The regularity problem for uniformly elliptic operators in weighted spaces

TL;DR

This paper extends regularity results for elliptic operators to weighted Lebesgue spaces with Muckenhoupt weights, employing weighted Hardy space theory and an inhomogeneous vertical square function.

Contribution

It generalizes previous unweighted regularity results to weighted spaces using advanced harmonic analysis techniques.

Findings

Weighted regularity results for elliptic operators established.

Introduction of an inhomogeneous vertical square function controlled by the gradient.

Extension of Hardy space theory to weighted elliptic operator analysis.

Abstract

This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an "inhomogeneous" vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.