

$$\begin{aligned} \int \limits _{{\mathbb {R}}^{q}_{\eta }}[\eta ]^{2s}\left\| ((I\times \chi _{\lambda })^{*})^{-1}\kappa ^{-1}_{[\eta ]}\varphi _{\lambda }\omega {\mathcal {F}}_{u\rightarrow \eta }(\phi _{j}{\mathcal {B}}_{k})(\eta )\right\| _{{\mathcal {H}}^{s,\gamma }({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}^{2}d\eta <\infty \end{aligned}$$

then, by (4.8), $g^{*}$ preserves ${\mathcal {W}}^{s,\gamma }(M)$ near the edge.

The push-forward $\theta _{*}$ is induced by a diffeomorphism $\theta :{\mathcal {X}}\longrightarrow S^{n-1}\subset {\mathbb {R}}^{n}$. Locally this push-forward acts on the components of vector fields by multiplications by the derivatives of the component functions $\theta _{k}$, hence the same argument as above applies and we conclude that $\omega \phi _{j}\tilde{{\mathcal {B}}_{k}}\in {\mathcal {W}}^{s,\gamma }(M).$

Now, the components $\tilde{{\mathcal {C}}}_{l}$ are obtained by applying $\tau _{*}g^{*}_{_{{\mathcal {E}}}}$ to the components ${\mathcal {C}}_{l}$. Given $g^{*}_{_{{\mathcal {E}}}}=\sum g^{ij}_{_{{\mathcal {E}}}}(u)\partial _{u_i}\otimes \partial _{u_j}$, it acts on the components ${\mathcal {C}}_{l}$ by multiplication by $g^{ij}_{_{{\mathcal {E}}}}(u)$.

In the same way the push-forward $\tau _{*}$ acts through multiplication by derivatives of its components $\frac{\partial \tau ^{i}}{\partial u_{l}}$. When composed with the coordinate function $\beta _{k}$, the maps

$$\begin{aligned} (\phi _{k}g^{ij}_{_{{\mathcal {E}}}})\circ \beta ^{-1}_{k}:{\mathbb {R}}^{q}\longrightarrow {\mathbb {R}}\end{aligned}$$

and

$$\begin{aligned} \left( \phi _{k}\frac{\partial \tau ^{i}}{\partial u_{l}}\right) \circ \beta ^{-1}_{k}:{\mathbb {R}}^{q}\longrightarrow {\mathbb {R}}\end{aligned}$$

belong to ${\mathcal {S}}({\mathbb {R}}^{q})$ as they have compact support. By [33, theorem 1.3.34], multiplication by an element in ${\mathcal {S}}({\mathbb {R}}^{q})$ defines a continuous operator on ${\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})$. Hence, by the same argument as in the first part of the proof, $\tau _{*}g^{*}_{_{{\mathcal {E}}}}$ preserves ${\mathcal {W}}^{s,\gamma }(M)$ near the edge and $\omega \phi _{k}\varphi _{\lambda }\tilde{{\mathcal {C}}}_{l} \in {\mathcal {W}}^{s,\gamma }(M)$.

Away from the edge on the compact manifold $M{\setminus }\left( (0,\varepsilon )\times \partial \mathbb {M}\right)$ take a finite covering of coordinate neighborhoods $\lbrace W_{i} \rbrace$ with a subordinate partition of unity $\lbrace \mu _{i} \rbrace$. Then

$$\begin{aligned} \left\| (1-\omega ){\mathcal {V}}_{\Xi }\right\| ^{2}_{H^{s}(2\mathbb {M},{\mathcal {N}}(M))} =\sum _{i}\left\| \mu _{i}{\mathcal {V}}_{\Xi }\right\| ^{2}_{H^{s}({\mathbb {R}}^{n},{\mathbb {R}}^{q})}. \end{aligned}$$

(4.10)

As on each of those patches of local coordinates the support of $\mu _i{\mathcal {V}}_{\Xi }$ is compact, clearly $\left\| (1-\omega ){\mathcal {V}}_{\Xi }\right\| ^{2}_{H^{s}(2\mathbb {M},{\mathcal {N}}(M))}<\infty$. $\square$

Proposition 4.7

Let $\Xi \in C^{\infty }_{0}(T^{*}_{\wedge }M)$ and ${\mathcal {V}}_{\Xi }=J(\Phi _{*}( g_{M}^{*}(\Xi )))\in C^{\infty }({\mathcal {N}}(M)).$ Then the pull-back of the standard Kähler form $\omega _{_{{\mathbb {C}}^{n}}}$ by the map $\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi$ is given in terms of edge-degenerate differential operators of order 1 by the following expression in a neighborhood $(0,\varepsilon )\times {\mathcal {U}}_\lambda \times \Omega _j$ near the edge:

$$\begin{aligned}&\left( \text {exp}({\mathcal {V}}_{\Xi })\circ \Phi \right) ^{*} (\omega _{{\mathbb {C}}^{n}}) \\&\quad =\sum \limits _{k=1}^{m}\Bigg ( \sum \limits _{i=1}^{n} \text {P}^{}_{k,i}({\mathcal {A}})+\text {Q}^{}_{k,i} (\tilde{{\mathcal {B}}}_{i})+(\text {S}^{}_{k,i}({\mathcal {A}}) +\text {T}^{}_{k,i}(\tilde{{\mathcal {B}}}_{i}))\text {R}^{}_{k,i} (\tilde{{\mathcal {C}}_{i}})\\&\qquad +(\text {L}^{}_{k,i}({\mathcal {A}})+\text {M}^{}_{k,i} (\tilde{{\mathcal {B}}}_{i}))\text {O}^{}_{k,i} (\tilde{{\mathcal {C}}}_{i}) \Bigg )rdr\wedge d\sigma _{k}\\&\qquad + \sum \limits _{l=1}^{q}\Bigg ( \sum \limits _{i=1}^{n} \text {P}^{}_{l,i}({\mathcal {A}})+\text {Q}^{}_{l,i} (\tilde{{\mathcal {B}}}_{i})+(\text {S}^{}_{l,i}({\mathcal {A}}) +\text {T}^{}_{l,i}(\tilde{{\mathcal {B}}}_{i}))\text {R}^{}_{l,i} (\tilde{{\mathcal {C}}_{i}})\\&\qquad +(\text {L}^{}_{l,i}({\mathcal {A}})+\text {M}^{}_{l,i} (\tilde{{\mathcal {B}}}_{i}))\text {O}^{}_{l,i} (\tilde{{\mathcal {C}}}_{i})\Bigg )dr\wedge d u_{l}\\&\qquad +\sum \limits _{k=1}^{m}\sum \limits _{l=1}^{q} \Bigg ( \sum \limits _{i=1}^{n} \text {P}^{}_{k,l,i}({\mathcal {A}}) +\text {Q}^{}_{k,l,i}(\tilde{{\mathcal {B}}}_{i})+\text {U}_{k,l,i} (\tilde{{\mathcal {C}}}_{i})+(\text {S}^{}_{k,l,i}({\mathcal {A}})\\&\qquad +\text {T}^{}_{k,l,i}(\tilde{{\mathcal {B}}}_{i})) \text {R}^{}_{k,l,i}(\tilde{{\mathcal {C}}_{i}})+(\text {L}^{}_{k,l,i}({\mathcal {A}})+\text {M}^{}_{k,l,i}(\tilde{{\mathcal {B}}}_{i}))\text {O}^{}_{k,l,i}(\tilde{{\mathcal {C}}}_{i})\Bigg ) rd\sigma _{k}\wedge d u_{l}\\&\qquad +\sum \limits _{k=1}^{m}\sum \limits _{j=1}^{m} \Bigg ( \sum \limits _{i=1}^{n}\text {P}^{}_{k,j,i}({\mathcal {A}}) +\text {Q}^{}_{k,j,i}(\tilde{{\mathcal {B}}}_{i}) +(\text {S}^{}_{k,j,i}({\mathcal {A}})+\text {T}^{}_{k,j,i} (\tilde{{\mathcal {B}}}_{i}))\text {R}^{}_{k,j,i}(\tilde{{\mathcal {C}}_{i}})\\&\qquad + (\text {L}^{}_{k,j,i}({\mathcal {A}}) +\text {M}^{}_{k,j,i}(\tilde{{\mathcal {B}}}_{i})) \text {O}^{}_{k,j,i}(\tilde{{\mathcal {C}}}_{i})\Bigg )r^{2}d\sigma _{k} \wedge d \sigma _{j}\\&\qquad + \sum \limits _{s=1}^{q}\sum \limits _{l=1}^{q} \Bigg ( \sum \limits _{i=1}^{n}\text {P}^{}_{s,j,i}(\tilde{{\mathcal {C}}}_i) +(\text {S}^{}_{s,j,i}({\mathcal {A}})+\text {T}^{}_{s,j,i} (\tilde{{\mathcal {B}}}_{i}))\text {R}^{}_{s,j,i}(\tilde{{\mathcal {C}}_{i}})\\&\qquad +(\text {L}^{}_{s,j,i}({\mathcal {A}}) +\text {M}^{}_{s,j,i}(\tilde{{\mathcal {B}}}_{i}))\text {O}^{}_{s,j,i} (\tilde{{\mathcal {C}}}_{i})\Bigg ) du_{s}\wedge d u_{l} \end{aligned}$$

where ${\mathcal {A}},\tilde{{\mathcal {B}}}_{i},\tilde{{\mathcal {C}}}_{i}$, $i=1,\dots ,n$ are the components of ${\mathcal {V}}_{\Xi }$ in a neighborhood of the edge as in (4.7) and the operators $\text {L}_{\bullet },\text {M}_{\bullet }, \text {O}_{\bullet },\text {P}_{\bullet }, \text {Q}_{\bullet },\text {R}_{\bullet },\text {S}_{\bullet }, \text {T}_{\bullet },\text {U}_{\bullet }$ belong to $\text {Diff}_{\text {edge}}^{1}(M).$

Proof

Let $\Xi ={\mathcal {A}}dr+\sum \nolimits _{k=1}^{m}{\mathcal {B}}_{k}rd \sigma _{k}+\sum \nolimits _{l=1}^{q}{\mathcal {C}}_{l}du_{l}$ in local coordinates near the edge, then

$$\begin{aligned} {\mathcal {V}}_{\Xi }=J\Phi _{*}( g_{M}^{*}(\Xi ))=\sum \limits _{i=1}^{n}-\tilde{{\mathcal {C}}}_{i} (r,\sigma ,u)\partial _{x_i}+({\mathcal {A}}(r,\sigma ,u)\theta ^{\circ }_{i}+ \tilde{{\mathcal {B}}}_{i}(r,\sigma ,u))\partial _{y_i} \end{aligned}$$

and

$$\begin{aligned}&(\text {exp}( {\mathcal {V}}_{\Xi })\circ \Phi )(r,\sigma ,u)\\&\quad =(r\theta _1-\tilde{{\mathcal {C}}_1} ,\dots ,r\theta _{n}-\tilde{{\mathcal {C}}}_{n}, \tau _1+{\mathcal {A}}\theta _{1}+\tilde{{\mathcal {B}}}_{1},\dots ,\tau _{n}+{\mathcal {A}}\theta _{n}+\tilde{{\mathcal {B}}}_{n}). \end{aligned}$$

The standard Kähler form in ${\mathbb {C}}^{n}$ is given by

$$\begin{aligned} \omega _{_{{\mathbb {C}}^{n}}}=\sum \limits _{i=1}^{n}dx_{i}\wedge dy_{i}, \end{aligned}$$

then a direct computation shows that

$$\begin{aligned}&(\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi )^{*}(\omega _{{\mathbb {C}}^{n}})= \sum \limits _{i=1}^{n} (\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi )^{*} ( dx_{i}\wedge dy_{i})\\&\quad =\sum \limits _{i=1}^{n}d\left( r\theta _{i}(\sigma )-\tilde{{\mathcal {C}}}_{i} (r,\sigma ,u)\right) \wedge d\left( \tau _i(u_l)+{\mathcal {A}}\theta _{i}(\sigma )+\tilde{{\mathcal {B}}}_{i}\right) \\&\quad =\sum \limits _{k=1}^{m}\bigg ( \sum \limits _{i=1}^{n}\frac{1}{r} \left( \theta _{i}\partial _{k}(\theta _i)+\theta _i\partial _{k}+\theta _{i} \partial _{k}(\theta _i)(-r\partial _{r}) \right) ({\mathcal {A}})\\&\qquad +\frac{1}{r}\left( \theta _{i}\partial _{k}+\partial _{k}(\theta _i) (-r\partial _{r})(\tilde{{\mathcal {B}}}_{i}) \right) - \frac{1}{r}\left( \left( \partial _{k}(\theta _i)+\theta _{i}\partial _{k}\right) ({\mathcal {A}})+\partial _{k}(\tilde{{\mathcal {B}}}_{i})\right) \partial _{r}(\tilde{{\mathcal {C}}}_{i})\\&\qquad +\frac{1}{r}\partial _{k}(\tilde{{\mathcal {C}}})\big (\theta _{i}\partial _{r}({\mathcal {A}}) + \partial _{r}(\tilde{{\mathcal {B}}}_{i}) \big )\bigg )rdr\wedge d\sigma _k\\&\qquad +\sum \limits _{l=1}^{q}\bigg (\sum \limits _{i=1}^{n} \theta ^{2}_{i}\partial _{u_{l}}({\mathcal {A}})+\theta _{i}\partial _{u_{l}}(\tilde{{\mathcal {B}}}_{i})-\partial _{u_{l}}(\tau _{i})\partial _{r}(\tilde{{\mathcal {C}}}_{i})\\&\qquad -\partial _{r}(\tilde{{\mathcal {C}}}_{i})\left( \theta _{i}\partial _{u_{l}}({\mathcal {A}}) +\partial _{u_{l}}(\tilde{{\mathcal {B}}}_i) \right) +\partial _{u_{l}}(\tilde{{\mathcal {C}}}_i)\left( \theta _{i}\partial _{r}({\mathcal {A}}) +\partial _{r}\tilde{{\mathcal {B}}}_i \right) \bigg )dr\wedge du_{l}\\&\qquad +\sum \limits _{k=1}^{m}\sum \limits _{l=1}^{q}\bigg (\sum \limits _{i=1}^{n} \frac{1}{r}\left( \partial _{k}(\theta _{i})\theta _i (r\partial _{u_{l}})\right) ({\mathcal {A}})+\frac{1}{r}\left( \partial _{k}(\theta _{i}) (r\partial _{u_{l}})\right) (\tilde{{\mathcal {B}}}_{i})\\&\qquad -\frac{1}{r}\partial _{u_{l}}(\tau _{i})\partial _{k}(\tilde{{\mathcal {C}}}_i)-\frac{1}{r}\partial _{k}(\tilde{{\mathcal {C}}}_i)\left( \theta _{i}\partial _{u_{l}}({\mathcal {A}})+\partial _{u_{l}}(\tilde{{\mathcal {B}}}_i)\right) \\&\qquad +\partial _{u_{l}}(\tilde{{\mathcal {C}}}_i)\left( (\partial _{k}(\theta _i) +\theta _{i}\partial _{k})({\mathcal {A}})+\partial _{k}(\tilde{{\mathcal {B}}}_i) \right) \bigg )rd\sigma _{k}\wedge du_{l}\\&\qquad +\sum \limits _{j=1}^{m}\sum \limits _{k=1}^{m}\Bigg (\sum \limits _{i=1}^{n}\frac{1}{r}\left( \theta _i \partial _{j}(\theta _i)\partial _{k}-\theta _i\partial _{k}(\theta _i)\partial _{j} \right) ({\mathcal {A}})\\&\qquad +\frac{1}{r}\left( \partial _{j}(\theta _i)\partial _{k}-\partial _{k}(\theta _i)\partial _{j} \right) (\tilde{{\mathcal {B}}}_i)\\&\qquad -\frac{1}{r}\partial _{j}(\tilde{{\mathcal {C}}}_i)\left( \frac{1}{r}\left( \partial _{k}(\theta _i)+\theta _i\partial _k \right) ({\mathcal {A}})+\frac{1}{r}\partial _k(\tilde{{\mathcal {B}}}_i) \right) \\&\qquad + \frac{1}{r}\partial _{k}(\tilde{{\mathcal {C}}}_i)\left( \frac{1}{r}\left( \partial _{j}(\theta _i)+\theta _i\partial _j \right) ({\mathcal {A}})+\frac{1}{r}\partial _j(\tilde{{\mathcal {B}}}_i) \right) \Bigg ) r^{2}d\sigma _{j}\wedge d \sigma _{k}\\&\qquad +\sum \limits _{\lambda =1}^{q}\sum \limits _{l =1}^{q}\Bigg (\sum \limits _{i=1}^{n} \left( -\partial _{u_{l}}(\tau _i)\partial _{u_{\lambda }}+ \partial _{u_{\lambda }}(\tau _i)\partial _{u_{l}} \right) (\tilde{{\mathcal {C}}}_i)\\&\qquad -\partial _{u_{\lambda }}(\tilde{{\mathcal {C}}}_i)\left( \theta _i\partial _{u_{l}}({\mathcal {A}})+\partial _{u_{l}}(\tilde{{\mathcal {B}}}_i) \right) + \partial _{u_{l}}(\tilde{{\mathcal {C}}}_i)\left( \theta _i\partial _{u_{\lambda }}({\mathcal {A}})+\partial _{u_{\lambda }}(\tilde{{\mathcal {B}}}_i) \right) \Bigg )du_{\lambda }\wedge du_{l}. \end{aligned}$$

Each of these terms are edge-degenerate differential operators acting on the components of $\Xi$ and products of these as it is claimed in the proposition. Note that we have used the fact that

$$\begin{aligned} \left( \text {exp}({\mathcal {V}}_{0})\circ \Phi \right) ^{*}\left( \omega _{{\mathbb {C}}^{n}} \right) =0 \end{aligned}$$

to remove products in each term that do not contain any of the component functions ${\mathcal {A}}$, $\tilde{{\mathcal {B}}}_{i}$ and $\tilde{{\mathcal {C}}}_{i}$. $\square$

Corollary 4.8

The map

$$\begin{aligned} \text {P}_{\omega _{{\mathbb {C}}^{n}}}: C^{\infty }_{0}(T^{*}_{\wedge }M) \longrightarrow C^{\infty }_{0}\left( M,\bigwedge \nolimits ^{2}T^{*}_{\wedge }M\right) \end{aligned}$$

defined by $\text {P}_{\omega _{_{{\mathbb {C}}^n}}}(\Xi ):=\left( \text {exp}({\mathcal {V}}_{\Xi })\circ \Phi \right) ^{*}(\omega _{_{{\mathbb {C}}^n}})$, extends to a continuous non-linear operator

$$\begin{aligned} \text {P}_{\omega _{{\mathbb {C}}^{n}}}: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M) \longrightarrow {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{2}T^{*}_{\wedge }M\right) \end{aligned}$$

for $s>\frac{\text {dim}{\mathcal {X}}+\text {dim}{\mathcal {E}}+3}{2}$ and $\gamma >\frac{\text {dim}{\mathcal {X}}+1}{2}.$

Proof

Let’s consider a sequence $\lbrace \Xi _{i}\rbrace _{i\in {\mathbb {N}}} \subset C^{\infty }_{0}(T^{*}_{\wedge }M)$ such that it is a Cauchy sequence in ${\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M).$ Then, in a neighborhood near the edge, the components of the elements of the sequence $\lbrace \Xi _{i} \rbrace _{i\in {\mathbb {N}}}$ define Cauchy sequences in ${\mathcal {W}}^{s,\gamma }(M)$ i.e.

$$\begin{aligned}&\left\| \omega \phi _{j}\varphi _{\lambda }{\mathcal {A}}^{i}-\omega \phi _{j}\varphi _{\lambda } {\mathcal {A}}^{j}\right\| _{{\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)}<\varepsilon ,\\&\left\| \omega \phi _{j}\varphi _{\lambda }\tilde{{\mathcal {B}}}_{k}^{i}- \omega \phi _{j}\varphi _{\lambda }\tilde{{\mathcal {B}}}_{k}^{j}\right\| _{{\mathcal {W}}^{s,\gamma } (M,T^{*}_{\wedge }M)}<\varepsilon \text { for all } k=1,2,\dots ,m \end{aligned}$$

and

$$\begin{aligned} \left\| \omega \phi _{j}\varphi _{\lambda }\tilde{{\mathcal {C}}}_{l}^{i} -\omega \phi _{j}\varphi _{\lambda }\tilde{{\mathcal {C}}}_{l}^{j}\right\| _{{\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)}<\varepsilon \text { for all } l=1,2,\dots ,q \end{aligned}$$

for all $i,j>N(\varepsilon ).$ Away from the edge we have

$$\begin{aligned} \left\| (1-\omega )(\Xi _{i}-\Xi _{j})\right\| _{H^{s}(2\mathbb {M},T^{*}_{}M)}< \varepsilon . \end{aligned}$$

We want to estimate

$$\begin{aligned} \left\| \text {P}_{\omega _{{\mathbb {C}}^{n}}}(\Xi _{i})-\text {P}_{\omega _{{\mathbb {C}}^{n}}} (\Xi _{j})\right\| _{{\mathcal {W}}^{s-1,\gamma -1}(M,T^{*}_{\wedge }M)}. \end{aligned}$$

Observe that the conditions on s and $\gamma$ imply that the edge-Sobolev spaces are Banach algebras (11.39), hence multiplication is well-defined and we have the estimate $\left\| fg\right\| _{s,\gamma }\le C_{s,\gamma }\left\| f\right\| _{s,\gamma }\left\| g\right\| _{s,\gamma }$ with a constant $C_{s,\gamma }$ depending only on s and $\gamma$. To simplify the notation we will use $\left\| \cdot \right\| _{s,\gamma }$ to denote $\left\| \cdot \right\| _{{\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)}.$

Now, near the edge, the components of the operator $\text {P}_{\omega _{{\mathbb {C}}^{n}}}$ are given by expressions of the form $\text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}$ where $\text {P},\text {Q},\text {R}, \text {S},\text {T}\in \text {Diff}^{1}_{\text {edge}}(M).$ By estimating one of these expressions we can apply the same argument to all components.

