Abstract
Let \((\Omega ,g)\) be a compact, real-analytic Riemannian manifold with real-analytic boundary \(\partial \Omega .\) The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle \(S^*\partial \Omega .\) These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations \(Pu=0\) near the characteristic set \(\{\sigma (P)=0\}\).
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1 Introduction
Let \((\Omega ,g)\) be an n-dimensional, compact \(C^{\infty }\) Riemannian manifold with boundary M and corresponding unit exterior normal \(\nu \). By some abuse of notation, we also let \(\nu \) denote a smooth vector field extension and \(\gamma _{M}: C^0(\Omega ) \rightarrow C^0(M)\) be the boundary restriction map. Let \({\mathcal {D}}: C^{\infty }(M) \rightarrow C^{\infty }(M)\) be the associated Dirichlet-to-Neumann (DtN) operator defined by
where u solves the Dirichlet problem
The operator \({\mathcal {D}}\) is an elliptic, first-order, self-adjoint pseudodifferential operator (see for example [21, Section 7.11]) with an \(L^2\)-normalized basis of eigenfunctions \(\varphi _{j}; j=1,2,....\) It is useful here to work in the semiclassical setting from the outset. Choosing \(h^{-1} \in \text {Spec} \, {\mathcal {D}},\) the corresponding eigenfunction \(\varphi _h\) then satisfies the semiclassical eigenfunction equation
The harmonic extension, \(u_{h} \in C^{\infty }(\Omega ),\) of a DtN eigenfunction \(\varphi _h\) is called a Steklov eigenfunction.
There has been a substantial amount of recent work devoted to the study of the asymptotic behavior of the DtN eigenvalues and both DtN and Steklov eigenfunctions, including the asymptotics of eigenfunction nodal sets (see for example [1, 3, 4, 6, 15, 16, 20, 26,27,28] and references therein).
For large eigenvalues, Steklov eigenfunctions possess both high oscillation inherited from the boundary DtN eigenfunctions and very sharp decay into the interior of \(\Omega .\) As a consequence, even though Steklov eigenfunctions decay rapidly, the oscillation implies, in particular, that the nodal sets have intricate structure. It has been conjectured [3] that the analogue of Yau’s conjecture [23, 24] for nodal volumes holds in the Steklov case. This was recently proved for real-analytic Riemann surfaces in [15].
The question of decay of Steklov eigenfunctions into the interior of M when (M, g) is real analytic was first raised by Hislop–Lutzer [6] where they conjecture that the Steklov eigenfunctions decay into the interior as \(e^{-d(x,\partial \Omega )/h}.\) In the special case where dim \(\Omega =2\) exponential decay with respect to \(d(x,\partial \Omega )\) was indeed proved in [15] and the eigenfunction decay is a key feature in their main results on nodal length. However, the analysis in [15] relies heavily on the assumption that the dimension equals two.
In Theorem 1, we prove an exponential decay result for Steklov eigenfunctions for general real-analytic metrics in arbitrary dimension that is sharp to first order at the boundary, \(\partial \Omega = M\), and we bound the quadratic error in the decay rate in terms of boundary curvature. In particular, we prove the conjecture due to Hislop–Lutzer [6].
One can heuristically view such exponential decay estimates for the Steklov eigenfunctions as describing the ‘tunnelling’ of the boundary DtN eigenfunctions on M into the interior of the manifold \(\Omega .\) In the Schrödinger case \(P(h)=-h^2\Delta _g+V-E\), one thinks of eigenfunctions as tunnelling from \(V\le E\) into the forbidden region \(V>E\). Since the interior eigenfunctions are harmonic, this decay is somewhat more subtle in the case of Steklov eigenfunctions. The boundary data \(\varphi _h\) concentrate microlocally on the cosphere bundle of the boundary, \(S^*\partial \Omega \), while the fact that \(-h^2\Delta _gu_h=0\) implies that \(u_h\) must concentrate at the zero section \(\{ (x,\xi ) \in T^*\Omega ; \xi = 0 \}.\) Thus, we think of \(u_h\) as tunnelling from boundary values with \(|\xi |_g=1\) to the interior where \(\xi =0\). For this reason, it is reasonable to view the decay estimates in Theorem 1 as a natural analogue of the well-known Agmon–Lithner estimates (see for example [7]) for Schrödinger eigenfunctions in classically forbidden regions. We note that the assumption that \((\Omega ,g)\) is \(C^{\omega }\) is necessary in Theorem 1 below because real-analyticity allows for accuracy up to exponential errors in h in the pseudodifferential calculus, whereas in the \(C^\infty \) case, one can only work to \(O(h^{\infty })\)-error. In particular, one can only microlocalize modulo such errors. Since Steklov eigenfunctions decay exponentially in h in the interior of \(\Omega \), the usual \(C^{\infty }\) semiclassical calculus of operators is not accurate enough to deal with these functions in a rigorous fashion. Similarly, our subsequent more general results in Theorems 2 and 3 hinge on the microlocal exponential weighted estimate in Proposition 2.5 which also requires real-analyticity to effectively control error terms.
Theorem 1
Let \((\Omega ^{n+1},g)\) be a compact, real-analytic (\(C^{\omega }\)) Riemannian manifold with \(C^{\omega }\) boundary \(\partial \Omega \) and \({\mathcal {D}}: C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(\partial \Omega )\) be the associated DtN operator. Then for all \(\delta >0\) there exist \(0<\varepsilon _0=\varepsilon _0(\Omega ^n,g,\delta )\) such that for \(\varphi _h\in C^\omega (\partial \Omega )\) with
the harmonic extension \(u_{h}(x)\) satisfies the exponential decay estimate
where \(C_{\alpha ,\delta }>0\) is a constant independent of h. In (1.3),
where \(d_{\partial \Omega }(x)\) is the Riemannian distance to the boundary and
Here \(Q(x',\xi ')\) is the symbol of the second fundamental form of the boundary \(\partial \Omega .\)
Remark 1.1
The estimate (1.3) is only valid in a small collar neighborhood of \(\partial \Omega \) and indeed, since \(C_{\Omega ,g}\) may be negative, the rate function d(x) may cease to give exponential decay outside a collar neighborhood of \(\partial \Omega \). However, the maximum principle for the Laplace equation on \(\Omega _\varepsilon =\{x\in \Omega \mid d_{\partial \Omega }(x)>\varepsilon \}\) together with Theorem 1 also implies that for any \(\varepsilon >0\), there exists \(C,c>0\) so that
Unfortunately, we lose control of the rate function outside a small collar neighborhood of the boundary.
We will see by examining the case of \(\Omega =B(0,R)\subset \mathbb {R}^2\) that the rate of decay in (1.3) is optimal to first order at the boundary. We do not expect to be able to obtain the optimal second order estimate since we are forced to throw away some of the oscillations in u when we apply the Cauchy–Schwarz inequality in (1.22), however, the behavior with respect to Q is optimal as we will see from several examples.
We point out that the estimate in Theorem 1 holds without change if one takes the \(\varphi _h\) to be \(L^2\)-normalized Laplace eigenfunctions on the boundary. Indeed, the bound in Theorem 1 can be adapted to the case of harmonic extensions of eigenfunctions of general elliptic, analytic, self-adjoint h-pseudodifferential operators on M. However, the bounds are somewhat cumbersome to state and we do not pursue this here.
It is worth noting that the proof of Theorem 1 is microlocal and thus the constant \(C_{\Omega ,g}\) can be made to depend on the nearest point in \(\partial \Omega \). In particular, let \((x',x_{{n+1}})\) be Fermi normal coordinates in a collar neighborhood of \(\partial \Omega \) so that \(x_{{n+1}}=d_{\partial \Omega }(x)\). Then the estimate (1.3) holds with d(x) replaced by
where
1.1 Examples of Steklov Eigenfunctions: Sharpness of d(x) to First Order
We now examine a few examples to illustrate the results of Theorem 1.
1.1.1 The Disk
Let \(\Omega =B(0,R)\subset \mathbb {R}^2\). Then the Steklov eigenvalues are precisely \(\sigma =0,\frac{1}{R},\frac{2}{R}\dots \) with corresponding Steklov eigenfunctions given by
In particular, letting \(h=\sigma ^{-1}={{k}}^{-1}R\),
Therefore, in this case
This shows that to first order, the results of Theorem 1 are sharp. Moreover, notice that the second fundamental form of \(\partial B(0,R)\) is given by \(R^{-1}\) and thus, for the disk, the optimal quadratic term is
hence, modulo the \(\frac{3}{2}\) in (1.4), the constant \(C_{\Omega ,g}\) sharp. In particular, as the curvature of the boundary increases, the decay into the interior becomes more rapid.
The case of spheres in higher dimensions is nearly identical if we replace \(e^{\pm in\theta }\) by a spherical harmonic.
1.1.2 Cylinders
Let (M, g) be a real-analytic manifold of dimension n without boundary and \(\Omega =(-1,1)_t\times M_x\) with metric \(dt^2+g(x).\) Then
Let \(\varphi _{{k}}\) be an orthonormal basis for \(L^2(M)\) with
Then the Steklov eigenfunctions are given by
with Steklov eigenvalues \(\sigma _{{k}}=\lambda _{{k}}\tanh (\lambda _{{k}})\) and \(\sigma '_{{k}}=\lambda _{{k}}\coth (\lambda _{{k}})\) respectively. Notice that for \(\lambda _{{k}}\gg 1\),
In particular, near \(|t|=1\),
Then, notice that
So, using Hörmander’s \(L^\infty \) bounds (see e.g., [29, Chapter 7]) we have the estimate
Indeed, it is not hard to construct examples (e.g., where M is the sphere) so that this estimate is sharp. The analysis is similar for \(v_h\).
In particular, the best possible decay rate is given by
and we again see that the first-order term in Theorem 1 is sharp and that the quadratic term is given by (1.6) since in this case the second fundamental form is 0.
1.1.3 The Annulus
Now, consider \(B(0,1){\setminus } B(0,r_0)\subset \mathbb {R}^2\). Then a simple computation shows that the Steklov eigenvalues are the roots, of
with corresponding eigenfunctions
It is easy to show that the roots of \(p_n(\sigma )\) have
Then,
The case of \(u_{\sigma _{{{k}},1}}\) is identical to that for the disk, so we focus on \(u_{\sigma _{{{k}},2}}\). Let \(h=\sigma _{{{k}},2}^{-1}=r_0{{k}}^{-1}+O(e^{-c{{k}}}).\) Then,
Therefore, in this case
This shows again that to first order, the results of Theorem 1 are sharp. Moreover, notice that the second fundamental form of \(\partial \Omega \) near \(\partial B(0,r_0)\) is given by \(-r_0^{-1}\) and thus, near this boundary component, the optimal quadratic term is again given by (1.6).
1.2 Microlocal Estimates
It is clear from the approximate Poisson formula for \(u_h(x)\) (see (3.1) below) that the exponential eigenfunction decay in the interior \(\Omega \) is closely related to the precise rate of h-microlocal exponential decay of the boundary DtN eigenfunctions \(\varphi _h\) off the cosphere bundle \(S^*\partial \Omega = \{(y,\eta ) \in T^*\partial \Omega ; |\eta |_g =1 \}.\) To derive the requisite bounds, we prove weighted exponential h-microlocal estimates for the associated wave packets \(T(h) \varphi _h\) where \(T(h): C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(T^*\partial \Omega )\) is a globally defined FBI transform in the sense of [18]. Since these estimates seem of independent interest, we prove them for a rather general class of analytic h-pseudodifferential operators. An important consequence is the following exponential decay estimate for \(T(h) \varphi _h\) off the characteristic variety when \(\varphi _h\) solves \(P(h)\varphi _h=O(e^{-c/h}).\)
We recall that an operator \(P(h) \in Op_{h}(S^{0,k})\) with principal symbol \(p(y,\eta )\) is said to have simple characteristics provided \(dp \ne 0\) on the set \(\{p=0 \}.\) Moreover, \(p(y,\eta )\) is classically elliptic if \(|p(y,\eta )| \ge C' \langle \eta \rangle ^{k}\) for \(|\eta | \ge C\) with constants \(C,C' >0.\) Let \(S^{m,k}_{cla}\) denote the class of classical analytic symbols (see Sect. 2).
