On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups

TL;DR

This paper establishes new horizontal weighted Hardy-Rellich, Caffarelli-Kohn-Nirenberg, and Poincaré inequalities on stratified groups, providing simpler proofs and extending known results to a broader setting.

Contribution

It introduces a unified approach to Hardy-Rellich and Caffarelli-Kohn-Nirenberg inequalities on stratified groups and proves the Badiale-Tarantello conjecture with a new simple proof.

Findings

Derived new weighted Hardy-Rellich inequalities on stratified groups

Proved the Badiale-Tarantello conjecture with a simplified approach

Established Poincaré inequalities involving $p$-sub-Laplacians

Abstract

In this paper, we present a version of horizontal weighted Hardy-Rellich type and Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale-Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincar\'e inequalities as well as inequalities involving the weighted and unweighted $p$-sub-Laplacians.