Neumann problems for $p$-harmonic functions, and induced nonlocal operators in metric measure spaces

Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-\Delta_p)^\theta$ with $0<\theta<1$ on any compact, doubling metric measure space $(Z,d,\nu)$, and prove existence, regularity and stability for the non-homogenous non-local equation $(-\Delta_p)^\theta u =f.$ These results, in turn, rest on the new existence, global H\"older regularity and stability theorems that we prove for the Neumann problem for $p$-Laplacians $\Delta_p$, $1更多