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

  • Michael RuzhanskyEmail author
  • Durvudkhan Suragan
  • Nurgissa Yessirkegenov
Open Access


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.


Sobolev inequality Hardy inequality Weighted Sobolev inequality Rellich inequality Euler–Hilbert–Sobolev space Sobolev–Lorentz–Zygmund space Homogeneous Lie group 

Mathematics Subject Classification

Primary 22E30 Secondary 46E35 



The third author was supported by the MESRK grant AP05133271.


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© The Author(s) 2018

Open AccessThis 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.

Authors and Affiliations

  • Michael Ruzhansky
    • 1
    Email author
  • Durvudkhan Suragan
    • 2
    • 3
  • Nurgissa Yessirkegenov
    • 1
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institute of Mathematics and Mathematical ModellingAlmatyKazakhstan
  3. 3.RUDN UniversityMoscowRussia

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