Theorem 2
Let \((M^{{n}},g)\) be a compact, closed, real-analytic manifold and \(P(h) \in Op_h (S^{0,k}_{cla})\) be an analytic, h-pseudodifferential operator with real, classically elliptic principal symbol \(p(x,\xi )\) having simple characteristics. Suppose that
Let \(T(h): C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) be a globally defined FBI transform as in (2.8) associated with an h-ellptic symbol \(a \in S^{3{{n}}/4,{{n}}/4}_{cla}\) and consider the weight function
Then, provided \(\delta >0\) is a sufficiently small positive constant depending on (M, g), it follows that for \(h \in (0,h_0(\delta )]\) with \(h_0(\delta )>0\) sufficiently small,
The main technical ingredient needed for the proof of Theorem 2 is given in Proposition 2.5; it is essentially the manifold analogue of the microlocal exponential weighted estimates in \({\mathbb R}^n\) proved by Martinez [12, 13] and Nakamura [14] (see also [22]). As a direct application of Theorem 2, we prove the first-order exponential decay estimate in Theorem 1 (1.3).
Remark 1.2
Although weaker than Theorem 2 (since no rate of decay is specified), we point out that the following exponential decay estimate is an immediate consequence of Theorem 2.
Corollary 1.3
For fixed \(\varepsilon _0 >0\) consider the cutoff \(\chi _{\varepsilon _0}(x,\xi ) := \chi \Big ( \frac{ p(x,\xi )}{\varepsilon _0} \Big ).\) Then, for any \(\varphi _h\) satisfying the assumptions in Theorem 2, there exists constant \(C(\varepsilon _0)>0\) such that for \(h \le h(\varepsilon _0)\),
We also note that Corollary 1.3 also follows from a rather standard parametrix construction for analytic h-pseudodifferential operators (see Sect. 2.2).
To prove Theorem 1 (1.4), one must estimate the constant \(C_{\Omega ,g}\) in the rate function \(d(x) = d_{\partial \Omega }(x) - C_{\Omega ,g} d^2_{\partial \Omega }(x).\) Unfortunately, the weighted \(L^2\) bound in Theorem 2 for \(T \varphi _h\) does not quite suffice for this since one needs an additional geometric bound for the constant \(\delta >0.\) To achieve this, we must h-microlocally refine the bound in Theorem 2. Global refinement of the decay estimate for \(T(h)\varphi _h\) by specifying sharp constant \(\delta >0\) in the Gaussian seems a rather intractable problem. However, with Corollary 1.3 in hand, we see that away from the characteristic variety, \(T(h)\varphi _h\) is exponentially small. Consequently, instead of attempting to refine the global estimate in Theorem 2, we h-microlocalize to a small neighborhood of the characteristic variety \(p^{-1}(0).\) In order to exploit the real-analyticity of (M, g) and P(h), we choose the neighborhood of M in \(T^*M\) to be the Grauert tube complexification \(M_{\tau }^{{\mathbb C}} \supset M,\) which we assume throughout contains the characteristic variety, \(p^{-1}(0).\) In addition, for the subsequent estimates, it will be important to choose a specific h-microlocal FBI transform \(T_{hol}(h): C^{\infty }(M) \rightarrow C^{\infty }(M_{\tau }^{{\mathbb C}})\) that is compatible with the complex structure on \(M_{\tau }^{{\mathbb C}}\). In view of [10, Theorem 0.1], it is natural to choose \(T_{hol}(h)\) to be the holomorphic continuation of the heat kernel on (M, g) at time \(t = \frac{h}{2}.\)
1.2.1 Motivating Example of FBI Transforms
Recall that the standard FBI transform on \({\mathbb R}^n\) is given by
We will introduce two globally defined FBI transforms below, \(T_{hol}\) and \(T_{geo}\). In the case of \({\mathbb R}^n\), these two FBI transforms agree and are given by (1.9). That is, on \({\mathbb R}^n\),
Before formally stating our next result, we give some motivation. Consider the simple example of the circle \({\mathbb R}/ 2\pi {\mathbb Z}\) with a flat metric \(g = dx^2.\) That is, consider the boundary for the examples in Sects. 1.1.1 and 1.1.3. The functions \(\varphi _h(x) = e^{ix/h}\) appearing in (1.5) and (1.7) satisfy \((hD_{x}-1)\varphi _h=0.\) In this case, the complexification is
We compute \(T_{hol}\varphi _h\) by extending \(\varphi _h\) smoothly to \({\mathbb R}\) as a solution of \((hD_{x}-1)\varphi _h=0\) or using the definition of \(T_{hol}\) in (1.12). Then,
In particular, with \(\psi _{hol}(\alpha ) = \frac{1}{2} (\alpha _\xi - 1)^2,\)
Thus, for any \(\gamma <1\),
We note that (1.10) is consistent with Theorem 2. However, it is useful to observe that with the precise weight function \(\psi _{hol} = \frac{1}{2}(\alpha _{\xi }-1)^2\) (with \(\gamma =1\)),
Thus, with the optimal weight function \(\psi _{hol}\), one has a polynomial gain of \(h^{-1/4}\) in the weighted \(L^2\) mass. The computations for eigenfunctions on higher-dimensional flat tori \({\mathbb R}^n/{\mathbb Z}^n\) are very similar.
To state the h-microlocal refinement of Theorem 2, we let M be a compact, closed, real-analytic manifold of dimension m and \(\widetilde{M}\) denote a Grauert tube complex thickening of M with M a totally real submanifold. By Bruhat-Whitney, \(\widetilde{M}\) can be identified with \(M^{{\mathbb C}}_{\tau } := \{ (\alpha _x, \alpha _\xi ) \in T^*M; \sqrt{2\rho }(\alpha _x,\alpha _\xi ) \le \tau \}\) where \(\sqrt{2\rho } = |\alpha _{\xi }|_g\) is the exhaustion function using the complex geodesic exponential map \( \kappa : M_{\tau }^{{\mathbb C}} \rightarrow \tilde{M}\) with \(\kappa (\alpha ) = \exp _{\alpha _x}( i \alpha _{\xi }).\)
Remark 1.4
We use the notation \(\alpha \) rather than \((x,\xi )\) because it is useful to think of \(\alpha \in \widetilde{M}\) at some times and \(\alpha \in T^*M\) at other times where we identify \(\widetilde{M}\) as a subset of \(T^*M\).
By possibly rescaling the semiclassical parameter h we assume without loss of generality that the characteristic manifold
Now, let \(e^{h\Delta _g/2}\) have Schwartz kernel E(x, y, h) and \(E^{\mathbb {C}}(\alpha ,y,h)\) denote the holomorphic continuation of E(x, y, h) to \(M^{\mathbb {C}}_\tau \) in the outgoing x-variables. It is proved in [10] Theorem 0.1 (see also Sect. 2) that the operator given by
is an \(L^2\)-normalized, h-microlocal FBI transform defined for \(\alpha \in {\widetilde{M}}.\)
Theorem 3
Under the same assumptions as in Theorem 2 and with \(T=T_{hol}\) is as in (1.12), there exists \(\varepsilon >0\) small enough so that with \(\gamma <1/2\),
where \(g_s\) is the Sasaki metric on TM (see for example [2, Chapter 9]).
Moreover, there exists \(\psi _{hol} \in C^{\infty }(M_{\tau }^{{\mathbb C}})\) with
where \(a_0\) is given in (1.13) such that for \(\varepsilon >0\) sufficiently small,
We note that the example of the Laplace eigenfunctions on the circle (see (1.10) and (1.11) above), shows that the upper bound in Theorem 3 (1.14) is sharp.
Remark 1.5
We use (1.13) instead of (1.14) in the proof of Theorem 1. Using (1.14) results in better estimates as soon as \(d(x)\gg ch\log h^{-1}\), but for simplicity and since the function d(x) that we obtain is still not sharp, we do not state these estimates here.
1.3 Sketch of the Proof of Theorem 1
Let \({\mathcal {P}}: C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(\Omega )\) be the Poisson operator for the boundary value problem in (1.2), so that \( u_h(x) = {\mathcal {P}} \varphi _h(x).\) The first step in the proof of Theorem 1 amounts to understanding the microlocal structure of the Poisson operator, \({\mathcal {P}}\) following the analysis in [19].
1.3.1 Microlocal Analysis of the Poisson Operator
In view of [19, Section 3] and the fact that \(\varphi _h\) is microlocally supported away from the zero section one can write
where \(U(h): C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(\Omega )\) is a semiclassical, complex-phase h-Fourier integral operator supported near diagonal. In terms of Fermi coordinates \((x_{{n+1}},x')\) in a collar neighborhood \(U= \{ (x',x_{{n+1}}); x_{{n+1}} \ge 0 \}\) of \(\partial \Omega = \{ x_{{n+1}} =0 \}\), U(h) has Schwartz kernel
Here,
satisfies a complex eikonal equation (3.3) and \(a(x,y',{\xi '},h)\) is a semiclassical analytic symbol as in (2.3). Note that the error is exponentially decreasing since we work in the analytic setting.
In the Euclidean case, we note that one can derive the semiclassical Poisson formula (1.15) in an elementary fashion directly from the potential layer formulas using residue computations. For the benefit of the reader, we outline the argument here. Let \(\Omega \subset {\mathbb R}^{n}\) be a bounded Euclidean domain with real-analytic boundary \(\partial \Omega .\) Let \(G(z,z') \in {\mathcal {D}}'({\mathbb R}^{n} \times {\mathbb R}^{n}) \) be the free Green’s functions with \( \Delta _{z'} G(z,z') = \delta (z-z').\) From Green’s formula and the DtN eigenfunction condition, one gets
with \(N(z,z') = \partial _{\nu (z')} G(z,z'),\,\,\, (z,z') \in {\mathbb R}^{{{n+1}}} \times \partial \Omega .\) Writing \(G(z,z')\) and \(N(z,z')\) as Fourier integrals and rescaling the frequency variables \(\xi \rightarrow h^{-1} \xi \) one rewrites (1.18) in the form
Let \(\chi \in C^{\infty }_0({\mathbb R})\) with \(\chi =1\) near the origin and supp \(\chi \subset [-\varepsilon _0,\varepsilon _0].\) We note that by making a change a change of contour \(\xi \mapsto \xi + i \delta \xi \) in (1.18) with \(0< \delta <1\) one can insert a spatial cutoff \(\chi (|z-z'|)\) in both integrals modulo an \(O(e^{-C/h})\) error. Next, we introduce convenient coordinates in a tubular neighborhood \(U_{\partial \Omega }\) of the boundary. Given a local \(C^{\omega }\) parameterization of the boundary \(q: U \rightarrow \partial \Omega \) with \(U \subset {\mathbb R}^n\) open, we write locally
By choosing \(\varepsilon _0>0\) sufficiently small, we can assume that z and \(z'\) lie in the same local coordinate chart. In terms of these new coordinates, one can rewrite the phase function
We make the affine change of variables in (1.18) given by \(\xi \mapsto (\eta ',\eta _{{{n+1}}})\) where
Then, for \(x= (x',x_{{n+1}}) \in U_{\partial \Omega },\) using the fact that the DtN eigenfunctions are h-microlocally \(O(e^{-C/h})\) near the zero section \({\eta }' = 0\) (see Proposition 4.6), one can write
where \( b(x,y',\eta ) = (1 + i \eta _{{n+1}}) ( \eta _{{n+1}}^2 + |\eta '|_x^2 + i 0)^{-1} a(x',y').\ \)
For \(\eta ' \ne 0,\) a residue computation gives \( \int _{{\mathbb R}} e^{i x_{{n+1}} \eta _{{n+1}}/h} ( \eta _{{n+1}}^2 + |\eta '|_{x} + i0)^{-1} \mathrm{d}\eta _{{n+1}} = \frac{ \pi e^{- x_{{n+1}} |\eta '|_x/h}}{ |\eta '|_x}\) and similarily, \( \int _{{\mathbb R}} e^{i x_{{n+1}} \eta _{{n+1}}/h} \eta _{{n+1}} ( \eta _{{n+1}}^2 + |\eta '|_{x} + i0)^{-1} \mathrm{d}\eta _{{n+1}} = \pi e^{- x_{{n+1}} |\eta '|_x/h}.\) Substitution of these integral formulas in (1.19) gives modulo \(O(e^{-C(\varepsilon )/h})\) error,
This is consistent with the general formula in (1.16).
