The Vladimirov-Taibleson Operator: Inequalities, Dirichlet Problem, Boundary Hölder Regularity

We study the Vladimirov-Taibleson operator, a model example of a pseudo-differential operator acting on real- or complex-valued functions defined on a non-Archimedean local field. We prove analogs of classical inequalities for fractional Laplacian, study the counterpart of the Dirichlet problem including the property of boundary Hölder regularity of solutions.