Pseudodifferential operators on ultra-modulation spaces

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

We consider pseudo-differential operators and Gabor frames on locally compact commutative Hausdorff groups. Let [Formula: see text] be an open subgroup of a locally compact commutative Hausdorff group [Formula: see text] and [Formula: see text] be a topological isomorphism, then we define the cross [Formula: see text]-ambiguity function [Formula: see text] and cross [Formula: see text]-Wigner function [Formula: see text] and a pseudo-differentiable operator [Formula: see text] with the symbol [Formula: see text]. Employing properties of [Formula: see text], we establish characteristics of pseudo-differentiable operators. Also, assume that [Formula: see text] and [Formula: see text], then the coefficient operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text]; and similarly, the adjoint operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text].

We prove optimal regularity results in Lp-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a (0<a<1) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian (-Δ)a, as well as elliptic operators (-∇·A(x)∇+b(x))a. The proofs are based on general results on maximal Lp-regularity and its relation to R-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Let \mathsf{G} be a unimodular type I second countable locally compact group and let \hat{\mathsf{G}} be its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on \mathsf{G}\times\hat{\mathsf{G}} , and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products C^\ast -algebras associated to certain C^\ast -dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-valued symbols.

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey–Stewartson II equation. We then use it to prove global well-posedness and scattering in $$L^2$$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderon in dimension 2, for conductivities $$\sigma >0$$ with $$\log \sigma \in \dot{H}^1$$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $$L^2$$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable is , where A is a finitely many orders smoothing linear hypoelliptic operator and is the noise magnitude. The covariance operator CU of U is smoothing of order , self-adjoint, injective and elliptic pseudodifferential operator. If was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem However, Gaussian white noise does not take values in L2 but in where is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

0. Introduction. 1. General Calculus of Pseudodifferential Operators. 2. Model Classes of Degenerate Elliptic Differential Operators. 3. General Classes of Degenerate Elliptic Differential Operators. 4. Degenerate Elliptic Operators in Non--Power--Like Degeneration Case. 5. Lp-Theory for Degenerate Elliptic Operators. 6. Coerciveness of Degenerate Quadratic Forms. 7. Some Classes of Hypoelliptic Pseudodifferential Operators on a Closed Manifold. 8. Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary. 9. General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations. 10. Spectral Asymptotics of Degenerate Elliptic Operators. 11. Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics. A Brief Review of the Bibliography. Bibliography. Index of Notations. Index.

Analytic K-theory on manifolds with corners

The paper deals with mixed boundary value problems and transmission problems for elliptic differential and pseudo-differential operators. These problems are considered as elements of an algebra (where Ω is the considered domain, x=∂Ω, Y⊂X a submanifold of codimension 1, where the conditions have a jump, and x∂ is a certain compactification of X/Y), the elements of which are described on a symbolic level. By means of additional conditions (of trace and potential type, with respect to Y) a given mixed problem is extended to an operator , and the Fredholm property of is established under an ellipticity condition, the parametrix is constructed on symbolic level and ind is expressed. In the second part the authors also study classes of special cases, namely differential operators of second and higher order with mixed boundary conditions for dim Ω≧2, where explicit index expressions are derived.

Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces

The main goal of this paper is to study properties of the linear canonical transform (LCT) on Schwartz-type space [Formula: see text]. The symbol class [Formula: see text] is introduced. The pseudo-differential operator (p.d.o.) involving LCT is defined and also some of its properties including boundedness are investigated in Sobolev-type space. Kernel and integral representation of p.d.o. are obtained. Some applications of LCT to generalized partial differential equations and canonical convolution integral equation have been solved.

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.

In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations.For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a survey of these methods).We consider in this paper just one of these methods. It is based on a study of the structure of the fundamental solution of the Cauchy problem for a hyperbolic differential (pseudodifferential) equation. The method we use is called “the method of geometrical optics”. For a system of first order differential equations it was originally developed in detail by Lax [8], and for pseudodifferential equations by Hörmander [2] and independently by Eskin [17], [18] and Maslov [19].In [2] Hörmander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley [5] one can then derive the asymptotic behaviour of the spectral function of an elliptic differential operator of arbitrary order.Similar methods have previously been applied by the author in [3], [4] for second order elliptic differential operators.In this article we give a partial account of the results of [8] and [2]. We also present some new results due to the author, concerning both the structure of the fundamental solution of the Cauchy problem and the asymptotic behaviour of the spectral function.