Given $\text {P},\text {Q},\text {R},\text {S},\text {T}$ in $\text {Diff}^{1}_{\text {edge}}(M)$ and ${\mathcal {A,B,C,A',B',C'}}$ in ${\mathcal {W}}^{s,\gamma }(M)$ by the continuity these operators (Proposition 3.6) and the elementary identity $ab-a'b'=\frac{1}{2}(a+a')(b-b')+\frac{1}{2}(a-a')(b+b')$ we have

$$\begin{aligned}&\parallel \text {P}({\mathcal {A}})+\text {Q}({\mathcal {B}}) +(\text {S}({\mathcal {A}})+\text {T}({\mathcal {B}}))\text {R}({\mathcal {C}}) -\left( \text {P}({\mathcal {A}}')+\text {Q}({\mathcal {B'}})\right. \\&\qquad \left. +(\text {S}({\mathcal {A}}')+\text {T}({\mathcal {B'}})) \text {R}({\mathcal {C'}})\right) \parallel _{s-1,\gamma -1}\\&\quad \le \left\| \text {P}\right\| \left\| {\mathcal {A}}-{\mathcal {A}}'\right\| _{s\gamma }+\left\| \text {Q}\right\| \left\| {\mathcal {B}}-{\mathcal {B}}'\right\| _{s,\gamma }\\&\qquad + \left\| (\text {S}({\mathcal {A}})+\text {T}({\mathcal {B}})) \text {R}({\mathcal {C}})-(\text {S}({\mathcal {A}}')+\text {T} ({\mathcal {B'}}))\text {R}({\mathcal {C'}})\right\| _{s-1,\gamma -1}\\&\quad \le \left\| \text {P}\right\| \left\| {\mathcal {A}}-{\mathcal {A}}'\right\| _{s,\gamma } +\left\| \text {Q}\right\| \left\| {\mathcal {B}}-{\mathcal {B}}'\right\| _{s,\gamma }\\&\qquad +\frac{1}{2}\left( \left\| \text {R}\right\| \left\| \text {S}\right\| \left\| {\mathcal {A}}-{\mathcal {A}}'\right\| _{s,\gamma }+\left\| \text {R}\right\| \left\| \text {T}\right\| \left\| {\mathcal {B}}-{\mathcal {B}}'\right\| _{s,\gamma }\right) \left\| {\mathcal {C}} +{\mathcal {C}}' \right\| _{s,\gamma }\\&\qquad + \frac{1}{2}\left( \left\| \text {R}\right\| \left\| \text {S}\right\| \left\| {\mathcal {A}}+{\mathcal {A}}'\right\| _{s,\gamma }+\left\| \text {R}\right\| \left\| \text {T}\right\| \left\| {\mathcal {B}}+{\mathcal {B}}'\right\| _{s,\gamma }\right) \left\| {\mathcal {C}} -{\mathcal {C}}' \right\| _{s,\gamma }. \end{aligned}$$

Therefore, if $\lbrace \Xi _{i} \rbrace$ is a Cauchy sequence in ${\mathcal {W}}^{s,\gamma }(M, T^{*}_{\wedge }M)$ then

$$\begin{aligned} \lbrace \text {P}({\mathcal {A}}_i)+\text {Q}({\mathcal {B}}_i) +(\text {S}({\mathcal {A}}_i)+\text {T}({\mathcal {B}}_i))\text {R}({\mathcal {C}}_i) \rbrace _{i\in {\mathbb {N}}} \end{aligned}$$

is a Cauchy sequence in ${\mathcal {W}}^{s-1,\gamma -1}(M)$ which implies that $\lbrace \omega (r) \text {P}_{\omega _{{\mathbb {C}}^{n}}}(\Xi _{i})\rbrace _i$ is Cauchy too.

Away from the edge we are in the setting of standard Sobolev spaces $H^{s}(2\mathbb {M},T^{*}_{\wedge }M)$ and the operator $\text {P}_{\omega _{{\mathbb {C}}^{n}}}$ is given by products of two differential operators of order 1. By using their continuity on $H^{s}(2\mathbb {M},T^{*}_{}M)$, the fact that standard Sobolev spaces are Banach algebras for $s>\frac{\text {dim}M}{2}$ and a similar argument give us that $\lbrace (1-\omega ) \text {P}_{\omega _{{\mathbb {C}}^{n}}}(\Xi _{i})\rbrace _i$ is Cauchy in $H^{s}(2\mathbb {M},T^{*}_{}M)$. Therefore the corollary follows immediately. $\square$

Proposition 4.9

Let $\Xi \in C^{\infty }_{0}(T^{*}_{\wedge }M)$ and ${\mathcal {V}}_{\Xi }=J\Phi _{*}( g_{M}^{*}(\Xi ))\in C^{\infty }({\mathcal {N}}(M)).$ Then, on a neighborhood $(0,\varepsilon )\times {\mathcal {U}}_{\lambda }\times \Omega _j$ near the edge, the pull-back of the imaginary part of the holomorphic volume form in ${\mathbb {C}}^{n}$

$$\begin{aligned} \text {Im}\Omega =\text {Im}(dz_1\wedge \dots \wedge dz_{n}), \end{aligned}$$

by the map $\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi$ is given as a sum of n products of the form

$$\begin{aligned} \text {P}_{i_{1}}(F_{i_{1}})\text {P}_{i_{2}}(F_{i_{2}})\cdot \dots \cdot \text {P}_{i_{n}}(F_{i_{n}}) \end{aligned}$$

where $\text {P}_{i_{j}}\in \text {Diff}^{1}_{\text {edge}}(M)$ and $F_{i_{j}}\in {\mathcal {W}}^{s,\gamma }(M)$.

Proof

The holomorphic volume form in ${\mathbb {C}}^{n}$ is given by

$$\begin{aligned} \text {Im}\Omega =\text {Im}(dz_1\wedge \dots \wedge dz_{n})&=\text {Im} d(x_1+iy_1)\wedge \dots \wedge d(x_{n}+iy_{n})\\&=\sum \limits _{\left| I\right| =odd} c_{I}dy_{I}\wedge dx_{1}\wedge \dots \wedge \widehat{dx_{I}}\wedge \dots dx_{n} \end{aligned}$$

where the sum is taken over all increasingly ordered multi-indexes I of odd length k, $I=(i_1 , \dots , i_k )$, the hat means that we omit the corresponding terms and $c_{I}=\pm 1.$ Then

$$\begin{aligned}&(\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi )^{*} \left( \text {Im}\Omega \right) \\&\quad =\sum \limits _{\left| I\right| =odd} c_{I}(\text {exp} ({\mathcal {V}}_{\Xi })\circ \Phi )^{*}\left( dy_{I}\wedge dx_{1}\wedge \dots \wedge \widehat{dx_{I}}\wedge \dots dx_{n}\right) \\&\quad =\sum \limits _{\left| I\right| =odd} c_{I}d(\tau _{i_{1}}+{\mathcal {A}}\theta _{i_{1}} +\tilde{{\mathcal {B}}}_{i_{1}})\wedge \dots \wedge d(\tau _{i_{k}}+{\mathcal {A}}\theta _{i_{k}}+\tilde{{\mathcal {B}}}_{i_{k}}) \wedge d (r\theta _1-\tilde{{\mathcal {C}}}_1)\wedge \dots \\&\qquad \wedge \widehat{d(r\theta _I-\tilde{{\mathcal {C}}}_I)} \wedge \dots \wedge d (r\theta _{n}-\tilde{{\mathcal {C}}}_{n}). \end{aligned}$$

Each of the terms in the sum is a n-form on $(0, \varepsilon )\times {\mathcal {U}}_{\lambda }\times \Omega _j$, hence

$$\begin{aligned}&c_{I}(\text {exp}({\mathcal {V}}_{\Xi })\circ \Phi )^{*} \left( dy_{I}\wedge dx_{1}\wedge \dots \wedge \widehat{dx_{I}}\wedge \dots dx_{n}\right) \\&\quad =F_{I}(r,\sigma ,u)r^{m}dr\wedge d\sigma _{1}\wedge \dots \wedge d\sigma _{m}\wedge du_{1}\wedge \dots \wedge du_{q}, \end{aligned}$$

where $F_{I}(r,\sigma ,u)$ is the determinant of the following matrix

$$\begin{aligned} \left[ \begin{array}{ccccccc} \partial _{r}(\tau _{i_{1}}+{\mathcal {A}}\theta _{i_{1}}+\tilde{{\mathcal {B}}}_{i_{1}}) &{} \dots &{} \partial _{r}(r\theta _{1}-\tilde{{\mathcal {C}}}_{1}) &{} \dots &{} \widehat{\partial _{r}(r\theta _{I}-\tilde{{\mathcal {C}}}_{I})} &{} \dots &{} \partial _{r}(r\theta _{n}-\tilde{{\mathcal {C}}}_{n}) \\ \vdots &{} \dots &{} \vdots &{} \dots &{} \vdots &{} \dots &{} \vdots \\ \frac{1}{r}\partial _{j}(\tau _{i_{1}}+{\mathcal {A}}\theta _{i_{1}}+\tilde{{\mathcal {B}}}_{i_{1}}) &{} \dots &{} \frac{1}{r}\partial _{j}(r\theta _{1}-\tilde{{\mathcal {C}}}_{1}) &{} \dots &{} \widehat{\frac{1}{r}\partial _{j}(r\theta _{I}-\tilde{{\mathcal {C}}}_{I})} &{} \dots &{} \frac{1}{r}\partial _{j}(r\theta _{n}-\tilde{{\mathcal {C}}}_{n}) \\ \vdots &{} \dots &{} \vdots &{} \dots &{} \vdots &{} \dots &{} \vdots \\ \partial _{u_{q}}(\tau _{i_{1}}+{\mathcal {A}}\theta _{i_{1}}+\tilde{{\mathcal {B}}}_{i_{1}}) &{}\dots &{} \partial _{u_{q}}(r\theta _{1}-\tilde{{\mathcal {C}}}_{1}) &{} \dots &{} \widehat{\partial _{u_{q}}(r\theta _{I}-\tilde{{\mathcal {C}}}_{I})} &{} \dots &{} \partial _{u_{q}}(r\theta _{n}-\tilde{{\mathcal {C}}}_{n}) \\ \end{array}\right] . \end{aligned}$$

Therefore $F_{I}(r,\sigma ,u)$ is the sum of products of the form

$$\begin{aligned} \text {P}_1(\tau _{i_{1}}+{\mathcal {A}}\theta _{i_{1}}+\tilde{{\mathcal {B}}}_{i_{1}})\dots \text {P}_{n}(r\theta _{n}-\tilde{{\mathcal {C}}}_{n}) \end{aligned}$$

with all the operators $\text {P}_k$, $k=1,\dots ,n,$ in $\text {Diff}^{1}_{\text {edge}}(M).$ By expanding these products and observing that the sum of all terms of the form

$$\begin{aligned} \text {P}_1(\tau _{i_{1}})\dots \text {P}_{n}(r\theta _{n}), \end{aligned}$$

i.e. all those products not containing any of the functions ${\mathcal {A}},\tilde{{\mathcal {B}}}_{\bullet }$ or $\tilde{{\mathcal {C}}}_{\bullet }$, is equal to zero as

$$\begin{aligned} \left( \text {exp}({\mathcal {V}}_{0})\circ \Phi \right) ^{*}\left( \text {Im}\Omega \right) = 0; \end{aligned}$$

we obtain that $F_{I}(r,\sigma ,u)$ is the sum of products of the form

$$\begin{aligned} \text {P}_{i_{1}}(F_{i_{1}})\text {P}_{i_{2}}(F_{i_{2}})\cdot \dots \cdot \text {P}_{i_{n}}(F_{i_{n}}) \end{aligned}$$

(4.11)

where $\text {P}_{i_{j}}\in \text {Diff}^{1}_{\text {edge}}(M)$ and $F_{i_{j}}\in {\mathcal {W}}^{s,\gamma }(M)$ as claimed. $\square$

Corollary 4.10

The map

$$\begin{aligned} \text {P}_{\text {Im}\Omega }: C^{\infty }_{0}(T^{*}_{\wedge }M) \longrightarrow C^{\infty }_{0}\left( M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M\right) \end{aligned}$$

given by $\text {P}_{\text {Im}\Omega }(\Xi ):=\left( \text {exp}({\mathcal {V}}_{\Xi })\circ \Phi \right) ^{*}(\text {Im}(\Omega ))$ extends to a continuous non-linear operator

$$\begin{aligned} \text {P}_{\text {Im}\Omega }: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M) \longrightarrow {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M\right) \end{aligned}$$

for $s>\frac{\text {dim}{\mathcal {X}}+\text {dim}{\mathcal {E}}+3}{2}$ and $\gamma >\frac{\text {dim}{\mathcal {X}}+1}{2}.$

Proof

Given $\Xi$ and $\Xi '$ in $C^{\infty }_{0}(T^{*}_{\wedge }M)$, near the edge, we have

$$\begin{aligned}&\text {P}_{\text {Im}\Omega }(\Xi ) -\text {P}_{\text {Im}\Omega }(\Xi ')\\&\quad =\left( \sum \limits _{I} \text {P}_{i_{1}}(F_{i_{1}})\dots \text {P}_{i_{n}}(F_{i_{n}}) -\text {P}_{i_{1}}(F_{i_{1}}')\dots \text {P}_{i_{n}}(F_{i_{n}}') \right) r^{m}dr\wedge d\sigma _{1}\dots \\&\qquad \dots \wedge d\sigma _{m}\wedge du_1\wedge \dots \wedge du_{q}. \end{aligned}$$

By applying the elementary identity $ab-a'b'=\frac{1}{2}(a+a')(b-b')+\frac{1}{2}(a-a')(b+b')$ we can decompose each term of the sum in factors, each of them containing one of the subtractions

$$\begin{aligned} {P}_{i_{n-k}}(F_{i_{n-k}})-\text {P}_{i_{n-k}}(F_{i_{n-k}}) \end{aligned}$$

(4.12)

for $k=0,\dots ,n$.

Now, if $\lbrace \Xi _{i}\rbrace _{i\in {\mathbb {N}}}\subset C^{\infty }_{0}(T^{*}_{\wedge }M)$ is a Cauchy sequence in the Sobolev space ${\mathcal {W}}^{s,\gamma }(M, T^{*}_{\wedge }M)$ as in Corollary 4.8, the sequences $\text {P}_{i_{j}}(F^{(k)}_{i_{j}})$ are Cauchy in ${\mathcal {W}}^{s-1,\gamma -1}(M)$ (by the continuity of edge-degenerate operators on edge-Sobolev spaces, see Proposition 3.6) and

$$\begin{aligned} \left\| \text {P}_{i_{n-k}}(F^{(k)}_{i_{n-k}}) -\text {P}_{i_{n-k}}(F^{(k')}_{i_{n-k}})\right\| _{s-1,\gamma -1} <\varepsilon \end{aligned}$$

for all $k,k'>N(\varepsilon ).$ Therefore, by the Banach algebra property (11.39) and (4.12), we have that $\left\{ \omega (r) \text {P}_{\text {Im}\Omega }(\Xi _{k}) \right\} _{k\in {\mathbb {N}}}$ is a Cauchy sequence in the Sobolev space ${\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M).$ Away from the edge a completely similar argument using the classical Sobolev spaces $H^{s}(2\mathbb {M},T^{*}M)$ implies that $\left\{ (1-\omega ) \text {P}_{\text {Im}\Omega }(\Xi _{k}) \right\} _{k\in {\mathbb {N}}}$ is Cauchy in the space $H^{s}(2\mathbb {M},T^{*}M)$ and the corollary follows immediately. $\square$

Corollaries 4.8 and  4.10 tell us that we have a continuous non-linear deformation operator:

$$\begin{aligned} \text {P}:=\begin{array}{ccc} \text {P}_{\omega _{_{{\mathbb {C}}^{n}}}} &{} &{} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{2}T^{*}_{\wedge }M\right) \\ \oplus &{}: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M) \longrightarrow &{}\oplus \\ \text {P}_{\text {Im}\Omega } &{} &{} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M\right) \end{array} \end{aligned}$$

for $s>\frac{\text {dim}{\mathcal {X}}+\text {dim}{\mathcal {E}}+3}{2}$ and $\gamma >\frac{\text {dim}{\mathcal {X}}+1}{2}.$

In fact, this operator is smooth as the following corollary shows.

Corollary 4.11

The non-linear deformation operator

$$\begin{aligned} \begin{array}{ccc} \text {P}_{\omega _{_{{\mathbb {C}}^{n}}}} &{} &{} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{2}T^{*}_{\wedge }M\right) \\ \oplus &{}: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M) \longrightarrow &{}\oplus \\ \text {P}_{\text {Im}\Omega } &{} &{} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M\right) . \end{array} \end{aligned}$$

is smooth.

Proof

Recall that from Proposition 4.7 the components of the operator $\text {P}_{\omega _{_{{\mathbb {C}}^n}}}$ are given by expressions of the form $\text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}$ where $\text {P}, \text {Q}, \text {R}, \text {S}, \text {T}\in \text {Diff}^{1}_{\text {edge}}(M).$ Therefore, given $f, \nu \in {\mathcal {W}}^{s,\gamma }(M)$ we have

$$\begin{aligned}&(\text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R})(f+\nu )-(\text {P} +\text {Q}+(\text {S}+\text {T})\cdot \text {R})(f)\\&\quad =\text {P}(\nu )+\text {Q}(\nu )+\text {R}(f)(\text {S}(\nu )+\text {T}(\nu )) +\text {S}(f)\text {R}(\nu )+\text {T}(f)\text {R}(\nu )\\&\qquad +\text {S}(\nu )\text {R}(\nu ) +\text {T}(\nu )\text {R}(\nu ). \end{aligned}$$

Observe that by the continuity of $\text {S}$

$$\begin{aligned} \left\| \frac{\text {S}(\nu )\text {R}(\nu )}{\left\| \nu \right\| }\right\| _{s-1,\gamma -1}\le \left\| \text {S}\right\| _{s,\gamma }\left\| \text {R}(\nu )\right\| _{s-1,\gamma -1}, \end{aligned}$$

hence by the continuity of $\text {R}$ we have $\lim _{\left\| \nu \right\| \rightarrow 0}\frac{\text {S}(\nu )\text {R}(\nu )}{\left\| \nu \right\| }=0$ and the same holds for $\text {T}(\nu )\text {R}(\nu )$. This implies that the Fréchet derivative of the operator $\text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}$ at f is given by

$$\begin{aligned} \text {D}\left( \text {P}+\text {Q}+(\text {S} +\text {T})\cdot \text {R}\right) [f]=\text {P}+\text {Q} +\text {R}(f)(\text {S}+\text {T}) +\text {S}(f)\text {R}+\text {T}(f)\text {R} \end{aligned}$$

and from this expression is clear that

$$\begin{aligned} \text {D}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [\bullet ]:{\mathcal {W}}^{s,\gamma }(M)\longrightarrow {\mathcal {L}}\left( {\mathcal {W}}^{s,\gamma }(M), {\mathcal {W}}^{s-1,\gamma -1}(M) \right) \end{aligned}$$

is a continuous operator. Moreover, observe that the second derivative is constant. More precisely, the second derivative of $\text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}$ at $f\in {\mathcal {W}}^{s,\gamma }(M)$ is the linear, continuous map

$$\begin{aligned} \text {D}^{2}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f]\in {\mathcal {L}}\left( {\mathcal {W}}^{s,\gamma }(M),{\mathcal {L}}\left( {\mathcal {W}}^{s,\gamma }(M), {\mathcal {W}}^{s-1,\gamma -1}(M) \right) \right) \end{aligned}$$

such that

$$\begin{aligned}&\left( \text {D}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f+\nu ]-\text {D}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f]\right. \\&\quad - \left. \text {D}^{2}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f](\nu )\right) \Big / {\left\| \nu \right\| } \end{aligned}$$

goes to zero in ${\mathcal {L}}\left( {\mathcal {W}}^{s,\gamma }(M), {\mathcal {W}}^{s-1,\gamma -1}(M) \right)$ when $\left\| \nu \right\| \rightarrow 0$.

