Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups

TL;DR

This paper introduces new Sobolev-type inequalities on homogeneous groups using Euler operators, providing sharp constants and exploring applications to Sobolev-Lorentz-Zygmund spaces, extending classical results to more general settings.

Contribution

It defines Euler-Hilbert-Sobolev spaces on homogeneous groups and derives sharp inequalities with best constants, extending classical Sobolev and Rellich inequalities.

Findings

Derived sharp Sobolev-Rellich inequalities with best constants.

Established analogues of classical Sobolev and Rellich inequalities.

Applied logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces.

Abstract

We define Euler-Hilbert-Sobolev spaces and obtain embedding results on homogeneous groups using Euler operators, which are homogeneous differential operators of order zero. Sharp remainder terms of $L^{p}$ and weighted Sobolev type and Sobolev-Rellich inequalities on homogeneous groups are given. Most inequalities are obtained with best constants. As consequences, we obtain analogues of the generalised classical Sobolev type and Sobolev-Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces. The obtained results are new already in the anisotropic $\mathbb R^{n}$ as well as in the isotropic $\mathbb R^{n}$ due to the freedom in the choice of any homogeneous quasi-norm.