Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

TL;DR

This paper establishes Hardy inequalities for fractional powers of the sublaplacian on the Heisenberg group, providing optimal constants and implications for uncertainty principles using integral representations and semigroup methods.

Contribution

It introduces new Hardy inequalities for fractional sublaplacians on the Heisenberg group with explicit integral representations and optimal constants, extending previous results.

Findings

Optimal Hardy inequality constants obtained.

Integral representations for fractional sublaplacian derived.

Heisenberg uncertainty inequalities extended to fractional case.

Abstract

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.