A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

TL;DR

This paper develops a regularity theory for initial value problems involving time-measurable pseudo-differential operators in weighted $L_p$-spaces, establishing existence, uniqueness, and maximal regularity estimates.

Contribution

It introduces a novel framework for analyzing such problems in weighted spaces with variable order Besov initial data, extending previous regularity results.

Findings

Proved existence and uniqueness of solutions.

Established maximal regularity estimates in weighted spaces.

Included Muckenhoupt weights in the regularity analysis.

Abstract

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.