Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces

Duván Cardona, William Obeng-Denteh, Frederick Opoku

TL;DR

The paper develops a Fourier-analytic framework for hyperbolic PDEs with complex characteristics on Fourier Lebesgue spaces, using Fourier integral operators with complex phases as the propagators. By imposing the spatial smooth factorization condition on the canonical relation, the authors establish precise boundedness results for FIOs between Fourier Lebesgue, Besov, and Triebel–Lizorkin spaces, including sharp regularity losses. These operator bounds are then applied to deduce well-posedness and regularity properties for first- and higher-order hyperbolic equations, with explicit dependence on the data regularity in $oldsymbol{ ext{FL}}^p_s$ spaces and associated index parameters $oldsymbol{eta}$. The work extends classical $L^p$-Sobolev results (notably Ruzhansky) to a broader functional-analytic setting and clarifies the role of complex phases and SSFC in ensuring sharp boundedness and stability. This has potential impact on microlocal analysis and PDE theory where complex characteristics arise, providing tools for precise propagation of singularities and regularity in Fourier-analytic spaces.

Abstract

The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}

Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces

TL;DR

The paper develops a Fourier-analytic framework for hyperbolic PDEs with complex characteristics on Fourier Lebesgue spaces, using Fourier integral operators with complex phases as the propagators. By imposing the spatial smooth factorization condition on the canonical relation, the authors establish precise boundedness results for FIOs between Fourier Lebesgue, Besov, and Triebel–Lizorkin spaces, including sharp regularity losses. These operator bounds are then applied to deduce well-posedness and regularity properties for first- and higher-order hyperbolic equations, with explicit dependence on the data regularity in spaces and associated index parameters . The work extends classical -Sobolev results (notably Ruzhansky) to a broader functional-analytic setting and clarifies the role of complex phases and SSFC in ensuring sharp boundedness and stability. This has potential impact on microlocal analysis and PDE theory where complex characteristics arise, providing tools for precise propagation of singularities and regularity in Fourier-analytic spaces.

Abstract

The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}
Paper Structure (16 sections, 6 theorems, 61 equations)

This paper contains 16 sections, 6 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Sharpness and regularity of the solution in Besov and in Triebel-Lizorkin spaces
  4. Methodology and bibliographic discussion
  5. Structure of the paper
  6. Preliminaries
  7. Notations
  8. Besov spaces $B^{s}_{p,q}$ and Triebel-Lizorkin Spaces $F^{s}_{p,q}$
  9. Fourier Integral Operators (FIOs)
  10. Fourier Lebesgue spaces vs $B^{s}_{p,q}$ and $F^{s}_{p,q}$ spaces
  11. Proof of Theorem \ref{['XXX:intro']}
  12. Boundedness properties of FIOs
  13. Well posedness properties
  14. First order hyperbolic partial differential equations
  15. Higher order hyperbolic partial differential equations
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $P$ be a classical pseudo-differential operator of order $m$ of the formeq:5.4.10:intro on a smooth compact manifold $X$ of dimension $d.$ Assume that $P$ has simple characteristics $\tau_j,$ satisfying $\operatorname{Im}\tau_j (t,x,\xi) \geq 0$ in $[0,T]\times (T^*X\setminus\{0\})$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 2.1: Spatial smooth factorization condition (SSFC)
  • Theorem 2.2: $B^s_{p,q} \hookrightarrow \mathscr{F}L^r$
  • Theorem 2.3: $F^{s}_{p,q} \hookrightarrow \mathscr{F}L^{r}$
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['Main:Theorem']}
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 1 more