We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel-Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition).
- Besov and Triebel-Lizorkin spaces
- Convex domain
- Green operator
- Lipschitz domain satisfying a uniform exterior ball condition
- Nontangential maximal function
- Poisson problem
- Semiconvex domain