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
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

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

[1]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[2]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005

[3]

Yuta Tanoue. Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 53-60. doi: 10.3934/puqr.2021003

[4]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[5]

Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011

[6]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[7]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[8]

Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021034

[9]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[10]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[11]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022

[12]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[13]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

[14]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[15]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[16]

Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023

[17]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[18]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[19]

Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063

[20]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (6)

[Back to Top]