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 Hp(∂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 Hölder spaces Cα(∂D) for α close to
zero.