Schwartz Kernel Research Articles

Understanding of linear operators through Wigner analysis

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S∈Sp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.

Propagation of anisotropic Gabor wave front sets

Abstract We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

BC-system, absolute cyclotomy and the quantized calculus

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the “zeta sector” of the latter space as the Scaling Site. The new result that we present concerns the description of the BC-system as the universal Witt ring (i.e., K -theory of endomorphisms) of the “algebraic closure” of the absolute base \mathbb{S} . In this way we attain a conceptual meaning of the BC dynamical system at the most basic algebraic level. Furthermore, we define an invariant of Schwartz kernels in one dimension and relate the Fourier transform (in one dimension) to its role over the algebraic closure of \mathbb{S} . We implement this invariant to prove that, when applied to the quantized differential of a function, it provides its Schwarzian derivative. Finally, we survey the roles of the quantized calculus in relation to Weil's positivity, and that of spectral triples in relation to the zeros of the Riemann zeta function.

Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case

Abstract We consider the solutions of the Klein–Gordon equation in the 2 + 1 2{+}1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ {\mathbb{R}_{t}\times\mathbb{R}_{+}\times\mathbb{R}/2\pi\alpha\mathbb{Z}} created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone.

Diffraction for the Dirac–Coulomb Propagator

The Dirac equation in $$\mathbb {R}^{1,3}$$ with potential $${\textsf{Z}}/r$$ is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is $$1-{\epsilon }$$ derivatives smoother than the main singularities, for all $${\epsilon }>0,$$ and is a conormal singularity.

Propagation of anisotropic Gelfand–Shilov wave front sets

We show a result on propagation of the anisotropic Gelfand–Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand–Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand–Shilov wave front set of the Schwartz kernel of the operator is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.

Resolvents and complex powers of semiclassical cone operators

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 2 Δ g + 1 $h^2\Delta _g+1$ on a manifold ( X , g ) $(X,g)$ of dimension n ≥ 3 $n\ge 3$ 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 × X × X $[0,1)_h\times X\times X$ of h-dependent integral kernels; the construction of complex powers relies on a calculus with a second semiclassical parameter. As an application, we characterize the domains of ( h 2 Δ g + 1 ) w / 2 ${\big (h^2\Delta _g+1\big )}^{w/2}$ for Re w ∈ − n 2 , n 2 $\operatorname{Re}w\in \left(-\tfrac{n}{2},\tfrac{n}{2}\right)$ and use this to prove the propagation of semiclassical regularity through a cone point on a range of weighted semiclassical function spaces.

Dynamical residues of Lorentzian spectral zeta functions

We define a dynamical residue which generalizes the Guillemin–Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott–Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.

The inner kernel theorem for a certain Segal algebra

The Segal algebra {mathbf{S}}_{0}(G) is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups G. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from {mathbf{S}}_{0}(G_1) to {mathbf{S}}'_{0}(G_2) by generalized functions in {mathbf{S}}'_{0}(G_1 times G_2) the “outer kernel theorem”. The main subject of this paper is to formulate what we call the “inner kernel theorem”. This is the characterization of those linear operators that have kernels in {mathbf{S}}_{0}(G_1 times G_2). Such operators are regularizing—in the sense that they map {mathbf{S}}'_{0}(G_1) into {mathbf{S}}_{0}(G_2) in a w^{*} to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies “integrate” to a Dirac delta distribution in a mathematically justifiable way.

Integral representations of isotropic semiclassical functions and applications

In a previous paper, we introduced a class of “semiclassical functions of isotropic type,” starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semiclassical and microlocal analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then, we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual cleanintersection hypothesis). The simplest examples of isotropic states are the “coherent states,” a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920’s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact.We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5).

Equivalent Norms for Modulation Spaces from Positive Cohen\u2019s Class Distributions

We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert–Schmidt operator. The main result is applied to Cohen’s class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen’s class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert–Schmidt operator must satisfy a nuclearity condition for these results to hold, and we investigate this condition in detail.

Decay and Strichartz estimates in critical electromagnetic fields

We study the L1→L∞-decay estimates for the Klein-Gordon equation in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the Schwartz kernels of the spectral measure and heat propagator of the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and prove that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the Lp-regularity property of wave propagation in the same setting.

