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

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Title: Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups
Authors: Ruzhansky, M
Suragan, D
Yessirkegenov, N
Item Type: Journal Article
Abstract: We define Euler–Hilbert–Sobolev spaces and obtain embed- ding 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 dis- cuss applications of logarithmic Hardy inequalities to Sobolev–Lorentz– Zygmund spaces. The obtained results are new already in the anisotropic R n as well as in the isotropic R n due to the freedom in the choice of any homogeneous quasi-norm.
Issue Date: 28-Feb-2018
Date of Acceptance: 16-Jan-2018
ISSN: 0378-620X
Publisher: Springer Verlag
Journal / Book Title: Integral Equations and Operator Theory
Volume: 90
Copyright Statement: © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/K039407/1
Keywords: Science & Technology
Physical Sciences
Sobolev inequality
Hardy inequality
Weighted Sobolev inequality
Rellich inequality
Euler-Hilbert-Sobolev space
Sobolev-Lorentz-Zygmund space
Homogeneous Lie group
22E30, 46E35
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
General Mathematics
Publication Status: Published
Article Number: 10
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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