Sharp affine weighted $L^p$ Sobolev type inequalities
HTML articles powered by AMS MathViewer
- by J. Haddad, C. H. Jiménez and M. Montenegro PDF
- Trans. Amer. Math. Soc. 372 (2019), 2753-2776 Request permission
Abstract:
We establish sharp affine weighted $L^p$ Sobolev type inequalities by using the $L_p$ Busemann–Petty centroid inequality proved by Lutwak, Yang, and Zhang. Our approach consists of combining the latter with a suitable family of sharp weighted $L^p$ Sobolev type inequalities obtained by Nguyen and allows us to characterize all extremizers in some cases. The new inequalities do not rely on any euclidean geometric structure.References
- Thierry Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598 (French). MR 448404
- Dominique Bakry, Ivan Gentil, and Michel Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348, Springer, Cham, 2014. MR 3155209, DOI 10.1007/978-3-319-00227-9
- William Beckner, Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), no. 1, 105–137. MR 1673903, DOI 10.1515/form.11.1.105
- S. G. Bobkov and M. Ledoux, From Brunn-Minkowski to sharp Sobolev inequalities, Ann. Mat. Pura Appl. (4) 187 (2008), no. 3, 369–384. MR 2393140, DOI 10.1007/s10231-007-0047-0
- X. Cabré and X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations, 255 (2013) 4312–4336.
- X. Cabré, X. Ros-Oton, and J. Serra, Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. 18 (2016) 2971–2998.
- S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, DOI 10.1006/aima.2001.2036
- Andrea Cianchi, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), no. 3, 419–436. MR 2551138, DOI 10.1007/s00526-009-0235-4
- D. Cordero-Erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), no. 2, 307–332. MR 2032031, DOI 10.1016/S0001-8708(03)00080-X
- Adriano Alves de Medeiros, The weighted Sobolev and mean value inequalities, Proc. Amer. Math. Soc. 143 (2015), no. 3, 1229–1239. MR 3293738, DOI 10.1090/S0002-9939-2014-12337-9
- Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875 (English, with English and French summaries). MR 1940370, DOI 10.1016/S0021-7824(02)01266-7
- Manuel Del Pino and Jean Dolbeault, The optimal Euclidean $L^p$-Sobolev logarithmic inequality, J. Funct. Anal. 197 (2003), no. 1, 151–161. MR 1957678, DOI 10.1016/S0022-1236(02)00070-8
- I. Fabbri, Remarks on some weighted Sobolev inequalities and applications, 2005. Thesis (Ph.D)–Università degli studi Roma Tre.
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
- Wendell H. Fleming and Raymond Rishel, An integral formula for total gradient variation, Arch. Math. (Basel) 11 (1960), 218–222. MR 114892, DOI 10.1007/BF01236935
- Ivan Gentil, The general optimal $L^p$-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations, J. Funct. Anal. 202 (2003), no. 2, 591–599. MR 1990539, DOI 10.1016/S0022-1236(03)00047-8
- Christoph Haberl and Franz E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641–658. MR 2530600, DOI 10.1016/j.jfa.2009.04.009
- Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028
- Christoph Haberl, Franz E. Schuster, and Jie Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann. 352 (2012), no. 3, 517–542. MR 2885586, DOI 10.1007/s00208-011-0640-9
- J. Haddad, C. H. Jiménez, and M. Montenegro, Sharp affine Sobolev type inequalities via the $L_p$ Busemann-Petty centroid inequality, J. Funct. Anal. 271 (2016), no. 2, 454–473. MR 3501854, DOI 10.1016/j.jfa.2016.03.017
- Michel Ledoux, Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics (Saint-Flour, 1994) Lecture Notes in Math., vol. 1648, Springer, Berlin, 1996, pp. 165–294. MR 1600888, DOI 10.1007/BFb0095676
- Monika Ludwig, Jie Xiao, and Gaoyong Zhang, Sharp convex Lorentz-Sobolev inequalities, Math. Ann. 350 (2011), no. 1, 169–197. MR 2785767, DOI 10.1007/s00208-010-0555-x
- Erwin Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), no. 1, 1–13. MR 840399
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370. MR 2067123, DOI 10.1090/S0002-9947-03-03403-2
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Optimal Sobolev norms and the $L^p$ Minkowski problem, Int. Math. Res. Not. , posted on (2006), Art. ID 62987, 21. MR 2211138, DOI 10.1155/IMRN/2006/62987
- Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
- V. G. Maz’ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960) 882–885.
- Bruno Nazaret, Best constant in Sobolev trace inequalities on the half-space, Nonlinear Anal. 65 (2006), no. 10, 1977–1985. MR 2258478, DOI 10.1016/j.na.2005.05.069
- Van Hoang Nguyen, Sharp weighted Sobolev and Gagliardo-Nirenberg inequalities on half-spaces via mass transport and consequences, Proc. Lond. Math. Soc. (3) 111 (2015), no. 1, 127–148. MR 3404778, DOI 10.1112/plms/pdv026
- C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547. MR 133733, DOI 10.2140/pjm.1961.11.1535
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
- Tuo Wang, The affine Sobolev-Zhang inequality on $BV(\Bbb R^n)$, Adv. Math. 230 (2012), no. 4-6, 2457–2473. MR 2927377, DOI 10.1016/j.aim.2012.04.022
- T. Wang, The affine Pólya-Szegö principle: Equality cases and stability, J. Funct. Anal. 265 (2013) 1728–1748.
- Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
- Giorgio Talenti, A weighted version of a rearrangement inequality, Ann. Univ. Ferrara Sez. VII (N.S.) 43 (1997), 121–133 (1998) (English, with English and Italian summaries). MR 1686750
- Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095
Additional Information
- J. Haddad
- Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, Brazil
- MR Author ID: 960778
- Email: jhaddad@mat.ufmg.br
- C. H. Jiménez
- Affiliation: Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente, 225 Edificio Cardeal Leme, Gávea 22.451-900, Rio de Janeiro, Brazil
- Email: hugojimenez@mat.puc-rio.br
- M. Montenegro
- Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, Brazil
- MR Author ID: 662861
- Email: montene@mat.ufmg.br
- Received by editor(s): September 27, 2017
- Received by editor(s) in revised form: May 8, 2018, and July 9, 2018
- Published electronically: December 7, 2018
- Additional Notes: The first author was partially supported by Fapemig (APQ-01454-15).
The second author was partially supported by CNPq (PQ 305650/2016-5) and the program Incentivo à produtividade em ensino e pesquisa of the PUC-Rio.
The third author was partially supported by CNPq (PQ 306855/2016-0) and Fapemig (APQ 02574-16). - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 2753-2776
- MSC (2010): Primary 46E35; Secondary 52A40, 52A05
- DOI: https://doi.org/10.1090/tran/7728
- MathSciNet review: 3988592