1.3.2 Microlocal Lift of the Poisson Representation (1.15)
Given the representation of \(u_h\) in (1.15) in terms of semiclassical, complex-phase h-Fourier integral operator \(U(h): C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(\Omega ),\) the key idea in the proof of Theorem 1 is to lift (1.15) to the cotangent bundle of the boundary \(T^*\partial \Omega \) and then apply the weighted estimate in Theorem 2 to give the first-order approximation for the Steklov decay rate function d(x) in Theorem 1 (1.3). The quadratic term in d(x) is then bounded from above to prove Theorem 1 (1.4) using the refined h-microlocal weighted estimates in Theorem 3.
Roughly speaking, we do this as follows: Viewing \(x \in \Omega \) as parameters, we consider the family of functions \(K_{x,h} \in C^{\infty }(\partial \Omega )\) with
Then, (1.15) can be written in the form
To lift (1.20) we let \(T(h): C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(T^*\partial \Omega )\) be an FBI transform in the sense of Sjöstrand [18] and \(S(h): C^{\infty }(T^*\partial \Omega ) \rightarrow C^{\infty }(\partial \Omega )\) be a left-parametrix with
and R(h) exponentially small in the sense that
Given the weight function \(\psi \in C^{\infty }(T^*\partial \Omega )\) in Theorem 2, one can write
Using the bound
in Theorem 2 and applying Cauchy–Schwarz in (1.21) one gets
Finally, a direct analysis of the first term on the RHS of (1.22) using the method of analytic stationary phase yields the bounds (1.3) and (1.4) in Theorem 1. We refer to Sect. 3 for a detailed proof of Theorem 1 using the weighted bounds in Theorems 2 and 3.
Remark 1.6
Both Theorems 2 and 3 have other applications to eigenfunction bounds, including the problem of obtaining geometric rates of decay for eigenfunctions in subdomains of configuration space M that correspond to classically forbidden regions that are geometrically more refined than the classical Agmon–Lithner estimates. Specific examples include (but are not limited to) joint eigenfunctions for quantum completely integrable (QCI) eigenfunctions. We hope to return to this elsewhere.
1.4 Outline of the Paper
In Sect. 2 we discuss the eigenfunction mass microlocalization results for the eigenfunctions \(\varphi _h\). The long range exponential decay estimates are proved in Proposition 2.3 and the short-range exponential weighted estimates near the characteristic variety (and inside the Grauert tube \(M_{\tau }^{{\mathbb C}}\)) are proved in Proposition 2.5. These estimates are combined to prove Theorem 2 in Sect. 2.6. The h-microlocally refined weighted estimates for \(T_{hol}(h) \varphi _h\) along with the proof of Theorem 3 are taken up in Sect. 2.7. In Sect. 3, the exponential weighted estimates in Sects. 2.6 and 2.7 are used to prove the decay estimates in Theorem 1 for the Steklov eigenfunctions. In Sect. 4, we prove the necessary h-microlocal exponential decay estimates for the Steklov eigenfunctions near the zero section of \(T^*\partial \Omega .\) This is necessary since the semiclassical DtN operator \( h {\mathcal {D}}: C^{\infty }(\partial \Omega ) \rightarrow C^{\infty }(\partial \Omega )\) fails to be an h-analytic pseudodifferential operator microlocally near the zero section.
2 Eigenfunction Mass Microlocalization
Let M be a compact, closed, real-analytic manifold of dimension m and \(\widetilde{M}\) denote a Grauert tube complex thickening of M with M a totally real submanifold. By Bruhat-Whitney, \(\widetilde{M}\) can be identified with \(M^{{\mathbb C}}_{\tau } := \{ (\alpha _x, \alpha _\xi ) \in T^*M; \sqrt{\rho }(\alpha _x,\alpha _\xi ) \le \tau \}\) where \(\sqrt{2\rho } = |\alpha _{\xi }|_g\) is the exhaustion function \(M^{{\mathbb C}}_{\tau }\), where we identify \(\widetilde{M}\) with \(M_{\tau }^{{\mathbb C}}\) using the complexified geodesic exponential map \( \kappa : M_{\tau }^{{\mathbb C}} \rightarrow \tilde{M}\) with \(\kappa (\alpha ) = \exp _{\alpha _x,{\mathbb C}}( i \alpha _{\xi })\) Viewed on \(\widetilde{M}\), the function \(\sqrt{\rho }(\alpha ) = \frac{-i}{2\sqrt{2}} r_{{\mathbb C}}(\alpha ,\bar{\alpha }),\) which satisfies homogeneous Monge–Ampere and its level sets exhaust the complex thickening \(\widetilde{M}\) (see Remark 2.6 and [5] for further details).
The example of the round sphere To illustrate these basic complex analytic entities, we consider the case of the n-dimensional round sphere
The complexification of M is the quadric
The Riemannian exponential map written in terms of affine ambient coordinates on \({\mathbb R}^{n+1}\) is
where \(\xi \in T_xM\), so \(\xi \in {\mathbb R}^{n+1}\) is orthogonal to x. The complexification of \(\exp _x(\xi )\) is then given by
The distance function \(r(x,y) = 2 \sin ^{-1} \Big ( \frac{|x-y|}{2} \Big )\) complexifies to
and the associated exhaustion function on the complexification is
Pulling \(r_{{\mathbb C}}(z,\bar{z})\) back to \(T^*{\mathbb {S}}^2\) via the complexified exponential map gives
In terms of local coordinates \(\alpha =(\alpha _x,\alpha _{\xi }) \in T^*{\mathbb {S}}^2,\) this just gives \(\sqrt{\rho }(\alpha ) = \frac{1}{\sqrt{2}} |\alpha _{\xi }|_g.\) Of course, the multiplicative factor of \(\frac{-i}{2\sqrt{2}}\) above is just a computationally convenient normalization that we choose to adopt here.
By possibly rescaling the semiclassical parameter h we assume without loss of generality that the characteristic manifold
We will also have to consider a complexification of \(T^*M\) of the form
where \(C \gg 1\) is a sufficiently large constant and \(T^*M \subset \widetilde{T^*M}\) is then a totally real submanifold.
We recall that a complex m-dimensional submanifold, \(\Lambda ,\) of \(\widetilde{T^*M}\) is said to be I-Lagrangian if it is Lagrangian with respect to
where \(\omega = d\alpha _{x} \wedge d \alpha _{\xi }\) is the complex symplectic form on \(\widetilde{T^*M}\).
Let \(U\subset T^*M\) be open. Following [18], we say that \(a \in S^{m,k}_{cla}(U)\) provided \(a \sim h^{-m} (a_0 + h a_1 + \dots )\) in the sense that
We sometimes write \(S^{m,k}_{cla}=S^{m,k}_{cla}(T^*M)\).
Following [19], we also define the notion of a homogeneous analytic symbol of order k and write \(a\in S^k_{ha}\) provided that there exist holomorphic functions \(a_k\) on a fixed complex conic neighborhood of \(T^*M{\setminus }\{0\}\) homogeneous of degree k in \(\xi \) so that there exists \(C_0>0\) so that
and for every \(C_1>0\) large enough, there exists \(C_2>0\) so that
We say that an operator A(h) is a semiclassical analytic pseudodifferential operator of order m, k if its kernel can be written as \({A}(x,y;h)=K_{{1}}(x,y;h)+R_{{1}}(x,y;h)\) where for all \(\alpha ,\beta \),
and
where \(\chi \in C_c^\infty (\mathbb {R})\) is 1 near 0 and \(a\in S^{m,k}_{cla}\). We say A is h-elliptic if \(|a_0(x,\xi )|>ch^{-m}\langle \xi \rangle ^k\) where \(a_0\) is from (2.3). Recall also that A is classically elliptic if there is \(C>0\) so that if \(|\xi |>C\), \(|a_0(x,\xi )|>C^{-1}h^{-m}|\xi |^k\).
We say that an operator, B is a homogeneous analytic pseudodifferential operator of orderk if its kernel can be written as \({B}(x,y)=K_{{2}}(x,y)+R_{{2}}(x,y)\) where \(R_{{2}}(x,y)\) is real analytic and
for some \({b}\in S^k_{ha}\).and \(\chi \in C_c^\infty (\mathbb {R})\) is 1 near 0. We say B is elliptic if there exists \(c>0\) so that \(b_k>c|\xi |^k\) on \(|\xi |\ge 1\) where \(b_k\) is from (2.4) and (2.5). For more details on the calculus of analytic pseudodifferential operators, we refer the reader to [17].
As in [18], given an h-elliptic, semiclassical analytic symbol \(a \in S^{3{{n}}/4,{{n}}/4}_{cla}(M \times (0,h_0]),\) we consider an intrinsic FBI transform \(T(h):C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) of the form
with \(\alpha = (\alpha _x,\alpha _{\xi }) \in T^*M\) in the notation of [18].
Remark 2.1
The normalization \(a\in S^{3n/4,n/4}_{cla}\) appears so that T is \(L^2\) bounded with uniform bounds as \(h\rightarrow 0\) [18].
The phase function is required to satisfy \(\varphi (\alpha ,\alpha _x) = 0, \, \partial _y \varphi (\alpha ,\alpha _x) = - \alpha _{\xi }\) and
Given \(T(h) :C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) it follows by an analytic stationary phase argument [18] that one can construct an operator \(S(h): C^{\infty }(T^*M) \rightarrow C^{\infty }(M)\) of the form
with \(b \in S^{3{{n}}/4,{{n}}/4}_{cla}\) such S(h) is a left-parametrix for T(h) in the sense that
We use two invariantly defined FBI transforms. The first transform \(T_{geo}(h): C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) is defined using only the Riemannian structure of (M, g) and has phase function
Here, \(r(\cdot ,\cdot )\) is geodesic distance and \(\chi (\alpha _x,y) = \chi _0(r(\alpha _x,y))\) where \(\chi _0: {\mathbb R}\rightarrow [0,1]\) is an even cutoff with supp \(\chi _0 \subset [-inj(M,g), inj(M,g)]\) and \(\chi _0(r) =1\) when \(|r| < \frac{1}{2} inj(M,g).\)
The transform \(T_{geo}(h)\) will be used to derive h-microlocal exponential decay outside the Grauert tube \(M_{\tau }^{{\mathbb C}}\) and far from the characteristic variety \(p^{-1}(0).\) We will refer to the corresponding estimates as long-range.
To estimate h-microlocal eigenfunction mass inside the Grauert tube \(M_{\tau }^{{\mathbb C}}\) containing \(p^{-1}(0)\) we use instead another FBI-transform \(T_{hol}(h)\) which is defined in terms of the holomorphic continuation of the heat operator \(e^{t \Delta _g}\) at time \(t = h/2.\) We refer to the corresponding estimates as short-range.
Before continuing, we briefly recall here some background on the operator \(T_{hol}(h): C^{\infty }(M) \rightarrow C^{\infty }(M_{\tau }^{{\mathbb C}})\) and refer the reader to [10] for further details.
