Harmonic measure and Riesz transform in uniform and general domains

TL;DR

This paper establishes a link between harmonic measure and the boundedness of the Riesz transform in general domains, extending classical results to non-doubling measures and uniform domains.

Contribution

It proves the boundedness of the Riesz transform under scale invariant conditions on harmonic measure without assuming measure doubling, and generalizes estimates for uniform domains.

Findings

Riesz transform is bounded in L^2 under certain conditions.

Harmonic measure relates to Green function estimates in uniform domains.

Extends classical results to non-doubling measures and broader domain classes.

Abstract

Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $\Omega$ satisfies some scale invariant $A_\infty$ type conditions with respect to $\mu$, then the $n$-dimensional Riesz transform $$R_\mu f(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,d\mu(y)$$ is bounded in $L^2(\mu)$. We do not assume any doubling condition on $\mu$. We also consider the particular case when $\Omega$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.