Sharp fractional Sobolev and related inequalities on H-type groups

TL;DR

This paper establishes the precise constants for fractional Sobolev inequalities on H-type groups, deriving related sharp inequalities and extending known results to this geometric setting.

Contribution

It determines the sharp constants for fractional Sobolev inequalities on H-type groups and extends Sobolev trace inequalities to this context.

Findings

Sharp constants for fractional Sobolev inequalities on H-type groups

Derivation of a sharp log-Sobolev inequality

Extension of Sobolev trace inequality results

Abstract

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers $\mathcal{L}_{s}(0<s<1)$ of the sublaplacian on H-type groups. From these inequalities we derive a sharp log-Sobolev inequality by considering a limiting case and a sharp Sobolev trace inequality. The later extends to this context the result of Frank, Gonz\'alez, Monticelli and Tan (Adv. Math, 2015).