A boundary formula for reproducing kernel Hilbert spaces of real harmonic functions in Lipschitz domains

Abstract

This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain . Such method involves some families of positive self-adjoint operators and makes use of characterizations of their trace data and of a special inner product on . We also establish boundary representation results for this family in terms of the -Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with -boundary data is provided. Reproducing kernels and orthonormal bases for the harmonic spaces are also found.