Global analytic hypoellipticity of involutive systems on compact manifolds

Abstract

Given M a compact, connected and orientable, real-analytic manifold, and closed, real-valued, real-analytic 1-forms $\omega _1, \ldots , \omega _m$ on M, we characterize the global analytic hypoellipticity of the first operator featuring in the differential complex over $M \times \mathbb {T}^m$ naturally associated to an involutive system of vector fields determined by them. Global Gevrey hypoellipticity is determined simultaneously.

1 Introduction

One of the foremost models of systems of linear PDEs is that of the so-called tube structures, whose global properties have long attracted the attention of several researchers; see e.g. [1,2,3,4,5,6, 8, 9, 11] as well as further references therein and subsequent works.

A straightforward way to define a corank m tube structure goes as follows: given a compact manifold M and closed 1-forms $\omega _1, \ldots , \omega _m$ on M, we look at the product manifold $M \times \mathbb {T}^m$ and the sub bundle $\mathcal {V}\subset \mathbb {C}T (M \times \mathbb {T}^m)$ annihilated by all the forms $\zeta _k \doteq \mathrm {d}x_k - \omega _k$, $k = 1, \ldots , m$; here $(x_1, \ldots , x_m)$ denote standard angular coordinates on the torus $\mathbb {T}^m$. Such a bundle is then involutive (in the sense of Frobenius) and therefore gives rise to a complex of first-order differential operators on $M \times \mathbb {T}^m$ [7, 17], whose first operator $\mathbb {L}$ maps (complex-valued) functions to 1-forms via the expression

$$\begin{aligned} \mathbb {L}f \doteq \mathrm {d}_t f + \sum _{k = 1}^m \omega _k \wedge (\partial _{x_k} f), \end{aligned}$$

(1.1)

where $t \in M$, $x \in \mathbb {T}^m$ and $\mathrm {d}_{t}$ stands for the exterior derivative on M.

The question of determining global hypoellipticity of $\mathbb {L}$ is then of interest:

$$\begin{aligned} u \in \mathcal {D}'(M \times \mathbb {T}^m) \ \text {and} \ \mathbb {L}u \in \Lambda ^1 C^\infty (M \times \mathbb {T}^m) \Longrightarrow u \in C^\infty (M \times \mathbb {T}^m), \end{aligned}$$

(1.2)

which was previously investigated mainly when $m = 1$, or for general corank when M is itself a torus. In the former situation, a classical result [1, Theorem 2.4] yields a complete characterization when $\omega = \omega _1$ is real: $\mathbb {L}$ is globally hypoelliptic in $M \times \mathbb {T}^1$ if and only if $\omega$ is neither rational nor Liouville – a Diophantine condition that aims to describe how $\omega$ can be approximated by rational forms in the Fréchet topology of $\Lambda ^1 C^\infty (M)$. This is our motivation and starting point.

In this work, we take M a compact (and for simplicity also connected and orientable) real-analytic manifold and real-analytic 1-forms $\omega _1, \ldots , \omega _m$ which we assume to be real-valued and closed, aiming to characterize, instead, global analytic hypoellipticity of $\mathbb {L}$:

$$\begin{aligned} u \in \mathcal {D}'(M \times \mathbb {T}^m) \ \text {and} \ \mathbb {L}u \in \Lambda ^1 C^\omega (M \times \mathbb {T}^m) \Longrightarrow u \in C^\omega (M \times \mathbb {T}^m). \end{aligned}$$

Besides some necessary conceptual adjustments for treating the case of arbitrary corank m, the actual difficulties arise from the more delicate nature of the spaces of real-analytic functions and forms. In Sect. 2, we define properly their natural (locally convex) topologies on a general compact, real-analytic manifold: from a functional analytic perspective, such topologies turn them in what one calls DFS spaces, which are well-understood [12] but have properties rather diverse than, say, Fréchet spaces. This fact is crucial to understand the structure of their bounded sets – a notion that is instrumental in the very definition of the number theoretic conditions (Definition 3.2) we impose on the system $\pmb {\omega } \doteq (\omega _1, \ldots , \omega _m)$.

The advantage of such an abstract approach is that we can abstain from adding any further hypotheses to M (such as the existence of global frames) and also make our definitions inherently coordinate-free; nevertheless, all of them admit characterizations by means of local data computed via suitable norms (which we actually needed in our proofs).

We prove our results in the more general framework of Gevrey classes of order $s \ge 1$ (the real-analytic case corresponds to $s = 1$) whose definition and essential properties we recall in Sect. 2 and along the way as needed; we refer the reader to [15] for more details. Next, we investigate the global s-hypoellipticity (3.1) of $\mathbb {L}$, whose characterization is the content of our main result (Theorem 3.4). For that matter, we can relax our assumptions and suppose that the 1-forms $\omega _1, \ldots , \omega _m$ are just Gevrey of order s (even though our base manifold M is always assumed real-analytic, mainly for simplicity). Actually, one is tempted to conjecture that our strategy can be carried out in ultradifferentiable settings of Roumieu type.

It must be pointed out that our results were previously obtained when M is a torus $\mathbb {T}^n$ [9, Theorem 8.3], where its strong geometric properties were used: for instance, its parallelizability and the possibility of doing “total” Fourier series (whilst in our case only a partial Fourier series in the x-variable makes sense, see Sect. 5), which in turn enables one to effectively reduce $\mathbb {L}$ to an operator with constant coefficients in $\mathbb {T}^n \times \mathbb {T}^m$; such properties make many technical issues a lot simpler. In the present work, however, we prove our results for a general M—in particular, we do not make use of symmetries or assume the existence of a global frame of vector fields for $\mathcal {V}$.

In [9], even the definition of the correct number theoretic condition is clearer: theirs and ours turn out, however, to be the same (Proposition 4.1); this is achieved through a concrete realization of our abstract Diophantine conditions by means of the so-called matrix of periods of $\pmb {\omega }$ (Sect. 4), which also happens to be critical in obtaining estimates throughout the proofs. Curiously, what plays a role in our conditions is the dimension d of the homology space $H_1(M; \mathbb {R})$, and not the dimension n of M as a manifold, thus revealing their true nature (when $M = \mathbb {T}^n$ these parameters are of course equal).

Moreover, although at a first glance there is no relationship between the definitions of s-exponential Liouville systems (Definition 3.2) for different values of s, the fact that these conditions can be read off as inequalities involving the matrix of periods of $\pmb {\omega }$ allows us to compare them, and conclude, for instance, that when $s_1 > s_2$ the global $s_1$-hypoellipticity of $\mathbb {L}$ implies its global $s_2$-hypoellipticity (provided the latter makes sense).

Finally, a similar condition can be obtained in classifying (smooth) global hypoellipticity of $\mathbb {L}$, thus generalizing [1, Theorem 2.4] to arbitrary corank. We state it in Sect. 7 (the proofs are omitted since can be easily obtained using our framework). This condition can be encoded in an inequality (7.1) involving the matrix of periods of $\pmb {\omega }$ as well (of “polynomial flavor”, as in [8]); which, in turn, imply the Gevrey ones (4.2) by simple comparison. A corollary of this reasoning is the following: if $\mathbb {L}$ is globally hypoelliptic in $M \times \mathbb {T}^m$ then it is also globally s-hypoelliptic in $M \times \mathbb {T}^m$ for every $s \ge 1$ for which $\omega _1, \ldots , \omega _m$ are $G^s$.

2 Spaces of Gevrey forms on compact manifolds and their topologies

The space $G^s(U)$ of Gevrey functions of order $s \ge 1$ over an open set $U \subset \mathbb {R}^n$ consists of all functions $f \in C^\infty (U)$ such that for each compact set $K \subset U$ one can find constants $C, h > 0$ for which

$$\begin{aligned} \sup _K |\partial ^\alpha f| \le C h^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n. \end{aligned}$$

Given $K \subset \mathbb {R}^n$ a regular compact set (i.e. K is the closure of a bounded open set with smooth boundary) and $h > 0$ we define

$$\begin{aligned} G^{s,h}(K) \doteq \left\{ f \in C^\infty (K) \ ; \ \Vert f \Vert _{s,h,K} \doteq \sup _{\alpha \in \mathbb {Z}_+^n} h^{-|\alpha |} \alpha !^{-s} \sup _K |\partial ^\alpha f| < \infty \right\} . \end{aligned}$$

This is a Banach space with respect to the norm $\Vert \cdot \Vert _{s,h,K}$, and for $h_+ > h$ the natural inclusion map $G^{s,h}(K) \hookrightarrow G^{s,h_+}(K)$ is compact, meaning that

$$\begin{aligned} G^s(K) \doteq \varinjlim _{h > 0} G^{s,h}(K) \end{aligned}$$

is a so-called DFS space.

Now let $\Omega$ be a real-analytic manifold. A function $f \in C^\infty (\Omega )$ is said to belong to $G^s(\Omega )$ if given an analytic atlas $\{ (U_i, \chi _i) \}_{i \in I}$ of $\Omega$ we have that $f \circ \chi _i^{-1} \in G^s(\chi _i(U_i))$ for every $i \in I$; this is a meaningful definition since Gevrey regularity is preserved by composition with real-analytic diffeomorphisms. It is also independent of our choice of the atlas $\{ (U_i, \chi _i) \}_{i \in I}$.

When $\Omega$ is further assumed to be compact (and, for simplicity, also connected) we endow $G^s(\Omega )$ with a locally convex topology as follows. We select a finite analytic atlas $\{ (U_i, \chi _i) \}_{i \in I}$ of $\Omega$ and regular compact sets $K_i \subset U_i$ whose interiors still cover $\Omega$, and endow $G^s(\Omega )$ with the coarsest topology which makes continuous each one of the linear maps

$$\begin{aligned} f \in G^s(\Omega ) \longmapsto f \circ \chi _i^{-1} \in G^s(\chi _i(K_i)), \quad i \in I. \end{aligned}$$

Or, equivalently, the coarsest topology that makes continuous their direct sum

$$\begin{aligned} f \in G^s(\Omega ) \longmapsto (f \circ \chi _i^{-1})_{i \in I} \in \bigoplus _{i \in I} G^s(\chi _i(K_i)), \end{aligned}$$

(2.1)

where we endow the right-hand side with the (finite) direct sum topology; this is also a DFS space, actually [12, Theorems 9 and 10]

$$\begin{aligned} \bigoplus _{i \in I} G^s(\chi _i(K_i)) = \varinjlim _{h > 0} \bigoplus _{i \in I} G^{s,h} (\chi _i(K_i)). \end{aligned}$$

(2.2)

Notice that the map (2.1) is injective since the family $\{ K_i \}_{i \in I}$ covers $\Omega$, and also has closed range (as one easily checks using (2.2) and [12, Theorem 6’]), being therefore a topological isomorphism onto its range (since closed subspaces of DFS spaces are also DFS [12, Theorem 7’] and the Open Mapping Theorem [13, p. 59] applies).

