Abstract
A sharp integral inequality is proved and used to obtain a Sobolev interpolation inequality. Further, a new proof of a Gross-Sobolev logarithmic inequality is constructed on the basis of the Sobolev interpolation inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. I. Babenko, “An inequality in the theory of Fourier integrals,” Izv. Akad. Nauk SSSR Ser. Mat. 25(4), 531–542 (1961).
W. Beckner, “Inequalities in Fourier analysis,” Ann. of Math. (2) 102(1), 159–182 (1975).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis and Self-Adjointness (Academic Press, New York, 1975; Mir, Moscow, 1978).
E. H. Lieb and M. Loss, Analysis, in Grad. Stud. Math. (Amer. Math. Soc., Providence, RI, 2001), Vol. 14.
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products (Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963; Academic Press, New York-London, 1965).
Sh. M. Nasibov, “Optimal constants in some Sobolev inequalities and their applications to the nonlinear Schroödinger equation,” Dokl. Akad. Nauk SSSR 307(3), 538–542 (1989). [Soviet Math. Dokl.].
W. Beckner and M. Pearson, “On sharp Sobolev embedding and the logarithmic Sobolev inequality,” Bull. London Math. Soc. 30(1), 80–84 (1998).
L. Gross, “Logarithmic Sobolev inequalities,” Amer. J. Math. 97(4), 1061–1683 (1975).
F. B. Weissler, “Logarithmic Sobolev inequalities for the heat-diffusion semigroup,” Tranc. Amer. Math. Soc. 237, 255–269 (1978).
W. Beckner, “Geometric asymptotics and the logarithmic Sobolev inequality,” Forum Math. 11(1), 105–137 (1999).
E. A. Carlen, “Superadditivity of Fisher’s information and logarithmic Sobolev inequalitie,” J. Funct. Anal. 101(1), 194–211 (1991).
Sh. M. Nasibov, “On an integral inequality and its application to the proof of the entropy inequality,” Mat. Zametki 84(2), 231–237 (2008) [Math. Notes 84 (2), 218–223 (2008)].
Sh. M. Nasibov, “On a generalization of the entropy inequality,” Mat. Zametki 99(2), 278–282 (2016) [Math. Notes 99 (2), 304–307 (2016)].
Acknowledgments
The author wishes to express gratitude to the referee for useful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 6, pp. 894–901.
Rights and permissions
About this article
Cite this article
Nasibov, S.M. A Sobolev Interpolation Inequality and a Gross-Sobolev Logarithmic Inequality. Math Notes 107, 977–983 (2020). https://doi.org/10.1134/S0001434620050296
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620050296