Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces

TL;DR

This paper investigates the well-posedness of hyperbolic PDEs with complex characteristics on Fourier Lebesgue spaces, using harmonic analysis techniques to establish boundedness of Fourier integral operators with complex phases.

Contribution

It introduces new boundedness results for Fourier integral operators with complex phases on Fourier Lebesgue, Besov, and Triebel-Lizorkin spaces, under the spatial smooth factorization condition.

Findings

Established well-posedness for hyperbolic PDEs with complex characteristics.

Proved boundedness of Fourier integral operators with complex phases.

Extended harmonic analysis tools to new classes of PDE problems.

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}