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Fourier integral operators on Orlicz modulation spaces

Serap Öztop, Rüya Üster, Joachim Toft

TL;DR

This work extends the theory of Fourier integral operators to the Orlicz modulation space setting, addressing operators with non-smooth amplitudes and phases. By placing amplitudes in $M^{\infty,1}_{(\omega)}$ and phases in $M^{\infty,1}_{(v)}$ under a nondegeneracy condition, it proves continuity on a wide class of Orlicz modulation spaces $M^{\Phi}_{(\omega)}$, including cases where $\Phi$ encodes Lebesgue-type norms. It then develops Schatten-von Neumann results for FIOs by analyzing kernels in Orlicz modulation spaces and establishing $\operatorname{Op}_\varphi(a) \in \mathscr I_{\Phi}(\omega_1,\omega_2)$, with extensions to kernel-based and $A$-modified FIOs. Overall, the paper broadens the applicability of FIOs to non-Lebesgue environments, enabling fine-grained analysis under subexponential weights and general Young functions, with potential impact on hyperbolic-type problems and related PDE analysis.

Abstract

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in Orlicz modulation spaces, when acting on other Orlicz modulation spaces themselves. The phase functions are non smooth and admit second order derivatives in suitable classes of modulation spaces.

Fourier integral operators on Orlicz modulation spaces

TL;DR

This work extends the theory of Fourier integral operators to the Orlicz modulation space setting, addressing operators with non-smooth amplitudes and phases. By placing amplitudes in and phases in under a nondegeneracy condition, it proves continuity on a wide class of Orlicz modulation spaces , including cases where encodes Lebesgue-type norms. It then develops Schatten-von Neumann results for FIOs by analyzing kernels in Orlicz modulation spaces and establishing , with extensions to kernel-based and -modified FIOs. Overall, the paper broadens the applicability of FIOs to non-Lebesgue environments, enabling fine-grained analysis under subexponential weights and general Young functions, with potential impact on hyperbolic-type problems and related PDE analysis.

Abstract

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in Orlicz modulation spaces, when acting on other Orlicz modulation spaces themselves. The phase functions are non smooth and admit second order derivatives in suitable classes of modulation spaces.
Paper Structure (11 sections, 15 theorems, 147 equations)

This paper contains 11 sections, 15 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gelfand-Shilov spaces
  4. Weight functions
  5. Orlicz Spaces
  6. Orlicz modulation spaces
  7. Kernel operators and Schatten-von Neumann classes
  8. Continuity for Fourier integral operators
  9. Mapping properties for certain kernel operators on Orlicz modulation spaces
  10. Continuity for Fourier integral operators, when acting on modulation spaces
  11. Schatten-von Neumann properties for Fourier integral operators

Key Result

Proposition 2.1

Let $s\ge \frac{1}{2}$ ($s>\frac{1}{2}$), $\phi \in \mathcal{S} _s(\mathbf R^{d})\setminus 0$ ($\phi \in \Sigma _s(\mathbf R^{d})\setminus 0$) and let $f$ be a Gelfand-Shilov distribution on $\mathbf R^{d}$. Then the following is true:

Theorems & Definitions (39)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 29 more