A Sharp Version of Price\u2019s Law for Wave Decay on Asymptotically Flat Spacetimes

We prove Price’s law with an explicit leading order term for solutions phi (t,x) of the scalar wave equation on a class of stationary asymptotically flat (3+1)-dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that phi (t,x)=c t^{-3}+{mathcal {O}}(t^{-4+}) for bounded |x|, where cin {mathbb {C}} is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise t^{-2 l-3} decay for waves with angular frequency at least l, and t^{-2 l-4} decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

Resolvent and spectral measure for Schrödinger operators on flat Euclidean cones

We construct the Schwartz kernel of resolvent and spectral measure for Schrödinger operators on the flat Euclidean cone (X,g), where X=C(Sσ1)=(0,∞)×Sσ1 is a product cone over the circle, Sσ1=R/2πσZ, with radius σ>0 and the metric g=dr2+r2dθ2. As products, we prove the dispersive estimates for the Schrödinger and half-wave propagators in this setting.

Spectral density for the Schrödinger operator with magnetic field in the unit complex ball: solutions of evolutionary equations and applications to special functions

The Schwartz kernel of the spectral density for the Schrodinger operator with magnetic field in the n-dimensional complex ball is given. As applications, we compute the heat, resolvent and the wave kernels. Moreover, the resolvent and wave kernels are used to establish two new formulas for the Gauss-hypergeometric function.

The convolution algebra of Schwartz kernels along a singular foliation

Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized functions on the underlying manifold, and on the leaves and their holonomy covers. This generalizes Schwartz kernel operators to singular foliations. We also define the algebra of smoothing operators in this context and prove that it is a two-sided ideal.

Distributions of anisotropic order and applications to H-distributions

We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.

Sharp Resolvent Estimates on Non-positively Curved Asymptotically Hyperbolic Manifolds

Abstract We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHMs). In the literature, estimates of $R(\lambda )$ on weighted Sobolev spaces of the order $O(|\lambda |^{N})$ were established for some $N> -1$, $|\lambda |$ large, and $\lambda \in{{\mathbb{C}}}$ in strips where $R(\lambda )$ is holomorphic. In this work, we prove the optimal bound $O(|\lambda |^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.

On Jacobi polynomials $${\mathscr {P}}_{k}^{(\alpha ,\beta )}$$ and coefficients $$c_{j}^{\ell }(\alpha ,\beta )$$ $$\left( k\ge 0, \ell =5,6; 1\le j\le \ell ; \alpha ,\beta > -1\right)$$

In their paper, Awonusika and Taheri proved a spectral identity that relates the even ( $$2\ell$$ )th, $$\ell \ge 1$$ , derivatives of Jacobi polynomials $$P_{k}^{(\alpha ,\beta )}(\cos \theta )$$ , $$k\ge 0; \alpha ,\beta >-1$$ (evaluated at $$\theta =0$$ ), to the $$\ell$$ th-degree polynomials $${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k}\right) :=R_{\ell}( \lambda _{k}^{\alpha ,\beta })$$ with constant coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ ( $$1\le j\le \ell$$ ; $$\alpha ,\beta >-1$$ ), called Jacobi coefficients; the numbers $$\lambda _{k}^{\alpha ,\beta }=k(k+\alpha +\beta +1)$$ ( $$k\ge 0$$ ) are the eigenvalues of the associated Jacobi operator. These Jacobi coefficients appear in the Maclaurin spectral expansion of the Schwartz kernels of functions of the Laplacian on rank one compact symmetric spaces. The first Jacobi coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ ( $$1\le j\le \ell \le 4$$ ) were computed using the qth, $$1\le q\le \ell ,$$ derivative formula for Jacobi polynomials $$P_{k}^{(\alpha ,\beta )}(t)$$ (evaluated at $$t=1$$ ) and the basic properties of the Gamma function. In this paper, we use the Jacobi differential equation to generate a recursion formula that explicitly computes the polynomials $${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k} \right)$$ and the coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ . To illustrate the recursion formula we compute the higher coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ for $$\ell =5,6;1\le j\le \ell$$ . Remarkably, the local Jacobi coefficients $${\mathsf {c}}^{q}_{j}(\alpha ,\beta )$$ $$(1\le j\le q\le \ell )$$ (which appear in the computation of $$c^{\ell }_{j}(\alpha ,\beta )$$ ) coincide with the Jacobi–Stirling numbers of the first kind $${{\mathsf {j}}}{{\mathsf {s}}}_{j}^{q}(\alpha ,\beta )$$ , and this is a new phenomenon in the Maclaurin spectral analysis of the Laplacian on rank one compact symmetric spaces. The Jacobi coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ are useful in the description of constants appearing in the power series expansion of any spectral functions involving Jacobi polynomials.

Lagrangian Distributions and Fourier Integral Operators with Quadratic Phase Functions and Shubin Amplitudes

We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as the composition of a Weyl pseudodifferential operator and a metaplectic operator and derive a characterization of its Schwartz kernel in terms of phase space estimates. Extending the conormal distributions in the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.