Rellich inequalities for sub-Laplacians with drift

TL;DR

This paper establishes new weighted Rellich inequalities for sub-Laplacians with drift on stratified groups, demonstrating how drift terms enhance classical inequalities and extending results to Euclidean spaces.

Contribution

It introduces novel weighted Rellich inequalities for sub-Laplacians with drift on stratified and Carnot groups, including Euclidean cases, with improved bounds due to drift effects.

Findings

Weighted Rellich inequalities proven for sub-Laplacians with drift.

Drift presence improves classical inequalities.

Results extend to Euclidean Laplacian, unifying classical and new inequalities.

Abstract

In this note we prove horizontal weighted Rellich inequalities for the sub-Laplacian and for sub-Laplacians with drift on general stratified groups. We show how the presence of a drift improves the known inequalities. Moreover, we obtain several versions of weighted Rellich inequalities for the sub-Laplacian with drift on the polarizable Carnot groups, also with the weights associated with the fundamental solution of the sub-Laplacian. The obtained results are already new for the Laplacian in the usual Euclidean setting of ${\mathbb R}^n$, embedding the classical Rellich inequality into a family of Rellich inequalities with parameter dependent drifts.