This device allows one to recast the topology on $G^s(\Omega )$ as follows: for each $h > 0$ we define

$$\begin{aligned} G^{s,h}(\Omega ) \doteq \{ f \in G^s(\Omega ) \ ; \ f \circ \chi _i^{-1} \in G^{s, h}(\chi _i(K_i)), \ \forall i \in I \} \end{aligned}$$

and endow it with the norm

$$\begin{aligned} \Vert f \Vert _{s, h, \Omega } \doteq \sum _{i \in I} \Vert f \circ \chi _i^{-1} \Vert _{s, h, \chi _i(K_i)}; \end{aligned}$$

then $G^s(\Omega ) = \varinjlim G^{s,h}(\Omega )$ as the direct limit of an injective sequence of Banach spaces with compact inclusion maps. By [12, Lemma 3 and Theorem 6’] we conclude:

Proposition 2.1

A subset $B \subset G^s(\Omega )$ is bounded if and only if B is contained in some $G^{s,h}(\Omega )$ and is bounded there. A sequence $\{ f_\nu \}_{\nu \in \mathbb {N}}$ converges to zero in $G^s(\Omega )$ if and only if there exists $h > 0$ such that either one of the following equivalent conditions hold:

• (1) $\{ f_\nu \}_{\nu \in \mathbb {N}} \subset G^{s, h}(\Omega )$ and $\Vert f_\nu \Vert _{s, h, \Omega } \rightarrow 0$;

• (2) $\{ f_\nu \circ \chi _i^{-1} \}_{\nu \in \mathbb {N}} \subset G^{s, h} (\chi _i(K_i))$ and $\Vert f_\nu \circ \chi _i^{-1} \Vert _{s, h, \chi _i(K_i)} \rightarrow 0$ for every $i \in I$.

It follows that the inclusion map $G^s(\Omega ) \hookrightarrow C^\infty (\Omega )$ is continuous. Moreover, the topology we endowed $G^s(\Omega )$ with is clearly independent of the coverings employed. Indeed, denote temporarily by $\tau$ the topology on $G^s(\Omega )$ defined above. Pick any analytic chart $(U_0, \chi _0)$ in $\Omega$ and $K_0 \subset U_0$ a compact set, and let $I_0 \doteq I \cup \{0\}$. Then $\{ (U_i, \chi _i)\}_{i \in I_0}$ is a new analytic atlas and $\{ \mathring{K}_i \}_{i \in I_0}$ is an open covering of $\Omega$. By definition, the topology induced on $G^s(\Omega )$ by this new choice is the coarsest one to make each assignment

$$\begin{aligned} f \in G^s(\Omega ) \longmapsto f \circ \chi _i^{-1} \in G^s(\chi _i(K_i)), \quad i \in I_0, \end{aligned}$$

continuous. Denote it by $\tau _0$: since $I \subset I_0$ we conclude that $\tau \subset \tau _0$ by definition, i.e. the identity map $(G^s(\Omega ), \tau _0) \rightarrow (G^s(\Omega ), \tau )$ is continuous, hence a homeomorphism (by the Open Mapping Theorem), meaning that $\tau _0 = \tau$. Proceeding inductively, any finite refinement of our choices yields that same topology $\tau$; since any two initial choices admit a common refinement (namely, their union) we are done.

The same basic construction works on the space of Gevrey sections of any real-analytic vector bundle over $\Omega$, but here we will only deal with the space of Gevrey 1-forms $\Lambda ^1 G^s(\Omega )$. A smooth 1-form $f \in \Lambda ^1 C^\infty (\Omega )$ can be written on each coordinate patch $U_i$ as

$$\begin{aligned} f = \sum _{j = 1}^n f_{ij} \ \mathrm {d}\chi _{ij}, \quad f_{ij} \in C^\infty (U_i), \end{aligned}$$

where $\chi _i = (\chi _{i1}, \ldots , \chi _{in}): U_i \rightarrow \mathbb {R}^n$. Since each $\chi _i$ is a real-analytic map, we have that $f \in \Lambda ^1 G^s(\Omega )$ if and only if $f_{ij} \circ \chi _i^{-1} \in G^s(\chi _i(U_i))$ for each $j \in \{1, \ldots , n\}$ and $i \in I$.If

$$\begin{aligned} f_i \doteq (f_{i1}, \ldots , f_{in}): U_i \longrightarrow \mathbb {R}^n \end{aligned}$$

then the condition above reduces to: $f_i \circ \chi _i^{-1} \in G^s (\chi _i(U_i))^n$ for every $i \in I$. We put on $G^s (\chi _i(K_i))^n$ the (finite) product topology, which turns it into a DFS space in the same manner as above; actually

$$\begin{aligned} G^s (\chi _i(K_i))^n = \varinjlim _{h > 0} G^{s,h} (\chi _i(K_i))^n, \end{aligned}$$

where $G^{s,h} (\chi _i(K_i))^n$ is a Banach space with norm

$$\begin{aligned} (g_1, \ldots , g_n) \in G^{s,h} (\chi _i(K_i))^n \longmapsto \sum _{j = 1}^n \Vert g_j \Vert _{s,h,\chi _i(K_i)}. \end{aligned}$$

We endow $\Lambda ^1 G^s(\Omega )$ with the coarsest topology that makes the linear map

$$\begin{aligned} f \in \Lambda ^1 G^s(\Omega ) \longmapsto (f_i \circ \chi _i^{-1} )_{i \in I} \in \bigoplus _{i \in I} G^s(\chi _i(K_i))^n \end{aligned}$$

continuous: again, this is the locally convex injective limit of the Banach spaces

$$\begin{aligned} \Lambda ^1 G^{s,h}(\Omega ) \doteq \{ f \in \Lambda ^1 G^s(\Omega ) \ ; \ f_i \circ \chi _i^{-1} \in G^{s, h}(\chi _i(K_i))^n, \ \forall i \in I \}, \quad h > 0, \end{aligned}$$

where the norm is defined, say, by

$$\begin{aligned} \Vert f \Vert _{s, h, \Omega } \doteq \sum _{i \in I} \sum _{j = 1}^n \Vert f_{ij} \circ \chi _i^{-1} \Vert _{s, h, \chi _i(K_i)}, \end{aligned}$$

hence turning $\Lambda ^1 G^s(\Omega )$ into a DFS space. One then easily derives the following criteria for boundedness and convergence of sequences there.

Proposition 2.2

A subset $B \subset \Lambda ^1 G^s(\Omega )$ is bounded if and only if there exists constants $C, h > 0$ such that

$$\begin{aligned} \Vert f_{ij} \circ \chi ^{-1}_i \Vert _{s, h, \chi _i(K_i)} \le C, \end{aligned}$$

for every $f \in B$, every $i \in I$ and $j \in \{1, \ldots , n\}$. A sequence $\{ f_\nu \}_{\nu \in \mathbb {N}}$ converges to zero in $\Lambda ^1 G^s(\Omega )$ if and only if there exists $h > 0$ such that for every $i \in I$ and $j \in \{1, \ldots , n\}$ we have $\{ (f_\nu )_{ij} \circ \chi _i^{-1} \}_{\nu \in \mathbb {N}} \subset G^{s, h} (\chi _i(K_i))$ and $\Vert (f_\nu )_{ij} \circ \chi _i^{-1} \Vert _{s, h, \chi _i(K_i)} \rightarrow 0$.

Again, one deduces that the inclusion map $\Lambda ^1 G^s(\Omega ) \hookrightarrow \Lambda ^1 C^\infty (\Omega )$ is continuous and that the topology on $\Lambda ^1 G^s(\Omega )$ just introduced is independent of the coverings chosen.

3 A class of real differential operators

Let M be a compact real-analytic manifold, which for simplicity we further assume to be connected and oriented. Given a system $\omega _1, \ldots , \omega _m$ of real, closed 1-forms belonging to $\Lambda ^1 G^s(M)$, our main purpose is to study global s-hypoellipticity of the differential operator $\mathbb {L}$ as defined in (1.1), by which we mean

$$\begin{aligned} u \in \mathcal {D}'(M \times \mathbb {T}^m) \ \text {and} \ \mathbb {L}u \in \Lambda ^1 G^s(M \times \mathbb {T}^m) \Longrightarrow u \in G^s (M \times \mathbb {T}^m). \end{aligned}$$

(3.1)

Our classification will be in terms of properties of the system $\pmb {\omega } \doteq (\omega _1, \ldots , \omega _m)$. Let us first recall [1, Definition 2.1]:

Definition 3.1

A real 1-form $\alpha \in \Lambda ^1 C^\infty (M)$ is integral if $\mathrm {d}\alpha = 0$ and $\int _\sigma \alpha \in 2 \pi \mathbb {Z}$ for every 1-cycle $\sigma$ in M. It is otherwise rational if $q \alpha$ is integral for some $q \in \mathbb {Z}{\setminus } \{0\}$.

Definition 3.2

We say that $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$ is:

• (1) a rational system if there exists$\xi \in \mathbb {Z}^m {\setminus } \{0\}$ such that

  $$\begin{aligned} \xi \cdot \pmb {\omega } \doteq \sum _{k = 1}^m \xi _k \omega _k \end{aligned}$$

  is an integral 1-form i.e.

  $$\begin{aligned} \frac{1}{2 \pi } \int _{\sigma } \xi \cdot \pmb {\omega } \in \mathbb {Z}\end{aligned}$$

  (3.2)

  for every 1-cycle $\sigma$ in M.

• (2) an s-exponential Liouville system if $\pmb {\omega }$ is not rational and there exist $\epsilon > 0$, a sequence of integral forms $\{ \theta _\nu \}_{\nu \in \mathbb {N}} \subset \Lambda ^1 G^s(M; \mathbb {R})$ and $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m$ such that $|\xi _\nu | \rightarrow \infty$ and

  $$\begin{aligned} \{ e^{\epsilon |\xi _\nu |^{\frac{1}{s}}} ( \xi _\nu \cdot \pmb {\omega } - \theta _\nu ) \}_{\nu \in \mathbb {N}} \text {is bounded in} \Lambda ^1 G^s(M). \end{aligned}$$

  (3.3)

Remark 3.3

The reader should notice that $\pmb {\omega }$ can be a rational system even if no $\omega _k$ is a rational form in the sense of Definition 3.1. Moreover, the conditions set forth in Definition 3.2 depend only on the cohomology classes of $\omega _1, \ldots , \omega _m$ in $H^1(M; \mathbb {R})$. In fact, suppose that $\pmb {\omega }^\bullet \doteq (\omega _1^\bullet , \ldots , \omega _m^\bullet )$ is another m-tuple of real, closed 1-forms in $\Lambda ^1 G^s(M)$ such that $[\omega _k^\bullet ] = [\omega _k]$ in $H^1(M; \mathbb {R})$ for every $k \in \{1, \ldots , m\}$, i.e. there exist $g_k \in C^\infty (M; \mathbb {R})$ such that $\omega _k^\bullet = \omega _k + \mathrm {d}g_k$ (hence $g_k$ are a posteriori $G^s$). It is then clear that $\xi \cdot \pmb {\omega }^\bullet$ and $\xi \cdot \pmb {\omega }$ are in the same cohomology class for every $\xi \in \mathbb {Z}^m$, in particular their integrals over an 1-cycle are the same; moreover, $\xi _\nu \cdot \pmb {\omega }^\bullet - \theta _\nu ^\bullet = \xi _\nu \cdot \pmb {\omega } - \theta _\nu$ provided we let

$$\begin{aligned} \theta _\nu ^\bullet \doteq \theta _\nu - \sum _{k = 1}^m (\xi _\nu )_k \ \mathrm {d}g_k \end{aligned}$$

which is obviously integral if so is $\theta _\nu$.

We are ready to state our main result. The next sections are dedicated to prove it.

Theorem 3.4

Let $\omega _1, \ldots , \omega _m \in \Lambda ^1 G^s(M)$ be real and closed. The operator $\mathbb {L}$ defined in (1.1) is globally s-hypoelliptic if and only if $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$ is neither a rational system nor an s-exponential Liouville system.

