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

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.

34 pages; The title and document style have been changed, to appear in Integral Equations and Operator Theory