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

Abstract

Let Omega subset of R (n+ 1) , n >= 2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary partial derivative 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 Hajlasz-Sobolev space M-1,M-1(partial derivative Omega) ) and the weak-A infinity infinity property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,1(partial derivative Omega) ) is equivalent to the solvability of the regularity problem in M1,'(partial derivative 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 derivative Omega supports a weak (1, 1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajlasz-Sobolev space M-1,M-1(partial derivative Omega) , is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.