Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

TL;DR

This paper proves square function and non-tangential maximal function estimates for solutions to elliptic operators in 1-sided NTA domains with capacity density, advancing understanding of elliptic measure behavior.

Contribution

It establishes new square function and non-tangential estimates for elliptic PDE solutions in complex domains, extending prior foundational work.

Findings

Bounded weak null-solutions satisfy Carleson measure estimates.

Square function controlled by non-tangential maximal function in Lebesgue spaces.

Results are key for perturbation theory of elliptic measures.

Abstract

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $\Omega$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that $\Omega$ satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider $L_0 u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$, two real (non-necessarily symmetric) uniformly elliptic operators in $\Omega$, and write $\omega_{L_0}$, $\omega_L$ for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $\omega_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $\omega_{L_0}$. In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in arXiv:1901.08261.