TY - JOUR
T1 - Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups
AU - Ruzhansky, Michael
AU - Suragan, Durvudkhan
AU - Yessirkegenov, Nurgissa
N1 - Funding Information:
The third author was supported by the MESRK grant AP05133271.
Funding Information:
The third author was supported by the MESRK grant AP05133271. 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.
Funding Information:
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.
Publisher Copyright:
© 2018, The Author(s).
PY - 2018/2/1
Y1 - 2018/2/1
N2 - 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.
AB - 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.
KW - Euler–Hilbert–Sobolev space
KW - Hardy inequality
KW - Homogeneous Lie group
KW - Rellich inequality
KW - Sobolev inequality
KW - Sobolev–Lorentz–Zygmund space
KW - Weighted Sobolev inequality
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U2 - 10.1007/s00020-018-2437-7
DO - 10.1007/s00020-018-2437-7
M3 - Article
AN - SCOPUS:85042762457
SN - 0378-620X
VL - 90
JO - Integral Equations and Operator Theory
JF - Integral Equations and Operator Theory
IS - 1
M1 - 10
ER -