The Vladimirov-Taibleson Operator: Inequalities, Dirichlet Problem, Boundary H\"older Regularity

TL;DR

This paper investigates the Vladimirov-Taibleson operator, establishing inequalities, analyzing the Dirichlet problem, and demonstrating boundary H"older regularity of solutions in the context of non-Archimedean local fields.

Contribution

It provides the first analogs of classical fractional Laplacian inequalities and boundary regularity results for the Vladimirov-Taibleson operator on non-Archimedean fields.

Findings

Established inequalities analogous to fractional Laplacian

Analyzed the Dirichlet problem for the operator

Proved boundary H"older regularity of solutions

Abstract

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 H\"older regularity of solutions.