2.1 Complexified Heat Operator on Closed, Compact Manifolds
Consider the heat operator of (M, g) defined at time h / 2 by
By a result of Zelditch [25, Section 11.1], the maximal geometric tube radius \(\tau _{\max }\) agrees with the maximal analytic tube radius in the sense that for all \( 0<\tau < \tau _{\max }\), all the eigenfunctions \(\varphi _j\) extend holomorphically to \(M_\tau ^{\mathbb C}\) (see also [10, Prop. 2.1]). In particular, the kernel \(E(\cdot ,\cdot ;h)\) admits a holomorphic extension to \(M_\tau ^{\mathbb C}\times M_\tau ^{\mathbb C}\) for all \(0<\tau < \tau _{\max }\) and \(h \in (0,1)\), [10, Prop. 2.4]. We denote the complexification by \(E_h^{\mathbb C}( \cdot , \cdot )\). To recall asymptotics for \(E^{{\mathbb C}}_h\) we note that the squared geodesic distance on M
holomorphically continues in both variables to \(M_{\tau } \times M_{\tau }\) in a straightforward fashion. More precisely, \(0<\tau <\tau _{\max }\), there exists a connected open neighborhood \(\tilde{\Delta }\subset M_\tau ^{\mathbb C}\times M_\tau ^{\mathbb C}\) of the diagonal \(\Delta \subset M \times M\) to which \(r^2(\cdot , \cdot )\) can be holomorphically extended [10, Corollary 1.24]. We denote the holomorphic extension by \(r_{\mathbb C}^2(\cdot ,\cdot ) .\) Moreover, one can easily recover the exhaustion function \(\sqrt{\rho _g}(\alpha _z)\) from \(r_{{\mathbb C}}\); indeed, \(\rho _g(\alpha _z)=-r^2_{\mathbb C}(\alpha _z, \bar{\alpha _z})\) for all \(\alpha _z \in M_\tau ^{\mathbb C}\).
To analyze the asymptotic behavior of \(E_h^{{\mathbb C}}(\alpha _z,y)\) with \((\alpha _z,y) \in M^{\mathbb C}_\tau \times M\), we split the kernel into two pieces where
-
(i)
the point \((\pi _{_M} \alpha _z,y) \in M \times M\) is close to the diagonal in terms of inj and the Grauert tube radius \(\tau \),
-
(ii)
the point \((\pi _{_M}\alpha _z ,y) \in M \times M\) is relatively far from the diagonal in terms of inj and \(\tau .\)
To control the behavior of the complexified heat kernel for a pair of points \((\pi _{_M} \alpha _z,y) \in M \times M\) that are relatively close or far from the diagonal, we need the following result [10].
Proposition 2.2
There exist \(0<\tau _0 \le \tau _{\max }\) and positive constants \(\beta , \delta _0, h_0\) and C, depending only on \(\tau _0>0\), such that for \(0<\tau \le \tau _0, 0<\delta \le \delta _0\) and \((\alpha _z,y)\in M_{\tau }^{{\mathbb C}}\times M\), the following is true:
-
(i)
When \(r(\pi _{M} \alpha _z, y)<\delta \) and \(h \in (0,h_0],\)
$$\begin{aligned} E_h^{\mathbb C}(\alpha _z,y)=e^{-\frac{r^2_{\mathbb C}(\alpha _z,y)}{2h}} a^{\mathbb C}(\alpha _z,y; h) + O (e^{-\beta /h}). \end{aligned}$$(2.9)Here, \(a^{\mathbb C}(\alpha _z,y;h)\) is the polyhomogeneous sum
$$\begin{aligned} a^{{\mathbb C}}(\alpha _z,y; h):=(2\pi h)^{-{{n}}/2} \sum _{0\le k\le D/h} a^{\mathbb C}_k(\alpha _z,y)h^k, \end{aligned}$$(2.10)where the \(a_k^{\mathbb C}\)’s denote the analytic continuation of the coefficients appearing in the formal solution of the heat equation on (M, g)
-
(ii)
There exists \(C>0\) so that when \(r(\pi _{M} \alpha _z, y)>\frac{\delta }{2}\) and \(h \in (0,1),\)
$$\begin{aligned} \left| E_h^{\mathbb C}(\alpha _z,y) \right| \le C \;e^{- \frac{\delta ^2}{{C} h}}, \end{aligned}$$(2.11)where C is a positive constant depending only on (M, g).
From now on, we always carry out our analysis in the complex Grauert tubes \(M_\tau ^{\mathbb C}\) with \(0<\tau \le \tau _{{0}},\) where in view of Proposition 2.2, we have good control of the complexified heat kernel, \(E^{\mathbb C}_h(\cdot ,y)\) for \(y \in M\).
For \((\alpha _z,y) \in M_{\tau }^{{\mathbb C}} \times M\) with \(r(\mathrm Rez,y) <\varepsilon \) with \(\varepsilon >0\) small, one can show that the function \(y \mapsto - \mathrm Rer^2_{{\mathbb C}}(\alpha _z,y)\) attains a non-degenerate maximum at \(y = \mathrm Rez\). The corresponding strictly plurisubharmonic weight is the square of the exhaustion function given by
where
Using this observation and the expansion in Proposition 2.2 it is proved in [10, Theorem 0.1] that the operator \(T_{hol}(h): C^{\infty }(M) \rightarrow C^{\infty }(M_{\tau }^{{\mathbb C}})\) given by
is also an FBI transform in the sense of (2.6) with amplitude \(a \in S^{m/2,0}_{cla}\) and phase function
In (2.12) the multiplicative factor \(h^{-{{n}}/4}\) is added to ensure \(L^2\)-normalization so that \(\Vert T_{hol} \varphi _h \Vert _{L^2(M_{\tau }^{{\mathbb C}})} \approx 1.\) The fact that the transform \(T_{hol}\) is compatible with the complex structure of the Grauert tube \(M_{\tau }^{{\mathbb C}}\) will be used a crucial way in the proof of the h-microlocal, short-range weighted \(L^2\) bounds in Proposition 2.5.
Since P(h) has simple characteristics and is classically elliptic \( p^{-1}(0)\) is a compact, real-analytic hypersurface and by assumption, \(p^{-1}(0) \subset M_{\tau }^{{\mathbb C}}.\)
2.2 Long-Range Estimates
Let \(p(\alpha , h)\sim \sum _{j=0}^\infty p_j(\alpha )h^j\in S^{m}\) be the full symbol of P(h) and assume that it lies in \(S^{0,m}_{cla}(W)\) where W is a neighborhood of \((x_0,\xi _0)\). Here, \(\xi _0\) is allowed to be a point at infinity in which case a neighborhood means a conic neighborhood of \(\xi _0\) near infinity. We say p is elliptic at \((x_0,\xi _0)\) if \(|p_0|\ge c\left\langle \xi \right\rangle ^m>0\) in a neighborhood of \((x_0,\xi _0)\).
Proposition 2.3
Suppose that P(h) is a semiclassical pseudodifferential operator analytic in a neighborhood of \((x_0,\xi _0)\) and elliptic at \((x_0,\xi _0)\). Suppose that
Then, for any FBI transform T(h), there exists \(c>0\) and W a neighborhood of \((x_0,\xi _0)\) so that
Proof
Let \(\chi _1\in C^{\infty }_0(T^*M)\) so that \(\chi _1\equiv 1\) near \((x_0,\xi _0)\) and p is elliptic and analytic on \({{\text {supp}}}\chi _1\). Let T(h) be an FBI transform with symbol \(r\in S_{cla}^{3n/4,n/4}\) and phase function \(\varphi \). An application of analytic stationary phase [18] gives
where
with
Here, \(r_0\) is the principal symbol of r from (2.3). Since
and \(|\alpha _x -y| < \delta \ll 1\) on supp \(\chi (\alpha _x-y)\) it follows that \(p_0(y,-d_y \varphi )\) is h-elliptic near \(\alpha =(x_0,\xi _0)\). In particular,
Then, from (2.14) and the eigenfunction equation \(P(h) \varphi _h =O(e^{-c/h}),\) it follows that
We claim that (2.17) is independent of FBI transform; in particular,
Since \(\chi _1(\alpha )b(\alpha ,y,h)\) is h-elliptic near \((x_0,\xi _0,x_0)\), [18, Proposition 6.2] proves the estimate. We review the proof here for the reader’s convenience. The operator given by
is an h-pseudodifferential operator with elliptic symbol near \((x_0,\xi _0)\). So, for any \(a(\alpha ,x)\), supported near \((x_0,\xi _0,x_0)\), we can find a classical analytic symbol, \(\tilde{b}\) defined near \((x_0,\xi _0,x_0)\) so that
modulo exponential errors. In particular, for W a small enough neighborhood of \((x_0,\xi _0)\),
where
By an application of analytic stationary phase,
where \( {{\text {Im}}\,}\Phi (\alpha , \beta ) \ge |\alpha - \beta |^2,\) and \(c \in S^{{n},{\infty }}_{cla}.\)
In particular, we can write
where K is tempered in h,
and \(\chi _1\equiv 1\) on \({{\text {supp}}}\chi _2\). Henceforward, we write \(\chi _2 \Subset \chi _1\) to denote this. Thus,
\(\square \)
2.3 Short-Range Estimates
Let \(\chi _{in} \in C^{\infty }_{0}(M_{\tau }^{{\mathbb C}};[0,1])\) and \(\tilde{\chi }_{in} \in C^{\infty }_{0} (M_{\tau }^{{\mathbb C}}; [0,1])\) be a cutoff with \(\tilde{\chi }_{in} \Supset \chi _{in}\).
