Micro-local analysis in Fourier Lebesgue and modulation spaces. Part II

TL;DR

This paper investigates local product operations in Fourier Lebesgue spaces, existence conditions for such products, and regularity results for solutions of semi-linear equations based on wave-front set analysis.

Contribution

It introduces new results on local product existence in Fourier Lebesgue spaces and establishes wave-front set regularity for solutions of semi-linear PDEs.

Findings

Existence of local products in Fourier Lebesgue spaces under wave-front conditions

Regularity transfer for solutions of semi-linear equations in Fourier Lebesgue spaces

Wave-front set propagation results for solutions based on non-characteristic conditions

Abstract

We consider different types of (local) products $f_1 f_2$ in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$ \qquad P(x,D)f = G(x,J_k f), $$ with appropriate polynomials $P $ and $G$. If the solution locally belongs to appropriate weighted Fourier Lebesgue space ${\mathscr F}L^q_{(\omega)} (\rr d)$ and $P$ is non-characteristic at $(x_0,\xi_0),$ then we prove that $(x_0,\xi_0)\not \in WF_{{\mathscr F}L^q_{(\widetilde {\omega})}} (f)$, where $\widetilde{\omega}$ depends on $\omega$, $P$ and $G$.