We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schrödinger operators $P=Δ+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null space and no resonance at 0. In particular, we show that the Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a desingularized version of $M\times M\times [0,1]$. Using this, we show that the Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that this range is optimal if $V$ is not identically zero or if $M$ has more than one end. We also analyze the case V=0 with one end. In a follow-up paper, we shall deal with the same problem in the presence of zero modes and zero-resonances.
On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+δ$ and $Δ_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+δ)Δ_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian…
Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform of the operator $P = Δ_g + V$, where $Δ_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function $V$ that is smooth on $M$ and vanishes …
We give resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations, using a uniform microlocal Fredholm analysis framework.
In this article, we shall construct the resolvent of Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.
We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.
We investigate the $L^p$-boundness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criteria for the $L^p$-unboundness of the Riesz transform are given. We recover known results about manifolds that are Euclidean or conical at infinity.
This paper is a continuation of the investigation of resolvents of elliptic operators on conic manifolds from math.AP/0410178 and math.AP/0410176 to the case of manifolds with boundary and realizations of operators under boundary conditions. We prove the existence of sectors of minimal growth for realizations of boundary value problems for cone operators under natural ellipticity conditions on the symbols. Special attention is devoted to the clarification of the analytic structure of the resolvent.
On a class of asymptotically conical manifolds, we prove two types of low frequency estimates for the resolvent of the Laplace-Beltrami operator. The first result is a uniform $ L^2 \rightarrow L^2 $ bound for $ \langle r \rangle^{-1} (- Δ_G - z)^{-1} \langle r \rangle^{-1} $ when $ \mbox{Re}(z) $ is small, with the optimal weight $ \langle r \rangle^{-1} $. The second one is about powers of the resolvent. For any integer $N$, we prove uniform $ L^2 \rightarrow L^2 $ bounds for $ \langle εr \rangle^{-N} (-ε^{-2} Δ_G - Z)^{-N} \langle εr \rangle^{-N} $ when $ \mbox{Re}(Z) $ belongs to a compact…
Let X be a compact manifold with boundary, and g a scattering metric on X, which may be either of short range or `gravitational' long range type. Thus, g gives X the geometric structure of a complete manifold with an asymptotically conic end. Let H be an operator of the form $H = Δ+ P$, where $Δ$ is the Laplacian with respect to g and P is a self-adjoint first order scattering differential operator with coefficients vanishing at the boundary of X and satisfying a `gravitational' condition. We define a symbol calculus for Legendre distributions on manifolds with codimension two corners and use …
We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a "very weak" Huygens' principle, which may be of independent interest. As applications of the estimate, we obtain…
We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.
We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (λ-A)^{-1} exists in a sector of the complex plane and decays like 1/|λ| as |λ| tends to infinity. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal r…
We consider a class of manifolds $\mathcal{M}$ obtained by taking the connected sum of a finite number of $N$-dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, δ) \times (\mathcal{M}_i, g)$, where $\mathcal{M}_i$ is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions $n_i$ are not all equal. This means that the ends have different `asymptotic dimension', and implies that the Riemannian manifold $\mathcal{M}$ is not a doubling space. In the first paper in this series, by the first and third authors, we considered the case wh…
We investigate $L^p$ boundedness of the maximal Bochner-Riesz means for self-adjoint operators of elliptic type. Assuming the finite speed of propagation for the associated wave operator, from the restriction type estimates we establish the sharp $L^p$ boundedness of the maximal Bochner-Riesz means for the elliptic operators. As applications, we obtain the sharp $L^p$ maximal bounds for the Schrödinger operators on asymptotically conic manifolds, the harmonic oscillator and its perturbations or elliptic operators on compact manifolds.
In this paper we are concerned with resolvent estimates for the Laplacian $Δ$ in Euclidean spaces. Uniform resolvent estimates for $Δ$ were shown by Kenig, Ruiz and Sogge \cite{KRS} who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $L^p$--$L^q$ bounds depending on $z$ has not been considered in a general framework which admits all possible $p,q$. In this paper, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$. We also obtain the sharp resolvent estimates for the f…
We consider the Laplacian $Δ$ on an asymptotically hyperbolic manifold $X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectral measure for the operator $(Δ- n^2/4)_+^{1/2}$ on such manifolds, under the assumptions that $X$ is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy (high energy estimates for a perturbation of the hyperbolic metric) and the present authors (see also the work by Yiran W…
Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $Δ$ be the positive Laplacian associated to $g$, and $P = Δ+ V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(λ) = (λ/πi) \big(R(λ+i0) - R(λ- i0) \big)$ of $P_+^{1/2}$, where $R(λ) = (P - λ^2)^{-1}$, as $λ\to 0$, in a manner similar to that done previously by the second author an…
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian $h^2Δ_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space $[0,1)_h\times X\times X$ of $h$-dependent integral kernels; …
We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L^p-spaces over B, 1 < p < \infty. Under suitable ellipticity assumptions we can define a family of complex powers A^z. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations.