To deal with the short-range case, using analytic stationary phase one constructs an h-pseudodifferential intertwining operator \(Q(h) \in Op_h(S^{0,\infty }(T^*M))\) that is h-microlocally analytic on the Grauert tube \(M_{\tau }^{{\mathbb C}} \subset T^*M\) and satisfies
To construct Q(h) in (2.19), using (2.14) we write
where \(b \in S^{3{{n}}/4,{{n}}/4+{m}}_{cla}(T^*M)\) is given by (2.15). Then, the symbol \(q(\alpha ,\alpha ^*;h) \sim _{h \rightarrow 0^+} \sum _{j=0}^{\infty } q_j(\alpha , \alpha ^*) h^j\) of Q(h) is determined by solving the equations
where \(\alpha ^*\) is the dual coordinate to \(\alpha \) and \(\tilde{p}_j\) are determined as in (2.15). To solve for the \(q_j\) in (2.20), we first consider the complexified equations
Let \(\widetilde{M_{\tau }^{{\mathbb C}}}\) be a complex extension of \(M_{\tau }^{{\mathbb C}}\). Here, \(^{\mathbb C}\) denotes holomorphic continuation to \((\alpha ,y) \in \widetilde{ M_{\tau }^{{\mathbb C}} } \times M_{\tau }^{{\mathbb C}}\) with \(|\alpha _x - y| < \varepsilon _0\) and \(M_{\tau }^{{\mathbb C}} = \{ \alpha ; \sqrt{\rho }(\alpha ) \le \tau \}\) is identified with the complex thickening \(\tilde{M}.\) Since
it follows that near \(\alpha _x=y\), \(\det \partial ^2_{y \, \alpha _\xi }\varphi \ne 0\) and so by the holomorphic implicit function theorem, \(d_{\alpha _{\xi }} \varphi (\alpha ,y)=w\) defines \(y=\beta _x^{{\mathbb C}}(\alpha ,w)\) with \(\beta _x^{{\mathbb C}}\) holomorphic in a neighborhood of \(\alpha _x=y.\) Hence, restricting to real points \((\alpha ,y) \in M_{\tau }^{{\mathbb C}} \times M\) with \(|\alpha _x -y| < \varepsilon _0,\) we can write
where \(\beta _x\) and \(\beta _{\xi }\) are locally \(C^{\omega }.\)
Since \(\tilde{p}_0 = p_0,\) for the principal symbols one gets
It will be useful to introduce the principal symbol of conjugated operator \(e^{\psi /h} Q(h) e^{-\psi /h}\) given by
Now, \(\partial ^2_{y, \alpha _\xi } \varphi |_{y=\alpha _x}=-{\text {Id}}\), and since \(\partial _y^2r(\alpha ,y)|_{y=\alpha _x}=2 {\text {Id}}\), \(\partial ^2_y\varphi |_{y=\alpha _x}=i{\text {Id}}\). Therefore, using also that
It follows from (2.22) that,
for \((\alpha ,y) \in M_{\tau }^{{\mathbb C}} \times M\) with \(r(\alpha _x,y) < \varepsilon _0.\) As an example, we note that in the \({\mathbb R}^n\) case with standard phase function \(\varphi (\alpha ,y) = (\alpha _x-y) \alpha _{\xi } + \frac{i}{2} |\alpha _x-y|^2,\) one has \(\beta _x = \alpha _x - \alpha _{\xi }^*\) and \(\beta _{\xi } = \alpha _{\xi }+i\alpha _{\xi }^*.\)
Given the intrinsic complex structure on the Grauert tube \(M_{\tau }^{{\mathbb C}},\) we denote the associated Cauchy–Riemann operators by \(\partial : C^{\infty }(M_{\tau }^{{\mathbb C}}) \rightarrow \Omega ^{1,0}(M_{\tau }^{{\mathbb C}})\) and \(\overline{\partial }: C^{\infty }(M_{\tau }^{{\mathbb C}}) \rightarrow \Omega ^{0,1}(M_{\tau }^{{\mathbb C}}).\) Moreover, given local coordinates \(\alpha _x\) in a chart \(U \subset M,\) the corresponding complex coordinates in \(U^{{\mathbb C}} \subset M_{\tau }^{{\mathbb C}}\) will be denoted by \(\alpha _{z}:= \exp _{\alpha _x}(-i \alpha _{\xi }), \,\, \bar{\alpha _z} = \exp _{\alpha _x}( i \alpha _{\xi }).\)
Given a smooth one-form \(\theta \in \Omega ^{1}(M_{\tau }^{{\mathbb C}})\) one can write it in local \((\alpha _x,\alpha _{\xi })\)-coordinates in the form
and in terms of complex coordinates \((\alpha _z, \bar{\alpha _z})\) as
Consequently, in terms of the Cauchy–Riemann operators, \(\alpha _z^* = \sigma (\partial )(\alpha )\) and \( \bar{\alpha _{z}}^* = \sigma (\bar{\partial })(\alpha ).\) Here, \(\sigma \) denotes the principal symbol of a pseudodifferential operator.
Given a weight function \(\psi \in C^{\infty }(M_{\tau }^{{\mathbb C}})\) and the strictly plurisubharmonic weight \(\rho (\alpha _{z},\bar{\alpha _z}) =\frac{ |\alpha _{\xi }|_g^2}{2},\) we consider the associated submanifold \(\Lambda \subset \widetilde{M_{\tau }^{{\mathbb C}}} \,\) given by
As we shall see below, the manifold \(\Lambda \) will play an important role in our main exponential weighted estimate in Proposition 2.5.
For future reference, we note that in terms of the local complex coordinates \((\alpha _z, \bar{\alpha }_z)\) in a geodesic normal coordinate chart U,
2.4 Complex Geometry of \(\Lambda \)
We first recall some basic complex symplectic geometry: Let X be a complex n-dimensional manifold with complex cotangent bundle \(T^*X.\) Viewing X as a real-analytic manifold, we let \(T^*X_{{{\mathbb R}}}\) denote the real 4n-dimensional cotangent bundle. There is a natural identification [9] of \( T^*X_{{\mathbb R}}\) with \(T^*X\) given as follows. Let \(v \in TX\) (a complex tangent vector) and \((z, \zeta ) \in T^*X\) (a complex covector). Then, the identification \(\iota : T^*X \rightarrow T^*X_{{\mathbb R}}\) is given by
In terms of local coordinates \((\mathrm Rez, {{\text {Im}}\,}z): X \rightarrow {\mathbb R}^{2n}\) and the corresponding dual coordinates \((\xi ,\eta ) \in T_{(\mathrm Rez, {{\text {Im}}\,}z)}^*X_{{\mathbb R}},\)
Let \(\Gamma \subset T^*X\) be I-Lagrangian with respect to the complex symplectic form \(\Omega = dz \wedge d \zeta .\) Then, for any contractible coordinate chart U, we recall that [9, Lemma 3.1] using the identification \(\iota ,\) one can locally characterize \(\Gamma \) as the graph of complex differential; that is,
with \(f \in C^{\infty }(U;{\mathbb R})\) and \( 2 \partial _z f = ( \partial _\mathrm{Rez} + i \partial _{{{\text {Im}}\,}z} ) f.\)
We claim that \(\Lambda \subset T^* ( M_{\tau }^{{\mathbb C}})\) in (2.27) can be naturally identified with an I-Lagrangian with respect to the canonical complex symplectic form. More precisely, consider
To see that \(\tilde{\Lambda }\) is indeed I-Lagrangian, we note that since \(d = \partial + \bar{\partial },\) one can write
and so,
Consequently, in view of (2.29), \(\tilde{\Lambda }\) is indeed I-Lagrangian. Moreover, since \(\rho \) is strictly plurisubharmonic with \( \partial \bar{\partial }\rho >0\), it follows that with \(\Vert \psi \Vert _{C^2}\) sufficiently small, \(\tilde{\Lambda }\) is also \({\mathbb R}\)-symplectic.
We note that clearly one can write the Toeplitz multiplier \(q_0 |_{\Lambda }\) on the RHS of Proposition 2.5 as \(\tilde{q_0} |_{\ \tilde{\Lambda }}\) where \(\tilde{q_0}(\alpha ,\alpha ^*) = q_0(\alpha , \alpha ^* + i d_{\alpha } \rho (\alpha ))\) and \(\tilde{\Lambda }\) is the I-Lagrangian above. However, we find working with \(q_0\) (from (2.24)) and \(\Lambda \) (from (2.27)) computationally simpler and so we continue to work throughout with these instead.
Remark 2.4
Here we call \(q_0\) a Toeplitz multiplier in reference to the corresponding Toeplitz operator \(S_{hol}q_0T_{hol}\) (see e.g., [29, Chapter 13.4]).
The following h-microlocal manifold version of the microlocal Agmon estimates in \({\mathbb R}^n\) [13, 14] is central to the proofs of Theorems 1, 2.
Proposition 2.5
Let \(\Lambda \subset \widetilde{M_{\tau }^{{\mathbb C}}}\) be as in (2.27). Then, for any \(P(h) \in Op_h (S^{0,\infty }_{cla})\) there exists \(\delta >0\) so that for \(\psi \in S^0(1)\) with \(\Vert \psi \Vert _{C^1}{<\delta }\),
Proof
The operator \(Q_{\psi }(h):= \chi _{in} e^{\psi /h} Q(h) e^{-\psi /h}\) has Schwartz kernel
By Taylor expansion,
with \(| \Psi |< \Vert \psi \Vert _{C^1} < \delta .\) Since \(q(\alpha ,\alpha ^*,h)\) is analytic, for \(\delta >0\) small it follows by Stokes formula one can make the contour deformation
in (2.30). Boundary terms as \(|\alpha ^*| \rightarrow \infty \) vanish and one gets that \(Q^{\psi }(h) \in Op_{h}(S^{0}(1))\) with symbol
and principal symbol
where \(q_0\) is defined in (2.24).
In view of (2.19) it follows that
Next, in analogy with [13], we observe that with \(\bar{D}_{\alpha _z} = \frac{1}{i} \bar{\partial }_{\alpha _z},\)
where we have written
The last line in (2.32) follows since \(r^{2}_{{\mathbb C}}(\cdot ,y)\) and \(a(\cdot ,y,h)\) are holomorphic and the exponential error arises from differentiation of the cutoff \(\chi (\alpha _x,y).\) In particular, when \(r(\alpha _x,y)\ge \varepsilon _0>0\), there exists \(C(\varepsilon _0)>0\) so that
Similarly, taking complex conjugates, one gets that
Taylor expansion of the principal symbol \(q^{\psi }_0\) of \(Q_{\psi }(h)\) around \(\bar{\alpha _z}^* = - i \overline{\partial }_{\alpha _z} \tilde{\psi }\) and \(\alpha _z^* = i \partial _{\alpha _z} \tilde{\psi }\) gives
Since \(r_1,r_2 \in S^{0}(1),\) it then follows from (2.31)–(2.34), \(L^2\)-boundedness of pseudodifferential operators, and that \([Op_h(S^{m_1}),Op_h(S^{m_2})]\in hOp_h(S^{m_1+m_2})\) (see for example [29, Chapters 4,9]) that
This finishes the proof of the Proposition. \(\square \)
2.5 Microlocal Eigenfunction Decay Estimates
2.5.1 Estimation of the Multiplier \(q_0 |_{\Lambda }\)
In order to prove Theorem 2 we give an invariant characterization of the Toeplitz multiplier \(q_0 |_{\Lambda }.\)
In terms of the local coordinates \((\alpha _z, \bar{\alpha }_z)\) the Toeplitz multiplier in Proposition 2.5 is
To give this in invariant meaning, we note that function \(q_{0} \in C^{\omega }(T^*M_{\tau }^{{\mathbb C}})\) and
are local coordinates for the point
Consequently, the Toeplitz multiplier equals
in view of the definition of \(\Lambda \) in (2.27), where
Here, \(\partial , \bar{\partial }: C^{\infty }(M_{\tau }^{{\mathbb C}}) \rightarrow \Omega ^{1}(M_{\tau }^{{\mathbb C}})\) are the intrinsic Cauchy–Riemann operators.
The \(\rho \) portion, \(i\bar{\partial }\rho -i\partial \rho \), of the argument on the RHS of (2.37) can be readily computed. Given the strictly plurisubharmonic weight function \(\rho (\alpha ) = \frac{1}{2} |\alpha _{\xi }|^2_{\alpha _x}\), we recall that [5, p. 568],
where \(\omega = \sum _j \alpha _{\xi _j} d\alpha _{x_j}\) is the canonical one-form. Since both sides are invariant, one can easily verify the identity (2.38) by computing in geodesic normal coordinates at the center of the coordinate chart.
It follows from (2.38) that
From (2.39) one gets
where in the last equality we have used (2.26).
Remark 2.6
We recall that \(\rho \) is a Kahler potential for the intrinsic Kahler form \(\Omega _g\) on the Grauert tube corresponding to the complex structure \(J_g\) induced by complexified Riemannian exponential map of the metric g; that is,
The corresponding exhaustion function \(\sqrt{2\rho }(\alpha ) = |\alpha _{\xi }|_g\) satisfies homogeneous Monge–Ampere,
We refer to [5, 11] for further details.
From Proposition 2.5 and (2.40) we get the following useful estimate.
Proposition 2.7
Under the same assumptions as in Proposition 2.5,
2.6 Microlocal Concentration of the Eigenfunctions: Proof of Theorem 2
In this section, we prove the global weighted decay estimate in Theorem 2.
Proof
We first prove the weighted estimate in (1.8) h-microlocally on support of \(\chi _{in} \in C^{\infty }_{0}(M_{\tau }^{{\mathbb C}})\) and for the h-microlocal FBI transform \(T_{hol}(h).\)
The transform \(T_{hol}(h)\) is only defined on the Grauert tube \(M_{\tau }^{{\mathbb C}},\) which is generally a proper, bounded subset of \(T^*M.\) We then need to prove that the weighted bound in (2.41) still holds for \(T(h) = T_{geo}(h)\), after possibly shrinking \(\delta >0\) somewhat (independently of h).
