The neumann problem on lipschitz domains in hardy spaces of order less than one

Recently, B.E.J. Dahlberg and C.E. Kenig considered the Neumann problem, Δu = 0 in D, ∂u/∂v = f on ∂D, for Laplace’s equation in a Lipschitz domain D. One of their main results considers this problem when the data lies in the atomic Hardy space H¹(∂D) and they show that the solution has gradient in L¹(∂D). The aim of this paper is to establish an extension of their theorem for data in the Hardy space H^p(∂D), 1-ε < p < 1, where 0 < ε < 1/n is a positive constant which depends only on m, the maximum of the Lipschitz constants of the functions which define the boundary of the domain. We also extend G. Verchota’s and Dahlberg and Kenig’s theorem on the potential representation of solutions of the Neumann problem to the range 1 -ε < p < 1. This has the interesting consequence that the double-layer potential is invertible on Holder spaces Cα (∂D) for a close to zero.