A new, rearrangement-free proof of the sharp hardy-littlewood-sobolev inequality

Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends

Volumn , Issue , 2012, Pages 55-67

  Frank, Rupert L ^a   Lieb, Elliott H ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

Hardy Littlewood Sobolev inequality; Sharp constants; Sobolev inequality

Indexed keywords

HARDY-LITTLEWOOD-SOBOLEV INEQUALITY; HEISENBERG GROUP; PARAMETER RANGE; SHARP CONSTANTS; SOBOLEV INEQUALITIES;

EID: 85014718935     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-0348-0263-5     Document Type: Chapter

References (25)

1. M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, New York, 1992.
2. Th. Aubin, Problémes isoperimétriques et espaces de Sobolev. J. Differ. Geometry 11 (1976), 573-598.
3. W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn. Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816-4819.
4. W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2) 138 (1993), no. 1, 213-242.
5. E. A. Carlen, J. A. Carrillo, M. Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA 107 (2010), no. 46, 19696-19701.
6. E. A. Carlen, M. Loss, Extremals of functionals with competing symmetries. J. Funct. Anal. 88 (1990), no. 2, 437-456.
7. E. A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on Sn. Geom. Funct. Anal. 2 (1992), no. 1, 90-104.
8. S-Y. A. Chang, P. C. Yang, Prescribing Gaussian curvature on S2. ActaMath. 159 (1987), no. 3-4, 215-259.
9. A. Erdélyi, M. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions. Vol. II. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Melbourne, FL, 1981.
10. R. L. Frank, E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality. Calc. Var. PDE 39 (2010), no. 1-2, 85-99.
11. R. L. Frank, E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group. Preprint (2010), arXiv:1009.1410.
12. I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007.
13. G. H. Hardy, J. E. Littlewood, Some properties of fractional integrals (1). Math. Z. 27 (1928), 565-606.
14. G. H. Hardy, J. E. Littlewood, On certain inequalities connected with the calculus of variations. J. London Math. Soc. 5 (1930), 34-39.
15. J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homoǵenes. C. R. Acad. Sci. Paris Sr. A-B 270 (1970), A1645-A1648.
16. Y. Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), no. 2, 383-417.
17. E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. 57 (1977), no. 2, 93-105.
18. E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349-374.
19. E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies inMathematics 14, Amer. Math. Soc., Providence, RI, 2001.
20. P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121.
21. E. Onofri, On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys. 86 (1982), no. 3, 321-326.
22. 3. SIAM J. Appl. Math. 21 (1971), 30-32.
23. S. L. Sobolev, On a theorem of functional analysis. Mat. Sb. (N.S.) 4 (1938), 471-479; English transl. in Amer. Math. Soc. Transl. Ser. 2 34 (1963), 39-68.
24. S. L. Sobolev, On a theorem of functional analysis English transl. in Amer. Math. Soc. Transl. Ser. 2 34 (1963), 39-68.
25. G. Talenti, Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353-372.