In the following, we let \(P(h) \in Op_h(S^{0,k}_{cla})\) be as in the statement of Theorem 2 and \(\varphi _h\) be an exponential quasimode with
2.6.1 h-Microlocal Bounds for \(T_{hol}(h) \varphi _h\)
Recall \(\psi =\frac{\delta p^2}{2\langle \xi \rangle ^{2k}}.\) For \(\delta >0\) small,
To see that (2.42) holds, we split into two cases:
-
Case (i)
\(\{ \alpha ; |p(\alpha )| \ll 1\}\) (near the characteristic variety). Here, we make a Taylor expansion to get
$$\begin{aligned} p^2(\alpha + O(|\partial _{\alpha }\psi |) = p^2(\alpha ) + O(\delta ) |2p\partial _{\alpha }p|^2 = p^2(\alpha ) + O(\delta \, p^2(\alpha )).\nonumber \\ \end{aligned}$$(2.43)Consequently, near \(p=0\) it follows from (2.43) that for \(\delta >0\) small,
$$\begin{aligned} \mathrm Rep^2(\alpha + O(|\partial _{\alpha }\psi |) ) \ge c p^2(\alpha ). \end{aligned}$$ -
Case (ii)
\(\{ \alpha ; |p(\alpha )| \gtrapprox 1 \}\) (far field). Here we use the fact that \(\psi \in S^{0}(1)\) and \( |\partial p^2| \lessapprox \langle \alpha _{\xi } \rangle ^{2k-1}\) and just make the first-order Taylor expansion
$$\begin{aligned} p^2(\alpha + O(|\partial _{\alpha }\psi |) ) = p^2(\alpha ) + O( \delta \langle \alpha _{\xi } \rangle ^{2k-1} ). \end{aligned}$$Since \( p^2(\alpha ) \gtrapprox \langle \alpha _{\xi } \rangle ^{2k}\) in this range, (2.42) is also satisfied in this case, provided one chooses \(\delta >0\) small.
Since \(P(h) \in Op_h(S^{0,\infty }_{cla})\) implies that also \(P^2(h) \in Op_h(S^{0,\infty }_{cla}),\) and so Proposition 2.7 applies just as well with the latter. We note that by (2.42), there is a constant \(C>0\) such that
and so, by an application of Proposition 2.7 with the globally defined weight function \(\psi \) in the statement of Theorem 2, it follows that
In the last line of (2.44) we have used the long-range estimate (2.3) yet again to write
To bound the LHS in (2.44) note that \(A_k(h):=S_{hol}(h) \left\langle \alpha _\xi \right\rangle ^{k}T_{hol}(h)\in Op_h(S^{0,k}_{cla})\). Hence, \(A_{-k}(h)P(h)\in {\text {Op}}_h(S^{0,0}_{cla})\) and since P(h) is classically elliptic with \(\Vert P(h) {\varphi _h}\Vert _{L^2}+\Vert {\varphi _h}\Vert _{L^2}=O(1)\), \(A_{k}(h){\varphi _h}\in L^2\). In particular,
since by assumption \(P(h) \varphi _h = O_{L^2}(e^{-C/h}).\)
Thus, from (2.44) one gets that for \(\delta \) small enough and \(C=C(\delta )>0\),
We note that for any \({{N}}>0,\)
so that \(e^{\psi (\alpha )/h} = O(1)\) when \(p^2(\alpha ) \le {{N}} h.\) It then follows from (2.45) that
Choosing \({{N}={N(\delta )}}>0\) large enough to absorb the O(h) term on the LHS of (2.46) (which is independent of \({{N}}>0\)), it follows that
Clearly, since \(\psi \approx p^2\) near \(p=0\) it also follows that
which finishes the proof of (2.41).
2.6.2 Weighted Bounds in Terms of \(T_{geo}(h)\)
Since \(T_{hol}(h)\) is only h-microlocally defined, we need to show that essentially the same weighted bound holds for the globally defined FBI transform \(T_{geo}(h).\) More precisely, in this section we show that with \(\varepsilon _0 <1\) sufficiently small (but independent of h) the analogue of (2.41) holds for the globally defined FBI transform. That is
Given (2.48), the global result in Theorem 2 follows (after possibly shrinking \(\delta >0\) further) since we have already established the long range bound in (2.18).
To prove (2.48), we write
Note in the second line we us that \(T_{geo}S_{hol}\) is pseudolocal modulo exponential errors. Then, by an application of analytic stationary phase, after possibly shrinking \(\delta >0\), the Schwartz kernel of the operator \(\chi _{in} e^{\varepsilon _0 \psi /h} T_{geo}(h) S_{hol}(h) e^{-\psi /h}\) can be written in the form
where,
\(c \in S^{{{n}},0}_{cla}\). To estimate \( \Phi (\alpha , \beta )\) when \(\alpha \) and \(\beta \) are near \(p^{-1}(0),\) we note that by Taylor expansion of \(p^2(\alpha )\) around \(\alpha = \beta \) one gets that
with some \(C_0 >0.\) Writing \(x = p(\alpha )\) and \(y = p(\beta )\) it therefore suffices to consider the function
An application of max/min shows that \(f(x,y) \ge 0\) provided \(\varepsilon _0 (C_0^2)>0\) is chosen sufficiently small and consequently, it follows that for small \(\varepsilon _0>0,\)
Then, from (2.49) and an application of the Schur‘s lemma, it follows that for \(\varepsilon _0>0\) sufficiently small,
and so,
Consequently, the h-microlocal bound (2.48) follows.
After possibly shrinking \(\varepsilon _0>0\) further, we know that by the long range bound in (2.18),
and so, Theorem 2 follows. \(\square \)
2.7 Refinement of the Weight Function: Proof of Theorem 3
Proof
We first turn to the proof of (1.13). Since the results of Theorem 3 are h-microlocal and, moreover, away from a neighborhood of \(\{p=0\}\) we know that any FBI transform applied to \(\varphi _h\) is exponentially small we work in a small neighborhood of \(\{p=0\}\). In particular, this implies that for \(\chi \in C_c^\infty (\mathbb {R})\) with \(\chi \equiv 1\) near 0, and for fixed arbitrarily small \(\varepsilon _0>0\), there exists \(c = c(\varepsilon _0)>0\) so that
Thus, we work exclusively with \(T_{hol}(h)\) here and start by reexamining the proof of (2.47). The key estimate is (2.44) where we use that \((P(h))^2\varphi _h=O_{H_h^{-k}}(e^{-c/h})\) (here \(H_h^{-k}\) is the semiclassical Sobolev space or order \(-k\); see for example [29, Chapter 14]). Notice that in order to conclude that \(e^{\psi /h}T_{hol}(h)\varphi _h\) is well controlled, we must have good control of the O(h) error term appearing in the right hand side of (2.44). In particular, we must have control of \(e^{\psi /h}T_{hol}(h)\varphi _h\) strictly away from \(\{p=0\}\). The long range estimates tell us that we have some exponential decay, however, we do not have any useful control over the constant. Therefore, in order to complete the arguments leading to (2.47), we must choose \(\psi \) so that
\(\square \)
Lemma 2.8
For all \(\delta _0>0\), \(0<\gamma <1/2\), there exists \(\psi _0\in C^\infty (M_\tau )\) so that \(\Vert \psi _0\Vert _{C^1}\le \delta _0\), \({{\text {supp}}}\psi _0\subset \{|p|\le \delta _0\}\),
and in a neighborhood of \(p=0\),
where \(g_s\) is the Sasaki metric on TM (see e.g., [2, Chapter 9]).
Proof
Motivated by the construction of the weight in Theorem 2, we let \(\chi \in C_c^\infty (\mathbb {R})\) with \(\chi \equiv 1\) on \([-1,1]\), \({{\text {supp}}}\chi \subset [-2,2]\), \(0\le \chi \le 1\), \(\chi '(x)\le 0\) on \(x\ge 0\), and make the ansatz
Note that throughout this proof, all \(O(\cdot )\) statements are uniform in \(\delta \).
By (2.26),
We work in geodesic normal coordinates centered at \(\alpha _x\) (i.e., \(\alpha _z=\exp _{\alpha _x}(-i\alpha _\xi )\)) so that \(\partial \psi _0=\frac{1}{2}(\partial _{\alpha _x}\psi _0+i\partial _{\alpha _\xi }\psi _0)(d\alpha _x-id\alpha _\xi )\). Therefore,
Now,
In particular, then
Therefore, choosing
proves the lemma since \(\chi '(x)\le 0\) on \(x\ge 0\) implies
and
\(\square \)
Using \(\psi _0\) from Lemma 2.8 in the analysis leading to (2.47) proves (1.13).
Next, we prove the bound in Theorem 3 (1.14). The fact that \((P(h))^2\varphi _h=O_{H_h^{-k}}(e^{-c/h})\) is a weaker condition than \(P(h)\varphi _h=O_{L^2}(e^{-c/h})\) and indeed, the example of \(e^{ix/h}\) on \(S^1\) shows that the weight \(\psi _0\) is not quite optimal. To remedy this, and obtain (2.41) we need to work directly with \(P(h)\varphi _h=O_{L^2}(e^{-c/h})\). Unlike \(p^2\), p does not have a fixed sign, so we need to work separately on \(p>0\) and \(p<0\). Therefore, we construct slightly different weights on \(p>0\) and \(p<0\). Since the weighted estimate naturally localizes to each region, we consider first the case \(p>0\) and then easily adapt the argument to the case \(p<0.\)
Lemma 2.9
For all \(\delta _0>0\), there exists \(\psi _+\in C^\infty (M_\tau )\) with \(\Vert \psi _+\Vert _{C^1}<\delta _0\), \({{\text {supp}}}\psi _+\subset \{|p|\le \delta _0\}\) so that
and in a neighborhood of \(p=0\),
where \(g_s\) is the Sasaki metric on TM.
Proof
First, let \(\chi \in C_c^\infty (\mathbb {R})\) with \(\chi \equiv 1\) on \([-1,1]\), \({{\text {supp}}}\chi \subset [-2,2]\), and \(\chi '(x)\le 0\) on \(x\ge 0\). Then fix \(\delta >0\) to be chosen small enough later and let
Note that throughout this proof, all \(O(\cdot )\) statements are uniform in \(\delta \).
Computing as in the proof of Lemma 2.8,
For convenience, let
Then,
Then, Taylor expansion of \(p\circ \beta (\alpha ,\alpha _\xi ^*(2i\partial \psi _+))\) around \(\alpha \) gives
where
Grouping terms by homogeneity in p, we have then
Now, let \({{N}}>0\) to be chosen large enough independently of \(\delta \) later
Then, notice that on \(p\ge 0\), \(\chi '(\delta ^{-1}p)\le 0\), so, since \(\chi \ge 0\),
Therefore, on \(\{\chi (\delta ^{-1}p)\ge \frac{1}{2}\}\cap \{p\ge 0\}\),
for N large enough. On the other hand, on \({{\text {supp}}}\chi (\delta ^{-1}p)\cap \{\chi (\delta ^{-1}p)\le \frac{1}{2}\}\cap \{p\ge 0\}\),
if \(\delta >0\) is chosen small enough.
This completes the proof of the lemma since
\(\square \)
Remark 2.10
It is possible to construct \(\psi _+\) satisfying the following: For every \({{N}}>0\), \({{M_0}}>0\), and \(\delta _0>0\), there exists \(\psi _+\in C^\infty (M_\tau )\) with \(\Vert \psi _+\Vert _{C^1}\le \delta _0\), and \({{\text {supp}}}\psi _+\subset \{|p|\le \delta _0\}\) so that
and in a neighborhood of \(p=0\),
Moreover, for any \(j<{{N}}-1\) and \(\delta >0\), there exist \(b\in C^\infty (M_\tau )\) with \(\Vert b\Vert _{C^1}<\delta \) so that if \(a_j\) is replaced by \(a_j+b,\) then
This allows one to improve the higher order terms in \(\psi _{hol}\) in Theorem 3 so that they are sharp for \(e^{ix/h}\) modulo \(p^{{N}}\) for any N.
