Sobolev Type Inequalities, Euler–Hilbert–Sobolev and Sobolev–Lorentz–Zygmund Spaces on Homogeneous Groups
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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.
Keywords
Sobolev inequality Hardy inequality Weighted Sobolev inequality Rellich inequality Euler–Hilbert–Sobolev space Sobolev–Lorentz–Zygmund space Homogeneous Lie groupMathematics Subject Classification
Primary 22E30 Secondary 46E35Notes
Acknowledgements
The third author was supported by the MESRK grant AP05133271.
References
- 1.Brezis, H., Lieb, E.: Inequalities with remainder terms. J. Funct. Anal. 62, 73–86 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Brezis, H., Marcus, M.: Hardy’s inequalities revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci 25(4), 217–237 (1997)MathSciNetzbMATHGoogle Scholar
- 3.Davies, E.B.: One-Parameter Semigroups. London Mathematical Society Monographs, vol. 15. Academic Press, Inc., London (1980)Google Scholar
- 4.Davies, E.B., Hinz, A.M.: Explicit constants for Rellich inequalities in \(L_p(\Omega )\). Math. Z. 227(3), 511–523 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Fischer, V., Ruzhansky, M.: Quantization on nilpotent Lie groups. Progress in Mathematics, vol. 314. Birkhäuser, Basel (2016). (open access book)CrossRefzbMATHGoogle Scholar
- 6.Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes, vol. 28. Princeton University Press, Princeton (1982)zbMATHGoogle Scholar
- 7.Ioku, N., Ishiwata, M., Ozawa, T.: Sharp remainder of a critical Hardy inequality. Arch. Math. (Basel) 106(1), 65–71 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Ioku, N., Ishiwata, M., Ozawa, T.: Hardy type inequalities in \({L}^p\) with sharp remainders. J. Inequal. Appl. 2017(5), 7 (2017)zbMATHGoogle Scholar
- 9.Komatsu, H.: Fractional powers of operators. Pacific J. Math. 19(2), 285–346 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Laptev, A., Ruzhansky, M., Yessirkegenov, N.: Hardy inequalities for Landau Hamiltonian and for Baouendi–Grushin operator with Aharonov–Bohm type magnetic field. Part I (2017). arXiv:1705.00062
- 11.Machihara, S., Ozawa, T., Wadade, H.: Remarks on the Hardy type inequalities with remainder terms in the framework of equalities. to appear in Adv. Stud. Pure Math. (2016). arXiv:1611.03580
- 12.Machihara, S., Ozawa, T., Wadade, H.: Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space. J. Inequal. Appl. 2015, 281 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Machihara, S., Ozawa, T., Wadade, H.: Remarks on the Rellich inequality. Math. Z. 286(3–4), 1367–1373 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Martinez, C.C., Sanz, M.A.: The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, vol. 187. North-Holland Publishing Co., Amsterdam (2001)Google Scholar
- 15.Ozawa, T., Sasaki, H.: Inequalities associated with dilations. Commun. Contemp. Math. 11(2), 265–277 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Ozawa, T., Ruzhansky, M., Suragan, D.: \(L^{p}\)-Caffarelli–Kohn–Nirenberg type inequalities on homogeneous groups (2016). arXiv:1605.02520
- 17.Ruzhansky, M., Suragan, D.: Critical Hardy inequalities (2016). arXiv:1602.04809v3
- 18.Ruzhansky, M., Suragan, D.: Anisotropic \(L^{2}\)-weighted Hardy and \(L^{2}\)-Caffarelli–Kohn–Nirenberg inequalities. Commun. Contemp. Math. 19(6), 1750014 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Ruzhansky, M., Suragan, D.: Local Hardy and Rellich inequalities for sums of squares of vector fields. Adv. Differ. Equ. 22, 505–540 (2017)MathSciNetzbMATHGoogle Scholar
- 20.Ruzhansky, M., Suragan, D.: Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups. Adv. Math. 317, 799–822 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Ruzhansky, M., Suragan, D.: On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups. J. Differ. Equ. 262(3), 1799–1821 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Ruzhansky, M., Suragan, D., Yessirkegenov, N.: Extended Caffarelli–Kohn–Nirenberg inequalities, and remainders, stability, and superweights for \(L^p\)-weighted Hardy inequalities. Trans. Am. Math. Soc. Ser. B 5, 32–62 (2018)CrossRefzbMATHGoogle Scholar
- 23.Ruzhansky, M., Tokmagambetov, N., Yessirkegenov, N.: Best constants in Sobolev and Gagliardo–Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations (2017). arXiv:1704.01490
- 24.Ruzhansky, M., Yessirkegenov, N.: Critical Sobolev, Gagliardo–Nirenberg, Trudinger and Brezis–Gallouet–Wainger inequalities, best constants, and ground states on graded groups (2017). arXiv:1709.08263
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