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

Abstract

Let $\Omega\subset \mathbb R^{n+1}$, $n\geq2$, 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 Haj{\l}asz-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 $(1,1)$-Poincar\'e inequality, we show that the solvability of the regularity problem in the Haj{\l}asz-Sobolev space $M^{1,1}(\partial\Omega)$ is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.