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

Abstract:

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.