Since we want to localize to \(p>0\), we insert a smooth cutoff \(\chi _+\) that approximates the indicator function \(\mathbf{1}_{[0, \varepsilon _0]}(p)\). However, in order to estimate error terms in the weighted \(L^2\) bounds corresponding to Proposition 2.5, we must ensure that \(\psi _+ = O(h)\) on supp \(\partial \chi _+\) so that, in particular,
where \({{\text {supp}}}\partial \chi _{+} \subset \{0 \le p \le h^{1/2} \}\cup \{p \ge \varepsilon _0 \}.\)
To construct \(\chi _+\), we let \(\chi _1\in C^\infty (\mathbb {R})\) with \({{\text {supp}}}\chi _1\subset (1,\infty )\) with \( \chi _1\equiv 1\) on \((2,\infty )\) and let
and \(\tilde{\chi }_{+}=\tilde{\chi }(\varepsilon _0^{-1}p^2/2)\tilde{\chi }_1(h^{-1/2}p)\) where \(\tilde{\chi }\in C_c^\infty (\mathbb {R})\) has \(\tilde{\chi }\equiv 1\) on \({{\text {supp}}}\chi \) and \(\tilde{\chi }_1\in C^\infty (\mathbb {R})\) with \(\tilde{\chi }_1\equiv 1\) on \({{\text {supp}}}\chi _1\) and \({{\text {supp}}}\tilde{\chi }_1\subset (1,\infty )\). Then \({{\text {supp}}}\chi _+\subset \{0\le p\le 2\varepsilon _0\}\) so that \(\psi _+=O(h)\) on \({{\text {supp}}}\partial \chi _+\).
We will need the following
Lemma 2.11
Under the same assumptions as in Proposition 2.5,
Proof
We follow very closely the argument in the proof of Proposition 2.5. The only difference here occurs in the integration by parts arguments with respect to \(\partial _{\alpha _z}\) and \(\bar{\partial }_{\alpha _z}\) in (2.32) and (2.33), precisely when the cutoff \(\chi _+\) is differentiated in the amplitude since it is a singular semiclassical symbol with \(\chi _+ \in S^{0}_{1/2}(1)\) [29, Chapter 4] i.e.,
Specifically, one needs to estimate a term of the form
Since \(\chi _+ \in S^{0}_{1/2}(1),\) it is in a singular symbol class. However, because it is a multiplier depending only the spatial coordinates \(\alpha \in M_{\tau }^{{\mathbb C}}\), it can be effectively composed with the standard h-pseudodifferential operators \(r(\alpha , hD_{\alpha }) \in Op_h (S^{0}(1)).\) In particular, symbols compose with \(h^{-j/2}\)-loss in the \(j^{\text {th}}\) term of the asymptotic expansion, \(L^2\)-boundedness and sharp Gårding still hold, as does h-pseudolocality. More precisely, for any \(R(h) \in Op_h (S^{0}(1))\) and spatial cutoff \(\chi _+ = \chi _+(\alpha ,h) \in S^{0}_{1/2}(1),\)
Since the cutoff \( \chi _1 \partial _{\alpha _z} \chi \in S^{0}_{1/2}(1),\) and depends only on the spatial \(\alpha \)-variables, it follows that for second term on the RHS of (2.53),
As for the first term on the RHS of (2.53), \(h \chi \partial _{\alpha _z} (\chi _1) \in h^{-\frac{1}{2}} S_{1/2}^{0}(1)\) since there is a loss of \(h^{1/2}\) coming from differentiation of the \(\chi _1\)-term. Thus,
\(\square \)
Since \( P(h) \varphi _h = O_{L^2}(e^{-c/h}),\) it follows from Lemma 2.3 that for \(\varepsilon \) small enough and \({{N}}>0,\)
In analogy with the arguments used to prove (2.47), we substitute the lower bound \((p\circ \beta )|_{\Lambda } \ge p^2\) on \(p \ge 0\) from Lemma 2.9 in (2.54). Then, since \(\psi _+ = O(h)\) when \(p^2 = O(h)\) (so that \(e^{\psi _+/h} = O(1)\) on the latter set), it follows that
In (2.55) we have also used that \(e^{\psi _+(\alpha )/h} = O(1)\) for all \(\alpha \in \text {supp} (\tilde{\chi }_+ - \chi _{+}) \cap \{ p > C h^{1/2} \}\) since by construction \(\psi _+ = O(h)\) on the latter set (see also Figure 1).
We show the various cutoffs used in the proof of Theorem 3
Choosing \({{N}}>0\) large enough to absorb the O(h) term on the LHS of (2.55) it follows that
The analysis on the set \(p<0\) follows in the same way as above, except one uses the reflected cutoff function \(\chi _{-}(p) = \chi _+(-p), \) and the ansatz for the corresponding weight function is
so that
3 Exponential Decay of the Harmonic Extensions of Boundary Eigenfunctions: Proof of Theorem 1
Proof
Recall that \(M:=\partial \Omega \) with \(\Omega \) the domain of the Steklov problem. Given the FBI transform \(T(h): C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) we construct a left-parametrix \(S(h): C^{\infty }(T^*M) \rightarrow C^{\infty }(M)\) and then given \(q\in S^{m}(1)\) we define the anti-Wick h-pseudodifferential quantization by
Let \({\mathcal {P}}: C^{\infty }(M) \rightarrow C^{\infty }(\Omega )\) be the Poisson operator and \(\chi \in C^{\infty }_{0}(T^*M; [0,1])\) be a cutoff supported near the zero section, with \(\chi ({x',\xi '}) =1\) for \(\{ 0 \le |{\xi '}| \le \varepsilon \}\) and \(\chi ({x',\xi '}) =0\) for \(|{\xi '}| > 2 \varepsilon .\) Here, \(\varepsilon >0\) is some arbitrarily small but fixed constant. Then, one can write
Here, \( x=(x_{{n+1}},x')\) denote Fermi coordinates in a neighborhood of the boundary with \(\Omega = \{ x_{{n+1}} \ge 0 \}\) and \(M = \{ x_{{n+1}} = 0 \}.\) We show in Sect. 4.2 that for \(\chi _{h}^{aw} \varphi _h\) (the piece supported near the zero section), one has the apriori bound
for some \( C>0.\)
It follows from [19] that \({\mathcal {P}} (1 - \chi _h^{aw}) \varphi _h (x)\) (the piece h-microlocally supported away from the zero section) has Schwartz kernel of the form
with \(c \in S^{0}(1)\) supported away from \(|{\xi '}|=0\) and in \(S^{0,0}_{cla}(|{\xi '}|>2\varepsilon )\), \({\Psi }\) solving
with \(r(a,x',\xi ')\) the symbol of the Laplacian induced on \(\{x_{{n+1}}=a\}.\) In particular,
where \(Q(x',\xi ')\) is the symbol of the second fundamental form. It follows by Taylor expansion in \(x_{{n+1}}\) that
where
is homogeneous of degree 1 in \(\xi '\). The near-diagonal cutoff \(\chi (x'-y')\) appears above since \(K(x,y',h)\) is the part of the Poisson operator arising h-microlocally from the complement of the zero section.
Consequently, from (3.1) and (3.2) we have
Next we make an analytic resolution of the identity and write
where \(T(h): C^{\infty }(M) \rightarrow C^{\infty }(T^*M)\) is an FBI transform with phase function \(\varphi \) (not to be confused with the Steklov eigenfunction \(\varphi _h\)) and \(S(h): C^{\infty }(T^*M \rightarrow C^{\infty }(M)\) is a left-parametrix. Notice that by Theorem 2 or Corollary 1.3 together with the analysis in Sect. 4.2 for \(\chi _1\in C_c(\mathbb {R})\) with \(\chi _1 \equiv 1\) on \([-1,1]\), and \({{\text {supp}}}\chi _1\subset [-2,2]\), for any \(\varepsilon >0\),
Let \(\chi _{1,\varepsilon }(\alpha )=\chi _1(\varepsilon ^{-1}(|\alpha _{\xi '}|_{\alpha _{x'}}-1)).\) Then,
and substitution in the integral formula for \(u_h\) gives
with
One can then rewrite the formula (3.5) in the form
Thus, \(\chi _{1,\varepsilon }(\alpha )S(h)^t K_{h,x}(\alpha )\) equals
with \(\alpha = (\alpha _{x'}, \alpha _{\xi '}) \in T^*M\) and \(c \in S^{0,0}_{cla}(|{\xi '}|>2\varepsilon ), \, a \in S^{3{{n}}/4, {{n}}/4}_{cla}.\) Next, we apply analytic stationary phase in the \((y',{\xi '})\) variables. We can do for \(x_{{n+1}}\) small since \(\psi (x_{{n+1}},x',\eta ')=\left\langle x',\eta '\right\rangle +O_{C^\infty }(x_{{n+1}})\). Writing
for the total phase and computing in geodesic normal coordinates centered at \(\alpha _{x'},\) the critical point equations are
It follows that, denoting the complex critical points by \(y'_c\), \(\eta '_c\), we have, using the fact that \(\psi _1\) is homogeneous of degree 1
Therefore,
Here, we implicitly analytically continue \(\overline{\varphi (\alpha ,y')}\). In particular, notice that since on the support of the integrand, \(\left| |\alpha _{\xi }|_g-1\right| \ll 1\), and \({\xi }_c'=\alpha _{\xi '}+O(|\alpha _{x'}-y_c'|)\), we have that \(||{\xi }_c'|_g-1|\ll 1\) and hence the fact that the amplitude is not analytic in \({\xi }'\) near \({\xi }'=0\), does not cause issues in the analytic stationary phase argument.
Consequently, the \((2 \pi h)^{-{{n}}}\) factor in front of (3.5) gets killed upon application of analytic stationary phase in \((y',\eta ')\) and we get the bound
Substitution of (3.7) together with the weighted \(L^2\) estimate \(\Vert e^{\psi /h} T(h) \varphi _h \Vert _{L^2} = O(1)\) with \(\psi = \gamma p^2, \, \gamma < \frac{1}{2}\) (see Theorem 3 (1.13) ) in (3.6) gives, by Cauchy–Schwarz,
Now, recall that we may work with \(T_{hol}\) and \(S_{hol}\) since for some \(\varepsilon >0\),
In this case the analytic continuation, of \(-\overline{\varphi }\) is given by
Computing in normal geodesic coordinates centered at \(\alpha _x\), observe that
Therefore,
Now, by Taylor expansion in \({\xi '}\),
so,
Written out explicitly, the last line in (3.8) says that for \(\gamma <1/2\),
Now, recall that \( {{\text {Im}}\,}y_c'=O(x_{{n+1}})\) and \(\partial _{\alpha }y_c'=O(x_{{n+1}})\) and hence applying the method of steepest descent, in the \(\alpha _{x'},r\) variables, the critical point of the phase occurs at
Using (3.9), we have that at the stationary point
Putting this into (3.10) gives
Now,
So, the critical point occurs at
Noting that \({{\text {Im}}\,}y_c'=x_{{n+1}}+O(x_{{n+1}}^2)\) and evaluating the exponential at this point yields
The same argument works for derivatives with each differentiation creating a power of \(h^{-1}.\)\(\square \)
Remark 3.1
We note that the first order bound with \(d(x) = d_{\partial \Omega }(x) + O( d_{\partial \Omega }^2(x))\) in Theorem 1 (1.3) follows from the weighted \(L^2\)-bound in Theorem 2 by essentially the same argument as above. The proof in that case is slightly simpler since one need only keep track of the analytic stationary phase computations to \(O(x_{{n+1}}^2)\) error (rather than \(O(x_{{n+1}}^3)\)).
4 Microlocal Estimates Near the Zero Section
While the Dirichlet-to-Neumann map is a homogeneous analytic pseudodifferential operator, it is not well behaved as a semiclassical analytic pseudodifferential operator near the zero section. In particular, when semiclassically rescaled, the full symbol of a homogeneous pseudodifferential operator, a, typically has singularities of the form
Apriori, this singular h dependence may result in the transport of semiclassical analytic wavefront sets away from the zero section under the action of a (homogeneous) pseudodifferential operator. The purpose of this section is to show that no such transport occurs and then to use this information to estimate Steklov eigenfunctions near the zero section.
