Boundary problems for harmonic functions and norm estimates for inverses of singular integrals in two dimensions

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L-p, 1 < p < infinity, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderon-Zygmund theory and Mellin transform techniques.