×

A generalization of a logarithmic Sobolev inequality to the Hölder class. (English) Zbl 1242.46041

Summary: In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality has originated from the Brezis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of the Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] H. Ibrahim, “A critical parabolic Sobolev embedding via Littlewood-Paley decomposition,” preprint. · Zbl 1266.42059
[2] T. Ogawa, “Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow,” SIAM Journal on Mathematical Analysis, vol. 34, no. 6, pp. 1318-1330, 2003. · Zbl 1036.35082
[3] H. Brézis and T. Gallouet, “Nonlinear Schrödinger evolution equations,” Nonlinear Analysis, vol. 4, no. 4, pp. 677-681, 1980. · Zbl 0451.35023
[4] H. Brézis and S. Wainger, “A note on limiting cases of Sobolev embeddings and convolution inequalities,” Communications in Partial Differential Equations, vol. 5, no. 7, pp. 773-789, 1980. · Zbl 0437.35071
[5] O. A. Lady\vzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, USA, 1967.
[6] H. Ibrahim, M. Jazar, and R. Monneau, “Global existence of solutions to a singular parabolic/Hamilton-Jacobi coupled system with Dirichlet conditions,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 346, no. 17-18, pp. 945-950, 2008. · Zbl 1166.35313
[7] H. Ibrahim and R. Monneau, “On a parabolic logarithmic Sobolev inequality,” Journal of Functional Analysis, vol. 257, no. 3, pp. 903-930, 2009. · Zbl 1180.46024
[8] H. Engler, “An alternative proof of the Brezis-Wainger inequality,” Communications in Partial Differential Equations, vol. 14, no. 4, pp. 541-544, 1989. · Zbl 0688.46016
[9] H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251-278, 2002. · Zbl 1055.35087
[10] H. Kozono, T. Ogawa, and Y. Taniuchi, “Navier-Stokes equations in the Besov space near L\infty and BMO,” Kyushu Journal of Mathematics, vol. 57, no. 2, pp. 303-324, 2003. · Zbl 1067.35064
[11] H. Kozono and Y. Taniuchi, “Limiting case of the Sobolev inequality in BMO, with application to the Euler equations,” Communications in Mathematical Physics, vol. 214, no. 1, pp. 191-200, 2000. · Zbl 0985.46015
[12] H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983. · Zbl 0546.46027
[13] H. Triebel, Theory of Function Spaces. III, vol. 100 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2006. · Zbl 1104.46001
[14] M. Bownik, “Anisotropic Triebel-Lizorkin spaces with doubling measures,” The Journal of Geometric Analysis, vol. 17, no. 3, pp. 387-424, 2007. · Zbl 1147.42006
[15] M. Bownik, “Duality and interpolation of anisotropic Triebel-Lizorkin spaces,” Mathematische Zeitschrift, vol. 259, no. 1, pp. 131-169, 2008. · Zbl 1213.42062
[16] M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces,” Journal of Functional Analysis, vol. 93, no. 1, pp. 34-170, 1990. · Zbl 0716.46031
[17] M. Bownik, “Anisotropic Hardy spaces and wavelets,” Memoirs of the American Mathematical Society, vol. 164, no. 781, p. vi+122, 2003. · Zbl 1036.42020
[18] W. Farkas, J. Johnsen, and W. Sickel, “Traces of anisotropic Besov-Lizorkin-Triebel spaces-a complete treatment of the borderline cases,” Mathematica Bohemica, vol. 125, no. 1, pp. 1-37, 2000. · Zbl 0970.46019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.