Browse > Article
http://dx.doi.org/10.14403/jcms.2011.24.2.12

FRACTIONAL POLYA-SZEGÖ INEQUALITY  

Park, Young Ja (Department of Mathematics, Hoseo University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.24, no.2, 2011 , pp. 267-271 More about this Journal
Abstract
Let 0 < s < 1. For $f^{\ast}$ representing the symmetric radial decreasing rearrangement of f, we build up a fractional version of Polya-$Szeg{\ddot{o}}$ inequality: $${\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f^{\ast}(x){\mid}^2dx{\leq}{\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f(x){\mid}^2dx$$.
Keywords
Polya-$Szeg{\ddot{o}}$ inequality; symmetric decreasing rearrangement; kinetic energy reduction; nonexpansivity;
Citations & Related Records
연도 인용수 순위
  • Reference
1 E. H. Lieb and M. Loss, Analysis, Volume 14 of Graduate Studies in Mathematics, AMS, 1997.
2 Y. J. Park, Logarithmic Sobolev trace inequality, Proc. Amer. Math. Soc. 132 (2004). 2067-2073.   DOI   ScienceOn
3 G. Polya and G. Szego, Inequalities for the capacity of a condenser, Amer. J. Math. 67 (1945), 1-32.   DOI   ScienceOn
4 Frederick J. Almgren and Elliott H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), 683-773.   DOI   ScienceOn
5 W. Beckner, Inequalities in Fourier analysis, Ann. Math. 102 (1975), 159-182.   DOI
6 W. Beckner, Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in honor of Elias M. Stein, Princeton University Press, 36-68, 1995.
7 W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere on Sn, Proc. Natl. Acad. Sci. 89 (1992), 4816-4819.   DOI   ScienceOn
8 H. Hajaiej and C.A. Stuart, Existence and non-existence of Schwarz symmet- ric ground states for elliptic eigenvalue problems, Ann. Mat. Pura Appl. 184 (2005), 297-314.   DOI   ScienceOn
9 B. Kawohl, Rearrangements and convexity of level sets in PDE, Springer- Verlag, 1985.
10 E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquards nonlinear equation, Studies in Appl. Math. 57 (1976/77), 93-105.   DOI