Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups

TL;DR

This paper introduces the Euler semigroup on homogeneous Lie groups to derive Hardy-Sobolev and Gagliardo-Nirenberg inequalities, including sharp remainder terms and weighted variants, advancing analysis on these groups.

Contribution

It develops a new framework using the Euler semigroup to establish various inequalities and sharp remainder terms on homogeneous groups.

Findings

Established the Euler semigroup on homogeneous groups.

Derived Hardy-Sobolev and Gagliardo-Nirenberg inequalities.

Proved sharp remainder and weighted inequalities.

Abstract

In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.