The heat kernel and frequency localized functions on the Heisenberg group

TL;DR

This paper investigates the heat operator on the Heisenberg group, characterizing Besov spaces via the heat kernel and deriving refined Sobolev inequalities, advancing understanding of harmonic analysis on non-commutative groups.

Contribution

It provides a novel characterization of Besov spaces on the Heisenberg group using heat kernel methods, extending to positive indices with Bernstein inequalities.

Findings

Characterization of Besov spaces of negative index on H^d

Extension to positive indices via Bernstein inequalities

Proof of refined Sobolev inequalities in W^{s,p} spaces

Abstract

The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to positive indexes using Bernstein inequalities. As a corollary we obtain a proof of refined Sobolev inequalities in W^{s,p} spaces.