Weighted inequalities for gradients on non-smooth domains

We prove weighted norm inequalities of integral type between the gradients of solutions u of elliptic equations and their boundary data f on bounded Lipschitz domains. 0. Introduction. We are interested in the following general question: To what extent is the interior smoothness of the solution of a PDE controlled by the size of its boundary values? To be more specific, suppose (for now) that Ω ⊂ R is a nice domain, μ is a positive measure supported in Ω, and v is a non-negative measurable function defined on ∂Ω. If f : ∂Ω → R is reasonable (say, continuous function with compact support), we let u : Ω → R be the solution of the classical Dirichlet problem with boundary values equal to f . (We are implicitly assuming that Ω is nice enough to have this make sense!) Let p and q be real numbers lying strictly between 1 and infinity. When is it the case that (∫ Ω |∇u|q dμ )1/q ≤ (∫