Remarks on Fourier multipliers and applications to the Wave equation

TL;DR

This paper investigates the use of Fourier multipliers on modulation and Wiener amalgam spaces to establish local well-posedness results for nonlinear wave equations with rough initial data.

Contribution

It introduces refined local well-posedness results for the nonlinear wave equation using modulation and Wiener amalgam spaces, extending previous frameworks.

Findings

Established local well-posedness for rough data in modulation spaces

Extended results to the nonlinear Klein-Gordon equation

Refined previous results in the literature

Abstract

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NLKG equation.