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.
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The third author was supported by the MESRK grant AP05133271.
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The authors were supported in parts by the EPSRC Grants EP/K039407/1 and EP/R003025/1, and by the Leverhulme Grants RPG-2014-02 and RPG-2017-151. The second author was also supported by the MESRF No. 02.a03.21.0008 and the MESRK Grant AP05130981. No new data was collected or generated during the course of research.
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Ruzhansky, M., Suragan, D. & Yessirkegenov, N. Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups. Integr. Equ. Oper. Theory 90, 10 (2018). https://doi.org/10.1007/s00020-018-2437-7
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DOI: https://doi.org/10.1007/s00020-018-2437-7
Keywords
- Sobolev inequality
- Hardy inequality
- Weighted Sobolev inequality
- Rellich inequality
- Euler–Hilbert–Sobolev space
- Sobolev–Lorentz–Zygmund space
- Homogeneous Lie group