On the Dirichlet problem on Lorentz and Orlicz spaces with applications to Schwarz-Christoffel domains

It is known (see [14]) that, for every Lipschitz domain Ω on the planeΩ={x+iy:y>ν(x)}, with ν a real valued Lipschitz function, there exists 1≤p0<2 so that the Dirichlet problem has a solution for every function f∈Lp(ds) and every p∈(p0,∞). Moreover, if p0>1, the result is false for every p≤p0. The purpose of this paper is to study in more detail what happens at the endpoint p0; that is, we want to find spaces X⊂Lp0 so that the Dirichlet problem is solvable for every f∈X. These spaces X will be either the Lorentz space Lp0,1(ds) or some type of logarithmic Orlicz space. Our results will be applied to the special case of Schwarz-Christoffel Lipschitz domains, among others, for which we explicitly compute the value of p0.