August  2014, 7(4): 695-724. doi: 10.3934/dcdss.2014.7.695

Improved interpolation inequalities on the sphere

1. 

Ceremade (UMR CNRS 7534), Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cédex 16

2. 

CEREMADE - UMR C.N.R.S. 7534, Université Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

3. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago

4. 

School of Mathematics, Skiles Building, Georgia Institute of Technology, Atlanta GA 30332-0160, United States

Received  September 2013 Revised  December 2013 Published  February 2014

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
Citation: Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
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show all references

References:
[1]

Commun. Math. Sci., 5 (2007), 971-979. doi: 10.4310/CMS.2007.v5.n4.a12.  Google Scholar

[2]

J. Funct. Anal., 225 (2005), 337-351. doi: 10.1016/j.jfa.2005.05.003.  Google Scholar

[3]

Comm. Partial Differential Equations, 26 (2001), 43-100. doi: 10.1081/PDE-100002246.  Google Scholar

[4]

Duke Math. J., 43 (1976), 245-268. doi: 10.1215/S0012-7094-76-04322-2.  Google Scholar

[5]

C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 161-164.  Google Scholar

[6]

C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778.  Google Scholar

[7]

in Séminaire de Probabilités, XIX, (1983/84), Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177-206. doi: 10.1007/BFb0075847.  Google Scholar

[8]

C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413.  Google Scholar

[9]

Duke Math. J., 85 (1996), 253-270. doi: 10.1215/S0012-7094-96-08511-7.  Google Scholar

[10]

Proc. Amer. Math. Soc., 105 (1989), 397-400. doi: 10.2307/2046956.  Google Scholar

[11]

Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816-4819. Google Scholar

[12]

Ann. of Math. (2), 138 (1993), 213-242. Google Scholar

[13]

C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187-190.  Google Scholar

[14]

in Séminaire de Probabilités, XXXVI, Lecture Notes in Math., 1801, Springer, Berlin, 2003, 230-250. Google Scholar

[15]

Proc. Japan Acad. Ser. A Math. Sci., 86 (2010), 55-59. doi: 10.3792/pjaa.86.55.  Google Scholar

[16]

Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971.  Google Scholar

[17]

J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.  Google Scholar

[18]

SIAM J. Math. Anal., 25 (1994), 859-875. doi: 10.1137/S0036141092230593.  Google Scholar

[19]

Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922.  Google Scholar

[20]

in Progress in Analysis and Its Applications, World Sci. Publ., Hackensack, NJ, 2010, 463-469. doi: 10.1142/9789814313179_0060.  Google Scholar

[21]

J. Math. Pures Appl., 93 (2010), 449-473. Google Scholar

[22]

Proc. Edinb. Math. Soc. (2), 46 (2003), 117-146. doi: 10.1017/S0013091501000426.  Google Scholar

[23]

Bull. Sci. Math., 127 (2003), 292-312. Google Scholar

[24]

SIAM J. Math. Anal., 34 (2002), 478-494. doi: 10.1137/S0036141001398435.  Google Scholar

[25]

Geom. Funct. Anal., to appear, (2013). Google Scholar

[26]

Indiana Univ. Math. J., 49 (2000), 113-142. doi: 10.1512/iumj.2000.49.1756.  Google Scholar

[27]

J. Math. Kyoto Univ., 44 (2004), 325-363.  Google Scholar

[28]

Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560.  Google Scholar

[29]

J. Eur. Math. Soc. (JEMS), 11 (2009), 1105-1139. doi: 10.4171/JEMS/176.  Google Scholar

[30]

Studia Sci. Math. Hungar., 2 (1967), 299-318  Google Scholar

[31]

J. Math. Pures Appl. (9), 81 (2002), 847-875. doi: 10.1016/S0021-7824(02)01266-7.  Google Scholar

[32]

Ph.D thesis, Université Paul Sabatier Toulouse 3, 2005. Google Scholar

[33]

J. Funct. Anal., 254 (2008), 593-611. Google Scholar

[34]

Chinese Annals of Mathematics, Series B, 34 (2013), 99-112. doi: 10.1007/s11401-012-0756-6.  Google Scholar

[35]

to appear in Analysis & PDE, (2013). Google Scholar

[36]

Tech. Rep., Ceremade, (2013). Google Scholar

[37]

Tech. Rep., Ceremade, (2013). Google Scholar

[38]

in Self-Similar Solutions of Nonlinear PDE, Banach Center Publ., 74, Polish Acad. Sci., Warsaw, 2006, 133-147. doi: 10.4064/bc74-0-8.  Google Scholar

[39]

Commun. Math. Sci., 6 (2008), 477-494. doi: 10.4310/CMS.2008.v6.n2.a10.  Google Scholar

[40]

Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.  Google Scholar

[41]

Bull. Sci. Math., 121 (1997), 71-96.  Google Scholar

[42]

in Séminaire de Probabilités, XXXII, Lecture Notes in Math., 1686, Springer, Berlin, 1998, 14-29. Google Scholar

[43]

Comm. Math. Phys., 298 (2010), 869-878. doi: 10.1007/s00220-010-1079-7.  Google Scholar

[44]

Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[45]

Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.  Google Scholar

[46]

IEEE Trans. Information Theory, IT-14 (1968), 765-766.  Google Scholar

[47]

in Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1745, Springer, Berlin, 2000, 147-168. doi: 10.1007/BFb0107213.  Google Scholar

[48]

C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337-1342.  Google Scholar

[49]

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 249-283.  Google Scholar

[50]

Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[51]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[52]

J. Funct. Anal., 48 (1982), 252-283. doi: 10.1016/0022-1236(82)90069-6.  Google Scholar

[53]

Comm. Math. Phys., 86 (1982), 321-326. doi: 10.1007/BF01212171.  Google Scholar

[54]

Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359-5361. doi: 10.1073/pnas.85.15.5359.  Google Scholar

[55]

J. Funct. Anal., 80 (1988), 212-234. Google Scholar

[56]

J. Funct. Anal., 80 (1988), 148-211. Google Scholar

[57]

Translated and edited by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.  Google Scholar

[58]

Monatsh. Math., 131 (2000), 235-253. doi: 10.1007/s006050070013.  Google Scholar

[59]

J. Funct. Anal., 37 (1980), 218-234. doi: 10.1016/0022-1236(80)90042-7.  Google Scholar

[60]

Proc. Amer. Math. Soc., 102 (1988), 773-774. doi: 10.2307/2047262.  Google Scholar

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