The Higher Order Regularity Problem for Elliptic Systems with Data in Banach Function Spaces

Abstract

We characterize the well-posedness of the higher order regularity problem in the upper half-space with data in Sobolev Banach function spaces by proving its equivalence to natural weighted estimates for the Hardy–Littlewood maximal operator. The generality of our framework allows for applications to Lebesgue spaces, rearrangement-invariant spaces such as Orlicz spaces, and variable exponent Lebesgue spaces, as well as their weighted counterparts, among others. This is established for the family of second-order, homogeneous, elliptic, constant complex coefficient systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} that admit a distinguished coefficient tensor, a natural condition that always holds in the scalar case and for the Lamé system of elasticity.