4.1 Cauchy Estimates and the Euclidean FBI Transform
For \(0\le \tilde{h}\le h\), let
Let also
Then for all \(u\in \mathcal {S}(\mathbb {R}^{{n}})\), \(u=S_{\tilde{h}}T_{\tilde{h}}u\).
Define
The next proposition is very similar to [8, Proposition 2.1]. The difference is that we do not require \(\mu \ge 1\). Instead, we keep track of the dependence of various estimates on \(\mu \ge \mu _0\).
Proposition 4.1
Suppose that a has tempered growth in \((x,y,\xi )\) and \(0<\mu _0\le \mu <\mu _1\). Then suppose that for W a neighborhood of \((x_0,\xi _0)\),
Then there is a neighborhood V of \((x_0,\xi _0)\)
Proof Sketch
In order to prove the lemma, one writes \(T'_{a,\mu ,\tilde{h}}u=T'_{a,\mu ,\tilde{h}}S_{\tilde{h}}T_{\tilde{h}}u\). Then, one can easily estimate the kernel of \(T'_{a,\mu ,\tilde{h}}S_{\tilde{h}}\) using a simple change of variables. \(\square \)
The next proposition is similar to [8, Proposition 2.2], except that we obtain a quantitative estimate on distances to the zero section.
Proposition 4.2
Fix \(x_0\in M\), \(\varepsilon >0\). Let X be a neighborhood of \(x_0\) in M. Suppose that \(\Vert u\Vert _{L^\infty ({X})}\le Ch^{-N}\). Then the following are equivalent:
-
there exist \(C,c>0\), \(h_0>0\) and an \(\varepsilon \)-independent neighborhood, W, of \(x_0\) so that for every \(0\le \tilde{h}\le h\le h_0\),
$$\begin{aligned} |T_{\tilde{h}}u(x,\xi ,h)|\le Ce^{-c /\tilde{h}}\Vert u\Vert _{L^\infty (X)},\quad x\in W ,\,|\xi |\ge \varepsilon , \end{aligned}$$(4.1) -
there exists \(C_1>0\), \(h_0>0\) and an \(\varepsilon \)-independent neighborhood, W, of \(x_0\) and constant \(c_1>0\) so that for \(0<h<h_0\)
$$\begin{aligned} \sup _\mathrm{Rex\in W,\,|{{\text {Im}}\,}x|\le c_1}|u|\le C_1e^{\frac{\varepsilon }{2h}}{\Vert u\Vert _{L^\infty (X)}} \end{aligned}$$(4.2) -
there exists \(C>0\), \({h_0>0}\) and an \(\varepsilon \)-independent neighborhood, W, of \(x_0\) so that for \(0<h<h_0\)
$$\begin{aligned} |(hD)^\alpha u{(x)}|\le C\Vert u\Vert _{L^\infty (X)}C^{|\alpha |}(h|\alpha |+\varepsilon )^{|\alpha |},\quad x\in W. \end{aligned}$$(4.3)
Proof Sketch
The fact that (4.2) implies (4.3) follows from basic Cauchy estimates, while that (4.3) implies (4.2) follows from writing out the Taylor formula with remainder.
The equivalence of (4.2) and (4.1) follows from writing down a resolution of the identity in terms of the FBI transform and deforming the contour into the complex domain. \(\square \)
The next estimate proves that the application of an analytic homogeneous pseudodifferential operator preserves estimates of the form (4.3). Roughly speaking, we show that the Sobolev mapping properties of such an operator behave like the Sobolev mapping properties of multiplication by an h-independent analytic function. Throughout the proof of the next proposition, we will use the following elementary estimates without comment
Proposition 4.3
Let P be a homogeneous analytic pseudodifferential operator of order k. Suppose that u satisfies (4.3) in a neighborhood, U of \(x_0\) with some constant C. Then there exists a neighborhood, W of \(x_0\) so that Pu satisfies
Proof
Let \(\chi \in C_c(\mathbb {R})\) with \(\chi \equiv 1\) on \([-1,1]\) and \({{\text {supp}}}\chi \subset [-2,2]\). Then for any \(\delta >0\), the kernel of P is given by \(K_\delta (x,y)+R_\delta (x,y)\) where \(R_\delta \) is real analytic and
with \({{p}}\in S^k_{ha}\).
The kernel of \(\partial _x^\beta P\) is given by \(\partial _x^\beta K+\partial _x^\beta R_\delta \). Since \(R_\delta \) is analytic,
so we need only consider \(\partial _x^\beta K_\delta \).
Observe that
Deforming the contour in \(\xi \) to
for some \(R>0\), we can, modulo an analytic error, replace the kernel by
Let \(\partial _x^\beta \tilde{P}\) be the operator with kernel as in (4.6). Then, let \(\psi \in C_c^\infty (M)\) have support on a neighborhood, W of \(x_0\) so that \(W\subset U\) and \(d(W,\partial U)\ge 2\delta \) with U as in the statement of the proposition. By [29, Theorem 4.23] there exists \(N>0\) a constant (independent of \(h,\beta ,P\)) so that
Relabeling \(|\beta |=|\beta |+{{n}}\) gives the desired estimates using Sobolev embeddings. The result follows from taking \(\psi \equiv 1\) on a slightly smaller neighborhood \(W'\subset W\). \(\square \)
Let \(\chi \in C_c^\infty (\mathbb {R})\) with \({{\text {supp}}}\chi \subset [-1,1]\), \(\varepsilon >0\), and define
Proposition 4.4
Let \(\chi \in C_c^\infty (\mathbb {R})\) with \({{\text {supp}}}\chi \subset [-1,1]\). Then for for all \(\varepsilon >0\), \(|\xi |\ge 2\varepsilon \), and \(0<\tilde{h}\le h\),
Proof
The kernel of \(T_{\tilde{h}}S_h^{\varepsilon }\) is given by
In particular, letting \(\mu =\frac{\tilde{h}}{h}\),
where
Shifting contours shows that
On \({{\text {supp}}}\chi (\varepsilon ^{-1}|\eta |)\), \(|\eta |\le \varepsilon \) and \(|\mu \eta |\le \varepsilon \). Therefore, taking \(|\xi |\ge 2\varepsilon \) proves the lemma. \(\square \)
Our next proposition is the key proposition for this section and shows that homogeneous analytic pseudodifferential operators do not transport semiclassical analytic wavefront sets away from the zero section.
We say that u is compactly microlocalized if there exists \(\chi \in C_c^\infty (\mathbb {R})\) so that for some global FBI transform \(\tilde{T}\),
Proposition 4.5
Let M be a compact, real-analytic manifold, and P be a homogeneous analytic pseudodifferential operator. Let \(\tilde{T}\) be a global FBI transform on M with left-parametrix \(\tilde{S}\). Let \(\chi _0,\chi _1\in C_c^\infty (\mathbb {R})\) have \(\chi _0\equiv 1\) on \([-1,1]\) with \({{\text {supp}}}\chi _0\subset [-2,2]\) and \(\chi _1\equiv 1\) on \([-{{N}},{{N}}]\) with \({{\text {supp}}}\chi _1\subset [-2{{N}},2{{N}}]\). Then, for \(\varepsilon >0\) small enough and \({{N}}>0\) large enough, and u compactly microlocalized, there exists \(c>0\) so that
Proof
Notice that
is microlocally vanishing for \(|\xi |\ge 2\varepsilon \). Therefore, for \(\psi \) with small enough support so that the Euclidean FBI transforms are well-defined,
Now, by Proposition 4.4, (4.1) holds for the image of \(S_h\chi _0((2\varepsilon )^{-1}|\xi |)\). In particular, Lemma 4.3 applies and hence taking a partition of unity, \(Q\tilde{S}\chi _0(\varepsilon ^{-1}|\alpha _\xi |_{\alpha _x}))\tilde{T}u\) satisfies (4.3) and hence, for any \(\tilde{\psi }\) supported in a small enough region so that the Euclidean FBI transform is well-defined,
is microlocally vanishing on \(|\xi |\ge 2\varepsilon ^{-1}\) and since this is independent of the choice of FBI transform, taking M large enough gives
Taking a partition of unity then proves the lemma. \(\square \)
4.2 Application to Eigenvalue Problems
Let Q be an (h-independent) homogeneous, elliptic, classical analytic pseudodifferential operator of order \(k>0\). Let \(\varphi _h\) denote a solution to
We are now in a position to analyze the behavior of \(\varphi _h\) near \(|\xi |=0\).
Notice that for any \(\tilde{\chi }\in C_c^\infty (\mathbb {R}^d)\) with \(\tilde{\chi }\equiv 1\) near 0, and \({{\text {supp}}}\tilde{\chi }\subset B(0,1)\), \((1-\tilde{\chi }(hD))h^kQ\) is a semiclassical pseudodifferential operator whose symbol is analytic in the interior of \(\{\tilde{\chi }= 0\}\). Thus, since Q is elliptic,
and hence by Proposition 2.3 for \(\chi \in C_c^\infty (\mathbb {R})\) with \(\chi \equiv 1\) on \([-2,2]\), for any \(\delta >0\), there exists \(c>0\) so that
In the next proposition, we estimate the eigenfunctions \(\varphi _h\) near the zero section by their norm in a small annulus around the zero section. In particular, this shows that the cutoff \((1-\chi (\delta ^{-1}|\alpha _\xi |_{\alpha _x}))\) can be removed from (4.7).
Proposition 4.6
Let \(\varphi _h\) be a solution to
For \(\varepsilon >0\), let \(\chi _\varepsilon =\chi (\varepsilon ^{-1}|\alpha _{\xi }|_{\alpha _x}).\) Then for \(\varepsilon >0\) small enough, there exists \(c>0\) so that
Proof
We first use the apriori estimate (4.7) together with Proposition 4.5 to decompose the eigenfunction equation (4.8) into two (essentially) orthogonal pieces—one supported near and one supported away from the zero section. We are then able to use a simple Neumann series argument to obtain estimates near the zero section.
Fix \(N>0\) to be chosen large enough later.
In (4.9), we used (4.7) to see that for any \(\psi \) with
In (4.10), we choose N large enough so that Proposition 4.5 implies
In order to invert \((\chi _{2^{N}\varepsilon }T_{geo}h^kQS_{geo}\chi _{2\varepsilon }-1)\) by Neumann series, we obtain estimates on \(\chi _{2^{N}\varepsilon }T_{geo}h^kQS_{geo}\chi _{2\varepsilon }\). Since \(S_{geo}\chi _{2\varepsilon }T_{geo}\) is the anti-Wick quantization of \(\chi _{2\varepsilon }\), it is a pseudodifferential operator with symbol \(\chi _{2\varepsilon }(\alpha ),\)
Therefore, \(\Vert Q\Vert _{H^k\rightarrow L^2}\le C\) implies
In particular,
Hence, applying (4.11) on the left of (4.10)
and, multiplying by \(\chi _{\varepsilon /2}\) on the left, we have
\(\square \)
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Acknowledgements
The authors would like to thank Iosif Polterovich and Steve Zelditch for their comments on an earlier version of this paper. Thanks also to Andras Vasy and Maciej Zworski for valuable suggestions. Finally, thanks to the anonymous referee for many helpful suggestions. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661. The research of J.T. was partially supported by NSERC Discovery Grant # OGP0170280 and an FRQNT Team Grant. J.T. was also supported by the French National Research Agency project Gerasic-ANR- 13-BS01-0007-0.
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Galkowski, J., Toth, J.A. Pointwise Bounds for Steklov Eigenfunctions. J Geom Anal 29, 142–193 (2019). https://doi.org/10.1007/s12220-018-9984-7
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DOI: https://doi.org/10.1007/s12220-018-9984-7