Solvability of the Poisson-Dirichlet problem with interior data in Lp' - Carleson spaces and its applications to the Lp' -regularity problem

We prove that the Lp' -solvability of the homogeneous Dirichlet problem for an elliptic operator L=-divA∇ with real and merely bounded coefficients is equivalent to the Lp' -solvability of the Poisson-Dirichlet problem Lw=H-divF, which is defined in terms of an Lp' -estimate on the non-tangential maximal function, assuming that dist(⋅,∂Ω)H and F lie in certain Lp' -Carleson-type spaces, and that the domain Ω⊂Rn+1, n≥2, satisfies the corkscrew condition and has n-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly n-rectifiable boundary that satisfies the corkscrew condition,Lp' -solvability of the homogeneous Dirichlet problem for an operator L=-divA∇ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the Lp -regularity problem for the adjoint operator L∗ =-divA T∇, where 1/p+ 1/p'=1 and AT is the transpose matrix of A. This result for Dahlberg-Kenig-Pipher operators is new even if Ω is the unit ball, despite the fact that the Lp' -solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001. Further novel applications include (i) new local estimates for the Green's function and its gradient in rough domains, (ii) a local T1-type theorem for the Lp -solvability of the "Poisson-regularity problem", itself equivalent to the Lp' -solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and (iii) new Lp -estimates for the eigenfunctions (and their gradients) of symmetric operators L on bounded rough domains.