4 The matrix of periods

The notions established in Definition 3.2 admit a more concrete characterization. Fix

and regard them as a real basis of $H_1(M; \mathbb {R})$. We may assume that these cycles are smooth (or even real-analytic [16, Theorem 5]). To $\pmb {\omega }$ (as defined in the previous section) we then assign a matrix of periods as follows: define ${\mathsf {A}}(\pmb {\omega }) \in {\mathsf {M}}_{d \times m} (\mathbb {R})$ by

$$\begin{aligned} {\mathsf {A}}(\pmb {\omega })_{\ell k} \doteq \frac{1}{2 \pi } \int _{\sigma _\ell } \omega _k, \quad \ell \in \{1, \ldots , d\}, \ k \in \{1, \ldots , m\}, \end{aligned}$$

that is¹

$$\begin{aligned} {\mathsf {A}}(\pmb {\omega }) \xi = \frac{1}{2 \pi } \left( \int _{\sigma _1} \xi \cdot \pmb {\omega }, \ldots , \int _{\sigma _d} \xi \cdot \pmb {\omega } \right) , \quad \xi \in \mathbb {Z}^m. \end{aligned}$$

(4.1)

Again, the definition of ${\mathsf {A}}(\pmb {\omega })$ clearly depends only on the classes $[\omega _1], \ldots , [\omega _m] \in H^1 (M; \mathbb {R})$ and we therefore have a linear map

$$\begin{aligned} {\mathsf {A}}: H^1(M; \mathbb {R})^m \longrightarrow {\mathsf {M}}_{d \times m} (\mathbb {R}). \end{aligned}$$

As in [9, Section 3], we say that a $d \times m$ matrix with real entries ${\mathbf {A}}$ satisfies condition $\mathrm {(DC)}_s^2$ if for every $\epsilon > 0$ there exists $C_\epsilon > 0$ such that

$$\begin{aligned} | \kappa + {\mathbf {A}} \xi | \ge C_\epsilon e^{-\epsilon (|\kappa | + |\xi |)^{\frac{1}{s}}}, \quad \forall (\kappa , \xi ) \in (\mathbb {Z}^d \times \mathbb {Z}^m) {\setminus } \{ (0,0) \}; \end{aligned}$$

(4.2)

or, equivalently, if for every $\epsilon > 0$ there exists $C_\epsilon > 0$ such that

$$\begin{aligned} | \kappa + {\mathbf {A}} \xi | \ge C_\epsilon e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall (\kappa , \xi ) \in (\mathbb {Z}^d \times \mathbb {Z}^m) {\setminus } \{ (0,0) \}. \end{aligned}$$

This condition implies [9, Lemma 8.1] that for every $\epsilon > 0$ there exists $C_\epsilon > 0$ such that

$$\begin{aligned} \max _{\ell } |e^{2 \pi i a_\ell \cdot \xi } - 1| \ge C_\epsilon e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m {\setminus } \{ 0 \}, \end{aligned}$$

(4.3)

where $a_\ell \in \mathbb {R}^m$ denotes the $\ell$-th row of ${\mathbf {A}}$.

Proposition 4.1

The system $\pmb {\omega }$ is rational if and only if ${\mathsf {A}}(\pmb {\omega })(\mathbb {Z}^m {\setminus } \{ 0 \}) \cap \mathbb {Z}^d \ne \emptyset$. It is an s-exponential Liouville system if and only if it is not rational and ${\mathsf {A}}(\pmb {\omega })$ does not satisfy $\mathrm {(DC)}_s^2$.

Proof

If $\pmb {\omega }$ is a rational system then it follows from (4.1) that ${\mathsf {A}}(\pmb {\omega }) \xi \in \mathbb {Z}^d$ for some $\xi \in \mathbb {Z}^m {\setminus } \{0\}$. Conversely, if ${\mathsf {A}}(\pmb {\omega }) \xi \in \mathbb {Z}^d$ then

$$\begin{aligned} \frac{1}{2 \pi } \int _{\sigma _\ell } \xi \cdot \pmb {\omega } \in \mathbb {Z}, \quad \forall \ell \in \{1, \ldots , d\}, \end{aligned}$$

which thanks to $\sharp$ is enough to ensure (3.2) for every 1-cycle $\sigma$ in M.

For the second statement, we need some preliminary remarks. By de Rham’s Theorem, one can identify $H^1(M; \mathbb {R})$ with the dual space of $H_1(M; \mathbb {R})$, via the pairing

$$\begin{aligned} ([\alpha ], [\sigma ]) \in H^1(M; \mathbb {R}) \times H_1(M; \mathbb {R}) \longmapsto \frac{1}{2 \pi } \int _\sigma \alpha \in \mathbb {R}, \end{aligned}$$

and hence consider

We can assume without loss of generality that each $\vartheta _\ell$ is actually real-analytic. Indeed, by endowing M with a real-analytic Riemannian metric (which is always possible thanks to Grauert’s embedding theorem [10]), the Laplace–Beltrami operator

$$\begin{aligned} \Delta \doteq \mathrm {d}\mathrm {d}^* + \mathrm {d}^* \mathrm {d}: \Lambda ^1 C^\infty (M) \longrightarrow \Lambda ^1 C^\infty (M) \end{aligned}$$

is an elliptic, real-analytic operator, and by Hodge theory every cohomology class in $H^1(M)$ has a representative $f \in \Lambda ^1 C^\infty (M)$ such that $\Delta f = 0$; such an f is therefore real-analytic thanks to the ellipticity of $\Delta$.

We may write for each $k \in \{1, \ldots , m\}$

$$\begin{aligned} \omega _k = \sum _{\ell = 1}^d \lambda _{\ell k} \vartheta _\ell + \mathrm {d}v_k, \end{aligned}$$

where each $\lambda _{\ell k} \in \mathbb {R}$ is uniquely determined by

$$\begin{aligned} \lambda _{\ell k} = \frac{1}{2 \pi } \int _{\sigma _\ell } \omega _k = {\mathsf {A}}(\pmb {\omega })_{\ell k}, \end{aligned}$$

and $v_k \in G^s(M; \mathbb {R})$ by ellipticity of the exterior derivative on M. Hence, for $\xi \in \mathbb {Z}^m$,

$$\begin{aligned} \xi \cdot \pmb {\omega } = \sum _{\ell = 1}^{d} \left( \sum _{k = 1}^{m} {\mathsf {A}}(\pmb {\omega })_{\ell k} \xi _{k} \right) \vartheta _\ell + \sum _{k = 1}^{m} \xi _{k} \mathrm {d}v_{k}. \end{aligned}$$

(4.4)

We assume from here until the end of the proof that $\pmb {\omega }$ is not a rational system; hence, as we have seen, ${\mathsf {A}}(\pmb {\omega }) \xi \notin \mathbb {Z}^d$ for every $\xi \in \mathbb {Z}^m {\setminus } \{0\}$.

Suppose first that ${\mathsf {A}}(\pmb {\omega })$ does not satisfy $\mathrm {(DC)}_s^2$. Then there exists $\epsilon > 0$ and a sequence $\{ (\kappa _{\nu }, \xi _{\nu }) \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^{d} \times \mathbb {Z}^{m} {\setminus } \{(0,0)\}$ such that

$$\begin{aligned} \lim _{\nu \rightarrow \infty } e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} | \kappa _{\nu } + {\mathsf {A}}(\pmb {\omega }) \xi _{\nu } | = 0. \end{aligned}$$

(4.5)

Suppose by contradiction that $\{ \xi _{\nu } \}_{\nu \in \mathbb {N}}$ is bounded: the same cannot hold for $\{ \kappa _{\nu } \}_{\nu \in \mathbb {N}}$ (otherwise the sequence $\{ (\kappa _{\nu }, \xi _{\nu }) \}_{\nu \in \mathbb {N}}$ would attain at most finitely many values, contradicting (4.5)). Moreover, in this case,

$$\begin{aligned} e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} | \kappa _{\nu } + {\mathsf {A}}(\pmb {\omega }) \xi _{\nu } | \ge | \kappa _{\nu } + {\mathsf {A}}(\pmb {\omega }) \xi _{\nu } | \ge | \kappa _{\nu } | - |{\mathsf {A}}(\pmb {\omega }) \xi _{\nu } | \ge | \kappa _{\nu } | - C, \end{aligned}$$

for some $C > 0$, now contradicting the unboundedness of $\{ \kappa _{\nu } \}_{\nu \in \mathbb {N}}$. We can therefore assume that $| \xi _{\nu } | \rightarrow \infty$ as $\nu \rightarrow \infty$. Now define

$$\begin{aligned} \theta _\nu \doteq - \sum _{\ell = 1}^{d} (\kappa _{\nu })_\ell \vartheta _{\ell } + \sum _{k = 1}^{m} (\xi _{\nu })_k \mathrm {d}v_{k}. \end{aligned}$$

(4.6)

Clearly each $\theta _{\nu }$ is an integral 1-form. Furthermore, using (4.4) and (4.6) we obtain

$$\begin{aligned} \rho _{\nu } \doteq e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} ( \xi _{\nu } \cdot \pmb {\omega } - \theta _{\nu } ) = e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \sum _{\ell = 1}^{d} \left( \sum _{k = 1}^{m} {\mathsf {A}}(\pmb {\omega })_{\ell k} (\xi _\nu )_{k} + (\kappa _\nu )_\ell \right) \vartheta _\ell . \end{aligned}$$

We fix a coordinate chart $(U; t_1, \ldots , t_n)$ in M and a compact set $K \subset U$. Hence, there exist $C_{1}, h_{1} > 0$ such that

$$\begin{aligned} \sup _K | \partial _t^{\alpha } \vartheta _\ell | \le C_{1} h_{1}^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \ell \in \{1, \ldots , d \}. \end{aligned}$$

Thus on K,

$$\begin{aligned} | \partial _t^{\alpha } \rho _{\nu } |&\le e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \sum _{\ell = 1}^{d} \left( \sum _{k = 1}^{m} |{\mathsf {A}}(\pmb {\omega })_{\ell k} (\xi _\nu )_{k} + (\kappa _\nu )_\ell | \right) |\partial _t^\alpha \vartheta _\ell | \\&\le C_{1} h_{1}^{|\alpha |} \alpha !^s e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \sum _{\ell = 1}^{d} \sum _{k = 1}^{m} |{\mathsf {A}}(\pmb {\omega })_{\ell k} (\xi _\nu )_{k} + (\kappa _\nu )_\ell | \\&\le C_{2} h_{1}^{|\alpha |} \alpha !^s e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} | {\mathsf {A}}(\pmb {\omega }) \xi _{\nu } + \kappa _{\nu } | \\&\le C_{3} h_{1}^{|\alpha |} \alpha !^s, \end{aligned}$$

for some constant $C_{3} > 0$ independent of $\nu$ thanks to (4.5). This proves that $\{ \rho _\nu \}_{\nu \in \mathbb {N}}$ is bounded in $\Lambda ^1 G^s(M)$; hence, $\pmb {\omega }$ is an s-exponential Liouville system.