A direct computation shows that

$$\begin{aligned}&\text {D}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f+\nu ]-\text {D}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f]\\&=\text {R}(\nu )(\text {S}+\text {T})+\text {S} (\nu )\text {R}+\text {T}(\nu )\text {R} \end{aligned}$$

hence by uniqueness of derivatives we have

$$\begin{aligned} \text {D}^{2}\left( \text {P}+\text {Q}+(\text {S}+\text {T})\cdot \text {R}\right) [f](\nu )= \text {R}(\nu )(\text {S}+\text {T}) +\text {S}(\nu )\text {R}+\text {T}(\nu )\text {R} \end{aligned}$$

for any $\nu \in {\mathcal {W}}^{s,\gamma }(M)$ and all $f\in {\mathcal {W}}^{s,\gamma }(M)$. Thus the second derivative is constant and given by

$$\begin{aligned} \text {D}^{2}\left( \text {P}+\text {Q} +(\text {S}+\text {T})\cdot \text {R}\right) [f](g,h)=\text {R}(g)(\text {S} +\text {T})(h)+(\text {S}+\text {T})(g)\text {R}(h) \end{aligned}$$

for any $f\in {\mathcal {W}}^{s,\gamma }(M)$. We conclude that $\text {P}_{\omega _{_{{\mathbb {C}}^n}}}$ is smooth.

Now, the operator $\text {P}_{\text {Im}\Omega }$ is given as a sum of products $\text {P}_{1}(F_1)\text {P}_{2}(F_2)\cdot \dots \cdot \text {P}_n(F_n)$ where $\text {P}_{i}\in \text {Diff}^{1}_{\text {edge}}(M)$ and $F_{i}\in {\mathcal {W}}^{s,\gamma }(M)$. Then we have that the expression

$$\begin{aligned} \text {P}_{1}(F_1+\nu )\text {P}_{2}(F_2+\nu )\cdot \dots \cdot \text {P}_n(F_n+\nu )-\text {P}_{1}(F_1)\text {P}_{2}(F_2)\cdot \dots \cdot \text {P}_n(F_n) \end{aligned}$$

is a sum of products of the operators $\text {P}_i$ evaluated at $F_i$ or at $\nu$ with at least one $\nu$ in each product. Those products containing 2 or more terms with $\nu$ does not contribute to the Frechet derivative as in the first part of the proof. Hence the Fréchet derivative is computed only with products having one term with $\nu$ which produce continuous linear operators of the form $\text {P}_{i_1}(F_{i_1})\text {P}_{i_2}(F_{i_2})\cdot \dots \cdot \text {P}_{i_{n-1}}(F_{i_{n-1}})\text {P}_{i_n}$. Furthermore, it is easily seen that the nth derivative of each of the products $\text {P}_{1}(F_1)\text {P}_{2}(F_2)\cdot \dots \cdot \text {P}_n(F_n)$ is constant and equal to

$$\begin{aligned} \sum \limits _{(i_1,\dots ,i_n)\in S_n}\text {P}_{i_{1}}\text {P}_{i_{2}}\cdot \dots \cdot \text {P}_{i_{n}} \end{aligned}$$

where the sum is taken over the group of permutations with n elements. From this it follows that $\text {P}_{\text {Im}\Omega }$ is smooth. $\square$

Observe that any element $\alpha \in {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M)$ is given as $\alpha =f\text {Vol}_{M}$ where $f\in {\mathcal {W}}^{s-1,\gamma -1}(M)$ and $f_{k}\text {Vol}_{M}\longrightarrow f_{}\text {Vol}_{M}$ in the edge-Sobolev space ${\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{n}T^{*}_{\wedge }M)$ if and only if $f_k\longrightarrow f$ in the space ${\mathcal {W}}^{s-1,\gamma -1}(M).$ Hence we can consider the operator $\text {P}_{\text {Im}\Omega }$ as an operator acting between the following spaces:

$$\begin{aligned} \text {P}_{\text {Im}\Omega }:{\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\longrightarrow {\mathcal {W}}^{s-1,\gamma -1}(M). \end{aligned}$$

5 The linear operator DP[0]

In this section we consider the operator $\text {DP}[0]$, the linearisation at zero of the deformation operator $\text {P}=\text {P}_{\omega _{_{{\mathbb {C}}^n}}}\oplus \text {P}_{\text {Im}\Omega }$. A careful analysis of this operator is necessary as we want apply the Implicit Function Theorem for Banach spaces to this linear operator in order to describe solutions (nearby zero) of the non-linear equation $\text {P}(f)=0$ in term of ${\text {Ker DP}}[0]$. McLean’s results in [26] implies that the linearisation of the deformation operator at zero acting on $C^{\infty }_{0}(M,T^{*}_{\wedge }M)$ is given by the Hodge–deRham operator i.e.

$$\begin{aligned} \text {DP}[0]\Big |_{C^{\infty }_{0}(M,T^{*}_{\wedge }M)}=d+d^{*}. \end{aligned}$$

In this section we analyse the extension of this operator to edge-Sobolev spaces, its ellipticity and the Fredholm property.

Observe that on a collar neighborhood any $\Xi \in C^{\infty }_{}\left( \bigwedge \nolimits ^{k}T^{*}_{\wedge }M\right)$ can be written as

$$\begin{aligned} \Xi =dr\wedge r^{k-1} \Theta _{{\mathcal {X}}} +dr\wedge \Theta _{{\mathcal {E}}}+dr\wedge \sum \limits _{i=1}^{k-2} r^{i}\Lambda ^{i}_{{\mathcal {X}},{\mathcal {E}}}+\sum \limits _{j=1}^{k-1}r^{j} \tilde{\Lambda }^{j}_{{\mathcal {X}},{\mathcal {E}}}+r^{k}\tilde{\Theta }_{{\mathcal {X}}}+\tilde{\Theta }_{{\mathcal {E}}}\quad \end{aligned}$$

(5.1)

where:

• (i) $\Theta _{{\mathcal {X}}}$ is a smooth section of the bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}(\bigwedge \nolimits ^{k-1} T^{*}{\mathcal {X}});$

• (ii) $\Theta _{{\mathcal {E}}}$ is a smooth section of the bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}(\bigwedge \nolimits ^{k-1}T^{*}{\mathcal {E}});$

• (iii) $\Lambda _{{\mathcal {X}},{\mathcal {E}}}^{i}$ is the wedge product of a smooth section of the pull-back bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}(\bigwedge \nolimits ^{i}T^{*}{\mathcal {X}})$ with a smooth section of the pull-back bundle

  $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}(\bigwedge \nolimits ^{k-1-i}T^{*}{\mathcal {E}});$

• (iv) $\tilde{\Lambda }_{{\mathcal {X}},{\mathcal {E}}}^{j}$ is the wedge product of a smooth section of the pull-back bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}(\bigwedge \nolimits ^{j}T^{*}{\mathcal {X}})$ with a smooth section of the bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}(\bigwedge \nolimits ^{k-j}T^{*}{\mathcal {E}});$

• (v) $\tilde{\Theta }_{{\mathcal {X}}}$ is a smooth section of the bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}(\bigwedge \nolimits ^{k} T^{*}{\mathcal {X}});$

• (vi) $\tilde{\Theta }_{{\mathcal {E}}}$ is a smooth section of the bundle $\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}(\bigwedge \nolimits ^{k} T^{*}{\mathcal {E}}).$

Recall that $\pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}$ and $\pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}$ are the corresponding projections

$$\begin{aligned}&\pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}:{\mathcal {X}}^{\wedge }\times {\mathcal {E}}\longrightarrow {\mathcal {E}}\\&\pi ^{}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}:{\mathcal {X}}^{\wedge }\times {\mathcal {E}}\longrightarrow {\mathcal {X}}. \end{aligned}$$

More generally, the bundle $\bigwedge ^{\bullet }T^{*}_{\wedge }M$ can be decomposed in the following way:

$$\begin{aligned} \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M&=dr\wedge \sum \limits _{k=0}^{m}r^{k} \pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}\left( \bigwedge \nolimits ^{k} T^{*}{\mathcal {X}}\right) \oplus dr\wedge \pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}}\left( \bigwedge \nolimits ^{\bullet }T^{*}{\mathcal {E}}\right) \\&\quad \oplus dr\wedge \sum \limits _{k=1}^{m} r^{k}\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}}\left( \bigwedge \nolimits ^{k}T^{*}{\mathcal {X}}\right) \wedge \pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}} \left( \bigwedge \nolimits ^{\bullet }T^{*}{\mathcal {E}}\right) \\&\quad \oplus \sum \limits _{k=1}^{m}r^{k}\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}} \left( \bigwedge \nolimits ^{k}T^{*}{\mathcal {X}}\right) \wedge \pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}} \left( \bigwedge \nolimits ^{\bullet }T^{*}{\mathcal {E}}\right) \\&\quad \oplus \sum \limits _{k=1}^{m}r^{k}\pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {E}}} (\bigwedge \nolimits ^{k} T^{*}{\mathcal {X}})\oplus \pi ^{*}_{{\mathbb {R}}^{+}\times {\mathcal {X}}} (\bigwedge \nolimits ^{\bullet } T^{*}{\mathcal {E}}). \end{aligned}$$

With respect to this splitting we can compute an explicit expression for $d+d^{*}$ acting on $C^{\infty }_{0}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M)$ to prove the following proposition.

Proposition 5.1

The operator $\text {DP}[0]\Big |_{C^{\infty }_{0}(M,T^{*}_{\wedge }M)}=d+d^{*}$ extends to a continuous linear operator acting between edge-Sobolev spaces

$$\begin{aligned} \begin{array}{cccc} \text {DP}[0]:&{\mathcal {W}}^{s,\gamma }_{}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M)\longrightarrow & {} {\mathcal {W}}^{s-1,\gamma -1}_{}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M) \end{array}. \end{aligned}$$

Proof

By using the splitting (5.1) and arranging in a vector the components of the differential form $\Xi \in C^{\infty }_{0}(M,\bigwedge ^{k}T^{*}_{\wedge }M)$ in the following way

$$\begin{aligned} \begin{pmatrix} \Theta _{{\mathcal {X}}},\Theta _{{\mathcal {E}}}, \Lambda ^{1}_{{\mathcal {X}},{\mathcal {E}}},\dots ,\Lambda ^{k-2}_{{\mathcal {X}},{\mathcal {E}}} \\ \tilde{\Lambda }^{1}_{{\mathcal {X}},{\mathcal {E}}},\dots , \tilde{\Lambda }^{k-1}_{{\mathcal {X}},{\mathcal {E}}},\tilde{\Theta }_{{\mathcal {X}}},\tilde{\Theta }_{{\mathcal {E}}} \end{pmatrix}, \end{aligned}$$

a direct computation shows that the Hodge–deRham operator acting on $\Xi$ is given by

$$\begin{aligned} d+d^{*}= \left[ \begin{array}{c|c} \mathbf A &{} \mathbf B \\ \hline \mathbf C &{} \mathbf D \end{array} \right] , \end{aligned}$$

where the operator matrices are given by

$$\begin{aligned} \mathbf A= & {} \left[ \begin{array}{ccccccc} \frac{1}{r}(d_{\mathcal {X}}+d^{*}_{{\mathcal {X}}}) &{} 0 &{} 0 &{} 0 &{} \cdots &{}\cdots &{} -d^{*}_{{\mathcal {E}}} \\ 0 &{} -\frac{1}{r}(d_{\mathcal {E}}+d^{*}_{{\mathcal {E}}}) &{} -\frac{1}{r}d^{*}_{{\mathcal {X}}} &{} 0 &{} \cdots &{} \cdots &{} 0 \\ 0 &{} -\frac{1}{r}d^{*}_{{\mathcal {X}}} &{} -(d_{{\mathcal {E}}}+\frac{1}{r}d_{{\mathcal {X}}}^{*}) &{} 0 &{} 0 &{} \cdots &{} \vdots \\ \vdots &{} 0&{} -(d^{*}_{{\mathcal {E}}}+\frac{1}{r}d^{}_{{\mathcal {X}}}) &{} -(d^{}_{{\mathcal {E}}}+\frac{1}{r}d^{*}_{{\mathcal {X}}}) &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} 0 &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} 0 &{} -(d^{*}_{{\mathcal {E}}}+\frac{1}{r}d^{}_{{\mathcal {X}}}) &{} -(d_{{\mathcal {E}}}+\frac{1}{r}d^{*}_{{\mathcal {X}}}) &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} 0 &{} -\frac{1}{r}d_{{\mathcal {X}}} &{} -d_{{\mathcal {E}}}\\ -d_{{\mathcal {E}}} &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} -\frac{1}{r}d_{{\mathcal {X}}} \\ \end{array}\right] \end{aligned}$$

$$\begin{aligned} \mathbf B= & {} \left[ \begin{array}{cccccc} 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad \frac{k}{r}+\partial _{r} &{}\quad 0 \\ 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad \partial _{r} \\ \frac{1}{r}+\partial _{r} &{}\quad 0 &{}\quad \cdots &{}\quad \vdots &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \ddots &{}\quad \ddots &{}\quad 0 &{}\quad \vdots &{}\quad 0 \\ \vdots &{}\quad \ddots &{}\quad &{}\quad 0 &{}\quad \vdots &{}\quad 0 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad \frac{k-1}{r}+\partial _{r} &{}\quad 0 &{}\quad 0 \\ \end{array}\right] \\ \mathbf C= & {} \left[ \begin{array}{cccccc} 0 &{}\quad 0 &{}\quad \frac{1}{r}-\partial _{r} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad \frac{k-2}{r}-\partial _{r} \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ \frac{k}{r}-\partial _{r} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad -\partial _{r} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \end{array}\right] \\ \mathbf D= & {} \left[ \begin{array}{ccccccc} d_{{\mathcal {E}}}+d^{*}_{{\mathcal {E}}} &{}\quad \frac{1}{r}d^{*}_{{\mathcal {X}}} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad \frac{1}{r}d_{{\mathcal {X}}} \\ \frac{1}{r}d_{{\mathcal {X}}} &{}\quad d_{{\mathcal {E}}}+d^{*}_{{\mathcal {E}}} &{}\quad \frac{1}{r}d^{*}_{{\mathcal {X}}} &{}\quad 0 &{}\quad \cdots &{}\quad \vdots &{}\quad 0 \\ 0 &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots &{}\quad 0 \\ \vdots &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \vdots &{}\quad \vdots \\ \vdots &{}\quad \cdots &{}\quad \frac{1}{r}d_{{\mathcal {X}}} &{}\quad d_{{\mathcal {E}}}+d^{*}_{{\mathcal {E}}} &{}\quad \frac{1}{r}d^{*}_{{\mathcal {X}}}&{}\quad 0 &{}\quad \vdots \\ \vdots &{}\quad \cdots &{}\quad 0 &{}\quad \frac{1}{r}d_{{\mathcal {X}}} &{}\quad d_{{\mathcal {E}}} &{}\quad 0 &{}\quad \vdots \\ \vdots &{}\quad \cdots &{}\quad \cdots &{}\quad 0&{}\quad \frac{1}{r}d_{{\mathcal {X}}} &{}\quad d_{{\mathcal {E}}} &{}\quad \vdots \\ 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad d^{*}_{{\mathcal {E}}}&{}\quad \frac{1}{r}(d_{{\mathcal {X}}}+d^{*}_{{\mathcal {X}}}) &{}\quad 0 \\ \frac{1}{r}d^{*}_{{\mathcal {X}}} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad d_{{\mathcal {E}}}+d^{*}_{{\mathcal {E}}} \end{array}\right] . \end{aligned}$$

Observe that each of the elements in these matrices is an element of the algebra $\text {Diff}^{1}_{\text {edge}}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M)$. From this, it follows that $d+d^{*}$ belongs to $\text {Diff}^{1}_{\text {edge}}\left( \bigwedge ^{\bullet }T^{*}_{\wedge }M\right)$ which implies that $\text {DP}[0]$ extends to a continuous linear operator between the corresponding edge-Sobolev spaces. $\square$

In order to verify some properties of the symbolic structure of $d+d^{*}$ it shall be useful to have similar explicit expressions for the Hodge-Laplacian associated with the edge metric $g_{_{M}}$ acting on $\bigwedge ^{\bullet }T^{*}_{\wedge }M$.

Proposition 5.2

The Hodge–Laplace operator $\Delta _{g_{_{M}}}\Big |_{C^{\infty }_{0}(M,T^{*}_{\wedge }M)}$ extends to a continuous linear operator acting between edge-Sobolev spaces

$$\begin{aligned} \begin{array}{cccc} \Delta _{g_{_{M}}}:&{\mathcal {W}}^{s,\gamma }_{}\left( M,\bigwedge ^{\bullet }T^{*}_{\wedge }M\right)\longrightarrow & {} {\mathcal {W}}^{s-2,\gamma -2}_{}\left( M,\bigwedge ^{\bullet }T^{*}_{\wedge }M\right) . \end{array} \end{aligned}$$

Proof

By arranging the components of $\Xi \in C^{\infty }_{0}(M,\bigwedge ^{k}T^{*}_{\wedge }M)$ in the same way as in the previous proposition, a direct computation shows that the Hodge-Laplacian acting on k-forms

$$\begin{aligned} \Delta _{g_{_{M}}}:C^{\infty }_{0}\left( M,\bigwedge \nolimits ^{k}T^{*}_{\wedge }M\right) \longrightarrow C^{\infty }_{0}\left( M,\bigwedge \nolimits ^{k}T^{*}_{\wedge }M\right) \end{aligned}$$

is given by

$$\begin{aligned} \Delta _{g_{_{M}}}= \left[ \begin{array}{c|c} \mathbf A ' &{} \mathbf B ' \\ \hline \mathbf C ' &{}\mathbf D ' \end{array} \right] , \end{aligned}$$

where the operator matrices are given by

Again each of the operators in the matrices is an element of the algebra $\text {Diff}^{2}_{\text {edge}}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M)$, hence the result follows.

5.1 Symbolic structure of edge-degenerate differential operators

In this section we collect the basic facts about the symbolic structure of edge-degenerate differential operators the we use in the following sections. The reader is referred to [33, chapters 2 and 3] for a full presentation.

Let’s consider a cone-degenerate differential operator (see (3.2))

$$\begin{aligned} \text {P}=r^{-l}\sum \limits _{i\le l}a_{i}(r)(-r\partial _{r})^{i}. \end{aligned}$$

As an element in $\text {Diff}(M)$ it has its classical principal homogeneous symbol given by

$$\begin{aligned} \sigma ^{l}(\text {P})(r,x,\rho ,\xi )=r^{-l}\sum \limits _{i+\left| \alpha \right| = l}a_{i,\alpha }(r,x)(- \sqrt{-1} r\rho )^{i}\xi ^{\alpha } \end{aligned}$$

with smooth coefficients $a_{i,\alpha }(r,x)$ up to $r=0$. In order to reflect the singular behavior on the symbolic structure two additional symbols are introduced. First we have the homogeneous boundary symbol

$$\begin{aligned} \sigma ^{l}_{b}(\text {P})(r,x,\tilde{\rho },\xi ):=\sum \limits _{i+\left| \alpha \right| = l}a_{i,\alpha }(r,x)(- \sqrt{-1} \tilde{\rho })^{i}\xi ^{\alpha } \end{aligned}$$

defined on the cotangent bundle of the stretched manifold $T^{*}\mathbb {M}$ and smooth up to $r=0$.