For certain Dirac operators $ð_ϕ$ associated to a fibered boundary metric $g_ϕ$, we provide a pseudodifferential characterization of the limiting behavior of $(ð_ϕ+kγ)^{-1}$ as $k\searrow 0$, where $γ$ is a self-adjoint operator anti-commuting with $ð_ϕ$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $ð_ϕ^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give …
We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to differential boundary conditions $T$. In general, these conditions ask for a particular pseudodifferential structure of the resolvent $(λ-\underline{A}_T)^{-1}$ in a sector $Λ\subset\mathbf{C}$. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem $(A,T)$. Examples concern the Dirichlet and Neumann Laplacians.
We consider Muckenhoupt weights $w$, and define weighted Hardy spaces $H^p_{\mathcal{T}}(w)$, where $\mathcal{T}$ denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second order divergence form elliptic operator $L$. In the range $0<p< 1$, we give a molecular characterization of these spaces. Additionally, in the range $p\in \mathcal{W}_w(p_-(L),p_+(L))$ we see that these spaces are isomorphic to the $L^p(w)$ spaces. We also consider the Riesz transform $\nabla L^{-\frac{1}{2}}$, associated with $L$, and show that…
In this paper, we investigate the long-time structure of the heat kernel on a Riemannian manifold M which is asymptotically conic near infinity. Using geometric microlocal analysis and building on results of Guillarmou and Hassell on the low-energy resolvent, we give a complete description of the asymptotic structure of the heat kernel in all spatial and temporal regimes. We apply this structure to define and investigate a renormalized zeta function and determinant of the Laplacian on M.
We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to $(p,2)$ restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the Bochner-Riesz summability. We also observe that this restriction estimate holds for operators satisfying dispersiv…
Consider a compact manifold with boundary $M$ with a scattering metric $g$ or, equivalently, an asymptotically conic manifold $(M^\circ, g)$. (Euclidean $\mathbb{R}^n$, with a compactly supported metric perturbation, is an example of such a space.) Let $Δ$ be the positive Laplacian on $(M,g)$, and $V$ a smooth potential on $M$ which decays to second order at infinity. In this paper we construct the kernel of the operator $(h^2 Δ+ V - (λ_0 \pm i0)^2)^{-1}$, at a nontrapping energy $λ_0 > 0$, uniformly for $h \in (0, h_0)$, $h_0 > 0$ small, within a class of Legendre distributions on manifolds w…
The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(λ)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $λ^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setti…
We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincaré operator), as a map on the boundary surface $Γ$ of a domain in $\mathbb{R}^3$ with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across $Γ$. In particular, if the domain is understood as an inclusion with complex permittivity $ε$, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when $(ε+1)/(ε-1)$ belongs to the resolvent set in energy norm. …
Consider the metric cone $X=C(Y)=(0,\infty)_r\times Y$ with the metric $g=\mathrm{d}r^2+r^2h$ where the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. Let $Δ_g$ be the Friedrich extension positive Laplacian on $X$ and let $Δ_h$ be the positive Laplacian on $Y$, and consider the operator $\LL_V=Δ_g+V_0 r^{-2}$ where $V_0\in\CC^\infty(Y)$ such that $Δ_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. In this paper, we prove the global-in-time Strichartz estimates without loss for the wave equation associated with the operator $\LL_V$ which verifies\ci…
Local time decay for fractional Schrödinger operators with slowly decaying potentials and a weaker Agmon type estimate in a classically forbidden region
We study energy propagation along line-defects (edges) in 2D continuous, energy preserving periodic media. The unperturbed medium (bulk) is modeled by a honeycomb Schroedinger operator, which is periodic with respect to the triangular lattice, invariant under parity, P, and complex-conjugation, C. A honeycomb operator has Dirac points: two dispersion surfaces touch conically at an energy level, $E_D$ [25,27]. Periodic perturbations which break P or C open a gap in the essential spectrum about energy $E_D$. Such operators model an insulator near energy $E_D$. Our edge operator is a small pertur…