Conversely, suppose that there exist $\epsilon > 0$, a sequence of integral forms $\{ \theta _{\nu } \}_{\nu \in \mathbb {N}} \subset \Lambda ^{1} G^{s}(M; \mathbb {R})$ and $\{ \xi _{\nu } \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^{m}$, such that $|\xi _{\nu }| \rightarrow \infty$ and (3.3) holds. Again, we may write for every $\nu \in \mathbb {N}$

$$\begin{aligned} \theta _{\nu } = \sum _{\ell = 1}^{d} \beta _{\nu \ell } \vartheta _\ell + \mathrm {d}g_{\nu }, \end{aligned}$$

(4.7)

where $\beta _{\nu \ell } \in \mathbb {Z}$ and $g_\nu \in G^s(M; \mathbb {R})$. By associating (4.4) with (4.7) we have

$$\begin{aligned} \rho _\nu&\doteq e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} ( \xi _{\nu } \cdot \pmb {\omega } - \theta _{\nu } ) \\&= e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \sum _{\ell = 1}^{d} \left( \sum _{k = 1}^{m} {\mathsf {A}}(\pmb {\omega })_{\ell k} (\xi _\nu )_{k} - \beta _{\nu \ell } \right) \vartheta _\ell + e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \left( \sum _{k = 1}^{m} (\xi _{\nu })_k \mathrm {d}v_{k} - \mathrm {d}g_{\nu } \right) . \end{aligned}$$

As a consequence of the exactness of both $\mathrm {d}v_{k}$ and $\mathrm {d}g_{\nu }$ we have

$$\begin{aligned} \frac{1}{2 \pi } \int _{\sigma _{\ell }} \rho _{\nu } = e^{\epsilon |\xi _{\nu }|^{\frac{1}{s}}} \left( \sum _{k = 1}^{m} {\mathsf {A}}(\pmb {\omega })_{\ell k} (\xi _\nu )_{k} - \beta _{\nu \ell } \right) , \quad \forall \ell \in \{1, \ldots , d \}, \ \forall \nu \in \mathbb {N}. \end{aligned}$$

Since by hypothesis $\{ \rho _\nu \}_{\nu \in \mathbb {N}}$ is bounded in $\Lambda ^1 G^s(M)$ we can find a constant $C > 0$ such that

$$\begin{aligned} \left| \int _{\sigma _{\ell }} \rho _{\nu } \right| \le C, \quad \forall \ell \in \{1, \ldots , d \}, \ \forall \nu \in \mathbb {N}. \end{aligned}$$

Indeed, only the sup norms of the local coefficients of the $\rho _\nu$ added across a finite covering of M play a role in the estimation of such integrals (no derivatives are required). Therefore, by setting $\kappa _\nu \doteq -(\beta _{\nu 1}, \ldots \beta _{\nu d}) \in \mathbb {Z}^d$ one obtains

$$\begin{aligned} | {\mathsf {A}}(\pmb {\omega })\xi _{\nu } + \kappa _\nu | \le C_1 e^{- \epsilon |\xi _{\nu }|^{\frac{1}{s}}}, \quad \forall \nu \in \mathbb {N}, \end{aligned}$$

for some $C_1 > 0$. Since $|\xi _{\nu }| \rightarrow \infty$, by possibly extracting a subsequence we may assume that $e^{-\frac{\epsilon }{2} |\xi _{\nu }|^{\frac{1}{s}}} < (C_1 \nu )^{-1}$, for every $\nu \in \mathbb {N}$, which allows us to obtain

$$\begin{aligned} | {\mathsf {A}}(\pmb {\omega })\xi _{\nu } + \kappa _\nu | \le \frac{1}{\nu } e^{- \frac{\epsilon }{2} |\xi _{\nu }|^{\frac{1}{s}}}, \quad \forall \nu \in \mathbb {N}; \end{aligned}$$

therefore ${\mathsf {A}}(\pmb {\omega })$ does not satisfy $\mathrm {(DC)}_s^2$. $\square$

Remark 4.2

If $d \doteq \dim H^1(M; \mathbb {R})$ then any ${\mathbf {A}} \in {\mathsf {M}}_{d \times m}(\mathbb {R})$ is the matrix of periods of some system of closed 1-forms $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$ on M. Indeed, defining

$$\begin{aligned} \omega _k \doteq \sum _{\ell = 1}^d {\mathbf {A}}_{\ell k} \vartheta _\ell , \quad k \in \{1, \ldots , m\}, \end{aligned}$$

where $\vartheta _1, \ldots , \vartheta _d$ are as in $\flat$, yields ${\mathsf {A}}(\pmb {\omega }) = {\mathbf {A}}$ by previous computations. We can use this fact to provide examples of systems $\pmb {\omega }$ on M that satisfy the number theoretic conditions in Definition 3.2, as these can be read directly from ${\mathsf {A}}(\pmb {\omega })$ by Proposition 4.1. For instance, in [9] many examples of matrices ${\mathbf {A}}$ that do (or do not) satisfy condition $\mathrm {(DC)}_s^2$ are discussed.

5 Partial Fourier series

Let $U \subset \mathbb {R}^n$ be an open set. Given $f \in C^\infty (U \times \mathbb {T}^m)$ we define for each $\xi \in \mathbb {Z}^m$ a function ${\hat{f}}_\xi \in C^\infty (U)$ by

$$\begin{aligned} {\hat{f}}_\xi (t) \doteq \int _{\mathbb {T}^m} e^{-i x \cdot \xi } f(t, x) \mathrm {d}x, \quad t \in U; \end{aligned}$$

more generally, if $f \in \mathcal {D}'(U \times \mathbb {T}^m)$ we define ${\hat{f}}_\xi \in \mathcal {D}'(U)$ by the rule

$$\begin{aligned} \phi \in C^\infty _c(U) \longmapsto \langle f, \phi \otimes e^{-i x \cdot \xi } \rangle \in \mathbb {C}. \end{aligned}$$

It is easy to see that this construction is local in U; that is

$$\begin{aligned} V \subset U \,\text {open} \Longrightarrow \widehat{(f|_{V \times \mathbb {T}^m})}_\xi = {\hat{f}}_\xi |_V. \end{aligned}$$

A related issue is the following formula that one checks at once:

$$\begin{aligned} \widehat{(\phi f)}_\xi = \phi {\hat{f}}_\xi , \quad \forall \phi \in C^\infty (U). \end{aligned}$$

(5.1)

Another important feature is its invariance under changes of variables. We take $\chi : U' \rightarrow U$ a diffeomorphism between open sets in $\mathbb {R}^n$ and define

$$\begin{aligned} X \doteq \chi \times \mathrm {id}_{\mathbb {T}^m}: U' \times \mathbb {T}^m \longrightarrow U \times \mathbb {T}^m, \end{aligned}$$

(5.2)

that is, $X(t',x) = (\chi (t'), x)$ for $(t',x) \in U' \times \mathbb {T}^m$. One checks easily that

$$\begin{aligned} \widehat{(X^*f)}_\xi = \chi ^* {\hat{f}}_\xi , \quad \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

(5.3)

whatever $f \in \mathcal {D}'(U \times \mathbb {T}^m)$.

Finally, we are able to define ${\hat{f}}_\xi \in \mathcal {D}'(M)$ for $f \in \mathcal {D}'(M \times \mathbb {T}^m)$, where M is now a smooth manifold. On a coordinate domain $U \subset M$ we must define ${\hat{f}}_\xi |_U \in \mathcal {D}'(U)$ through the following steps:

• (1) take $\chi _1: U'_1 \rightarrow U$ a diffeomorphism where $U_1' \subset \mathbb {R}^n$ is an open set;

• (2) define $X_1 \doteq \chi _1 \times \mathrm {id}_{\mathbb {T}^m}$;

• (3) let $f_1 \doteq X_1^* (f|_{U \times \mathbb {T}^m}) \in \mathcal {D}'(U_1' \times \mathbb {T}^m)$;

• (4) take its Fourier coefficient $\widehat{(f_1)}_\xi \in \mathcal {D}'(U_1')$ (using the former definition);

• (5) define${\hat{f}}_\xi |_U \doteq (\chi _1^{-1})^* \widehat{(f_1)}_\xi \in \mathcal {D}'(U)$.

This definition is independent of our choice of parametrization on U: if $\chi _2: U'_2 \rightarrow U$ is another such diffeomorphism then we let, in accordance with (5.2),

$$\begin{aligned} \chi \doteq \chi _2^{-1} \circ \chi _1: U_1' \longrightarrow U_2' \Longrightarrow X \doteq \chi \times \mathrm {id}_{\mathbb {T}^m} = X_2^{-1} \circ X_1 : U_1' \times \mathbb {T}^m \longrightarrow U_2' \times \mathbb {T}^m \end{aligned}$$

so that, by (5.3),

$$\begin{aligned} \chi ^* \widehat{(f_2)}_\xi = \widehat{(X^* f_2)}_\xi = \widehat{(f_1)}_\xi , \end{aligned}$$

where we have used that

$$\begin{aligned} X^* f_2 = X^* X_2^* (f|_{U \times \mathbb {T}^m}) = (X_2 \circ X)^* (f|_{U \times \mathbb {T}^m}) = X_1^* (f|_{U \times \mathbb {T}^m}) = f_1, \end{aligned}$$

and from which it follows that

$$\begin{aligned} (\chi _1^{-1})^* \widehat{(f_1)}_\xi = (\chi _1^{-1})^* \chi ^* \widehat{(f_2)}_\xi = (\chi _2^{-1})^* \widehat{(f_2)}_\xi . \end{aligned}$$

Notice that this procedure yields, for $f \in L^1_{\mathrm {loc}}(M \times \mathbb {T}^m)$:

$$\begin{aligned} f_1(t',x) = f(\chi _1(t'), x)&\Longrightarrow \widehat{(f_1)}_\xi (t') = \int _{\mathbb {T}^m} e^{-i x \cdot \xi } f(\chi _1(t'), x) \mathrm {d}x \\&\Longrightarrow {\hat{f}}_\xi (t) = \int _{\mathbb {T}^m} e^{-i x \cdot \xi } f(t, x) \mathrm {d}x \end{aligned}$$

as expected.

Let again $U \subset \mathbb {R}^n$ be an open set.

Lemma 5.1

An $f \in \mathcal {D}'(U \times \mathbb {T}^m)$ is zero if and only if ${\hat{f}}_\xi = 0$ for every $\xi \in \mathbb {Z}^m$.

Proof

Suppose all the partial Fourier coefficients of f are zero. Thanks to (5.1) we may assume that $f \in \mathcal {E}'(U \times \mathbb {T}^m)$, which we regard as a continuous linear functional on $C^\infty (U \times \mathbb {T}^m)$, whose vanishing we proceed to check. By $\mathbb {C}$-linearity, it is sufficient to show that it vanishes on the space of real-valued functions $C^\infty (U \times \mathbb {T}^m; \mathbb {R})$.

Let $\psi \in C^\infty (\mathbb {T}^m; \mathbb {R})$. Using Fourier series we can write

$$\begin{aligned} \psi = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} {\hat{\psi }}_\xi \ e^{i x \cdot \xi }, \quad {\hat{\psi }}_\xi \in \mathbb {C}, \end{aligned}$$

with convergence in $C^\infty (\mathbb {T}^m)$. We have that

$$\begin{aligned} \langle f, \phi \otimes \psi \rangle = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} {\hat{\psi }}_\xi \ \langle f, \phi \otimes e^{i x \cdot \xi } \rangle = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} {\hat{\psi }}_\xi \ \langle {\hat{f}}_{(-\xi )}, \phi \rangle = 0, \end{aligned}$$

whatever $\phi \in C^\infty (U; \mathbb {R})$; hence, by passing to finite sums of simple tensors we prove that f vanishes on $\mathcal {A} \doteq C^\infty (U; \mathbb {R}) \otimes C^\infty (\mathbb {T}^m; \mathbb {R})$. This is a real subalgebra of $C^\infty (U \times \mathbb {T}^m; \mathbb {R})$ that satisfies:

• (1) given distinct $(t,x), (t', x') \in U \times \mathbb {T}^m$ there exists $g \in \mathcal {A}$ such that $g(t,x) \ne g(t',x')$;

• (2) given $(t,x) \in U \times \mathbb {T}^m$ there exists $g \in \mathcal {A}$ such that $g(t,x) \ne 0$; and

• (3) given $(t,x) \in U \times \mathbb {T}^m$ and a non-zero $(v,w) \in T_t U \oplus T_x \mathbb {T}^m \cong T_{(t,x)} (U \times \mathbb {T}^m)$ there exists $g \in \mathcal {A}$ such that $\mathrm {d}g_{(t,x)}(v, w) \ne 0$.

