The regularity problem for degenerate elliptic operators in weighted spaces

TL;DR

This paper investigates the solvability of the regularity problem for degenerate elliptic operators in weighted spaces, providing new bounds and extending results to weighted and unweighted cases with novel methods.

Contribution

It introduces a direct approach to establish non-tangential bounds for degenerate elliptic operators in weighted spaces, extending classical results to more general weighted settings.

Findings

Established non-tangential bounds for solutions in weighted spaces

Extended solvability results to degenerate elliptic operators with weights

Provided new methods based on change of angles in weighted estimates

Abstract

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let $L_w$ be a degenerate elliptic operator with degeneracy given by a fixed weight $w\in A_2(dx)$ in $\mathbb{R}^n$, and consider the associated block second order degenerate elliptic problem in the upper-half space $\mathbb{R}_+^{n+1}$. We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces $L^p(vdw)$ where $v$ is a Muckenhoupt weight with respect to the underlying natural weighted space $(\mathbb{R}^n, wdx)$. We recover earlier results in the non-degenerate case (when $w\equiv 1$, and with or without weight $v$). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) $L^2(dx)$-solvability of the regularity problem for the block operator \[ \mathbb{L}_\alpha u(x,t) = -|x|^{\alpha} \mathrm{div}_x \big(|x|^{-\alpha }\,A(x) \nabla_x u(x,t)\big)-\partial_{t}^2 u(x,t) \] for any complex-valued uniformly elliptic matrix $A$ and for all $-\epsilon<\alpha<\frac{2\,n}{n+2}$, where $\epsilon$ depends just on the dimension and the ellipticity constants of $A$.