Fundamental solution of the Vladimirov-Taibleson operator on noncommutative Vilenkin 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 -adic Heisenberg group and the -adic Engel group, and discuss possible extensions of our results to more general homogeneous operators on graded -adic Lie groups.

Hardy inequalities on constant-order noncommutative Vilenkin groups

“…For the heat kernel of the Vladimirov Laplacian on the groups ℍ or 4 the following properties are proven in [2]: (iv) The heat semigroup − 2 is symmetric and Markovian. Moreover, the following estimate holds for its kernel:…”

“…See [2] for all the details. Consequently, we get the following corollary of Theorem 2.34: Corollary 2.41.…”