These happen to be the hypotheses of Nachbin’s extension of the Stone-Weierstrass Theorem [14, Theorem 1.2.1], by virtue of which $\mathcal {A}$ is dense in $C^\infty (U \times \mathbb {T}^m; \mathbb {R})$. Continuity of f entails our conclusion. $\square$

5.1 Gevrey type estimates

Back to an open set $U \subset \mathbb {R}^n$, for an $f \in \mathcal {D}'(U \times \mathbb {T}^m)$ we take a closer look at the following couple of properties:

• (1) for each $\xi \in \mathbb {Z}^m$ we have that ${\hat{f}}_\xi \in C^\infty (U)$ and

• (2) for each compact set $K \subset U$ there exist constants $C, h, \epsilon > 0$ such that

  $$\begin{aligned} \sup _K | \partial _t^\alpha {\hat{f}}_\xi | \le C h^{|\alpha |} \alpha !^s e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

  (5.4)

Let us investigate how they behave under a change of variables $\chi : U' \rightarrow U$. First, concerning condition 1, it is clear from (5.3) that ${\hat{f}}_\xi$ is smooth in U if and only if $\widehat{(X^*f)}_\xi$ is smooth in $U'$. As for condition 2, we will be interested only in the case when $\chi$ is a real-analytic diffeomorphism, in which case the same is true for its associated diffeomorphism X defined in (5.2). Keeping in mind (5.3), a careful inspection in the proof of [15, Proposition 1.4.6] shows that if (5.4) holds for some constants $C, h, \epsilon > 0$ (provided of course each ${\hat{f}}_\xi$ is smooth) then

$$\begin{aligned} \sup _{K'} \left| \partial _{t'}^\alpha \widehat{(X^*f)}_\xi \right| \le C' (h')^{|\alpha |} \alpha !^s e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

where $K' \doteq \chi ^{-1}(K) \subset U'$ and $C', h' > 0$ depend only on C, h and $\chi$. Notice that every compact set $K' \subset U'$ is of that form. We summarize our conclusions in the first statement of the next result.

Proposition 5.2

Conditions 1–2 are invariant by real-analytic changes of variables. They hold if and only if $f \in G^s(U \times \mathbb {T}^m)$.

Proof

We prove the equivalence stated above. If$f \in G^s(U \times \mathbb {T}^m)$ then given $K \subset U$ a compact set there exist $C, h > 0$ such that

$$\begin{aligned} \sup _{K \times \mathbb {T}^m} |\partial _t^\alpha f| \le C h^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n; \end{aligned}$$

hence for any $t \in K$ and $\alpha \in \mathbb {Z}_+^n$,

$$\begin{aligned} | \partial _t^\alpha {\hat{f}}_\xi (t)| = \left| \int _{\mathbb {T}^m} e^{-i x \cdot \xi } \partial _t^\alpha f(t, x) \mathrm {d}x \right| \le \int _{\mathbb {T}^m} |\partial _t^\alpha f(t, x)| \mathrm {d}x \le C (2 \pi )^m h^{|\alpha |} \alpha !^s. \end{aligned}$$

For the converse, conditions 1–2 ensure that the series

$$\begin{aligned} \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} {\hat{f}}_\xi (t) e^{i x \cdot \xi } \end{aligned}$$

converges uniformly on compact sets to a continuous function $g: U \times \mathbb {T}^m \rightarrow \mathbb {C}$, which is actually smooth and, moreover, satisfies

$$\begin{aligned} \partial _t^\alpha \partial _x^\beta g (t,x) = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} \partial _t^\alpha {\hat{f}}_\xi (t) (i \xi )^\beta e^{i x \cdot \xi }, \end{aligned}$$

with uniform convergence on compact sets for every $(\alpha , \beta ) \in \mathbb {Z}_+^n \times \mathbb {Z}_+^m$ thanks to 2. In particular, for $t \in K$ we have

$$\begin{aligned} |\partial _t^\alpha \partial _x^\beta g (t,x)| \le \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} |\partial _t^\alpha {\hat{f}}_\xi (t)| |\xi |^{|\beta |} \le \frac{1}{(2 \pi )^m} C h^{|\alpha |} \alpha !^s \sum _{\xi \in \mathbb {Z}^m} |\xi |^{|\beta |} e^{-\epsilon |\xi |^{\frac{1}{s}}}, \end{aligned}$$

which easily yields $g \in G^s(U \times \mathbb {T}^m)$. Since we are allowed to integrate under the summation sign we obtain

$$\begin{aligned} {\hat{g}}_\eta (t) = \frac{1}{(2 \pi )^m} \sum _{\xi \in \mathbb {Z}^m} \int _{\mathbb {T}^m} e^{-i x \cdot \eta } {\hat{f}}_\xi (t) e^{i x \cdot \xi } \mathrm {d}x = {\hat{f}}_\eta (t), \quad \forall \eta \in \mathbb {Z}^m; \end{aligned}$$

hence, by Lemma 5.1 we conclude that $f = g$ is $G^s$. $\square$

5.2 Forms of type (0, 1)

Let M be a smooth manifold. Given $U \subset M$ the domain of a coordinate system $(t_1, \ldots , t_n)$ we denote by $\Lambda ^{0,1} C^\infty (U \times \mathbb {T}^m)$ the space of 1-forms f on $U \times \mathbb {T}^m$ with no $\mathrm {d}x$ component i.e.

$$\begin{aligned} f = \sum _{j = 1}^n f_j \ \mathrm {d}t_j, \end{aligned}$$

(5.5)

where $f_j \in C^\infty (U \times \mathbb {T}^m)$. We then define ${\hat{f}}_\xi \in \Lambda ^1 C^\infty (U)$ by

$$\begin{aligned} {\hat{f}}_\xi \doteq \sum _{j = 1}^n \widehat{(f_j)}_\xi \ \mathrm {d}t_j. \end{aligned}$$

(5.6)

One can prove that these definitions are independent of the choice of coordinates on U (recall (5.1)), which allows us to define the space $\Lambda ^{0,1} C^\infty (M \times \mathbb {T}^m)$ of all $f \in \Lambda ^1 C^\infty (M \times \mathbb {T}^m)$ such that $f|_{U \times \mathbb {T}^m} \in \Lambda ^{0,1} C^\infty (U \times \mathbb {T}^m)$ for every coordinate open set $U \subset M$, as well as their partial Fourier coefficients ${\hat{f}}_\xi \in \Lambda ^1 C^\infty (M)$. We also let $\Lambda ^{0,1} G^s(U \times \mathbb {T}^m) \doteq \Lambda ^{0,1} C^\infty (U \times \mathbb {T}^m) \cap \Lambda ^1 G^s(U \times \mathbb {T}^m)$.

More generally, for $U \subset M$ we define $\Lambda ^{0,1} \mathcal {D}'(U \times \mathbb {T}^m)$ as the space of currents $f \in \Lambda ^1 \mathcal {D}'(U \times \mathbb {T}^m)$ which can be written as (5.5), where now $f_j \in \mathcal {D}'(U \times \mathbb {T}^m)$ for each $j \in \{1, \ldots , n\}$, in which case we define ${\hat{f}}_\xi \in \Lambda ^1 \mathcal {D}'(U)$ by (5.6). Again, this is independent of the coordinates $(t_1, \ldots , t_n)$ so we can define the space of currents $\Lambda ^{0,1} \mathcal {D}'(M \times \mathbb {T}^m)$ and their partial Fourier coefficients, which are elements of $\Lambda ^1 \mathcal {D}'(M)$. One can apply the results in the previous section to each local coefficient $f_j$ in order to retrieve Gevrey regularity of f from local estimates on ${\hat{f}}_\xi$; more precisely, on their local coefficients $\widehat{(f_j)}_\xi$.

Concerning our operator $\mathbb {L}$ defined in (1.1), notice that for $u \in C^\infty (M \times \mathbb {T}^m)$ we have that $\mathbb {L}u \in \Lambda ^{0,1} C^\infty (M \times \mathbb {T}^m)$ and, moreover,

$$\begin{aligned} \widehat{(\mathbb {L}u)}_\xi = \mathrm {d}{\hat{u}}_\xi + i {\hat{u}}_\xi (\xi \cdot \pmb {\omega }) \doteq \mathbb {L}_\xi {\hat{u}}_\xi , \end{aligned}$$

(5.7)

thus defining a differential operator $\mathbb {L}_\xi = \mathrm {d}+ i (\xi \cdot \pmb {\omega }) \wedge \cdot : C^\infty (M) \rightarrow \Lambda ^1 C^\infty (M)$; identity (5.7) also holds for $u \in \mathcal {D}'(M \times \mathbb {T}^m)$. It is enough to check this locally: we reason in a coordinate chart $(U; t_1, \ldots , t_n)$, where

$$\begin{aligned} \omega _k = \sum _{j = 1}^n \omega _{jk} \ \mathrm {d}t_j, \quad \omega _{jk} \in C^\infty (U), \end{aligned}$$

hence, by (1.1), we have that

$$\begin{aligned} \mathbb {L}u = \sum _{j = 1}^n \left( \partial _{t_j} u + \sum _{k = 1}^m \omega _{jk} \partial _{x_k} u \right) \mathrm {d}t_j \doteq \sum _{j = 1}^n (\mathrm {L}_j u) \ \mathrm {d}t_j \doteq \sum _{j = 1}^n f_j \ \mathrm {d}t_j \end{aligned}$$

belongs to $\Lambda ^{0,1} \mathcal {D}'(U \times \mathbb {T}^m)$, where $\mathrm {L}_j$ is a complex vector field with smooth coefficients on $U \times \mathbb {T}^m$ for each $j \in \{1, \ldots , n\}$. We take as a parametrization of U the inverse of the chart map $\chi \doteq (t_1, \ldots , t_n)^{-1}: U' \rightarrow U$ where $U' \subset \mathbb {R}^n$ is an open set and let $X \doteq \chi \times \mathrm {id}_{\mathbb {T}^m}$. Notice that

$$\begin{aligned} X^* f_j = X^* (\mathrm {L}_j u) = (X^* \mathrm {L}_j) (X^* u) = \partial _{t_j} (X^* u) + \sum _{k = 1}^m (\omega _{jk} \circ \chi ) \partial _{x_k} (X^* u), \end{aligned}$$

where by abuse of notation $(t_1, \ldots , t_n)$ also denotes the standard Euclidean coordinates on $U'$ so that $\partial _{t_j}$ is simply a partial derivative. Using the local definition of the Fourier coefficients we take an arbitrary $\phi \in C^\infty _c(U')$ and evaluate

$$\begin{aligned} \langle \widehat{(X^* f_j)}_\xi , \phi \rangle&= \langle X^* f_j, \phi \otimes e^{-i x \cdot \xi } \rangle \\&= \langle \partial _{t_j} (X^* u), \phi \otimes e^{-i x \cdot \xi } \rangle + \sum _{k = 1}^m \langle (\omega _{jk} \circ \chi ) \partial _{x_k} (X^* u), \phi \otimes e^{-i x \cdot \xi } \rangle \\&= - \langle X^* u, ( \partial _{t_j} \phi ) \otimes e^{-i x \cdot \xi } \rangle - \sum _{k = 1}^m \langle X^* u, (\omega _{jk} \circ \chi ) \phi \otimes (-i \xi _k e^{-i x \cdot \xi }) \rangle \\&= - \langle \widehat{(X^* u)}_\xi , \partial _{t_j} \phi \rangle + i \sum _{k = 1}^m \xi _k \langle \widehat{(X^* u)}_\xi , (\omega _{jk} \circ \chi ) \phi \rangle \\&= \langle \partial _{t_j} \widehat{(X^* u)}_\xi , \phi \rangle + i \sum _{k = 1}^m \xi _k \langle (\omega _{jk} \circ \chi ) \widehat{(X^* u)}_\xi , \phi \rangle , \end{aligned}$$