The second symbol is the Mellin symbol $\sigma ^{l}_{M}(\text {P})$. Recall that any cone-degenerate operator $\text {P}=r^{-l}\sum \nolimits _{i\le l}a_{i}(r)(-r\partial _{r})^{i}$ is given in terms of the Mellin transformation as follows

$$\begin{aligned} \text {P}=r^{-l}{\mathcal {M}}^{-1}h(r,z){\mathcal {M}}, \end{aligned}$$

where $h(r,z)=\sum \nolimits _{i\le l}a_{i}(r)z^{i}.$ At the singular set i.e. when $r=0$ we have a holomorphic family of operators

$$\begin{aligned} h(0,z):{\mathbb {C}}\longrightarrow {\mathcal {L}}\left( H^{s}({\mathcal {X}}),H^{s-l}({\mathcal {X}})\right) . \end{aligned}$$

This holomorphic operator-valued function is the Mellin symbol of $\text {P}$ i.e. $\sigma ^{l}_{M}(\text {P})(z):=h(0,z).$

Now consider an edge-degenerate differential operator (see (3.4) above)

$$\begin{aligned} \text {P}=r^{-l}\sum \limits _{i+\left| \alpha \right| \le l}a_{i,\alpha }(r,u)(-r\partial _{r})^{i}(rD_u)^{\alpha }. \end{aligned}$$

In the same way as the cone-degenerate case it has a classical homogeneous principal symbol and a homogeneous boundary symbol

$$\begin{aligned} \sigma ^{l}(\text {P})(r,x,u,\rho ,\xi ,\eta )= & {} r^{-l}\sum \limits _{i+\left| \alpha \right| +\left| \beta \right| = l}a_{i,\alpha ,\beta }(r,x,u)(-\sqrt{-1} r\rho )^{i}(r\eta )^{\alpha }\\ \sigma ^{l}_{b}(\text {P})(r,x,u,\tilde{\rho },\xi ,\tilde{\eta })= & {} \sum \limits _{i+\left| \alpha \right| +\left| \beta \right| = l}a_{i,\alpha ,\beta }(r,x,u)(-\sqrt{-1} \tilde{\rho })^{i}\tilde{\eta }^{\alpha } \end{aligned}$$

satisfying the relation

$$\begin{aligned} \sigma ^{l}_{b}(\text {P})(r,x,u,\rho ,\xi ,\eta )=r^{l}\sigma ^{l}(\text {P})(r,x,r^{-1}\rho ,\xi ,r^{-1}\eta ). \end{aligned}$$

In the edge setting we have the edge symbol $\sigma ^{l}_{\wedge }(\text {P})$ that is defined as an operator-valued function acting on spaces one level below in the singular hierarchy, in this case those operators act on the cone-Sobolev spaces ${\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ as cone-degenerate operators. More precisely we have

$$\begin{aligned} \sigma ^{l}_{\wedge }(\text {P}):T^{*}{\mathcal {E}}{\setminus }\{ 0\}\longrightarrow {\mathcal {L}}\left( {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }) ,{\mathcal {K}}^{s-l,\gamma -l}({\mathcal {X}}^{\wedge }) \right) \end{aligned}$$

given by

$$\begin{aligned} \sigma ^{l}_{\wedge }(\text {P})(u,\eta )=r^{-l}\sum \limits _{i+\left| \alpha \right| \le l}a_{i,\alpha }(0,u)(-r\partial _{r})^{i}(r\eta )^{\alpha }. \end{aligned}$$

This is a family of cone-degenerate operators parametrized by the cotangent bundle of the edge ${\mathcal {E}}$.

5.2 Ellipticity on manifolds with conical and edge singularities

In this subsection we present the definition and principal implications of ellipticity for cone and edge-degenerate differential operators. We use these results to study the regularity of our moduli spaces in Sect. 7. Further details can be found in [33, chapter 2 and 3].

Definition 5.3

A cone-degenerate operator of order l, $\text {P}\in \text {Diff}^{l}_{\text {cone}}(M)$, is called elliptic with respect to the weight $\gamma$ if

• (i) $\sigma ^{l}_{b}(\text {P})(\tilde{\rho },\xi )\ne 0$ on $T^{*}_{}\mathbb {M}{\setminus }\lbrace 0 \rbrace$

• (ii) the Mellin symbol $\sigma ^{l}_{M}(\text {P})(z)=h(0,z)$ defines a family of Banach space isomorphisms in ${\mathcal {L}}\left( H^{s}({\mathcal {X}}),H^{s-l}({\mathcal {X}})\right)$ for some $s\in {\mathbb {R}}$ and all $z\in \Gamma _{\frac{\text {dim}{\mathcal {X}}+1}{2}-\gamma }$.

Definition 5.4

If $\gamma \in {\mathbb {R}}$ such that the Mellin symbol is invertible along $\Gamma _{\frac{\text {dim}{\mathcal {X}}+1}{2}-\gamma }$ we say that $\gamma$ is an admissible weight.

Now we have some important implications of being elliptic.

Theorem 5.5

Let $\text {P}$ be a cone-degenerate operator of order l.

• (i) (Fredholm property) $\text {P}$ is elliptic with respect to the weight $\gamma$ if and only if

  $$\begin{aligned} \text {P}: {\mathcal {H}}^{s,\gamma }(M)\longrightarrow {\mathcal {H}}^{s-l,\gamma -l}(M) \end{aligned}$$

  is Fredholm for every $s\in {\mathbb {R}}$.

• (ii) (Parametrix) If $\text {P}$ is elliptic then there exists a parametrix of order $-l$ with asymptotics.

• (iii) (Elliptic regularity) If $\text {P}$ is elliptic with respect to $\gamma$ and $\text {P}f=g$ with $g\in {\mathcal {H}}^{s-l,\gamma -l}_{O}(M)$ and $f\in {\mathcal {H}}^{-\infty ,\gamma }(M)$ for some $s\in {\mathbb {R}}$ and some asymptotic type O associated with $\gamma -l$ then $f\in {\mathcal {H}}^{s,\gamma }_{Q}(M)$ with an asymptotic type Q associated with $\gamma$.

Remark 5.6

Let’s consider a cone-degenerate operator $\text {P}$ and $f\in {\mathcal {H}}^{s,\gamma }(M)$ such that $\text {P}f=0$. The parametrix with asymptotics implies that $({\mathfrak {P}}\text {P}-I)(f)$ $=f\in {\mathcal {H}}^{\infty ,\gamma }_{O}(M)$. Therefore solutions of cone-degenerate differential equations are smooth on the regular part of M and have conormal asymptotics expansions near the vertex of the cone.

In order to introduce the notion of ellipticity in the edge singular setting we need to make some assumptions. Assume that there exist vector bundles $J^{+}$ and $J^{-}$ over ${\mathcal {E}}$ and operator families parametrized by $T^{*}{\mathcal {E}}{\setminus }\{ 0\}$ acting as follows

$$\begin{aligned}&\sigma ^{l}_{\wedge }(\text {T})(u,\eta ):{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })\longrightarrow J^{+}_{u},\\&\sigma ^{l}_{\wedge }(\text {C})(u,\eta ): J^{-}_{u} \longrightarrow {\mathcal {K}}^{s-l,\gamma -l}({\mathcal {X}}^{\wedge }),\\&\sigma ^{l}_{\wedge }(\text {B})(u,\eta ): J^{-}_{u} \longrightarrow J^{+}_{u}; \end{aligned}$$

such that

$$\begin{aligned} \begin{array}{cccc} \begin{bmatrix} \sigma ^{1}_{\wedge }(\text {P}) (u,\eta ) &{}\quad \sigma ^{l}_{\wedge }(\text {C})(u,\eta ) \\ \sigma ^{l}_{\wedge }(\text {T})(u,\eta ) &{}\quad \sigma ^{l}_{\wedge }(\text {B})(u,\eta ) \end{bmatrix}: &{} \begin{array}{c} {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }) \\ \oplus \\ J^{-}_{u} \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {K}}^{s-l,\gamma -l}({\mathcal {X}}^{\wedge }) \\ \oplus \\ J^{+}_{u} \end{array} \end{array} \end{aligned}$$

(5.2)

is a family of continuous operators for every $(u,\eta )\in T^{*}{\mathcal {E}}{\setminus }\{ 0\}.$

The existence of the vector bundles $J^{\pm }$ and operators acting between the fibers and cone-Sobolev spaces will be discussed in Sect. 5.3. Here we only mention that there is a topological obstruction (see Theorem 5.12) that must be satisfied in order to guarantee the existence of $J^{\pm }$ and the operators.

Definition 5.7

An edge-degenerate differential operator $\text {P}\in \text {Diff}^{l}_{\text {edge}}(M)$ of order l for which (5.2) exists, is called elliptic with respect to the weight $\gamma$ if

• (i) $\sigma ^{l}_{b}(\text {P})\ne 0$ on $T^{*}_{}\mathbb {M}{\setminus }\lbrace 0 \rbrace$

• (ii) the operator matrix (5.2) defines an invertible operator for some $s\in {\mathbb {R}}$ and each $(u,\eta )\in T^{*}{\mathcal {E}}{\setminus }\{0\}$.

Theorem 5.8

Let $\text {P}$ be an edge-degenerate operator of order l.

• (i) (Fredholm property) $\text {P}$ is elliptic with respect to the weight $\gamma$ if and only if the operator

  $$\begin{aligned} \mathbf {A}_{_{\text {P}}}:=\begin{bmatrix} \text {P}&\quad \text {C} \\ {\mathcal {T}}&\quad \text {B} \end{bmatrix}={\mathcal {F}}^{-1}_{\eta \rightarrow u} \begin{bmatrix} \sigma ^{1}_{\wedge }(\text {P}) (u,\eta )&\quad \sigma ^{l}_{\wedge }(\text {C})(u,\eta ) \\ \sigma ^{l}_{\wedge }(\text {T})(u,\eta )&\quad \sigma ^{l}_{\wedge }(\text {B})(u,\eta ) \end{bmatrix}{\mathcal {F}}_{u' \rightarrow \eta } \end{aligned}$$

  (5.3)

  acting on the spaces

  $$\begin{aligned} \mathbf {A}_{\text {P}}: \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M) \\ \oplus \\ H^{s}({\mathcal {E}},J^{-}) \end{array}\longrightarrow \begin{array}{c} {\mathcal {W}}^{s-l,\gamma -l}(M) \\ \oplus \\ H^{s-l}({\mathcal {E}},J^{+}) \end{array} \end{aligned}$$

  is Fredholm for every $s\in {\mathbb {R}}$.

• (ii) (Parametrix) If $\text {P}$ is elliptic then there exists a parametrix of order $-l$ with asymptotics for $\mathbf {A}_{\text {P}}$.

• (iii) (Elliptic regularity) If $\text {P}$ is elliptic with respect to $\gamma$ and $\mathbf {A}_{\text {P}}f=g$ with

  $$\begin{aligned} g\in \begin{array}{c} {\mathcal {W}}^{s-l,\gamma -l}_{O}(M) \\ \oplus \\ H^{s-l}({\mathcal {E}},J^{+}) \end{array}\quad \text {and}\quad f\in \begin{array}{c} {\mathcal {W}}^{-\infty ,\gamma }_{}(M) \\ \oplus \\ H^{-\infty }({\mathcal {E}},J^{-}) \end{array} \end{aligned}$$

  for some $s\in {\mathbb {R}}$ and some asymptotic type O associated with $\gamma -l$ then

  $$\begin{aligned} f\in \begin{array}{c} {\mathcal {W}}^{s,\gamma }_{Q}(M) \\ \oplus \\ H^{s}({\mathcal {E}},J^{-}) \end{array} \end{aligned}$$

  with an asymptotic type Q associated with $\gamma$.

Remark 5.9

Analogously to Remark 5.6 we have that solutions to the edge-degenerate equation $\mathbf {A}_{\text {P}}f=0$ belong to ${\mathcal {W}}^{\infty ,\gamma }_{O}(M)$, hence they are smooth and have conormal asymptotics near the edge ${\mathcal {E}}$.

5.3 The symbolic structure of DP[0]

Recall from Sect. 5.1 that the symbolic structure of the edge-degenerate operator $\text {DP}[0]$ is given by the pair

$$\begin{aligned} \left( \begin{array}{cc} \sigma ^{1}_{b}\left( \text {DP}[0]\right) (r,\sigma ,u,\tilde{\rho },\xi ,\tilde{\eta }),&\sigma ^{1}_{\wedge } \left( \text {DP}[0]\right) (u,\eta ) \end{array} \right) \end{aligned}$$

where $\sigma ^{1}_{b}\left( \text {DP}[0]\right)$ is a bundle map on $\pi ^{*}_{_{T^{*}\mathbb {M}}}(\bigwedge ^{\bullet }T^{*}_{\wedge }\mathbb {M}).$

The edge symbol $\sigma ^{1}_{\wedge }\left( \text {DP}[0]\right)$ is a family of continuous linear operators acting on cone-Sobolev spaces and parametrized by the cosphere bundle over ${\mathcal {E}}$:

$$\begin{aligned} \sigma ^{1}_{\wedge }\left( \text {DP}[0]\right) :S^{*}{\mathcal {E}}\longrightarrow {\mathcal {L}}\left( {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M), {\mathcal {K}}^{s-1,\gamma -1}({\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)\right) . \end{aligned}$$

The ellipticity of the operator $\text {DP}[0]$ shall require the invertibility of its symbolic structure. From Propositions 5.1 and 5.2 we obtain the first part of the desired result.

Proposition 5.10

$$\begin{aligned} \sigma ^{1}_{b}(\text {DP}[0])(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta }): \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma , u)}\mathbb {M}\longrightarrow \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma , u)}\mathbb {M}\end{aligned}$$

is a bundle isomorphism for every non-zero $(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })\in T^{*}\mathbb {M}$ up to $r=0.$

Proof

To prove this result we shall use the symbolic structure of the Hodge–Laplace operator

$$\begin{aligned} \left( \begin{array}{cc} \sigma ^{2}_{b}(\Delta _{g_{_{M}}}) (r,\sigma ,u,\tilde{\rho },\xi ,\tilde{\eta }),&\sigma ^{2}_{\wedge }(\Delta _{g_{_{M}}}) (u,\eta ) \end{array} \right) . \end{aligned}$$

From [33, theorem 3.4.56] we have the natural expected symbolic relation

$$\begin{aligned} \sigma ^{2}_{b}(\Delta _{g_{_{M}}})= & {} \sigma ^{1}_{b}(\text {DP}[0])\circ \sigma ^{1}_{b}(\text {DP}[0]) \end{aligned}$$

(5.4)

$$\begin{aligned} \sigma ^{2}_{\wedge }(\Delta _{g_{_{M}}})= & {} \sigma ^{1}_{\wedge }(\text {DP}[0])\circ \sigma ^{1}_{\wedge }(\text {DP}[0]) \end{aligned}$$

(5.5)

as $\text {DP}[0]=d+d^{*}$ and $\Delta _{g_{M}}=(d+d^{*})\circ (d+d^{*})$. Now, from the matrices representing $\Delta _{g_{_{M}}}$ in Proposition 5.2 we have that the elements in $\mathbf B '$ and $\mathbf C '$ are operators of order 1 hence they do not intervene in the computation of $\sigma ^{2}_{b}(\Delta _{g_{M}})$. Hence let’s focus on the operators in $\mathbf A '$ and $\mathbf D '$.

Observe that for any $\alpha \in \bigwedge ^{\bullet }T^{*}_{\wedge }M$

$$\begin{aligned} \sigma ^{2}_{ b}(d^{*}_{{\mathcal {E}}}d_{{\mathcal {X}}}+d_{{\mathcal {X}}}d^{*}_{{\mathcal {E}}})(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })(\alpha )&= \tilde{\eta }^{*}\lrcorner (\xi \wedge \alpha )+\xi \wedge ( \tilde{\eta }^{*}\lrcorner \alpha )\\&=(\tilde{\eta }^{*}\lrcorner \xi )\wedge \alpha - \xi \wedge ( \tilde{\eta }^{*}\lrcorner \alpha )+ \xi \wedge ( \tilde{\eta }^{*}\lrcorner \alpha )\\&=0 \end{aligned}$$

as $\tilde{\eta }^{*}\in T{\mathcal {E}}$ and $\xi \in T^{*}{\mathcal {X}}.$ Moreover

$$\begin{aligned} \sigma ^{2}_{ b}(d^{*}_{{\mathcal {E}}}d_{{\mathcal {E}}})( \tilde{\eta })(\Theta _{{\mathcal {X}}})&= \tilde{\eta }^{*}\lrcorner (\tilde{\eta }\wedge \Theta _{{\mathcal {X}}})\\&=\Theta _{{\mathcal {X}}}+\tilde{\eta }^{}\wedge (\tilde{\eta }^{*}\lrcorner \Theta _{{\mathcal {X}}})\\&=\Theta _{{\mathcal {X}}} \end{aligned}$$

and in the same way

$$\begin{aligned} \sigma ^{2}_{ b}(d^{*}_{{\mathcal {X}}}d_{{\mathcal {X}}})( \xi )(\Theta _{{\mathcal {E}}})=\Theta _{{\mathcal {E}}}. \end{aligned}$$

Therefore $\sigma ^{2}_{b}(\Delta _{g_{_{M}}})(r,\sigma ,u,\tilde{\rho },\xi ,\tilde{\eta })$ is a diagonal matrix with entries given by

$$\begin{aligned}&\sigma ^{2}_{b}(\Delta _{{\mathcal {X}}}+d^{*}_{{\mathcal {E}}}d_{{\mathcal {E}}}-\partial ^{2}_{r})(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })=\left| \xi \right| _{g_{_{{\mathcal {X}}}}}^{2}+1+\left| \tilde{\rho }\right| ^{2}\\&\sigma ^{2}_{b}(\Delta _{{\mathcal {E}}}+d^{*}_{{\mathcal {X}}}d_{{\mathcal {X}}}-\partial ^{2}_{r}) (r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })=\left| \tilde{\eta }\right| _{g_{_{{\mathcal {E}}}}}^{2}+1+\left| \tilde{\rho }\right| ^{2} \end{aligned}$$

and

$$\begin{aligned} \sigma ^{2}_{b}(\Delta _{{\mathcal {X}}}+\Delta _{{\mathcal {E}}}-\partial ^{2}_{r})(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })=\left| \xi \right| _{g_{_{{\mathcal {X}}}}}^{2}+\left| \tilde{\eta }\right| _{g_{_{{\mathcal {E}}}}}^{2}+\left| \tilde{\rho }\right| ^{2}. \end{aligned}$$

Hence

$$\begin{aligned} \sigma ^{2}_{b}(\Delta _{g_{_{M}}})(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta }):\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\longrightarrow \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\end{aligned}$$

is an isomorphism for every non-zero $(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })\in T^{*}\mathbb {M}$ up to $r=0$. By (5.4) we have that

$$\begin{aligned} \sigma ^{1}_{b}(\text {DP}[0])(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta }): \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\longrightarrow \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\end{aligned}$$

has the same property. $\square$

In order to obtain information about the invertibility of the edge symbol $\sigma ^{1}_{\wedge }(\text {DP}[0])$ we will use Proposition 5.10 together with theorem 2.4.18 and theorem 3.5.1 in [33]. These theorems state the existence of admissible weights $\gamma \in {\mathbb {R}}$ such that $\sigma ^{1}_{\wedge }(\text {DP}[0])$ is a Fredholm operator on the corresponding cone-Sobolev spaces of any order. We adapt those theorems to our setting in the following result. Its proof follows immediately from theorem 2.4.18 and theorem 3.5.1 in [33].

Theorem 5.11

The condition that

$$\begin{aligned} \sigma ^{1}_{b}(\text {DP}[0])(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta }): \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\longrightarrow \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge ,(r,\sigma ,u)}\mathbb {M}\end{aligned}$$

is an isomorphism for every non-zero $(r,\sigma , u, \tilde{\rho },\xi , \tilde{\eta })\in T^{*}_{\wedge }\mathbb {M}$ up to $r=0$, implies that there exists a countable set $\Lambda \subset {\mathbb {C}}$, where $\Lambda \cap K$ is finite for every $K\subset \subset {\mathbb {C}}$, such that

$$\begin{aligned} \sigma ^{1}_{M}\left( \sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta ) \right) (z): H^{s}({\mathcal {X}}, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)\longrightarrow H^{s-1}({\mathcal {X}}, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \end{aligned}$$

is an isomorphism (invertible, linear, continuous operator) for every $z\in {\mathbb {C}}{\setminus }\Lambda$ and all $s\in {\mathbb {R}}$. This implies that there is a countable subset $D\subset {\mathbb {R}}$ given by $D=\Lambda \cap {\mathbb {R}}$, with the property that $D\cap \lbrace z : a\le \text {Re}z \le b \rbrace$ is finite for every $a\le b$, such that

$$\begin{aligned} \sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta ): {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)\longrightarrow {\mathcal {K}}^{s-1,\gamma -1}({\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \end{aligned}$$

is a family of Fredholm operators for each $\gamma \in {\mathbb {R}}{\setminus } D$ and $(u,\eta )\in S^{*}{\mathcal {E}}$ with $\eta \ne 0.$

Theorem 5.11 tell us that for an admissible weight $\gamma$, the wedge symbol $\sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta )$ defines a Fredholm operator for each $(u,\eta )\in S^{*}{\mathcal {E}}$. However, if we require the ellipticity of $\text {DP}[0]$ we need to have a family of invertible operators. In some cases this can be achieved by adding boundary and coboundary operators that defines an elliptic edge boundary value problem. This can be done in the following way.

