Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X, d(X), mu(X)) satisfying a 2-Poincare inequality. Given a bounded domain Omega subset of X with mu(X) (X \ Omega) > 0, and a function f in the Besov class B-2,2(theta)(X) boolean AND L-2(X), we study the problem of finding a function u is an element of B-2,2(theta)(X) such that u = f in X \ Omega and epsilon(theta)(u, u) <= epsilon(theta)(h, h) whenever h is an element of B-2,2(theta)(X) with h = f in X \ Omega. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Holder continuous on Omega, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups. (C) 2021 The Authors. Published by Elsevier Inc.