which implies that

$$\begin{aligned} \widehat{(X^* f_j)}_\xi = \partial _{t_j} \widehat{(X^* u)}_\xi + i \sum _{k = 1}^m \xi _k (\omega _{jk} \circ \chi ) \widehat{(X^* u)}_\xi \end{aligned}$$

as elements of $\mathcal {D}'(U')$. Finally, pulling everything back to U via $\chi ^{-1}$ we obtain, by definition,

$$\begin{aligned} \widehat{(f_j)}_\xi = \partial _{t_j} {\hat{u}}_\xi + i \sum _{k = 1}^m \xi _k \omega _{jk} {\hat{u}}_\xi \quad \text {in}\, U, \end{aligned}$$

for each $j \in \{1, \ldots , n\}$; hence, by (5.6),

$$\begin{aligned} {\hat{f}}_\xi = \sum _{j = 1}^n \left( \partial _{t_j} {\hat{u}}_\xi + i {\hat{u}}_\xi \sum _{k = 1}^m \xi _k \omega _{jk} \right) \mathrm {d}t_j = \mathrm {d}{\hat{u}}_\xi + i {\hat{u}}_\xi \sum _{k = 1}^m \xi _k \omega _k = \mathrm {d}{\hat{u}}_\xi + i {\hat{u}}_\xi (\xi \cdot \pmb {\omega }). \end{aligned}$$

6 Proof of Theorem 3.4

Before we start, we state a technical lemma whose proof follows closely that of [9, Lemma 4.3].

Lemma 6.1

Let $U \subset \mathbb {R}^n$ be an open set and $\pmb {\phi } = (\phi _1, \ldots , \phi _m): U \rightarrow \mathbb {R}^m$ be a smooth map satisfying the following condition: for some compact set $K \subset U$ there exist $C_1, h_1 > 0$ such that

$$\begin{aligned} \sup _K | \partial ^\alpha \phi _k | \le C_1 h_1^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall k \in \{1, \ldots , m\}. \end{aligned}$$

Then for every $\epsilon > 0$ we can find $h_2 > 0$ depending on $C_1, h_1, m$ and $\epsilon$ such that

$$\begin{aligned} \sup _K | \partial ^\alpha e^{i \xi \cdot \pmb {\phi }} | \le h_2^{|\alpha |} \alpha !^s e^{\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

We will make use of the universal covering space $\Pi : {\widetilde{M}} \rightarrow M$ of M. One can prove that since M is a real-analytic manifold then so is ${\widetilde{M}}$, and that $\Pi$ is a real-analytic map that satisfies the following property: a map $f: M \rightarrow N$ is real-analytic (resp. $G^s$) – N being an arbitrary real-analytic manifold – if and only if the same holds for $f \circ \Pi : {\widetilde{M}} \rightarrow N$.

It is also helpful to endow M with a real-analytic Riemannian metric, $\mathbb {T}^m$ with the standard (flat) metric and $M \times \mathbb {T}^m$ with the product metric, whose volume forms we denote by $\mathrm {d}\mu$, $\mathrm {d}x$ and $\mathrm {d}\mu \wedge \mathrm {d}x$, respectively. We assume without loss of generality that $\int _M \mathrm {d}\mu = 1$, and for $\Omega \in \{ M, \mathbb {T}^m, M \times \mathbb {T}^m \}$ we consider the space of square integrable functions $L^2(\Omega )$.

6.1 Step 1

Suppose that $\pmb {\omega }$ is a rational system. Then there exists $\eta \in \mathbb {Z}^m {\setminus } \{0\}$ such that $\eta \cdot \pmb {\omega }$ is integral. It is well-known that if we take a $\psi \in C^\infty ({\widetilde{M}}; \mathbb {R})$ such that $\mathrm {d}\psi = \Pi ^* (\eta \cdot \pmb {\omega })$ (recall that integral forms on M are by definition closed, and therefore exact on ${\widetilde{M}}$) then

$$\begin{aligned} \text {for every} p, q \in {\widetilde{M}} \,\text {such that}\, \Pi (p) = \Pi (q) \,\text {we have} \,\psi (p) - \psi (q) \in 2 \pi \mathbb {Z}, \end{aligned}$$

which is the condition one needs to descend $e^{i \psi }$ to M via $\Pi$ to a function $g \in C^\infty (M)$. Notice that

$$\begin{aligned} \Pi ^* (\mathrm {d}g) = \mathrm {d}\Pi ^* g = \mathrm {d}e^{i \psi } = i e^{i \psi } \mathrm {d}\psi = \Pi ^* [i g (\eta \cdot \pmb {\omega })] \Longrightarrow \mathrm {d}g = i g (\eta \cdot \pmb {\omega }) \quad \text {on}\, M, \end{aligned}$$

thanks to injectivity of $\Pi ^*$ (since $\Pi$ is a submersion). By ellipticity of the exterior derivative on ${\widetilde{M}}$ we have $\psi \in G^s({\widetilde{M}}; \mathbb {R})$, which in turn yields $g \in G^s(M)$. We define

$$\begin{aligned} u\doteq \frac{1}{(2 \pi )^m} \sum _{\nu = 1}^\infty \nu ^{-2} g^\nu e^{-i x \cdot (\nu \eta )} \in L^2(M \times \mathbb {T}^m), \end{aligned}$$

which satisfies ${\hat{u}}_\xi = \nu ^{-2} g^\nu$ if $\xi = - \nu \eta$ and ${\hat{u}}_\xi = 0$ otherwise. Notice that u does not belong to $G^s(M \times \mathbb {T}^m)$: indeed, if it did (Proposition 5.2) there would exist constants $C, \epsilon > 0$ such that

$$\begin{aligned} \sup _{M} |{\hat{u}}_\xi | \le C e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

contradicting that $|{\hat{u}}_{(-\nu \eta )}(t)| = \nu ^{-2} |g(t)|^\nu = \nu ^{-2}$, for every $\nu \in \mathbb {N}$. However,

$$\begin{aligned} \mathbb {L}_{(-\nu \eta )} {\hat{u}}_{(-\nu \eta )} = \mathrm {d}(\nu ^{-2} g^\nu ) + i (\nu ^{-2} g^\nu )(- \nu \eta \cdot \pmb {\omega }) = \nu ^{-1} g^{\nu - 1} [\mathrm {d}g - ig (\eta \cdot \pmb {\omega })]. \end{aligned}$$

Hence, $\widehat{(\mathbb {L}u)}_\xi = \mathbb {L}_\xi {\hat{u}}_\xi = 0$, for every $\xi \in \mathbb {Z}^m$; i.e. $\mathbb {L}u = 0$ (which follows by applying Lemma 5.1 locally), showing that $\mathbb {L}$ is not globally s-hypoelliptic.

6.2 Step 2

We proceed to the case where $\pmb {\omega }$ is an s-exponential Liouville system. By hypothesis there exist $\epsilon > 0$, a sequence of integral forms $\{ \theta _\nu \}_{\nu \in \mathbb {N}} \subset \Lambda ^1 G^s(M; \mathbb {R})$ and a sequence $\{ \xi _\nu \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m$ such that $|\xi _\nu | \rightarrow \infty$ and (3.3) holds. For each $\nu \in \mathbb {N}$ we select $\psi _\nu \in C^\infty ({\widetilde{M}}; \mathbb {R})$ such that $\mathrm {d}\psi _\nu = \Pi ^* \theta _\nu$, and once again integrality of $\theta _\nu$ allows us to descend $e^{i \psi _\nu }$ to a function $g_\nu \in C^\infty (M)$ that satisfies $\mathrm {d}g_\nu = i g_\nu \theta _\nu$ on M; as before, $\psi _\nu$ and $g_\nu$ are a posteriori $G^s$ on their respective domains.

Assume without loss of generality $\xi _\nu \ne \xi _{\nu '}$ if $\nu \ne \nu '$, and set

$$\begin{aligned} u \doteq \frac{1}{(2 \pi )^m} \sum _{\nu = 1}^\infty \nu ^{-2} g_\nu e^{-i x \cdot \xi _\nu }; \end{aligned}$$

by similar arguments as in the previous step we have $u \in L^2(M \times \mathbb {T}^m) {\setminus } G^s(M \times \mathbb {T}^m)$. We will prove that $f \doteq \mathbb {L}u \in \Lambda ^1 G^s(M \times \mathbb {T}^m)$. Notice that

$$\begin{aligned} \mathbb {L}_{(-\xi _\nu )} {\hat{u}}_{(-\xi _\nu )}= & {} \mathrm {d}(\nu ^{-2} g_\nu ) + i (\nu ^{-2} g_\nu )(- \xi _\nu \cdot \pmb {\omega }) = i \nu ^{-2} g_\nu (\theta _\nu - \xi _\nu \cdot \pmb {\omega }) \\= & {} -i \nu ^{-2} e^{-\epsilon |\xi _\nu |^{\frac{1}{s}}} g_\nu \rho _\nu , \end{aligned}$$

where $\{ \rho _\nu \}_{\nu \in \mathbb {N}}$ is bounded in $\Lambda ^1 G^s(M)$; hence,

$$\begin{aligned} {\hat{f}}_\xi = {\left\{ \begin{array}{ll} -i \nu ^{-2} e^{-\epsilon |\xi _\nu |^{\frac{1}{s}}} g_\nu \rho _\nu , &{} if \xi = \xi _\nu ; \\ 0, &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

(6.1)

We proceed to estimate the $G^s$ norms of these terms.

