Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups

TL;DR

This paper establishes sharp fractional subelliptic Sobolev and Gagliardo-Nirenberg inequalities on stratified Lie groups, providing explicit constants and ground state solutions for related nonlinear equations, extending classical results to a non-Euclidean setting.

Contribution

It introduces new sharp inequalities and ground state existence results for fractional subelliptic equations on stratified Lie groups, using novel methods suitable for non-commutative geometries.

Findings

Sharp fractional subelliptic Sobolev inequalities with explicit constants.

Existence of ground state solutions for nonlinear fractional Schrödinger equations.

Extension of classical inequalities to stratified Lie groups, including the Heisenberg group.

Abstract

In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional $p$-sublaplacian ($1<p<\infty$) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schr\"odinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676 (1982/1983)) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. (2023)). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for $p=2$ even in the context of the Heisenberg group.