Transmission problems and spectral theory for singular integral operators on Lipschitz domains

Abstract

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential $\text{K}{}^{\mathrm{\ast }}$ in $\text{L}{}^{\mathrm{p}}{}_0\text{(∂}\text{Ω}\text{)}$ is less than $\text{1}\text{2}$, whenever $\text{Ω}$ is a bounded convex domain and 1<p⩽2.