We address $g_\nu$ first, focusing our attention on a coordinate domain $V \subset M$ so small that $\Pi : {\widetilde{V}} \rightarrow V$ is a real-analytic diffeomorphism for some open set ${\widetilde{V}} \subset {\widetilde{M}}$. Let $\phi _\nu \in G^s(V; \mathbb {R})$ be given by $\phi _\nu \circ \Pi = \psi _\nu |_{{\widetilde{V}}}$; hence,

$$\begin{aligned} g_\nu = e^{i \phi _\nu } \quad \text {on}\, V \end{aligned}$$

(6.2)

and therefore

$$\begin{aligned} \Pi ^*(\mathrm {d}\phi _\nu ) = \mathrm {d}\Pi ^* \phi _\nu = \mathrm {d}\psi _\nu = \Pi ^* \theta _\nu \Longrightarrow \mathrm {d}\phi _\nu = \theta _\nu \, \text {on}\, V. \end{aligned}$$

(6.3)

Since DFS spaces are regular injective limits of Banach spaces, one can reduce the property of boundedness in $\Lambda ^1 G^s(M)$ to boundedness in some normed space (Proposition 2.2): this one piece of information (the actual definition of the norm is irrelevant in this argument) can be used to prove that, in the topology of $\Lambda ^1 G^s(M)$,

$$\begin{aligned} e^{\epsilon |\xi _\nu |^{\frac{1}{s}}} ( \xi _\nu \cdot \pmb {\omega } - \theta _\nu )\, \text {bounded} \Longrightarrow |\xi _\nu |^{-1} ( \xi _\nu \cdot \pmb {\omega } - \theta _\nu )\, \text {bounded} \Longrightarrow |\xi _\nu |^{-1} \theta _\nu \, \text {bounded}, \end{aligned}$$

where we have used that $|\xi _\nu | \rightarrow \infty$. Hence, by Proposition 2.2, given a compact set $K \subset V$ there exist $C_1, h_1 > 0$ such that

$$\begin{aligned} \sup _K \left| \partial _t^\alpha ( |\xi _\nu |^{-1} \theta _\nu ) \right| \le C_1 h_1^{|\alpha |} \alpha !^s, \quad \sup _K \left| \partial _t^\alpha \rho _\nu \right| \le C_1 h_1^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \nu \in \mathbb {N},\nonumber \\ \end{aligned}$$

(6.4)

with the abuse of notation of treating a 1-form as a n-tuple of functions on V, on which the partial derivatives act, and whose indices we omit. It follows from the former inequality, together with identity (6.3), that there exist $C_2, h_2 > 0$ such that

$$\begin{aligned} \sup _K \left| \partial _t^\alpha ( |\xi _\nu |^{-1} \phi _\nu ) \right| \le C_2 h_2^{|\alpha |} \alpha !^s, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \nu \in \mathbb {N}; \end{aligned}$$

hence, by the scalar version of Lemma 6.1 applied to (6.2) we conclude that one can find $h_3 > 0$ (depending only on $C_2, h_2, \epsilon$) such that

$$\begin{aligned} \sup _K | \partial _t^\alpha g_\nu | = \sup _K \left| \partial _t^\alpha e^{i \phi _\nu } \right| = \sup _K \left| \partial _t^\alpha e^{i |\xi _\nu | ( |\xi _\nu |^{-1} \phi _\nu )} \right| \le h_3^{|\alpha |} \alpha !^s e^{\frac{\epsilon }{2} |\xi _\nu |^\frac{1}{s}}, \end{aligned}$$

(6.5)

for every $\alpha \in \mathbb {Z}_+^n$ and $\nu \in \mathbb {N}$. From (6.4) and (6.5) one then deduces that

$$\begin{aligned} \sup _K | \partial _t^\alpha (g_\nu \rho _\nu )| \le C_1 e^{\frac{\epsilon }{2} |\xi _\nu |^\frac{1}{s}} \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) h_3^{|\alpha | - |\beta |} (\alpha - \beta )!^s h_1^{|\beta |} \beta !^s \le C_4 h_4^{|\alpha |} \alpha !^s e^{\frac{\epsilon }{2} |\xi _\nu |^\frac{1}{s}}, \end{aligned}$$

for some constants $C_4, h_4 > 0$. This ultimately proves, in view of (6.1), that

$$\begin{aligned} \sup _K | \partial _t^\alpha {\hat{f}}_\xi | \le C_4 h_4^{|\alpha |} \alpha !^s e^{-\frac{\epsilon }{2} |\xi _\nu |^\frac{1}{s}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

ensuring that $f \in \Lambda ^1 G^s(M \times \mathbb {T}^m)$ by Proposition 5.2.

6.3 Step 3

Finally, we consider the case when the system $\pmb {\omega }$ is neither rational nor s-exponential Liouville. In this situation, take $u \in \mathcal {D}'(M \times \mathbb {T}^m)$ such that $f \doteq \mathbb {L}u \in \Lambda ^1 G^s(M \times \mathbb {T}^m)$. Then, for every $\xi \in \mathbb {Z}^m$ we have

$$\begin{aligned} {\hat{f}}_\xi = \mathbb {L}_\xi {\hat{u}}_\xi = \mathrm {d}{\hat{u}}_\xi + i {\hat{u}}_\xi (\xi \cdot \pmb {\omega }). \end{aligned}$$

For each $k \in \{1, \ldots , m\}$ we take a function $\psi _k \in G^s({\widetilde{M}}; \mathbb {R})$ such that $\mathrm {d}\psi _k = \Pi ^* \omega _k$. It follows that

$$\begin{aligned}&\mathrm {d}[e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{u}}_\xi )] = e^{i \xi \cdot \pmb {\psi }} \mathrm {d}\Pi ^* ({\hat{u}}_\xi ) + \Pi ^* ({\hat{u}}_\xi ) \mathrm {d}e^{i \xi \cdot \pmb {\psi }} \nonumber \\&= e^{i \xi \cdot \pmb {\psi }} \{ \Pi ^* ({\hat{f}}_\xi ) - i \Pi ^* [{\hat{u}}_\xi (\xi \cdot \pmb {\omega })] \} + \Pi ^* ({\hat{u}}_\xi ) i e^{i \xi \cdot \pmb {\psi }} \Pi ^*(\xi \cdot \pmb {\omega }) = e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi )\nonumber \\ \end{aligned}$$

(6.6)

on ${\widetilde{M}}$, where we employed the notation $\pmb {\psi } \doteq (\psi _1, \ldots , \psi _m)$. For each $\ell \in \{1, \ldots , d\}$ we denote by ${\tilde{\sigma }}_\ell : [0, 2 \pi ] \rightarrow {\widetilde{M}}$ a lift of the 1-cycle $\sigma _\ell$ described in $\sharp$ to ${\widetilde{M}}$. If we fix a base point $t_0 \in M$ and some ${\tilde{t}}_0 \in \Pi ^{-1}(t_0)$ we can assume that $\sigma _\ell (0) = t_0 = \sigma _\ell (2 \pi )$ and that ${\tilde{\sigma }}_\ell (0) = {\tilde{t}}_0$ for every $\ell \in \{1, \ldots , d\}$. Then, by (6.6),

$$\begin{aligned}&\int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) = \int _{{\tilde{\sigma }}_\ell } \mathrm {d}[e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{u}}_\xi )] \\&\quad = e^{i \xi \cdot \pmb {\psi }({\tilde{t}}_\ell )} {\hat{u}}_\xi (\Pi ({\tilde{t}}_\ell )) - e^{i \xi \cdot \pmb {\psi }({\tilde{t}}_0)} {\hat{u}}_\xi (\Pi ({\tilde{t}}_0)) = (e^{i \xi \cdot \pmb {\psi }({\tilde{t}}_\ell )} - e^{i \xi \cdot \pmb {\psi }({\tilde{t}}_0)}) {\hat{u}}_\xi (t_0), \end{aligned}$$

where ${\tilde{t}}_\ell \doteq {\tilde{\sigma }}_\ell (2 \pi ) \in \Pi ^{-1}(t_0)$, which allows us to deduce that

$$\begin{aligned} {\hat{u}}_\xi (t_0) = e^{i \xi \cdot [\pmb {\psi }({\tilde{t}}_0) - \pmb {\psi }({\tilde{t}}_\ell )]} {\hat{u}}_\xi (t_0) + e^{-i \xi \cdot \pmb {\psi }({\tilde{t}}_\ell )} \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ). \end{aligned}$$

(6.7)

Observe that

$$\begin{aligned} \psi _k ({\tilde{t}}_\ell ) - \psi _k ({\tilde{t}}_0) = \int _{{\tilde{\sigma }}_\ell } \mathrm {d}\psi _k = \int _{{\tilde{\sigma }}_\ell } \Pi ^* \omega _k = \int _{\Pi \circ {\tilde{\sigma }}_\ell } \omega _k = \int _{\sigma _\ell } \omega _k = 2 \pi {\mathsf {A}}(\pmb {\omega })_{\ell k}; \end{aligned}$$

hence,

$$\begin{aligned} {\mathsf {A}}(\pmb {\omega }) \xi = \frac{1}{2 \pi } \left( \xi \cdot [\pmb {\psi } ({\tilde{t}}_1) - \pmb {\psi }({\tilde{t}}_0)], \ldots ,\xi \cdot [\pmb {\psi } ({\tilde{t}}_d) - \pmb {\psi }({\tilde{t}}_0)] \right) . \end{aligned}$$

Due to our assumptions on $\pmb {\omega }$, we have by Proposition 4.1 that ${\mathsf {A}}(\pmb {\omega })$ satisfies condition $\mathrm {(DC)}_s^2$. Hence, by (4.3) for every $\epsilon > 0$ there exists $C_\epsilon > 0$ such that

$$\begin{aligned} \max _{\ell } \left| 1 - e^{i \xi \cdot [\pmb {\psi } ({\tilde{t}}_0) - \pmb {\psi }({\tilde{t}}_\ell )]} \right| = \max _{\ell } \left| 1 - e^{-2 \pi i ({\mathsf {A}}(\pmb {\omega }) \xi )_\ell } \right| \ge C_\epsilon e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m {\setminus } \{ 0 \}. \end{aligned}$$

Therefore, for a given $\xi \in \mathbb {Z}^m {\setminus } \{ 0 \}$ we pick that $\ell \in \{1, \ldots , d\}$ at which the maximum in the left-hand side is attained; in particular, that term does not vanish, allowing us to conclude from (6.7) that

$$\begin{aligned} |{\hat{u}}_\xi (t_0)| = \frac{|e^{-i \xi \cdot \pmb {\psi }({\tilde{t}}_\ell )}|}{\left| 1 - e^{i \xi \cdot [\pmb {\psi } ({\tilde{t}}_\ell ) - \pmb {\psi }({\tilde{t}}_0)]} \right| } \left| \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) \right| \le C_\epsilon ^{-1} e^{\epsilon |\xi |^{\frac{1}{s}}} \left| \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) \right| .\nonumber \\ \end{aligned}$$

(6.8)

We claim that there exist $C, \delta > 0$ such that

$$\begin{aligned} \left| \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) \right| \le C e^{-\delta |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

(6.9)

Indeed, first of all we have, by definition,

$$\begin{aligned} \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) = \int _0^{2\pi } e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ {\tilde{\sigma }}_\ell ^* \Pi ^* ({\hat{f}}_\xi ) = \int _0^{2\pi } e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ). \end{aligned}$$

As in Sect. 2, we fix a finite family $\{ K_i \}_{i \in I}$ of coordinate compact subsets of M whose interiors form an open covering of M, and a partition $0 = \tau _0< \tau _1< \cdots < \tau _N = 2 \pi$ such that for each $r \in \{1, \ldots , N\}$ (that will remain fixed until (6.11) and upon which our choices will depend without explicit mention) $\sigma _\ell ([\tau _{r - 1}, \tau _r])$ is contained in the interior of a single $K_i$ (depending on r). Denoting by $(t_1, \ldots , t_n)$ a fixed set of real-analytic coordinates in a neighborhood of that $K_i$, we write f as (5.5) and ${\hat{f}}_\xi$ as (5.6), and conclude that on $[\tau _{r - 1}, \tau _r] \subset \mathbb {R}$ we have

$$\begin{aligned} \sigma _\ell ^* ({\hat{f}}_\xi ) = \sum _{j = 1}^n (\widehat{(f_j)}_\xi \circ \sigma _\ell ) \ \mathrm {d}(t_j \circ \sigma _\ell ) = \sum _{j = 1}^n (\widehat{(f_j)}_\xi \circ \sigma _\ell ) g_j \ \mathrm {d}\tau , \end{aligned}$$

(6.10)

for some continuous function²$g_j$ in $[\tau _{r - 1}, \tau _r]$.