For each $(u,\eta )\in S^{*}{\mathcal {E}}$ we have that $\sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta )$ is a Fredholm operator, then

$$\begin{aligned} \text {Ker}\left( \sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta )\right) \subset {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \end{aligned}$$

is a finite dimensional subspace. Let $N{(u,\eta )}={\text {dim Ker}}\left( \sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta )\right)$ and choose an isomorphism

$$\begin{aligned} \text {k}(u,\eta ):{\mathbb {C}}^{N(u,\eta )}\longrightarrow \text {Ker}\left( \sigma ^{1}_{\wedge }(\text {DP}[0])(u,\eta )\right) , \end{aligned}$$

then

$$\begin{aligned}&\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {k} \end{pmatrix}(y,\eta ):\\&\begin{array}{ccc} {\mathcal {K}}^{s,\gamma }\left( {\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) &{} &{} \\ \oplus &{} \longrightarrow &{} {\mathcal {K}}^{s-1,\gamma -1}\left( {\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ {\mathbb {C}}^{N(u,\eta )} &{} &{} \end{array} \end{aligned}$$

is a surjective operator. Now, because the set of surjective operators is an open set and the space $S^{*}{\mathcal {E}}$ is compact, there exists $N^{+}\in {\mathbb {N}}$ and $\text {c} \in {\mathcal {L}}\left( {\mathbb {C}}^{N^{+}}, {\mathcal {K}}^{s-1,\gamma -1}({\mathcal {X}}^{\wedge }, \bigwedge \nolimits ^{\bullet }\right. \left. T^{*}_{\wedge }M)\right)$ such that

$$\begin{aligned} \begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta ): \begin{array}{c} {\mathcal {K}}^{s,\gamma }\left( {\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ {\mathbb {C}}^{N^{+}} \end{array} \longrightarrow {\mathcal {K}}^{s-1,\gamma -1}\left( {\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \end{aligned}$$

(5.6)

is Fredholm and surjective for each $(y,\eta )\in S^{*}{\mathcal {E}}$ (see [33, theorem 1.2.30] for further details). Because $\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ is Fredholm and surjective we have that the kernel of $\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ has constant dimension equal to its index for every $(y,\eta )\in S^{*}{\mathcal {E}}$:

$$\begin{aligned} {\text {dim Ker}}\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )=\text {Ind}\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta ):=N^{-} \end{aligned}$$

for all $(u,\eta )\in S^{*}{\mathcal {E}}.$ The finite dimensional spaces $\text {Ker}\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ define a smooth vector bundle over $S^{*}{\mathcal {E}}$ (see section 1.2.4 in [33]).

Now consider the trivial bundle of dimension $N^{+}$ over $S^{*}{\mathcal {E}}$, here we denote it simply as ${\mathbb {C}}^{N^{+}}.$ The formal difference of these vector bundles defines an element in the K-theory of $S^{*}{\mathcal {E}}$

$$\begin{aligned} \left[ \text {Ker} \begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix} \right] -\left[ {\mathbb {C}}^{N^{+}} \right] \in K(S^{*}{\mathcal {E}}). \end{aligned}$$

This element in the K-group represents a topological obstruction to the existence of an elliptic edge boundary value problem for the operator $\text {DP}[0]$. More precisely we have the following theorem. For its proof and more details about the obstruction of ellipticity in the edge calculus see [29, section 6.2].

Theorem 5.12

A necessary and sufficient condition for the existence of an elliptic edge problem for $\text {DP}[0]$ is given by

$$\begin{aligned} \left[ \text {Ker} \begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix} \right] -\left[ {\mathbb {C}}^{N^{+}} \right] \in \pi ^{*}_{_{S^{*}{\mathcal {E}}}}K({\mathcal {E}}) \end{aligned}$$

(5.7)

where $\pi ^{}_{_{S^{*}{\mathcal {E}}}}:S^{*}{\mathcal {E}}\longrightarrow {\mathcal {E}}$ is the natural projection and $\pi ^{*}_{_{S^{*}{\mathcal {E}}}}K({\mathcal {E}})$ is the subgroup of $K(S^{*}{\mathcal {E}})$ generated by vector bundles lifted from ${\mathcal {E}}$ by means of $\pi ^{}_{_{S^{*}{\mathcal {E}}}}$.

Now, assume for the moment that the condition in Theorem 5.12 is satisfied. Then the bundle defined by $\text {Ker} \begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ is stably equivalent to a vector bundle $J^{-}$ lifted from ${\mathcal {E}}$. Then, by adding zeros to $\text {c}$ if needed, we can assume that the vector bundle given by $\text {Ker} \begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ is isomorphic to $J^{-}.$ By extending this isomorphism by zero on the orthogonal complement of $\text {Ker}\begin{pmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0])&\text {c} \end{pmatrix}(y,\eta )$ we obtain a map

$$\begin{aligned} \begin{pmatrix} \text {t}(u,\eta )&\text {b}(u,\eta ) \end{pmatrix}: \begin{array}{c} {\mathcal {K}}^{s,\gamma }\left( {\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ {\mathbb {C}}^{N^{+}} \end{array} \longrightarrow J^{-}_{(u,\eta )} \end{aligned}$$

(5.8)

such that

$$\begin{aligned}&\begin{bmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0]) (u,\eta )&\quad \text {c}(u,\eta ) \\ \text {t}(u,\eta )&\quad \text {b}(u,\eta ) \end{bmatrix}:\\&\begin{array}{ccc} \begin{array}{c} {\mathcal {K}}^{s,\gamma }\left( {\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ {\mathbb {C}}^{N^{+}} \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {K}}^{s-1,\gamma -1}\left( {\mathcal {X}}^{\wedge },\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ J^{-}_{(u,\eta )} \end{array} \end{array} \end{aligned}$$

is an invertible, linear operator for every $\eta \ne 0$.

Then, the operator

$$\begin{aligned} \mathbf {A}_{_{\text {DP}[0]}}=\begin{bmatrix} \text {DP}[0]&\quad \text {C} \\ {\mathcal {T}}&\quad \text {B} \end{bmatrix}={\mathcal {F}}^{-1}_{\eta \rightarrow u} \begin{bmatrix} \sigma ^{1}_{\wedge }(\text {DP}[0]) (u,\eta )&\quad \text {c}(u,\eta ) \\ \text {t}(u,\eta )&\quad \text {b}(u,\eta ) \end{bmatrix}{\mathcal {F}}_{u' \rightarrow \eta } \end{aligned}$$

acting on the spaces

$$\begin{aligned} \begin{array}{cccc} \begin{bmatrix} \text {DP}[0] &{}\quad \text {C} \\ {\mathcal {T}} &{} \quad \text {B} \end{bmatrix}: &{} \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array} \end{array} \end{aligned}$$

is an elliptic edge operator ( see Definition 5.7) for all $s\in {\mathbb {R}}$ and $\gamma$ the admissible weight chosen at the beginning.

In order to prove the claim that condition (5.7) is satisfied we have the following theorem which is contained in [29, theorem 6.30].

Theorem 5.13

If the Atiyah-Bott obstruction vanishes for an edge-degenerate operator $\text {A}$ on the stretched manifold $\mathbb {M}$, then there exists an elliptic edge problem for $\text {A}$.

In our case $\text {DP}[0]$ is the Hodge–deRham operator, the vanishing of the Atiyah-Bott obstruction for this operator was proved by Atiyah et al. [4], hence the topological condition (5.7) is satisfied.

6 Conormal deformations and regularity

6.1 Conormal asymptotics

In this subsection we recall the basic facts of conormal asymptotics. For a complete presentation see [33, section 2.3].

A sequence $O= \lbrace (p_j , m_j) \rbrace _{j\in {\mathbb {N}}}$ in ${\mathbb {C}}\times {\mathbb {Z}}^{+}$ is called an asymptotic type for the weight data $\gamma \in {\mathbb {R}}$ if

$$\begin{aligned} \text {Re}p_j <\frac{\text {dim}{\mathcal {X}}+1}{2}-\gamma \end{aligned}$$

and $\text {Re}p_j \rightarrow -\infty$ when $j\rightarrow \infty .$

Definition 6.1

Let $O= \lbrace (p_j , m_j) \rbrace _{j\in {\mathbb {N}}}$ be an asymptotic type for the weight $\gamma \in {\mathbb {R}}.$ The cone-Sobolev space with conormal asymptotics O, denoted by ${\mathcal {K}}^{s,\gamma }_{O}({\mathcal {X}}^{\wedge }),$ is defined as the set of all $f\in {\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ such that for every $l\in {\mathbb {N}}$ there is $N(l)\in {\mathbb {N}}$ such that

$$\begin{aligned} f(r,\sigma ,y)-\omega (r)\sum \limits _{j=0}^{N(l)}\sum \limits _{k=0}^{m_j} c_{j,k}(\sigma ) r^{-p_j}\text {log}^{k}(r)\in {\mathcal {K}}^{s, \gamma +l}({\mathcal {X}}^{\wedge }) \end{aligned}$$

with $c_{j,k}(\sigma )\in C^{\infty }({\mathcal {X}}).$

The space ${\mathcal {K}}^{s,\gamma }_{O}({\mathcal {X}}^{\wedge })$ has the structure of a Fréchet space given as an inductive limit of spaces with asymptotics of finite type ${\mathcal {K}}^{s,\gamma }_{O_k}({\mathcal {X}}^{\wedge })$ where $O_k=\lbrace (p_j,m_j)\in O: \frac{\text {dim}{\mathcal {X}}+1}{2}-\gamma -k<\text {Re}p_j< \frac{\text {dim}{\mathcal {X}}+1}{2}-\gamma \rbrace$, see [9, sec. 8.1.1] for details. By using this inductive limit structure we define the edge-Sobolev space with conormal asymptotics O as the inductive limit of Fréchet spaces

$$\begin{aligned} {\mathcal {W}}^{s,\gamma }\left( {\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }_{O}({\mathcal {X}}^{\wedge })\right) :=\varprojlim \limits _{k} {\mathcal {W}}^{s,\gamma }\left( {\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }_{O_k}({\mathcal {X}}^{\wedge })\right) . \end{aligned}$$

In particular, if $f\in {\mathcal {W}}^{\infty ,\gamma }_O (M)$ then for every $l\in {\mathbb {N}}$ there is $N(l)\in {\mathbb {N}}$ such that

$$\begin{aligned} f(r,\sigma ,y)-\omega (r)\sum \limits _{j=0}^{N(l)}\sum \limits _{k=0}^{m_j} c_{j,k}(\sigma ) v_{j,k}(y) r^{-p_j}\text {log}^{k}(r)\in {\mathcal {W}}^{\infty , \gamma +l}(M) \end{aligned}$$

with $c_{j,k}(\sigma )\in C^{\infty }({\mathcal {X}})$ and $v_{j,k}(y)\in H^{\infty }({\mathcal {E}})$, see [33, proposition 3.1.33].

6.2 Conormal asymptotic embeddings

Given a special Lagrangian submanifold of ${\mathbb {C}}^{n}$ with edge singularity, $(M,\Phi )$ (see 3.1), in this section we define the moduli space of special Lagrangian deformations of $(M,\Phi )$. Broadly speaking, we want to have in the moduli space all nearby special Lagrangian submanifolds with edge singularity. This rough idea has two aspects that must be considered for the moduli space. First, as the manifold M is non-compact, the important aspect to consider when defining the concept of nearby submanifold is the behavior on the collar neighborhood $(0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}.$ Here we shall define the concept of nearby submanifold by means of its asymptotic behavior with respect to the conormal variable r and weight $\gamma$. Second, the property of being special Lagrangian is completely determined by the equations (2.5). As we mentioned in Remarks 5.6 and 5.9 in Sect. 5.2, solutions of the linearised deformation equation have conormal asymptotics (Sect. 6.1). This asymptotic behavior is transferred to the induced metric of the deformed submanifold making the induced metric asymptotic to the original edge metric $g_{_{M}}$ in a very special way that reflects the fact it comes from the solution of an edge-degenerate PDE on a singular space. All of these considerations are formalized in the following definition.

Definition 6.2

Given an embedding $\Upsilon : M\longrightarrow {\mathbb {C}}^{n}$ we say that $\Upsilon$ is conormal asymptotic to $(M,\Phi )$ with rate $\gamma$ if:

• (i) For every multi-index $\alpha$ we have

  $$\begin{aligned} \left| \partial ^{\alpha }_{(r,\sigma ,u)}\left( \Upsilon (r,\sigma ,u) -\Phi (r,\sigma ,u)\right) \right| =O(r^{\gamma - \left| \alpha \right| })\quad \forall (r,\sigma ,u)\in (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}; \end{aligned}$$

• (ii) $\Upsilon ^{*}g_{_{{\mathbb {C}}^{n}}}=r^{2}g_{_{{\mathcal {X}}}}+dr^{2}+g_{_{{\mathcal {E}}}}+\beta$ where $\beta$ is a symmetric 2-tensor on $\Upsilon (M)=M_{\Upsilon }$ such that their components $\beta _{ij}$ have conormal asymptotic expansions on the collar neighborhood $(0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$ with respect to some asymptotic type associated to $\gamma$.

Because we want to describe a small neighborhood of $(M,\Phi )$ in the moduli space by means of the Implicit Function Theorem 4.2 applied to a neighborhood of zero in edge-Sobolev spaces, we want to make sure that smooth elements in edge-Sobolev spaces with small norm will produce submanifolds. In order to show this, we will define a neighborhood of deformations i.e. we will define a small neighborhood of M in ${\mathbb {C}}^{n}$ such that small deformations will be inside this neighborhood. Because of the geometric singularities of the manifold M and the behavior of the elements in edge-Sobolev spaces, this neighborhood will be constructed as an edge neighborhood to guarantee that small submanifolds induced by edge-degenerate forms will fit inside it.

Proposition 6.3

There exists an open edge neighborhood of the zero section in the normal bundle ${\mathcal {N}}\left( (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}\right)$ such that it is given by $V\times W$ where $V\subset {\mathcal {N}}((0,\varepsilon )\times {\mathcal {X}})\subset T{\mathbb {R}}^{n}_x$ is an open conical set and $W\subset {\mathcal {N}}({\mathcal {E}})\subset T{\mathbb {R}}^{n}_y$ is an open set both of them being neighborhoods of the zero section in the corresponding normal bundles and diffeomorphic to an open edge set $\tilde{V}\times \tilde{W}\subset {\mathbb {R}}_{x}^{n}\oplus {\mathbb {R}}_{y}^{n}\cong {\mathbb {C}}^{n}$ with diffeomorphism given by the exponential map $\text {exp}_{g_{_{{\mathbb {C}}^{n}}}}$. Moreover, for every $\gamma >\frac{m+3}{2}$ and $s>\frac{q+m+1}{2}+ {\mathfrak {c}}_{\gamma }$, where ${\mathfrak {c}}_{\gamma }$ is the positive constant defined in (10.7), there exists $\vartheta >0$ depending on s and $\gamma$ such that

$$\begin{aligned} \left\{ {\mathcal {V}}_{\Xi }\big |_{(0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}}:\Xi \in {\mathcal {W}}^{s,\gamma }(M, T^{*}_{\wedge }M) \text { and } \left\| \Xi \right\| _{s,\gamma }<\vartheta \right\} \subset V\times W. \end{aligned}$$

Proof

First let’s define the conical open neighborhood $V\subset {\mathcal {N}}((0,\varepsilon )\times {\mathcal {X}})\subset T{\mathbb {R}}^{n}_x$. The tubular neighborhood Theorem 4.1 applied to ${\mathcal {X}}$ as a compact submanifold in ${\mathbb {R}}^{n}$ gives us an open neighborhood of the zero section in ${\mathcal {N}}({\mathcal {X}})$. Take $l_0>0$ to be the maximum l such that the uniform neighborhood $\lbrace X\in {\mathcal {N}}({\mathcal {X}}):\left| X\right| _{g_{_{{\mathbb {R}}^n}}}<l \rbrace$ is inside the tubular neighborhood. By applying the ${\mathbb {R}}^{+}$-action defined on the cone ${\mathcal {X}}^{\wedge }$ to this uniform neighborhood we can obtain the desired open conical neighborhood V of the zero section in the normal bundle ${\mathcal {N}}((0,\varepsilon )\times {\mathcal {X}})\subset T{\mathbb {R}}^{n}_x$. Now, for any section ${\mathcal {V}}$ of the normal bundle ${\mathcal {N}}({\mathcal {X}})$ lying in V we have $\left| {\mathcal {V}}(r,\sigma ,u)\right| _{g_{_{{\mathbb {C}}^n}}}< C_{1}\cdot r$ for all $(r,\sigma ,u)\in (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$, where the constant $C_1$ is independent of ${\mathcal {V}}$. The constant $C_1$ can be taken to be the maximum $l>0$ chosen above. Now choose a uniform tubular neighborhood of the zero section in the normal bundle ${\mathcal {N}}({\mathcal {E}})$ given by $\lbrace Y\in {\mathcal {N}}({\mathcal {E}}):\left| Y\right| _{g_{_{{\mathbb {R}}^n}}}<\vartheta \rbrace$ for some $\vartheta >0$. Clearly this is possible because ${\mathcal {E}}$ is compact. If necessary we can choose a smaller $\varepsilon$ such that $\vartheta >C_{1}\varepsilon$. Define W as this uniform neighborhood of the zero section $W=\lbrace Y\in {\mathcal {N}}({\mathcal {E}}):\left| Y\right| _{g_{_{{\mathbb {R}}^n}}}<\vartheta \rbrace$. Then $V\times W$ is our open edge neighborhood of the zero section in ${\mathcal {N}}\left( (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}\right)$.

To prove the second part of the proposition let $\Xi \in {\mathcal {W}}^{s,\gamma }(M, T^{*}_{\wedge }M)$ with $s>\frac{q+m+1}{2}+ {\mathfrak {c}}_{\gamma }$ and consider its local expression in a neighborhood $(0,\varepsilon )\times {\mathcal {U}}\times \Omega \subset (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$ given by

$$\begin{aligned} \Xi (r,\sigma ,u)={\mathcal {A}}(r,\sigma ,u)dr+\sum \limits _{k=1}^{m}{\mathcal {B}}_k(r,\sigma ,u)rd\sigma _k+\sum \limits _{l=1}^{q}{\mathcal {C}}_l(r,\sigma ,u)du_{l}, \end{aligned}$$

where $\omega \phi _{j}\varphi _{\lambda }{\mathcal {A}}$, $\omega \phi _{j}\varphi _{\lambda }{\mathcal {B}}_k$ and $\omega \phi _{j}\varphi _{\lambda }{\mathcal {C}}_l$ belong to ${\mathcal {W}}^{s,\gamma }(M)$ as in Lemma 4.5. Then by Proposition 10.6 there exists a constant $C>0$ depending only on s and $\gamma$ such that

$$\begin{aligned} \left| {\mathcal {A}}(r,\sigma ,u)\right|\le & {} C\left\| \Xi \right\| _{s,\gamma }r^{\gamma -\frac{m+1}{2}} \end{aligned}$$

(6.1)

$$\begin{aligned} \left| {\mathcal {B}}(r,\sigma ,u)\right|\le & {} C\left\| \Xi \right\| _{s,\gamma }r^{\gamma -\frac{m+1}{2}} \end{aligned}$$

(6.2)

$$\begin{aligned} \left| {\mathcal {C}}(r,\sigma ,u)\right|\le & {} C\left\| \Xi \right\| _{s,\gamma }r^{\gamma -\frac{m+1}{2}} \end{aligned}$$

(6.3)

for all $(r,\sigma ,u)\in (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$. Hence by Lemma 4.5 there exists a constant $C'$ depending only on s and $\gamma$ such that

$$\begin{aligned} \left| \tilde{{\mathcal {C}}}(r,\sigma ,u)\right|\le & {} C'\left\| \Xi \right\| _{s,\gamma }r^{\gamma -\frac{m+1}{2}} \end{aligned}$$

(6.4)

$$\begin{aligned} \left| {\mathcal {A}}(r,\sigma ,u)\theta _i+\tilde{{\mathcal {B}}_i}(r,\sigma ,u)\right|\le & {} C'\left\| \Xi \right\| _{s,\gamma }r^{\gamma -\frac{m+1}{2}}. \end{aligned}$$

(6.5)

Then, by (6.4 6.5) and because $0<\varepsilon <1$, we have that

$$\begin{aligned} \left| \tilde{{\mathcal {C}}}(r,\sigma ,u)\right| \le C_{1}r \end{aligned}$$

(6.6)

for all $(r,\sigma ,u)\in (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$ if

$$\begin{aligned} \gamma >\text {log}\left( \frac{C_1}{C'\left\| \Xi \right\| _{s,\gamma }} \right) \frac{1}{\text {log}(r)}+\frac{m+3}{2}. \end{aligned}$$

(6.7)

Note that if $\left\| \Xi \right\| _{s,\gamma }$ is small enough, then (6.7) is satisfied and this implies (6.6). More precisely, if $\frac{C_1}{C'}\ge \left\| \Xi \right\| _{s,\gamma }$ then (6.7) is satisfied as $\text {log}(r)<0$ for $r\le \varepsilon$. Therefore $\frac{C_1}{C'}\ge \left\| \Xi \right\| _{s,\gamma }$ implies

$$\begin{aligned} \left| \tilde{{\mathcal {C}}}(r,\sigma ,u)\right| \le C_{1}r. \end{aligned}$$

Analogously, $\frac{C_1}{C'}>\left\| \Xi \right\| _{s,\gamma }$ implies that

$$\begin{aligned} \left| {\mathcal {A}}(r,\sigma ,u)\theta _i+\tilde{{\mathcal {B}}_i}(r,\sigma ,u)\right| \le C_{1}r \end{aligned}$$

for any $\gamma >\frac{m+3}{2}$. Then it follows from our chose of $\vartheta$ that

$$\begin{aligned} \left| {\mathcal {A}}(r,\sigma ,u)\theta _i+\tilde{{\mathcal {B}}_i}(r,\sigma ,u)\right| \le \vartheta \end{aligned}$$

for all $(r,\sigma ,u)\in (0,\varepsilon )\times {\mathcal {X}}\times {\mathcal {E}}$. $\square$

Observe that the manifold $M{\setminus }\left( (0,\varepsilon )\times \partial \mathbb {M}\right)$ is compact hence we can extend our edge neighborhood $V\times W$ to this compact space to get a open neighborhood of the zero section in ${\mathcal {N}}(M)$ such that near the edge this neighborhood corresponds to the edge open neighborhood constructed above. We denote this neighborhood as ${\mathfrak {A}}$. Moreover this proposition implies that any smooth form $\Xi \in {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)$ as above with $\left\| \Xi \right\| _{s,\gamma }<\vartheta$ produces a smooth embedded submanifold inside the neighborhood of deformations ${\mathfrak {A}}$. This submanifold is defined by the embedding $\text {exp}_{g_{_{{\mathbb {C}}^{n}}}}({\mathcal {V}}_{\Xi })\circ \Phi$.

