Off-singularity bounds and Hardy spaces for Fourier integral operators

In this paper, we study Hardy spaces associated with non-negative self-adjoint operators and develop their vector-valued theory. The complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint p=1. The holomorphic functional calculus of L is also shown to be bounded on the associated Hardy space H1L(X). These results, along with the atomic decomposition for the aforementioned space, rely on boundedness of certain integral operators on the tent space T1(X).

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].

Some new function spaces and their applications to Harmonic analysis

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.

We show that the Hardy spaces for Fourier integral operators form natural spaces of initial data when applying ℓp-decoupling inequalities to local smoothing for the wave equation. This yields new local smoothing estimates which, in a quantified manner, improve the bounds in the local smoothing conjecture on Rn for p≥2(n+1)/(n−1), and complement them for 2<p<2(n+1)/(n−1). These estimates are invariant under application of Fourier integral operators, and they are essentially sharp.

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, potential theory and operator theory, and has found essential applications in robust control engineering. For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane. With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

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.

Some new tent spaces and duality theorems for fractional Carleson measures and [formula omitted

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.

Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces

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 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.

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.

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