Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be an open set satisfying the corkscrew condition with $n$-Ahlfors regular boundary $\partial \Omega$, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space $M^{1,1}(\partial \Omega )$ and the weak-${\mathcal{A}}_{\infty }$ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in $M^{1,1}(\partial \Omega )$ is equivalent to the solvability of the regularity problem in $M^{1,p}(\partial \Omega )$ for some $p>1$. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that $\partial \Omega$ supports a weak $(1,1)$-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space $M^{1,1}(\partial \Omega )$ is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

Revised version, we now address extrapolation for general elliptic operators and for the Poisson regularity problem as well