7 Regularity of deformations

Let’s consider an elliptic edge problem (see Sect. 5.3) for the operator $\text {DP}[0]$ acting on edge-Sobolev spaces with admissible weigh $\gamma >\frac{m+1}{2}$,

$$\begin{aligned} \mathbf {A}_{\text {DP}[0]}= & {} \begin{bmatrix} \text {DP}[0]&\quad \text {C} \\ {\mathcal {T}}&\quad \text {B} \end{bmatrix}:\\&\begin{array}{ccc} \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array} \end{array}. \end{aligned}$$

Then, by augmenting the deformation operator $\text {P}=\text {P}_{\omega _{_{{\mathbb {C}}^{n}}}}\oplus \text {P}_{\text {Im}\Omega }$ with the trace operator

$$\begin{aligned} {\mathcal {T}}: {\mathcal {W}}^{s,\gamma }\left( M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \longrightarrow H^{s-1}({\mathcal {E}}, J^{-}), \end{aligned}$$

we obtain a non-linear boundary value problem for $\text {P}$:

$$\begin{aligned} \begin{bmatrix} \text {P}_{\omega _{_{{\mathbb {C}}^{N}}}}\oplus P_{\text {Im}\Omega } \\ {\mathcal {T}} \end{bmatrix}: {\mathfrak {A}}\subset {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\longrightarrow \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array} \end{aligned}$$

whose linearisation at zero is given by

$$\begin{aligned} \begin{bmatrix} \text {DP}[0]\\ {\mathcal {T}} \end{bmatrix}: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\longrightarrow \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}\left( M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M\right) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array}. \end{aligned}$$

In this section we consider some properties of solutions of the equation

$$\begin{aligned} \begin{bmatrix} \text {P}_{\omega _{_{{\mathbb {C}}^{N}}}}\oplus \text {P}_{\text {Im}\Omega } \\ {\mathcal {T}} \end{bmatrix}\left( \Xi \right) =0 \end{aligned}$$

(7.1)

where $\Xi \in {\mathcal {W}}^{s,\gamma }_{}(M,T^{*}_{\wedge }M).$ We are mainly interested in those solutions given by the Implicit Function Theorem for Banach spaces (when applicable) i.e. we assume that $\Xi =\Xi _1+\Xi _2$ where $\Xi _1$ is solution of the linear boundary value problem

$$\begin{aligned} \begin{bmatrix} \text {DP}[0]\\ {\mathcal {T}} \end{bmatrix}(\Xi _1)=0 \end{aligned}$$

(7.2)

and $\Xi _2$ belongs to the Banach space complement in ${\mathcal {W}}^{s,\gamma }_{}(M,T^{*}_{\wedge }M)$ defined by a splitting (not unique) induced by the finite dimensional space $\text {Ker}\mathbf {A}_{\text {DP}[0]}$. First we have some straightforward observations. The ellipticity of the operator $\mathbf {A}_{\text {DP}[0]}$ (Theorem 5.13) and the fact that

$$\begin{aligned} \mathbf {A}_{\text {DP}[0]}\begin{bmatrix} \Xi _1 \\ 0 \end{bmatrix}=0 \end{aligned}$$

implies that $\Xi _1\in {\mathcal {W}}^{\infty ,\gamma }(M,T^{*}_{\wedge }M)$ by elliptic regularity (Theorem 5.8). Moreover, because ${\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\subset H^{s}_{loc}(M,T^{*}_{\wedge }M)$ for all $s\in {\mathbb {R}}$ ([9, section 9.3, proposition 5]), standard Sobolev embeddings (Theorem 10.1) imply that $\Xi _1$ is smooth. The ellipticity of $\mathbf {A}_{\text {DP}[0]}$ implies the existence of a parametrix $\text {B}_{\text {DP}[0]}$ with asymptotics O (Theorem 5.8) i.e.

$$\begin{aligned} \text {B}_{\text {DP}[0]}\mathbf {A}_{\text {DP}[0]}-\text {I}:\begin{array}{cccc} &{} \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {W}}^{\infty ,\gamma }_{O}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{\infty }({\mathcal {E}}, J^{-}) \end{array} \end{array}. \end{aligned}$$

Consequently, any element in the kernel of the operator $\mathbf {A}_{\text {DP}[0]}$ belongs to ${\mathcal {W}}^{\infty ,\gamma }_{O}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)$ for some asymptotic type O associated to $\gamma$. In particular

$$\begin{aligned} \Xi _1\in {\mathcal {W}}^{\infty ,\gamma }_{O}(M,T^{*}_{\wedge }M). \end{aligned}$$

(7.3)

Now let’s consider the regularity properties of $\Xi _2$.

Let $\Xi \in {\mathcal {W}}^{s,\gamma }_{}(M,T^{*}_{\wedge }M)$ such that

$$\begin{aligned} (\text {P}_{\omega _{_{{\mathbb {C}}^{n}}}}\oplus \text {P}_{\text {Im}\Omega })(\Xi )=0. \end{aligned}$$

(7.4)

Hence $\text {exp}_{g_{_{{\mathbb {C}}^{n}}}}({\mathcal {V}}_{\Xi })\circ \Phi :M\longrightarrow {\mathbb {C}}^{n}$ is a special Lagrangian submanifold. Harvey and Lawson pointed out in [10, theorem 2.7] that $C^{2}$ special Lagrangian submanifolds in ${\mathbb {C}}^{n}$ are real analytic, in particular they are smooth. Therefore, by choosing s large enough, (7.4) implies that $\Xi \in C^{\infty }(M,T^{*}_{\wedge }M)$ which, together with (7.3), allow us to conclude that $\Xi _2$ is smooth.

Even though $\Xi _1+\Xi _2$ is solution of the non-linear edge boundary value problem (7.1) we cannot conclude immediately that $\Xi$ has a conormal asymptotic expansion near the singular set ${\mathcal {E}}$. The edge calculus tell us that solutions of the linearised equation (7.2), here denoted by $\Xi _1$, have such asymptotics. It turns out that it is possible to prove that $\Xi _2$ also has a conormal expansion i.e. the whole solution of the non-linear edge boundary value problem has conormal expansion. In order to prove this we follow and adapt to our very specific setting in the next two propositions the general argument in [7, theorem 5.1]. The author thanks Frédéric Rochon for pointing out and explaining his work.

Observe that

$$\begin{aligned} \begin{bmatrix} \text {P}_{\omega _{_{{\mathbb {C}}^{N}}}}\oplus \text {P}_{\text {Im}\Omega } \\ {\mathcal {T}} \end{bmatrix}(\Xi _1+\Xi _2)= \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{aligned}$$

implies that ${\mathcal {T}}(\Xi _2)=0$ because ${\mathcal {T}}(\Xi _1)=0$ due to the fact that $\Xi _1$ is solution of the linearised equation (7.2). By writing the non-linear equation as $\text {P}_{\omega _{_{{\mathbb {C}}^{N}}}}\oplus \text {P}_{\text {Im}\Omega }=\text {DP}[0]+\text {Q}$ (see Proposition 4.7 and (4.11)) where $\text {Q}$ is a non-linear operator locally defined by the sum of products of 2 or more operators in $\text {Diff}^{1}_{\text {edge}}(M)$ acting on $\Xi _1$ or $\Xi _2$ we have

$$\begin{aligned} (\text {DP}[0]+\text {Q})(\Xi _2)=-\text {Q}(\Xi _1)-\sum \limits _{j\ge 2}\text {Q}_{i_{1}}(\Xi _{\bullet })\cdots \text {Q}_{i_{j}}(\Xi _{\bullet }). \end{aligned}$$

(7.5)

In order to avoid cumbersome notation to keep track of the specific asymptotic types, we say that an element belongs to ${\mathcal {W}}^{s,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$ if it belongs to the edge-Sobolev space with some asymptotic type associated to $\gamma$.

Proposition 7.1

Let $\xi _1\in {\mathcal {W}}^{\infty ,\gamma -1}_{\text {As}}(M,\bigwedge ^{\bullet } T^{*}_{\wedge }M)$ and $\Xi _2\in {\mathcal {W}}^{\infty ,\gamma }(M,T^{*}_{\wedge }M)$ such that

$$\begin{aligned} (\text {DP}[0]+\text {Q})(\Xi _2)=\xi _1. \end{aligned}$$

(7.6)

Assume that $\gamma$ is an admissible weight for $\text {DP}[0]$ and there exists $\beta >0$ such that $\text {Q}(\Xi _2)\in {\mathcal {W}}^{\infty ,\gamma +\beta }(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M)$ and $\gamma +\beta +1$ is an admissible weight. Then $\Xi _2=E_1+E_2$ with $E_2\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$ and $E_1\in {\mathcal {W}}^{\infty ,\gamma +\beta +1}(M,T^{*}_{\wedge }M).$

Proof

Let’s consider the Fredholm operator defined by $\text {DP}[0]$ acting on edge-Sobolev spaces with weight $\gamma +\beta +1$

$$\begin{aligned}&\mathbf {A}_{\text {DP}[0],\gamma +\beta +1}:\\&\begin{array}{ccc} \begin{array}{c} {\mathcal {W}}^{s+1,\gamma +\beta +1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s+1}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {W}}^{s,\gamma +\beta }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, J^{-}) \end{array} \end{array}. \end{aligned}$$

Because $\text {Coker}\mathbf {A}_{\text {DP}[0],\gamma +\beta +1}$ is finite dimensional and $C^{\infty }_{0}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)$ is a dense subset of the edge-Sobolev spaces we have

$$\begin{aligned} \begin{bmatrix} -\text {Q}(\Xi _2)\\ 0 \end{bmatrix}=\mathbf {A}_{\text {DP}[0],\gamma +\beta +1}\begin{bmatrix} E_1 \\ e_1 \end{bmatrix}+\begin{bmatrix} {\mathfrak {F}}\\ f \end{bmatrix} \end{aligned}$$

with

$$\begin{aligned} \begin{bmatrix} E_1 \\ e_1 \end{bmatrix}\in \begin{array}{c} {\mathcal {W}}^{s+1,\gamma +\beta +1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)\\ \oplus \\ H^{s+1}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} \end{aligned}$$

and ${\mathfrak {F}}\in C^{\infty }_{0}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M)$, $f\in C^{\infty }_{0}({\mathcal {E}},{\mathbb {C}}^{N^{+}}).$ Observe that this implies

$$\begin{aligned} \mathbf {A}_{\text {DP}[0],\gamma +\beta +1}\begin{bmatrix} E_1 \\ e_1 \end{bmatrix}\in \begin{array}{c} {\mathcal {W}}^{\infty ,\gamma +\beta }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{\infty }({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} \end{aligned}$$

as $\text {Q}(\Xi _2)\in {\mathcal {W}}^{\infty ,\gamma +\beta }(M,T^{*}_{\wedge }M)$.

Hence by elliptic regularity (Theorem 5.8)

$$\begin{aligned} \begin{bmatrix} E_1 \\ e_1 \end{bmatrix}\in \begin{array}{c} {\mathcal {W}}^{\infty ,\gamma +\beta +1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{\infty }({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array}. \end{aligned}$$

Now define $\begin{bmatrix} E_2 \\ e_2 \end{bmatrix}:=\begin{bmatrix} \Xi _2\\ 0 \end{bmatrix}-\begin{bmatrix} E_1 \\ e_1 \end{bmatrix}$, then by (7.6)

$$\begin{aligned} \mathbf {A}_{\text {DP}[0],\gamma }\begin{bmatrix} E_2 \\ e_2 \end{bmatrix}=\begin{bmatrix} \xi _1 -\text {Q}(\Xi _2)\\ 0 \end{bmatrix}-\mathbf {A}_{\text {DP}[0],\gamma } \begin{bmatrix} E_1 \\ e_1 \end{bmatrix}. \end{aligned}$$

Observe that as $\mathbf {A}_{\text {DP}[0],\gamma }$ and $\mathbf {A}_{\text {DP}[0],\gamma +\beta +1}$ are $2\times 2$ operator matrices with $\text {DP}[0]$ in the upper left corner they differ by Green operators with asymptotics acting on the corresponding spaces (see [33, theorem 3.4.3]). Hence we can write $\mathbf {A}_{\text {DP}[0],\gamma }=\mathbf {A}_{\text {DP}[0],\gamma +\beta +1}-G_{\text {DP}[0],\gamma +\beta +1}+G_{\text {DP}[0],\gamma }$, where $G_{\text {DP}[0],\gamma +\beta +1}$ is the Green operator matrix with the elliptic boundary conditions for $\text {DP}[0]$ acting on spaces with weight $\gamma +\beta +1$ and analogously for $G_{\text {DP}[0],\gamma }$. This implies

$$\begin{aligned} \mathbf {A}_{\text {DP}[0],\gamma }\begin{bmatrix} E_2 \\ e_2 \end{bmatrix}=\begin{bmatrix} \xi _1\\ 0 \end{bmatrix}+\begin{bmatrix} {\mathfrak {F}}\\ {f} \end{bmatrix}-(-G_{\text {DP}[0],\gamma +\beta +1}+G_{\text {DP}[0],\gamma }) \begin{bmatrix} E_1 \\ e_1 \end{bmatrix} \end{aligned}$$

therefore, by the mapping properties of Green operators (see [33, theorem 3.4.3]) we have

$$\begin{aligned} \mathbf {A}_{\text {DP}[0],\gamma }\begin{bmatrix} E_2 \\ e_2 \end{bmatrix}\in {\mathcal {W}}^{\infty ,\gamma -1}_{\text {As}} (M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M). \end{aligned}$$

By elliptic regularity we conclude

$$\begin{aligned} \begin{bmatrix} E_2 \\ e_2 \end{bmatrix}\in \begin{array}{c} {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,\bigwedge ^{\bullet } T^{*}_{\wedge }M) \\ \oplus \\ H^{\infty }({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} \end{aligned}$$

and $\Xi _2=E_1+E_2$ as claimed. $\square$

Proposition 7.2

Let $\Xi _1\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$ and $\Xi _2\in {\mathcal {W}}^{\infty ,\gamma }(M,T^{*}_{\wedge }M)$ with an admissible weight $\gamma > \frac{m+5}{2}$ such that

$$\begin{aligned} (\text {DP}[0]+\text {Q})(\Xi _1+\Xi _2)=0 \end{aligned}$$

(7.7)

and $\text {DP}[0](\Xi _1)=0.$ Then $\Xi _2\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M).$

Proof

Equation (7.7) can be written as

$$\begin{aligned} (\text {DP}[0]+\text {Q})(\Xi _2)=-\text {Q}(\Xi _1)-\sum \limits _{j\ge 2}\text {Q}_{i_{1}}(\Xi _{\bullet })\cdots \text {Q}_{i_{j}}(\Xi _{\bullet }), \end{aligned}$$

(7.8)

(see (7.5)). The right hand side of (7.8) consists of products where at least one of the operators in each of these products is acting on $\Xi _1$, let’s say $\text {Q}_{1}(\Xi _1)$. Now, the term $\text {Q}_{1}(\Xi _1)$ has asymptotics (associated to $\gamma -1$) and the other elements $\text {Q}_{i_{k}}(\Xi _{\bullet })$ in the products satisfy the estimate (10.10) near the edge. Hence by [33, theorem 2.3.13], multiplication by elements in $C^{\infty }_{0}(\mathbb {M})$ induces a continuous operator on Sobolev spaces with asymptotics (with possibly different asymptotic type but associated to the same weight). Therefore we conclude that the right hand side in (7.8) belongs to ${\mathcal {W}}^{\infty ,\gamma -1}_{\text {As}}(M,T^{*}_{\wedge }M).$

Now, the fact that $\gamma > \frac{m+5}{2}$ together with (11.40) implies that $\text {Q}(\Xi _2)\in {\mathcal {W}}^{\infty ,\gamma +\beta }_{}(M,T^{*}_{\wedge }M)$ for some $\beta >0$. If necessary we can choose $\beta$ small enough such that $\gamma +\beta +1$ is an admissible weight for $\text {DP}[0]$. Then, Proposition 7.1 implies $\Xi _2=E_1+E_2$ with $E_1$ belonging to ${\mathcal {W}}^{\infty ,\gamma +\beta +1}(M,T^{*}_{\wedge }M)$ and $E_2\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M).$

Define $S_1:=\Xi _2-E_2=E_1$ and observe that (7.8) and a similar argument as above implies

$$\begin{aligned} (\text {DP}[0]+\text {Q})(S_1):=\xi _2\in {\mathcal {W}}^{\infty ,\gamma -1}_{\text {As}}(M,T^{*}_{\wedge }M). \end{aligned}$$

Moreover, (11.40) and $\gamma >\frac{m+5}{2}$ imply that $\text {Q}(S_1)$ belongs to the space ${\mathcal {W}}^{\infty ,\gamma +\beta +1+\beta '}(M,T^{*}_{\wedge }M)$ for some $\beta '>0$. Then by following the same argument as in Proposition 7.1 we have

$$\begin{aligned} S_1=E_3+E_4 \end{aligned}$$

with $E_3\in {\mathcal {W}}^{\infty ,\gamma +\beta +\beta '+2}(M,T^{*}_{\wedge }M)$ and $E_4={\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M).$ Hence we have found an element $E_2+E_4\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$ such that

$$\begin{aligned} \Xi _2-(E_2+E_4)=E_3\in {\mathcal {W}}^{\infty ,\gamma +\beta +\beta '+2}(M,T^{*}_{\wedge }M). \end{aligned}$$

We continue this recursion argument by setting $S_2=S_1-E_4=\Xi _2-E_2-E_4=E_3$ and

$$\begin{aligned} (\text {DP}[0]+\text {Q})(S_2)=\xi _3. \end{aligned}$$

Therefore by means of this iterative process we conclude that for every $l>0$ there exists $E\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$ such that $\Xi _2-E\in {\mathcal {W}}^{\infty ,\gamma +l}(M,T^{*}_{\wedge }M).$ Therefore $\Xi _2\in {\mathcal {W}}^{\infty ,\gamma }_{\text {As}}(M,T^{*}_{\wedge }M)$. $\square$

8 The moduli space

In this section we define the moduli space we are interested in and prove the main result of this paper. Let $(M,\Phi )$ be a special Lagrangian submanifold with edge singularity i.e. M is a manifold with edge singularity (Sect. 3.1) and $\Phi :M\longrightarrow {\mathbb {C}}^{n}$ is an edge special Lagrangian embedding (see Definition 4.4).

