The optimal constant in the $L^{2}$ Folland-Stein inequality on the H-type group

TL;DR

This paper finds the best constant in the $L^{2}$ Folland-Stein inequality on H-type groups, confirming part of a long-standing conjecture and building on recent mathematical advances.

Contribution

It determines the optimal constant in the $L^{2}$ Folland-Stein inequality on H-type groups, partially confirming Garofalo and Vassilev's conjecture.

Findings

Optimal constant in the inequality is identified.

Partial confirmation of the conjecture by Garofalo and Vassilev.

Method inspired by Frank and Lieb, Hang and Wang.

Abstract

We determine the optimal constant in the $L^{2}$ Folland-Stein inequality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (Duke Math. J., 2001). The proof is inspired by the work of Frank and Lieb (Ann. of Math., 2012) and Hang and Wang.