Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Factoring Sobolev inequalities through classes of functions


Authors: David Alonso-Gutiérrez, Jesús Bastero and Julio Bernués
Journal: Proc. Amer. Math. Soc. 140 (2012), 3557-3566
MSC (2010): Primary 46E35; Secondary 46E30, 26D10, 52A40
DOI: https://doi.org/10.1090/S0002-9939-2012-11355-3
Published electronically: February 24, 2012
MathSciNet review: 2929024
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, while the second one relies on tools from Convex Geometry. In this paper we prove a (sharp) connection between them.


References [Enhancements On Off] (What's this?)

  • [1] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), 573-598. MR 0448404 (56:6711)
  • [2] S. Bobkov and M. Ledoux, From Brunn-Minkowski to sharp Sobolev inequalities, Ann. Mat. Pura Appl.(4) 187 (2008), 369-384. MR 2393140 (2010i:46052)
  • [3] J. Bastero, M. Milman and F.J. Ruiz, A note on $ L(\infty , q)$ spaces and Sobolev embeddings, Indiana U. Math. J. 52 (2003), 1215-1229. MR 2010324 (2004h:46025)
  • [4] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Diff. Eq. 5 (1980), 773-789. MR 579997 (81k:46028)
  • [5] I. Chavel, Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives in Mathematical Physics, Cambridge Tracts in Mathematics 145, Cambridge Univ. Press, 2001. MR 1849187 (2002h:58040)
  • [6] A. Cianchi, E. Lutwak, D. Yang and G. Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differ. Equ. 36 (2009), 419-436. MR 2551138 (2010h:46041)
  • [7] D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), 307-332. MR 2032031 (2005b:26023)
  • [8] H. Federer and W. H. Fleming, Normal integral currents. Ann. of Math. (2) 72 (1960), 458-520. MR 0123260 (23:A588)
  • [9] C. Haberl and F.E. Schuster, Asymmetric affine $ L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), 641-658. MR 2530600 (2010j:46068)
  • [10] C. Haberl and F.E. Schuster, General $ L_p$ affine isoperimetric inequalties, J. Differential Geom. 83 (2009), 1-26. MR 2545028 (2010j:52026)
  • [11] C. Haberl, F.E. Schuster and J. Xiao, An asymmetric affine Pólya-Szegö principle, preprint.
  • [12] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77-102. MR 567435 (81j:31007)
  • [13] S. Kesavan, Symmetrization & Applications. Series in Analysis, vol. 3. World Scientific Publ., Hackensack, NJ, 2006. MR 2238193 (2008a:35005)
  • [14] V. Kolyada, Rearrangements of functions and embedding theorems, Russ. Math. Surveys Eq. 44 (1989), 73-117. MR 1040269 (91i:46029)
  • [15] E. Lutwak, The Brunn-Minkowski-Firey Theory I: Mixed volumes and the Minkowski problem. J. Differential Geom. 38 (1993), 131-150. MR 1231704 (94g:52008)
  • [16] E. Lutwak, D. Yang and G. Zhang, On $ L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111-132. MR 1863023 (2002h:52011)
  • [17] E. Lutwak, D. Yang and G. Zhang, Sharp affine $ L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), 17-38. MR 1987375 (2004d:46039)
  • [18] W. G. Maz$ '$ja, Classes of domains and embedding theorems for functional spaces, Soviet Math. Dokl. 1 (1961), 882-885. MR 0126152 (23:A3448)
  • [19] W. G. Maz$ '$ja, Sobolev spaces, Springer Series in Soviet Mathematics 145, Springer-Verlag, Berlin, 1985. MR 817985 (87g:46056)
  • [20] J. Malý and L. Pick, An elementary proof of sharp Sobolev embeddings, Proc. Amer. Math. Soc. 13 (2002), 555-563. MR 1862137 (2002j:46042)
  • [21] J. Martin and M. Milman, Pointwise symmetrization inequalities for Sobolev functions and applications, Adv. Math. 225 (2010), 121-199. MR 2669351
  • [22] J. Martin, M. Milman, and E. Pustylnik, Sobolev inequalities: symmetrization and self-improvement via truncation, J. Funct. Anal. 252 (2) (2007), 677-695. MR 2360932 (2009a:46059)
  • [23] J. Moser, A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. MR 0301504 (46:662)
  • [24] J. Mossino, Inégalités Isopérimétriques et Applications en Physique, Travaux en Cours, Hermann, Paris, 1984. MR 733257 (85k:49002)
  • [25] C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547. MR 0133733 (24:A3558)
  • [26] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stu. 27, Princeton University Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
  • [27] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • [28] G. Talenti, Best constants in Sobolev inequality, Ann. Math. Pura Appl. 110 (1976), 353-372. MR 0463908 (57:3846)
  • [29] G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear Analysis, Function Spaces and Applications 5, Prometheus, Prague, 177-230, 1994. MR 1322313 (96a:46062)
  • [30] L. Tartar, Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8, 1 (1998), 479-500. MR 1662313 (99k:46060)
  • [31] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. MR 0216286 (35:7121)
  • [32] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), 183-202. MR 1776095 (2001m:53136)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46E35, 46E30, 26D10, 52A40

Retrieve articles in all journals with MSC (2010): 46E35, 46E30, 26D10, 52A40


Additional Information

David Alonso-Gutiérrez
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: daalonso@unizar.es

Jesús Bastero
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: bastero@unizar.es

Julio Bernués
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: bernues@unizar.es

DOI: https://doi.org/10.1090/S0002-9939-2012-11355-3
Keywords: Sobolev inequality, sharp constants, affine isoperimetric inequalities
Received by editor(s): October 20, 2010
Received by editor(s) in revised form: April 15, 2011
Published electronically: February 24, 2012
Additional Notes: The three authors were partially supported by MCYT Grant(Spain) MTM2010-16679 and DGA E-64
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society