Fundamental solution of the Vladimirov-Taibleson operator on noncommutative Vilenkin groups

TL;DR

This paper derives the fundamental solution and heat semigroup for the Vladimirov-Taibleson operator on noncommutative Vilenkin groups, extending results to p-adic Lie groups and providing heat kernel estimates.

Contribution

It provides explicit fundamental solutions and heat semigroup analysis for Vladimirov-Taibleson operators on noncommutative Vilenkin groups and p-adic Lie groups, a novel extension in p-adic analysis.

Findings

Explicit fundamental solutions for the operator on Vilenkin groups.

Heat kernel estimates for the associated heat semigroup.

Existence results for the Vladimirov Laplacian on p-adic groups.

Abstract

The fundamental solution and the heat semigroup of the Vladimirov-Taibleson operator on constant-order noncommutative Vilenkin groups are obtained, together with some estimates on the associated heat kernel. We also show the existence of a fundamental solution for the "Vladimirov Laplacian" on the $p$-adic Heisenberg group and the $p$-adic Engel group, and discuss possible extensions of our results to more general homogeneous operators on graded $p$-adic Lie groups.