Definition 8.1

Given an admissible weight $\gamma >\frac{m+5}{2}$, we define the moduli space of conormal asymptotic special Lagrangian deformations of $(M,\Phi )$ with rate $\gamma$ and elliptic boundary trace condition ${\mathcal {T}}$ as the space of smooth embeddings $\Upsilon :M\longrightarrow {\mathbb {C}}^{n}$, isotopic to $\Phi$ and conormal asymptotic to $(M,\Phi )$ with rate $\gamma$, such that they satisfy the boundary condition ${\mathcal {T}} (\Upsilon ) =0$ where ${\mathcal {T}}$ is a trace pseudo-differential operator:

$$\begin{aligned} {\mathcal {T}}:{\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\longrightarrow H^{s-1}({\mathcal {E}},J^{-}) \end{aligned}$$

(8.1)

with $s>\text {max}\left\{ \frac{m+1+q}{2}+{\mathfrak {c}}_{\gamma }, \frac{m+3+q}{2}\right\}$ such that ${\mathcal {T}}$ belongs to a set of boundary conditions for an elliptic edge boundary value problem for the operator $\text {DP}[0]$ on the edge-Sobolev space ${\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)$. We denote this moduli space as ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$.

Given a special Lagrangian submanifold with edge singularity $(M,\Phi )$, the moduli space of conormal asymptotic special Lagrangian deformations depends on the parameters $\gamma$ and ${\mathcal {T}}$. The role of the weight $\gamma$ is explained in the definition of conormal asymptotic embedding (see Definition 6.2). The existence of a trace operator ${\mathcal {T}}$ and its role in the elliptic theory of edge degenerate equations was discussed in Sect. 5.3. Here we want to include further details about ${\mathcal {T}}$. The edge symbol defining ${\mathcal {T}}$ was defined in (5.8) as a family of continuous linear maps $\text {t}(u,\eta ):{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })\longrightarrow J^{-}_{(u,\eta )}$ continuously parametrized by $T^{*}{\mathcal {E}}{\setminus }\{ 0\}$. Note that as $J^{-}$ is a finite rank vector bundle, the operators $\text {t}(u,\eta )$ are finite rank operators (in particular compact operators). The trace operator ${\mathcal {T}}$ was locally defined by

$$\begin{aligned} {\mathcal {T}}=\text {Op}(\text {t}(u,\eta ))={\mathcal {F}}^{-1}_{\eta \rightarrow u}\text {t}(u,\eta ){\mathcal {F}}^{}_{u'\rightarrow \eta }. \end{aligned}$$

Now recall from Sect. 5.3 that the fibers of the vector bundle $J^{-}$ consist mainly of isomorphic images of finite dimensional kernels of Fredholm operators acting on the extension of cone-Sobolev spaces defined by (5.6). The operator-valued symbols $\text {t}(u,\eta )$ correspond to the projection of the cone-Sobolev space ${\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ onto the finite dimensional kernel of (5.6). By considering a local trivialization of $J^{-}$ over an open subset $\Omega \subset {\mathcal {E}}$ and using the fact that $\text {t}(u,\eta )$ are projections, it is possible to prove the following proposition.

Proposition 8.2

Locally on $\Omega \subset {\mathcal {E}}$, the trace operator ${\mathcal {T}}$ in (8.1) acts on each of the components of the stretched cotangent bundle $T^{*}_{\wedge }M$ as an integral operator with kernel in $C^{\infty }(\Omega \times {\mathcal {X}}^{\wedge }\times \Omega )\otimes {\mathbb {C}}^{N^{-}}$.

This proposition and its proof is contained in the more general result presented in [33, proposition 3.4.6]. The reader is referred to that book for details.

Given a operator-valued trace symbol $\text {t}(u,\eta )$, the trace operator ${\mathcal {T}}$ is unique module negligible operators from the point of view of ellipticity and smoothness. See [29, section 6.1] for details.

Theorem 8.3

Locally near M the moduli space ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ is homeomorphic to the zero set of a smooth map ${\mathfrak {G}}$ between smooth manifolds ${\mathcal {M}}_1$, ${\mathcal {M}}_2$ given as neighborhoods of zero in finite dimensional Banach spaces. The map ${\mathfrak {G}}: {\mathcal {M}}_1 \longrightarrow {\mathcal {M}}_2$ satisfies ${\mathfrak {G}}(0)=0$ and ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ near M is a smooth manifold of finite dimension when ${\mathfrak {G}}$ is the zero map.

Proof

As $\gamma$ is an admissible weight we have that

$$\begin{aligned}&\mathbf {A}_{\text {DP}[0]}= \begin{bmatrix} \text {DP}[0]&\quad C \\ {\mathcal {T}}&\quad B \end{bmatrix}:\\&\begin{array}{ccc} \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array} &{} \longrightarrow &{} \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array} \end{array} \end{aligned}$$

is a Fredholm operator. Thus the cokernel is a finite dimensional space and it can be identified with a finite dimensional subspace in the codomain of $\mathbf {A}_{\text {DP}[0]}$ (denoted as $\text {Coker}\mathbf {A}_{\text {DP}[0]}$) such that it splits the codomain in the following way

$$\begin{aligned} \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array}=\text {Im}\mathbf {A}_{\text {DP}[0]} \oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}. \end{aligned}$$

(8.2)

Now consider the Banach space

$$\begin{aligned} \left( \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array}\right) \oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}. \end{aligned}$$

Consider the following extension $\hat{\text {P}}$ of the deformation operator to this space

$$\begin{aligned} \hat{\text {P}}:\left( \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array}\right) \oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}\longrightarrow \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array} \end{aligned}$$

given by

$$\begin{aligned} \hat{\text {P}}\left( \begin{bmatrix} \Xi \\ g \end{bmatrix}, \begin{bmatrix} v\\ w \end{bmatrix} \right) =\mathbf {A}_{\text {P}}\begin{bmatrix} \Xi \\ g \end{bmatrix}+\begin{bmatrix} v\\ w \end{bmatrix}, \end{aligned}$$

where we are using the notation $\mathbf {A}_{\text {P}}$ for the operator $\begin{bmatrix} \text {P}&\quad \text {C} \\ {\mathcal {T}}&\quad \text {B} \end{bmatrix}.$

Hence

$$\begin{aligned} \text {D}\hat{\text {P}}[0]\left( \begin{bmatrix} \Xi \\ g \end{bmatrix}, \begin{bmatrix} v\\ w \end{bmatrix} \right) =\begin{bmatrix} \text {DP}[0]&\quad \text {C} \\ {\mathcal {T}}&\quad \text {B} \end{bmatrix}\begin{bmatrix} \Xi \\ g \end{bmatrix}+\begin{bmatrix} v\\ w \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} {\text {Ker D}}\hat{\text {P}}[0]=\text {Ker}\mathbf {A}_{\text {DP}[0]}\times \lbrace 0 \rbrace . \end{aligned}$$

Observe that $\text {D}\hat{\text {P}}[0]$ is surjective and ${\text {Ker D}}\hat{\text {P}}[0]$ is finite dimensional. Then

$$\begin{aligned} \left( \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,\bigwedge \nolimits ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s}({\mathcal {E}}, {\mathbb {C}}^{N^{+}}) \end{array}\right) \oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}=\left( {\text {Ker D}}\hat{\text {P}}[0]\oplus N\right) \oplus \text {Coker}\mathbf {A}_{\text {DP}[0]} \end{aligned}$$

for some closed subspace N.

By the Implicit Function Theorem 4.2 there exists ${\mathcal {U}}_1\subset \text {Ker}\mathbf {A}_{\text {DP}[0]}$, ${\mathcal {U}}_2={\mathcal {U}}'_2\times {\mathcal {U}}''_2\subset N\oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}$ and a smooth map ${\mathfrak {G}}_1\times {\mathfrak {G}}_2:{\mathcal {U}}_1\longrightarrow {\mathcal {U}}'_2\times {\mathcal {U}}''_2$ such that

$$\begin{aligned}&\hat{\text {P}}^{-1}(0)\cap ({\mathcal {U}}_1\times {\mathcal {U}}_2)=\left\{ \left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) ,{\mathfrak {G}}_2\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) : \begin{bmatrix} a \\ b \end{bmatrix}\in {\mathcal {U}}_1 \right\} \\&\subset {\text {Ker D}}\hat{\text {P}}[0]\oplus N\oplus \text {Coker}\mathbf {A}_{\text {DP}[0]}. \end{aligned}$$

This give us a description of the elements in the null set of the non-linear operator $\hat{\text {P}}$ in a neighborhood of zero in terms of elements in ${\text {Ker D}}\hat{\text {P}}[0].$ In order to pass to solutions of the deformation operator $\text {P}$ in terms of ${\text {Ker DP}}[0]$ we have the following.

Observe that

$$\begin{aligned} \left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) ,{\mathfrak {G}}_2\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) \in \hat{\text {P}}^{-1}(0)\cap ({\mathcal {U}}_1\times {\mathcal {U}}_2), \end{aligned}$$

implies

$$\begin{aligned}&\hat{P}\left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) ,{\mathfrak {G}}_2\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) \\&\quad =\mathbf {A}_{P}\left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) +{\mathfrak {G}}_2\left( \begin{bmatrix} a\\ b \end{bmatrix}\right) =0. \end{aligned}$$

Hence the term ${\mathfrak {G}}_2\left( \begin{bmatrix} a \\ b \end{bmatrix}\right)$ (that belongs to $\text {Coker}\mathbf {A}_{\text {DP}[0]}$) represents an obstruction to lifting the infinitesimal solution $\begin{bmatrix} a \\ b \end{bmatrix}$ to an authentic solution

$$\begin{aligned} \left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) \end{aligned}$$

of the non-linear operator $\mathbf {A}_{\text {P}}$. Therefore if all obstructions vanish i.e. if

$$\begin{aligned} {\mathfrak {G}}_2:{\mathcal {U}}_1\subset \text {Ker}\mathbf {A}_{\text {DP}[0]}\longrightarrow {\mathcal {U}}''_2\subset \text {Coker}\mathbf {A}_{\text {DP}[0]} \end{aligned}$$

(8.3)

is the zero map we have

$$\begin{aligned} \mathbf {A}_{\text {P}}^{-1}(0) \cap ({\mathcal {U}}_1\times {\mathcal {U}}'_2)=\left\{ \left( \begin{bmatrix} a \\ b \end{bmatrix},{\mathfrak {G}}_1\left( \begin{bmatrix} a \\ b \end{bmatrix}\right) \right) : \begin{bmatrix} a \\ b \end{bmatrix}\in {\mathcal {U}}_1 \right\} . \end{aligned}$$

and the set $\mathbf {A}_{\text {P}}^{-1}(0)\cap ({\mathcal {U}}_1\times {\mathcal {U}}'_2)$ is diffeomorphic to ${\mathcal {U}}_1\subset \text {Ker}\mathbf {A}_{\text {DP}[0]}.$

Consequently, if the obstructions vanish, small solutions of the non-linear boundary value problem (7.1) are given by

$$\begin{aligned}&\mathbf {A}_{\text {P}}^{-1}(0)\cap ({\mathcal {U}}_1\times {\mathcal {U}}'_2)\cap \left( \begin{array}{c} {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M) \\ \oplus \\ \lbrace 0 \rbrace \end{array}\right) \\ {}&=\left\{ \left( \Xi _1,{\mathfrak {G}}_1(\Xi _1)\right) : \begin{bmatrix} \text {P}_{\omega _{_{{\mathbb {C}}^{N}}}}\oplus \text {P}_{\text {Im}\Omega } \\ {\mathcal {T}} \end{bmatrix}(\Xi _1+{\mathfrak {G}}_1(\Xi _1))=0 \right\} . \end{aligned}$$

This is a non-empty open neighborhood of zero in ${\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)$ diffeomorphic to an open set of the finite dimensional space

$$\begin{aligned} \text {Ker}\begin{bmatrix} \text {DP}[0] \\ {\mathcal {T}} \end{bmatrix}. \end{aligned}$$

(8.4)

Thus we can conclude that when ${\mathfrak {G}}_2$ is the zero map the moduli space is a smooth manifold of finite dimension less or equal to the dimension of the kernel of the linear boundary value problem

$$\begin{aligned} \begin{bmatrix} \text {DP}[0] \\ {\mathcal {T}} \end{bmatrix}: {\mathcal {W}}^{s,\gamma }(M,T^{*}_{\wedge }M)\longrightarrow \begin{array}{c} {\mathcal {W}}^{s-1,\gamma -1}(M,\bigwedge ^{\bullet }T^{*}_{\wedge }M) \\ \oplus \\ H^{s-1}({\mathcal {E}}, J^{-}) \end{array}. \end{aligned}$$

$\square$

9 Conclusions and final remarks

In this paper we have put the problem of deforming special Lagrangain submanifolds with edge singularities into the framework of the edge calculus developed by B.-W. Schulze. Our main theorem, Theorem 8.3 in Sect. 8, says that when the map (8.3)

$$\begin{aligned} {\mathfrak {G}}_2:{\mathcal {U}}_1\subset \text {Ker}\mathbf {A}_{\text {DP}[0]}\longrightarrow {\mathcal {U}}''_2\subset \text {Coker}\mathbf {A}_{\text {DP}[0]} \end{aligned}$$

is the zero map, the moduli space ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$ is a smooth manifold of finite dimension. For every small solution $\Xi _1$ of the linearised boundary value problem $\mathbf {A}_{\text {DP}[0]}$, the map ${\mathfrak {G}}_2$ gives us an obstruction

$$\begin{aligned} {\mathfrak {G}}_2(\Xi _2)\in {\mathcal {U}}''_2\subset \text {Coker}\mathbf {A}_{\text {DP}[0]}, \end{aligned}$$

to lift the linearised solution to a solution of the non-linear deformation operator with boundary condition. When the obstruction space ${\mathcal {U}}''_2$ vanishes it follows immediately that there are no obstructions, as the map ${\mathfrak {G}}_2$ is trivially the zero map, and the moduli space is smooth and finite dimensional.

A careful analysis of the obstruction space is needed to determine under which conditions it vanishes. In Joyce [13] analyzed the obstruction space of the moduli space of deformations of special Lagrangian submanifolds with conical singularities. He found that the obstruction space depends only on the cones that model the singularities. In the edge singular case we expect a similar result i.e. the obstruction space depends only on the geometric structures that model the singularity, namely, the cone ${\mathcal {X}}^{\wedge }$ and the edge ${\mathcal {E}}$.

If the obstruction space vanishes (therefore $\text {Coker}\mathbf {A}_{\text {DP}[0]}=\{ 0\}$) the moduli space is a smooth manifold of finite dimension. The next step is to determine its expected dimension. From Theorem 8.3 we only know that $\text {dim}{\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )\le {\text {dim Ker}}\mathbf {A}_{\text {DP}[0]}=\text {Ind}\mathbf {A}_{\text {DP}[0]}$. In order to compute the dimension we need to consider the index of edge-degenerate operators and Hodge theory in the edge singular context. In this direction we consider the material related to index theory in [29, chapter 5] quite relevant for this purpose. Moreover, some elements of Hodge theory on manifolds with edge singularities have been studied in [11, 34]. These references might be helpful to compute the expected dimension of our moduli space ${\mathfrak {M}}(M, \Phi , {\mathcal {T}},\gamma )$.

Appendices

Appendix 1: Vector-valued Sobolev embeddings

In Sects. 4 and 6 we used results from this “Appendix”. Most of the results here are based on vector-valued Sobolev embeddings. In the first section of this “Appendix” we recall the basics of vector-valued Sobolev embeddings and derive some consequences related to cone and edge-Sobolev spaces. In the second part we prove the estimate (11.1) following a similar result of Dreher and Witt [8]. This estimate implies the Banach Algebra property of our edge-Sobolev spaces on M, (11.39), and the regularity of the product of elements in ${\mathcal {W}}^{s,\gamma }(M)$, (11.40). In order to simplify the notation we will denote $a\approx b$ and $a\lesssim b$ if $a=\kappa b$ or $a\le \kappa b$ respectively with a positive constant $\kappa$ depending only on s and $\gamma$.

Let’s consider the classical Sobolev spaces $W^{m,p}({\mathbb {R}}^{q})$ (see [5] for a detailed introduction). A classical tool in the analysis of partial differential equations on ${\mathbb {R}}^{q}$ is the set of Sobolev embeddings, see [5, section 9.3].

Theorem 10.1

Let $m\in {\mathbb {Z}}$, $m>1$ and $p\in [1,+\infty ).$

$$\begin{aligned}&\text {If}\quad \frac{q}{p}>m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{k}({\mathbb {R}}^{q}) \quad \text {where } \frac{1}{q}=\frac{1}{p}-\frac{m}{q}. \end{aligned}$$

(10.1)

$$\begin{aligned}&\text {If}\quad \frac{q}{p}=m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{k}({\mathbb {R}}^{q}) \quad \text {for all }\quad k\in [p,+\infty ). \end{aligned}$$

(10.2)

$$\begin{aligned}&\text {If}\quad \frac{q}{p}<m \quad \text {then }\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow L^{\infty }({\mathbb {R}}^{q})\text { and}\quad W^{m,p}({\mathbb {R}}^{q})\hookrightarrow {\mathcal {C}}^{r}({\mathbb {R}}^{q}) \end{aligned}$$

(10.3)

where $r=[s-\frac{q}{2}]$ i.e. r is the integer part of $s-\frac{q}{2}$.

In this “Appendix” we are interested in the vector-valued version of this theorem i.e. given a Banach space ${\mathfrak {B}}$ we want a version for the ${\mathfrak {B}}$-valued Sobolev spaces $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}}).$ There are many books and monographs dealing with vector-valued spaces of all kinds like $L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}),{\mathcal {C}}^{k}({\mathbb {R}}^{q},{\mathfrak {B}})$ and ${\mathcal {S}}({\mathbb {R}}^{q},{\mathfrak {B}})$, see [3, 12, 36]. In many cases they work in the more general context where ${\mathfrak {B}}$ is a Fréchet or locally convex Hausdorff space. For our specific purposes we follow closely [23]. Here Kreuter analyzes carefully the validity of Theorem 10.1 for the spaces $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}})$ where ${\mathfrak {B}}$ is a Banach space.

Recall that the vector-valued space of distributions is defined as the space of continuous operators from $C^{\infty }_{0}({\mathbb {R}}^{q})$ to ${\mathfrak {B}}$ i.e. we have ${\mathcal {D'}}({\mathbb {R}}^{q},{\mathfrak {B}}):={\mathcal {L}}(C^{\infty }_{0}({\mathbb {R}}^{q}),{\mathfrak {B}}).$ The vector-valued $L^{p}$-spaces, $L^{p}({\mathbb {R}}^{q},{\mathfrak {B}})$, are defined by means of the Bochner integral. The Bochner integral is constructed by means of ${\mathfrak {B}}$-valued step functions in a similar way to the standard Lebesgue integral. See [1, Appendix A.4] for details. The vector-valued ${\mathcal {C}}^{k}$-spaces, ${\mathcal {C}}^{k}({\mathbb {R}}^{q},{\mathfrak {B}})$, are defined with respect to the Fréchet derivative. The vector-valued Sobolev space is defined by

$$\begin{aligned} W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}}):=\left\{ f\in L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}):\partial ^{\alpha }f\in L^{p}({\mathbb {R}}^{q},{\mathfrak {B}}) \quad \forall \quad \left| \alpha \right| \le m \right\} \end{aligned}$$

(10.4)

where the derivatives of f are taken in the distribution sense i.e. weak derivatives.

Here we recall the definition of the Radon-Nikodym property and some results related to it. It turns out that the key property that ${\mathfrak {B}}$ must satisfy in order to have vector-valued Sobolev embeddings for $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {B}})$ is the Radon-Nikodym property. For extended details the reader is referred to [23, chapter 2].

Definition 10.2

A Banach space ${\mathfrak {B}}$ has the Radon-Nikodym property if every Lipschitz continuous function $f:I\longrightarrow {\mathfrak {B}}$ is differentiable almost everywhere, where $I\subset {\mathbb {R}}$ is an arbitrary interval.

Proposition 10.3

Every reflexive space has the Radon-Nikodym property. In particular the spaces $L^{p}({\mathbb {R}}^{q})$ with $1<p<\infty$ and Hilbert spaces have the Radon-Nikodym property.

Corollary 10.4

The Sobolev embeddings in Theorem 10.1 are valid for the spaces $W^{m,p}({\mathbb {R}}^{q},L^{p}({\mathbb {R}}^{q}))$ with $1<p<\infty$ and $W^{m,p}({\mathbb {R}}^{q},{\mathfrak {H}})$ where ${\mathfrak {H}}$ is a Hilbert space.

As a consequence of these vector-valued results we have the following applications to cone and edge-Sobolev spaces.

