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

Michael Ruzhansky, Durvudkhan Suragan, Nurgissa Yessirkegenov

Published 2016-10-11Version 1

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 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 and Sobolev-Rellich inequalities. We also discuss applications of logarithmic Hardy inequalities to Sobolev-Lorentz-Zygmund spaces.