Absolute continuity of degenerate elliptic measure

TL;DR

This paper establishes a comprehensive set of equivalent conditions relating the absolute continuity of degenerate elliptic measures to solvability and regularity properties of boundary value problems for degenerate elliptic operators in complex geometric settings.

Contribution

It provides new equivalences connecting elliptic measure absolute continuity, boundary problem solvability, and square function estimates in degenerate elliptic PDEs with irregular boundaries.

Findings

Equivalence between elliptic measure in A_infinity class and boundary problem solvability.

Characterization of absolute continuity via local L^2 estimates of conical square functions.

Finiteness of conical square functions almost everywhere characterizes absolute continuity.

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that $\Omega$ satisfies the quantitative openness and connectedness, and there exist doubling measures $m$ on $\Omega$ and $\mu$ on $\partial \Omega$ with appropriate size conditions. Let $Lu=-\mathrm{div}(A\nabla u)$ be a real (not necessarily symmetric) degenerate elliptic operator in $\Omega$. Write $\omega_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) $\omega_L \in A_{\infty}(\mu)$, (ii) the Dirichlet problem for $L$ is solvable in $L^p(\mu)$ for some $p \in (1, \infty)$, (iii) every bounded null solution of $L$ satisfies Carleson measure estimates with respect to $\mu$, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q(\mu)$ for all $q \in (0, \infty)$ for any null solution of $L$, and (v) the Dirichlet problem for $L$ is solvable in $\mathrm{BMO}(\mu)$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of $\omega_L$ with respect to $\mu$ in terms of local $L^2(\mu)$ estimates of the truncated conical square function for any bounded null solution of $L$. This is also equivalent to the finiteness $\mu$-almost everywhere of the truncated conical square function for any bounded null solution of $L$.