Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups

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 Lp 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 Rn as well as in the isotropic Rn due to the freedom in the choice of any homogeneous quasi-norm. © 2018, The Author(s).

Authors
Ruzhansky M.1 , Suragan D. 2, 3 , Yessirkegenov N.1, 2
Publisher
Springer Basel
Number of issue
1
Language
English
Status
Published
Number
10
Volume
90
Year
2018
Organizations
  • 1 Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, United Kingdom
  • 2 Institute of Mathematics and Mathematical Modelling, 125 Pushkin str., Almaty, 050010, Kazakhstan
  • 3 RUDN University, 6 Miklukho-Maklay str., Moscow, 117198, Russian Federation
Keywords
Euler–Hilbert–Sobolev space; Hardy inequality; Homogeneous Lie group; Rellich inequality; Sobolev inequality; Sobolev–Lorentz–Zygmund space; Weighted Sobolev inequality
Date of creation
19.10.2018
Date of change
19.10.2018
Short link
https://repository.rudn.ru/en/records/article/record/6877/