Sobolev Estimates for Fractional and Singular Radon Transforms

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.

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1<p<\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0<p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1<p<\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0<p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

The fundamental solution of the Cauchy problem is given in the form of Fourier integral operator, and when the problem is not C∞ well-posed, Cattabriga-Zanghirati2 have proved that the symbols of Fourier integral operators in the fundamental solution are not able to be estimated by finite order but able to be estimated by infinite order of the following type $$\begin{array}{*{20}{c}} {C\exp [c{{\xi }^{{1/\kappa }}}],} & {c > 0.} \\ \end{array}$$ (1) These symbols are estimeted from above by (1) for some constant tc satisfying k >1. Foureir integral operators and pseudo-differential operators with symbols estimated by (1) are investigated in Zanghirati26, Rodino-Zanghirati16, Cattabriga-Zanghirati2 and Aokil, and it is proved that they have the following properties: i) They map the ultradistributions into themselves; ii) pseudo-differential operators with symbols estimated from above by (1) make algebras; and iii) they propagate Gevrey wave front sets canonically.

We discuss the recent achievements in the application of asymptotic methods based on Fourier Integral Operators (FIOs) for inversion and direct modeling of radio occultations. We show that FIOs can be derived from the first principles: stationary phase principle and energy conservation. We discuss accurate and approximate solutions for the kernel of the FIOs. The approximations can be used for designing very efficient numerical algorithms, where the FIOs are reduced to a composition of multiplying with reference signals and Fourier transforms. Another application is inversion algorithms using an FIO that retrieves the geometric optical ray structure of wave fields. Asymptotic methods of forward modeling are based on inverse FIOs that map the geometric optical ray structure to wave fields. Such algorithms are very fast and significantly reduce numerical inaccuracies, which arise in computation of multiple diffractive integrals.

Fourier integral operators (FIOs) are used for constructing asymptotic solutions of wave problems and for the generalization of the geometrical optics. Geometric optical rays are described by the canonical Hamilton system, which can be written in different canonical coordinates in the phase space. The theory of FIOs generalizes the formalism of canonical transforms for solving wave problems. The FIO associated with a canonical transform maps the wave field to a different representation. Mapping to the representation of ray impact parameter was used in the formulation of the canonical transform (CT) method for processing radio occultation data. The full‐spectrum inversion (FSI) method can also be looked at as an FIO associated with a canonical transform of a different type. We discuss the general principles of the theory of FIOs and formulate a generalization of the CT and FSI techniques. We derive the FIO that maps radio occultation data measured along the low Earth orbiter orbit without first applying back propagation. This operator is used for the retrieval of refraction angles and atmospheric absorption. We give a closed derivation of the exact phase function of the FIO obtained in the “phase matching” approach by Jensen et al. [2004] We derive a novel FIO algorithm denoted CT2, which is a modification and improvement of FSI. We discuss the use of FIOs for asymptotic direct modeling of radio occultation data. This direct model is numerically much faster then the multiple phase screen technique. This is especially useful for simulating LEO‐LEO occultations at frequencies of 10–30 GHz.

This paper addresses the cone transform that integrates a function over the surfaces of circular cones. We derive an inversion formula for the cone transform in n-dimensional Euclidean space for cones having a fixed central axis direction, but having all opening angles and vertices. We also show that the cone transform and its dual are Fourier integral operators, and the normal operator is a pseudo-differential operator. We finally present a differential equation that the cone transform satisfies, which can be seen as a first step in describing the range of the cone transform.

I am grateful to Lars Hormander and to my collaborator, Hans Duistermaat, for allowing us to reprint the two classical articles on Fourier integral operators: F.I.O., I (Acta Mathematica, Vol. 127, pp. 77–183) and II (Acta Mathematica, Vol. 128, pp. 184–269). Much of the material in these articles is now available in book form. In particular many would regard the definitive treatment of Fourier integral operators as being Volumes Ill and IV of Hormander’s book [26]. Nevertheless I feel there are two very good reasons for going to the trouble of reprinting these articles as an ensemble. One is that they provide a relatively concise introduction to this subject. For someone who is trying to get a sense of the lay of the land they are probably a better place to start than Hormander’s book or any other standard text book. Though these two papers (particularly F.I.O. I) would probably pose problems for someone who had never seen pseudodifferential operators before, they are essentially self-contained, and, considering the fact that they cover a lot of ground very quickly, are quite readable as well.

Microlocal analysis of a seismic linearized inverse problem

We define a scale of Hardy spaces H F I O p ( R n ) \mathcal {H}^{p}_{FIO}(\mathbb {R}^n) , p ∈ [ 1 , ∞ ] p\in [1,\infty ] , that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for p = 1 p=1 [J. Geom. Anal. 8 (1998), pp. 629–653]. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of R n \mathbb {R}^n , and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about L p L^{p} -boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.

We consider Fourier integral operators on non-compact manifolds and their applications, in particular in spectral theory. Fourier integral operators appear naturally as the solution operators of certain pseudodifferential evolution equations, such as the Schrödinger equation or the wave equation. For Euclidean space there are two important global pseudodifferential calculi: First there is the isotropic calculus, which contains the quantum harmonic oscillator, its inverse, and similar operators. We consider the solution operator to the dynamical Schrödinger equation with an isotropic pseudodifferential operator of order two and show how singularities and growth evolve with time. Moreover we show that for generic lower order perturbations of the harmonic oscillator the eigenvalues are more equally distributed then in the case of the unperturbed operator. The second important calculus, the scattering calculus, contains the Laplacian plus a bounded potential on asymptotically Euclidean manifolds. We define a class of geometric distributions that are related to the solution operators of the Klein-Gordon equation of quantum field theory and contain certain distributions that are appear in the scattering theory of the Laplacian. We show that these distributions have a symbol structure that admits an invariantly defined order and the existence of a principal symbol.

Second Microlocalization and Propagation of Singularities for Semi-Linear Hyperbolic Equations

In this article, we extend the space of rapidly decaying functions to a space of rapidly decaying Boehmians. We provide convolution products, convolution theorems and generate their associated spaces of Boehmian. Then, we define the short‐time Fourier integral operator on the Boehmian spaces. Moreover, we show that the short‐time Fourier integral operator of the Boehmian is a sequentially continuous mapping that preserves certain desired properties. An inversion formula and some injections have also been obtained.

We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces.We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces.We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces.Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel-Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.

The theory of the Sobolev spacesH p m (R n) (m∈R,p polyhedron in R 2n)of [BG]is revisited here in the frame of new classes of pseudodifferential operators related to the same polyhedron p.These operators generalize to corresponding classes of Fourier integral operators, for which we present the main lines of a symbolic calculus and results of continuity on the H p m (R n) spaces.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.