A weighted $L_p$-theory for second-order elliptic and parabolic partial differential systems on a half space

TL;DR

This paper develops advanced harmonic analysis tools and establishes existence and uniqueness results for second-order elliptic and parabolic PDE systems in weighted Sobolev spaces, enhancing the theoretical framework for such equations.

Contribution

It introduces a weighted $L_p$-theory for PDE systems on half spaces, including new theorems and estimates in weighted Sobolev spaces, which were not previously available.

Findings

Established a Fefferman-Stein theorem in weighted Sobolev spaces

Proved Hardy-Littlewood theorem and sharp function estimates in weighted contexts

Provided existence and uniqueness results for elliptic and parabolic PDE systems in weighted Sobolev spaces

Abstract

In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.