Proposition 10.5

If $f\in {\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ (see (3.7)) and $s>\frac{m+1}{2}$ then there exists $C>0$ depending only on s and $\gamma$ such that we have the following estimate on $(0,1)\times {\mathcal {X}}$

$$\begin{aligned} \left| \partial ^{\alpha '}_{r}\partial ^{\alpha ''}_{\sigma }f(r,\sigma )\right| \le C\left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}r^{\gamma -\frac{m+1}{2}-\left| \alpha '\right| } \end{aligned}$$

(10.5)

for all $(r,\sigma )\in (0,1)\times {\mathcal {X}}$ and $\left| \alpha '\right| +\left| \alpha ''\right| \le [s-\frac{m+1}{2}] .$

Proof

We can work locally on ${\mathbb {R}}^{+}\times {\mathcal {U}}_{\lambda }$ where $\lbrace {\mathcal {U}}_{\lambda }\rbrace$ is a finite open covering of ${\mathcal {X}}$, $\lbrace \varphi _{\lambda }\rbrace$ is a subbordinate partition of unity and we consider $\omega \varphi _{\lambda }f$. For simplicity we write just f instead of $\omega \varphi _{\lambda }f$. At the end we take the smallest constant among those obtained for each element in the finite covering. Take $f\in {\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ then by (3.6) we have $S_{\gamma -\frac{m}{2}}f\in H^{s}({\mathbb {R}}^{1+m}).$ Therefore if $s>\frac{m+1}{2}$ by (10.3) we have $S_{\gamma -\frac{m}{2}}f\in L^{\infty }({\mathbb {R}}^{1+m})$ and

$$\begin{aligned} \sup _{(t,\sigma )\in {\mathbb {R}}^{1+m}}\left| \partial ^{\alpha }_{(t,\sigma )}(S_{\gamma -\frac{m}{2}}f)(t,\sigma )\right| \lesssim \left\| S_{\gamma -\frac{m}{2}}f\right\| _{H^{s}({\mathbb {R}}^{1+m})}\lesssim \left\| f\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })} \end{aligned}$$

(10.6)

for all $\left| \alpha \right| \le [s-\frac{m+1}{2}]$. Now by definition (see (3.6))

$$\begin{aligned} (S_{\gamma -\frac{m}{2}}f)(t,x)=e^{-(\frac{1}{2}-(\gamma -\frac{m}{2}))t}f(e^{-t},x) \end{aligned}$$

with $r=e^{-t}$. Thus (10.5) follows immediately. $\square$

In general, if ${\mathfrak {B}}$ is a Banach space and $\lbrace \kappa _{\lambda }\rbrace _{\lambda \in {\mathbb {R}}^{+}}\in {\mathcal {C}}\left( {\mathbb {R}}^{+},{\mathcal {L}}({\mathfrak {B}})\right)$ is a continuous one-parameter group of invertible operators we have that there exist positive constants $K,{\mathfrak {c}}$ such that

$$\begin{aligned} \left\| \kappa _{\lambda }\right\| _{{\mathcal {L}}({\mathfrak {B}})} \le \left\{ \begin{array}{ll} K\lambda ^{{\mathfrak {c}}}&{}\quad \text { for } \lambda \ge 1 \\ K\lambda ^{-{\mathfrak {c}}}&{}\quad \text { for } 0<\lambda \le 1 \end{array} \right. . \end{aligned}$$

(10.7)

See [33, proposition 1.3.1] for details.

When ${\mathfrak {B}}={\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ and $(\kappa _{\lambda }f)(r,\sigma )=\lambda ^{\frac{m+1}{2}}f(\lambda r,\sigma )$ we can use 3.6 to compute $\left\| \kappa _{\lambda }\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}=\lambda ^{\gamma }$ (see [32, section 1.1]). By the proof of proposition 1.3.1 in [33] it is easy to see that the constant ${\mathfrak {c}}$ in (10.7) depends only on the weight $\gamma$. When ${\mathfrak {B}}={\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ we denote this constant by ${\mathfrak {c}}_{\gamma }$.

As a consequence of (10.7), we have the following continuous embeddings

$$\begin{aligned}&{\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q}) \longrightarrow H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })) \end{aligned}$$

(10.8)

$$\begin{aligned}&H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))\longrightarrow {\mathcal {W}}^{s+{\mathfrak {c}}_{\gamma },\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q}) \end{aligned}$$

(10.9)

for all $s\in {\mathbb {R}}$ where $H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))$ is the standard vector-valued Sobolev space with norm given by

$$\begin{aligned} \left\| f\right\| _{H^{s}({\mathbb {R}}^{q},{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge }))}:=\left( \,\,\int \limits _{{\mathbb {R}}^q}[\eta ]^{2s}\left\| {\mathcal {F}}_{u\rightarrow \eta }f(\eta )\right\| ^{2}_{{\mathcal {K}}^{s,\gamma }({\mathcal {X}}^{\wedge })}d\eta \right) ^{\frac{1}{2}}. \end{aligned}$$

The reader is refer to [33, proposition 1.3.1 and remark 1.3.21] for details.

Proposition 10.6

If $g\in {\mathcal {W}}^{s,\gamma }(M)$ (see (3.9)) and $s>\frac{m+1+q}{2}+{\mathfrak {c}}_{\gamma }$ where ${\mathfrak {c}}_{\gamma }$ is the constant defined in (10.7), then there exists $C'>0$ depending only on s and $\gamma$ such that we have the following estimate on $(0,1)\times {\mathcal {X}}\times {\mathcal {E}}$

$$\begin{aligned} \left| \partial ^{\alpha '}_{r}\partial ^{\alpha ''}_{(\sigma ,u)}g(r,\sigma ,u)\right| \le C'\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}r^{\gamma -\frac{m+1}{2}-\left| \alpha '\right| } \end{aligned}$$

(10.10)

for all $(r,\sigma ,u)\in (0,1)\times {\mathcal {X}}\times {\mathcal {E}}$ and $\left| \alpha '\right| +\left| \alpha ''\right| \le [s-\frac{m+1}{2}].$

Proof

We work locally on $(0,1)\times {\mathcal {U}}_{\lambda } \times \Omega _{j}$ as in Proposition 4.6 Sect. 4.4. Take $g\in {\mathcal {W}}^{s,\gamma }(M)$. Again by (10.3) and $s>\frac{m+1}{2}$ we have that for each $u\in {\mathbb {R}}^{q}$

$$\begin{aligned} (S_{\gamma -\frac{m}{2}}g)(u)\in H^{s}({\mathbb {R}}^{1+m}) \end{aligned}$$

and

$$\begin{aligned} \left\| (S_{\gamma -\frac{m}{2}}g)(u)\right\| _{L^{\infty }({\mathbb {R}}^{m+1})}\lesssim \left\| g(u)\right\| _{{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })}. \end{aligned}$$

(10.11)

Now (10.8) implies $g\in H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))$ and $s>\frac{q}{2}+{\mathfrak {c}}_{\gamma }$ together with (10.3) and Corollary 10.4 implies that we have a continuous embedding

$$\begin{aligned} H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))\hookrightarrow L^{\infty }({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })) \end{aligned}$$

as ${\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge })$ is a Hilbert space, see Definition 3.2 in Sect. 3.2.

Consequently

$$\begin{aligned} \left\| g\right\| _{L^{\infty }({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))}&=\sup _{u\in {\mathbb {R}}^{q}}\left\{ \left\| g(u)\right\| _{{\mathcal {H}}^{s,\gamma }} \right\} \end{aligned}$$

(10.12)

$$\begin{aligned}&\lesssim \left\| g\right\| _{H^{s-{\mathfrak {c}}_{\gamma }}({\mathbb {R}}^{q},{\mathcal {H}}^{s,\gamma }({\mathcal {X}}^{\wedge }))} \end{aligned}$$

(10.13)

$$\begin{aligned}&\lesssim \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }({\mathcal {X}}^{\wedge }\times {\mathbb {R}}^{q})}. \end{aligned}$$

(10.14)

Hence (10.11), (10.14) and the change of variable $r=e^{-t}$ implies (10.10) as in Proposition 10.5. $\square$

Appendix 2: Banach algebra property of edge-Sobolev spaces

In [8] Dreher and Witt used a variant of the edge-Sobolev spaces we use in this paper. In that paper they are interested in applications to weakly hyperbolic equations. Their spaces are defined on $(0,T)\times {\mathbb {R}}^{n}$. In this context they proved (proposition 4.1 in [8]) that their edge-Sobolev spaces have the structure of a Banach algebra. With some modifications and by using vector-valued Sobolev embeddings it is possible to extend their result to our edge-Sobolev spaces on M. This extension follows closely the proof of Witt and Dreher. For completeness we include the details of this extension in our context as we used the estimates (11.40) and (11.39) in Chapter 3 and 4.

Proposition 11.1

Let $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ with $s\in {\mathbb {N}}$ and $s>\frac{q+m+3}{2}$. Then $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)$ and we have the following estimate

$$\begin{aligned} \left\| fg\right\| _{{\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)}\le C\left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.1)

with a constant $C>0$ depending only on s and $\gamma$.

Proof

By means of finite open covers and partitions of unity on ${\mathcal {X}}$ and ${\mathcal {E}}$ we need to estimate in terms of $\omega \varphi _{\lambda }\phi _{j}f$ and $\omega \varphi _{\lambda }\phi _{j}g$ as in Proposition 4.7 Sect. 4.4. To avoid unnecessary long expressions we will denote them simply by f and g. To save space in long expressions we use the notation $\hat{f}$ to denote the Fourier transformation with respect to the conormal variable $\eta$ i.e. $\hat{f}={\mathcal {F}}_{u\rightarrow \eta }f.$ We will estimate on an open set $(0,\varepsilon )\times {\mathcal {U}}_{\lambda }\times \Omega _{j}$. Then the global estimate is obtained by adding these terms. Take $f,g\in C^{\infty }_{0}(M)$. By definition of our edge-Sobolev (3.8) norm and by (3.6) we have

$$\begin{aligned}&\left\| fg\right\| ^{2}_{{\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}(M)} \approx \int \limits _{{\mathbb {R}}^{q}_{\eta }} [\eta ]^{2s}\left\| \kappa ^{-1}_{[\eta ]} {\mathcal {F}}_{u\rightarrow \eta }(fg)(\eta )\right\| _{{\mathcal {H}}^{s,2\gamma -\frac{m+1}{2}} ({\mathbb {R}}^{+}\times {\mathbb {R}}^{m})}^{2}d\eta \end{aligned}$$

(11.2)

$$\begin{aligned}&\quad \approx \sum \limits _{\left| \alpha \right| \le s} \int \limits _{{\mathbb {R}}^{q}_{\eta }} \int \limits _{{\mathbb {R}}^{}_{t}}\int \limits _{{\mathbb {R}}^{m}_{\sigma }}[\eta ]^{2s-(m+1)}\Big |\partial ^{\alpha }\big (e^{-(\frac{m+1}{2}-\gamma )2t}{\mathcal {F}}_{u\rightarrow \eta }(fg)([\eta ]^{-1}e^{-t}, \end{aligned}$$

(11.3)

$$\begin{aligned}&\qquad \sigma ,\eta )\big )\Big |^{2} d\sigma dt d\eta \end{aligned}$$

(11.4)

Here we will estimate the term with $\alpha =0$. The estimates on the other terms $\alpha \ne 0$ are similar. For each term in (11.4) we have

$$\begin{aligned}&\int \limits _{{\mathbb {R}}^{q}_{\eta }}\int \limits _{{\mathbb {R}}^{}_{t}}\int \limits _{{\mathbb {R}}^{m}_{\sigma }} [\eta ]^{2s-(m+1)}\left| e^{-(\frac{m+1}{2}-\gamma )2t} {\mathcal {F}}_{u\rightarrow \eta }(fg)\right| ^{2}d\eta \end{aligned}$$

(11.5)

$$\begin{aligned}&\quad = \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} \left\| [\eta ]^{s-\frac{m+1}{2}}{\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt. \end{aligned}$$

(11.6)

Now, the hypothesis $s>\frac{q+m+3}{2}$ allows us to use lemma 4.6 in [8]. Basically, this lemma implies that for fixed $(t,\sigma )$ we have the following estimate

$$\begin{aligned}&\left\| \Lambda (\eta )\widehat{fg(t,\sigma )}(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\le \left\| f(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\\&\quad +\left\| g(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(r,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\\&\quad +C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

with $C_0>0$ and $\Lambda (\eta )=[\eta ]^{s-\frac{m+1}{2}}$.

Applying this estimate to (11.6) we have

$$\begin{aligned}&\left( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }} e^{-(\frac{m+1}{2}-\gamma )4t}\left\| [\eta ]^{s-\frac{m+1}{2}} {\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt \right) ^{\frac{1}{2}} \end{aligned}$$

(11.7)

$$\begin{aligned}&\quad \le \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }} \bigg (e^{-(\frac{m+1}{2}-\gamma )2t}\left\| f(t,\sigma )(u)\right\| _{L^{\infty }({\mathbb {R}}^{q}_{u})} \cdot \left\| \Lambda (\eta ) \hat{g}(t,x)(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })}+\quad \end{aligned}$$

(11.8)

$$\begin{aligned}&\qquad + e^{-(\frac{m+1}{2}-\gamma )2t}\left\| g(t,\sigma )(u)\right\| _{L^{\infty } ({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.9)

$$\begin{aligned}&\qquad + e^{-(\frac{m+1}{2}-\gamma )2t} C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| _{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.10)

$$\begin{aligned}&\qquad \cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| _{L^{2} ({\mathbb {R}}^{q}_{\eta })}\Bigg )^{2} d\sigma dt \Bigg )^{\frac{1}{2}}. \end{aligned}$$

(11.11)

By the Minkowski inequality we have that (11.7) is less or equal to the following terms

$$\begin{aligned}&\Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t}\left\| f(t,\sigma ) (u)\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}d\sigma dt \Bigg )^{\frac{1}{2}} \end{aligned}$$

(11.12)

$$\begin{aligned}&\quad + \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t}\left\| g(t,\sigma ) (u)\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}\cdot \left\| \Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}d\sigma dt \Bigg )^{\frac{1}{2}} \end{aligned}$$

(11.13)

$$\begin{aligned}&\quad + \Bigg ( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} C_{0}\left\| \Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })} \end{aligned}$$

(11.14)

$$\begin{aligned}&\quad \cdot \left\| \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })} d\sigma dt \Bigg )^{\frac{1}{2}}, \end{aligned}$$

(11.15)

hence, by the inequality in (11.7), we have

$$\begin{aligned}&\left( \int \limits _{{\mathbb {R}}^{}_{t}} \int \limits _{{\mathbb {R}}^{m}_{\sigma }}e^{-(\frac{m+1}{2}-\gamma )4t} \left\| [\eta ]^{s-\frac{m+1}{2}}{\mathcal {F}}_{u\rightarrow \eta }(fg)\right\| _{L^{2} ({\mathbb {R}}^{q}_{\eta })}^{2}d\sigma dt \right) ^{\frac{1}{2}} \end{aligned}$$

(11.16)

$$\begin{aligned}&\quad \le \left\| e^{-(\frac{m+1}{2}-\gamma )t}f(t,\sigma ,u)\right\| ^{}_{_{L^{\infty } ({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})}} \end{aligned}$$

(11.17)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.18)

$$\begin{aligned}&\qquad + \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma ,u)\right\| ^{}_{_{L^{\infty }({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})}} \end{aligned}$$

(11.19)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.20)

$$\begin{aligned}&\qquad + C_{0}\left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f}(t,\sigma )(u)\slash [\eta ]\right\| ^{}_{_{L^{\infty }({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}} \end{aligned}$$

(11.21)

$$\begin{aligned}&\qquad \cdot \left\| e^{-(\frac{m+1}{2}-\gamma )t} \Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}}. \end{aligned}$$

(11.22)

The edge-Sobolev norm of f and g written as in (11.6) implies that

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{g}(t,\sigma )(\eta )\right\| ^{}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}\le \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.23)

and

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta ) \hat{f}(t,\sigma )(\eta )\right\| ^{}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))}\le \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}, \end{aligned}$$

(11.24)

hence we only need to deal with the $L^{\infty }$ terms.

To analyze the $L^{\infty }$ terms recall that by hypothesis $s>\frac{q}{2}$ so we have the standard continuous Sobolev embedding $H^{s}({\mathbb {R}}^{q})\hookrightarrow L^{\infty }({\mathbb {R}}^{q})$. Consequently for fixed $(t,\sigma )$ we have

$$\begin{aligned} \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{q}_{u})}&\lesssim \left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{H^{s}({\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.25)

$$\begin{aligned}&=\left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t}\hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{L^{2}({\mathbb {R}}^{q}_{\eta })}, \end{aligned}$$

(11.26)

therefore

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}g(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1} \times {\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.27)

$$\begin{aligned}&\quad \lesssim \left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t}\ hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.28)

$$\begin{aligned}&\quad \lesssim \left\| \langle \eta \rangle ^{s} e^{-(\frac{m+1}{2}-\gamma )t} \hat{g}(t,\sigma )(\eta ) \right\| ^{2}_{W^{s,2}({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.29)

$$\begin{aligned}&\quad =\sum _{\left| \beta \right| \le s} \left\| \langle \eta \rangle ^{s} \partial ^{\beta }_{(t,\sigma )}\Big ( e^{-(\frac{m+1}{2}-\gamma )t}\hat{g}(t,\sigma ) (\eta ) \Big )\right\| ^{2}_{L^{2}({\mathbb {R}}^{m+1}, L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.30)

$$\begin{aligned}&\quad \le \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.31)

In (11.29) we have used the vector-valued version of the standard Sobolev embedding (see Sect. 10). In the same way we obtain

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}f(t,\sigma )(\eta ) \right\| ^{2}_{L^{\infty } ({\mathbb {R}}^{m+1}\times {\mathbb {R}}^{q}_{u})} \end{aligned}$$

(11.32)

$$\begin{aligned}&\quad \lesssim \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.33)

Then (11.31) and (11.33) implies that (11.18) and (11.20) are bounded by

$$\begin{aligned} C(s,\gamma ) \left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2} \left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)}^{2}. \end{aligned}$$

(11.34)

Thus the only term remaining is (11.22).

Again using the vector-valued Sobolev embedding we have

$$\begin{aligned}&\left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f}(t,\sigma ) (u)\slash [\eta ]\right\| ^{2}_{L^{\infty }({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.35)

$$\begin{aligned}&\quad \lesssim \left\| e^{-(\frac{m+1}{2}-\gamma )t}\Lambda (\eta )\hat{f} (t,\sigma )(u)\slash [\eta ]\right\| ^{2}_{W^{s,2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.36)

$$\begin{aligned}&\quad =\sum _{\left| \beta \right| \le s}\left\| \frac{\Lambda (\eta )}{ [\eta ]} \partial ^{\beta }_{(t,x)}\bigg ( e^{-(\frac{m+1}{2}-\gamma )t}\hat{f}(t,\sigma ) (u)\bigg )\right\| ^{2}_{L^{2}({\mathbb {R}}^{m+1},L^{2}({\mathbb {R}}^{q}_{\eta }))} \end{aligned}$$

(11.37)

$$\begin{aligned}&\quad \lesssim \left\| f\right\| ^{2}_{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.38)

as $\left| \frac{\Lambda (\eta )}{[\eta ]}\right| ^{2}=[\eta ]^{2s-(m+1)}\cdot [\eta ]^{-2}\lesssim [\eta ]^{2s-(m+1)}.$ Thus (11.38) and (11.23) implies that (11.22) is bounded by (11.34). $\square$

Corollary 11.2

If $s\in {\mathbb {N}}$ with $s>\frac{q+m+3}{2}$ and $\gamma \ge \frac{m+1}{2}$ then the edge Sobolev space ${\mathcal {W}}^{s,\gamma }(M)$ is a Banach algebra under point-wise multiplication i.e. given $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ we have

$$\begin{aligned} \left\| fg\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\le C'\left\| f\right\| _{{\mathcal {W}}^{s,\gamma }(M)}\left\| g\right\| _{{\mathcal {W}}^{s,\gamma }(M)} \end{aligned}$$

(11.39)

with a constant $C'$ depending only on s and $\gamma$.

Proof

$\square$

By (11.1) we have $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}$. Note that $\gamma \ge \frac{m+1}{2}$ if and only if $2\gamma -\frac{m+1}{2}\ge \gamma$ from which the corollary follows immediately.

Corollary 11.3

Let $f,g\in {\mathcal {W}}^{s,\gamma }(M)$ such that $s\in {\mathbb {N}}$ with $s>\frac{q+m+3}{2}$ and $\gamma >\frac{m+1}{2}$. Then

$$\begin{aligned} f g\in {\mathcal {W}}^{s,\gamma +\beta }(M) \end{aligned}$$

(11.40)

for $\beta >0$ given by $\beta =\gamma -\frac{m+1}{2}$.

Proof

By (11.1) we have $fg\in {\mathcal {W}}^{s,2\gamma -\frac{m+1}{2}}$. Moreover $\gamma >\frac{m+1}{2}$ implies $2\gamma -\frac{m+1}{2}=\gamma +\beta$ with $\beta =\gamma -\frac{m+1}{2}>0$.