Sobolev and Besov Space Estimates for Solutions to Second Order Pde on Lipschitz Domains in Manifolds with Dini or Hölder Continuous Metric Tensors

We examine solutions u = PI f to ∆u − V u = 0 on a Lipschitz domain Ω in a compact Riemannian manifold M , satisfying u = f on ∂Ω, with particular attention to ranges of (s, p) for which one has Besov-to-L-Sobolev space results of the form PI : B s (∂Ω) −→ Lps+1/p(Ω), and variants, when the metric tensor on M has limited regularity, described by a Hölder or a Dini-type modulus of continuity. We also discuss related estimates for solutions to the Neumann problem.