A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Let $\mathbb{G}_D$ be the solution operator for $\Delta u = f$ in $\Omega$, Tr $u = 0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$. B. E. J. Dahlberg proved that for a bounded Lipschitz domain $\Omega, \nabla \mathbb{G}_D$ maps $L^1 (\Omega)$ boundedly into weak- $L^1(\Omega)$ and that there exists $p_n > 1$ such that $\nabla\mathbb{G}_D : L^p (\Omega)\rightarrow L^{p^{*}} (\Omega)$ is bounded for $1 < p < n, \frac{1}{p^*} = \frac {1}{p} - \frac {1}{n}$. In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator $\mathbb{G}_N$ corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.