Hence,

$$\begin{aligned} \int _{\tau _{r - 1}}^{\tau _r} e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ) = \sum _{j = 1}^n \int _{\tau _{r - 1}}^{\tau _r} e^{i \xi \cdot (\pmb {\psi } ({\tilde{\sigma }}_\ell (\tau )))} \widehat{(f_j)}_\xi (\sigma _\ell (\tau )) g_j(\tau ) \ \mathrm {d}\tau . \end{aligned}$$

Now, since $f \in \Lambda ^1 G^s(M \times \mathbb {T}^m)$ one can apply Proposition 5.2 to find constants $C', \delta > 0$ independent of r such that

$$\begin{aligned} \sup _{K_i} |\widehat{(f_j)}_\xi | \le C' e^{-\delta |\xi |^{\frac{1}{s}}}, \quad \forall j \in \{1, \ldots , n\}, \ \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

and, therefore,

$$\begin{aligned} \left| \int _{\tau _{r - 1}}^{\tau _r} e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ) \right| \le \sum _{j = 1}^n \int _{\tau _{r - 1}}^{\tau _r} |\widehat{(f_j)}_\xi (\sigma _\ell (\tau ))| |g_j(\tau )| \ \mathrm {d}\tau \le C'' e^{-\delta |\xi |^{\frac{1}{s}}},\nonumber \\ \end{aligned}$$

(6.11)

for some $C'' > 0$ independent of $\xi \in \mathbb {Z}^m$ and $r \in \{1, \ldots , N\}$. Since

$$\begin{aligned} \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) = \int _0^{2\pi } e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ) = \sum _{r = 1}^N \int _{\tau _{r - 1}}^{\tau _r} e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ), \end{aligned}$$

we conclude that

$$\begin{aligned} \left| \int _{{\tilde{\sigma }}_\ell } e^{i \xi \cdot \pmb {\psi }} \Pi ^* ({\hat{f}}_\xi ) \right| \le \sum _{r = 1}^N \left| \int _{\tau _{r - 1}}^{\tau _r} e^{i \xi \cdot (\pmb {\psi } \circ {\tilde{\sigma }}_\ell )} \ \sigma _\ell ^* ({\hat{f}}_\xi ) \right| \le C'' N e^{-\delta |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

thus proving (6.9).

It follows from (6.8) and (6.9), by taking $\epsilon \doteq \delta / 2$ in the former and $C_1 \doteq C_\epsilon ^{-1} C$, that

$$\begin{aligned} |{\hat{u}}_\xi (t_0)| \le C_1 e^{- \frac{\delta }{2} |\xi |^{\frac{1}{s}}}, \quad \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

(6.12)

Next we must estimate the derivatives of ${\hat{u}}_\xi$. Once again we take $V \subset M$ a coordinate ball centered at $t_0$, so small that $\Pi : {\widetilde{V}} \rightarrow V$ is a real-analytic diffeomorphism for an open set ${\widetilde{V}} \subset {\widetilde{M}}$; this time, for each $k \in \{1, \ldots , m\}$ there exists $\phi _k \in G^s(V; \mathbb {R})$ so that $\phi _k \circ \Pi = \psi _k$ on ${\widetilde{V}}$; notice that $\mathrm {d}\phi _k = \omega _k$ on V, and thus

$$\begin{aligned} \mathrm {d}(e^{i \xi \cdot \pmb {\phi }} {\hat{u}}_\xi )= & {} e^{i \xi \cdot \pmb {\phi }} \mathrm {d}{\hat{u}}_\xi + {\hat{u}}_\xi \mathrm {d}e^{i \xi \cdot \pmb {\phi }} = e^{i \xi \cdot \pmb {\phi }} [{\hat{f}}_\xi - i {\hat{u}}_\xi (\xi \cdot \pmb {\omega })] + {\hat{u}}_\xi [i e^{i \xi \cdot \pmb {\phi }} (\xi \cdot \pmb {\omega })] \\= & {} e^{i \xi \cdot \pmb {\phi }} {\hat{f}}_\xi . \end{aligned}$$

Therefore, by integrating over the segment $\{ \tau t_0 + (1 - \tau ) t \ ; \ \tau \in [0,1] \} \subset V$ we deduce that

$$\begin{aligned} e^{i \xi \cdot \pmb {\phi }(t)} {\hat{u}}_\xi (t) - e^{i \xi \cdot \pmb {\phi }(t_0)} {\hat{u}}_\xi (t_0) = \int _{t_0}^{t} e^{i \xi \cdot \pmb {\phi }} {\hat{f}}_\xi , \end{aligned}$$

which implies that

$$\begin{aligned} {\hat{u}}_\xi (t) = \underbrace{e^{i \xi \cdot [\pmb {\phi }(t_0) - \pmb {\phi }(t)] } {\hat{u}}_\xi (t_0)}_{\doteq \varUpsilon ^1_\xi (t)} + \underbrace{e^{-i \xi \cdot \pmb {\phi }(t)} \int _{t_0}^{t} e^{i \xi \cdot \pmb {\phi }} {\hat{f}}_\xi }_{\doteq \varUpsilon ^2_\xi (t)}, \quad \forall t \in V, \ \forall \xi \in \mathbb {Z}^m.\nonumber \\ \end{aligned}$$

(6.13)

It follows from (6.12) and Lemma 6.1 that on a given compact set $K \subset V$

$$\begin{aligned} |\partial _t^\alpha \varUpsilon ^1_\xi | \le C_1 e^{- \frac{\delta }{2} |\xi |^{\frac{1}{s}}} | \partial _t^\alpha e^{-i \xi \cdot \pmb {\phi }} | \le C_1 h_2^{|\alpha |} \alpha !^s e^{- \frac{\delta }{4} |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m,\nonumber \\ \end{aligned}$$

(6.14)

for some $h_2 > 0$. We further write

$$\begin{aligned} \varUpsilon ^2_\xi (t) = e^{-i \xi \cdot \pmb {\phi }(t)} \int _{t_0}^{t} e^{i \xi \cdot \pmb {\phi }} {\hat{f}}_\xi \doteq e^{-i \xi \cdot \pmb {\phi }(t)} Y_\xi (t), \end{aligned}$$

(6.15)

so that

$$\begin{aligned} \mathrm {d}Y_\xi = e^{i \xi \cdot \pmb {\phi }} {\hat{f}}_\xi . \end{aligned}$$

It follows from the hypothesis that $f \in \Lambda ^1 G^s(M \times \mathbb {T}^m)$ and Proposition 5.2 that there exist $C_3, h_3, \theta > 0$ such that

$$\begin{aligned} \sup _K |\partial _t^\alpha {\hat{f}}_\xi | \le C_3 h_3^{|\alpha |} \alpha !^s e^{-\theta |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

(6.16)

where again we treat 1-forms on V as n-tuples of functions. Therefore, it follows from (6.16) and Lemma 6.1 that for some $h_4 > 0$ we have on K:

$$\begin{aligned} | \partial _t^\alpha (\mathrm {d}Y_\xi ) |&\le \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) | \partial _t^{\alpha - \beta } e^{i \xi \cdot \pmb {\phi }} | | \partial _t^\beta {\hat{f}}_\xi | \\&\le \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) | \partial _t^{\alpha - \beta } e^{i \xi \cdot \pmb {\phi }} | C_3 h_3^{|\beta |} \beta !^s e^{-\theta |\xi |^{\frac{1}{s}}} \\&\le \sum _{\beta \le \alpha } \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) h_4^{|\alpha | - |\beta |} (\alpha - \beta )!^s C_3 h_3^{|\beta |} \beta !^s e^{-\frac{\theta }{2} |\xi |^{\frac{1}{s}}} \\&\le C_3 h_5^{|\alpha |} \alpha !^s e^{-\frac{\theta }{2} |\xi |^{\frac{1}{s}}}, \end{aligned}$$

whatever $\alpha \in \mathbb {Z}_+^n$, for some $h_5 > 0$. By possibly increasing $C_3$ and $h_5$, one obtains

$$\begin{aligned} \sup _K | \partial _t^\alpha Y_\xi | \le C_3 h_5^{|\alpha |} \alpha !^s e^{-\frac{\theta }{2} |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

(6.17)

Finally, we return to (6.15); one repeats the previous argument using Lemma 6.1 and now (6.17) to conclude that

$$\begin{aligned} \sup _K | \partial _t^\alpha \varUpsilon ^2_\xi | \le C_7 h_7^{|\alpha |} \alpha !^s e^{-\frac{\theta }{4} |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m, \end{aligned}$$

(6.18)

for some constants $C_7, h_7 > 0$. Using (6.13), (6.14) and (6.18), we deduce the existence of C, h and $\epsilon > 0$ such that

$$\begin{aligned} \sup _K | \partial _t^\alpha {\hat{u}}_\xi | \le C h^{|\alpha |} \alpha !^s e^{-\epsilon |\xi |^{\frac{1}{s}}}, \quad \forall \alpha \in \mathbb {Z}_+^n, \ \forall \xi \in \mathbb {Z}^m. \end{aligned}$$

Since M can be covered by finitely many V with the aforementioned properties and $K \subset V$ is arbitrary, this last estimate shows that u is $G^s$ in $M \times \mathbb {T}^m$ by a final application of Proposition 5.2.

7 Final remarks

Using the tools developed above, one can derive the following characterizations of (smooth) global hypoellipticity of the operator $\mathbb {L}$, thus extending [1, Theorem 2.4] to arbitrary corank. Of course, in that case M and the 1-forms $\omega _{1}, \ldots , \omega _{m}$ can be assumed just smooth.

In order to properly state them, we need some preliminary definitions.

Definition 7.1

We say that $\pmb {\omega } = (\omega _{1}, \ldots , \omega _{m})$ is a Liouville system if $\pmb {\omega }$ is not rational and there exist a sequence of integral forms $\{ \theta _{\nu } \}_{\nu \in \mathbb {N}} \subset \Lambda ^{1} C^\infty (M; \mathbb {R})$ and $\{ \xi _{\nu } \}_{\nu \in \mathbb {N}} \subset \mathbb {Z}^m$ such that $|\xi _\nu | \rightarrow \infty$ and

$$\begin{aligned} \{ |\xi _\nu |^\nu ( \xi _\nu \cdot \pmb {\omega } - \theta _\nu ) \}_{\nu \in \mathbb {N}}\, \text {is bounded in}\, \Lambda ^1 C^\infty (M). \end{aligned}$$

Definition 7.2

We say that a matrix ${\mathbf {A}} \in {\mathsf {M}}_{d \times m}(\mathbb {R})$ satisfies condition $(\mathrm {DC})^2$ if there exist $C, \rho > 0$ such that

$$\begin{aligned} | \kappa + {\mathbf {A}} \xi | \ge C (|\kappa | + |\xi |)^{-\rho }, \quad \forall (\kappa , \xi ) \in (\mathbb {Z}^{d} \times \mathbb {Z}^{m}) {\setminus } \{(0,0) \}. \end{aligned}$$

(7.1)

Notice that the latter condition implies condition $\mathrm {(DC)}$ used in [8] to study global solvability of corank m tube structures when $M = \mathbb {T}^d$. It also implies condition $\mathrm {(DC)}^2_s$ for every $s \ge 1$.

Theorem 7.3

The following are equivalent:

• (1) $\mathbb {L}$ is globally hypoelliptic; i.e. (1.2) holds.

• (2) $\pmb {\omega } = (\omega _1, \ldots , \omega _m)$ is neither a rational system nor a Liouville system.

• (3) The matrix of periods ${\mathsf {A}}(\pmb {\omega })$ satisfies condition $\mathrm {(DC)}^2$.

We omit the proof. As a consequence, using the theorem above together with Theorem 3.4 and Proposition 4.1 we conclude that when $\omega _1, \ldots , \omega _m$ are $G^s$ we have:

Corollary 7.4

If $\mathbb {L}$ is globally hypoelliptic then it is globally s-hypoelliptic.