Skip to main content
SpringerLink
Log in

The Affine Convex Lorentz–Sobolev Inequalities

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

The Lorentz-Sobolev inequality on \(C_{0}^{\infty }(\mathbb {R}^n)\) was established by Alvino and Maz’ya independently, and the convex Lorentz–Sobolev inequalities on \(C_{0}^{\infty }(\mathbb {R}^n)\) were showed by Ludwig, Xiao and Zhang. In this paper, we prove the affine Lorentz–Sobolev inequalities on \(W^{1, p}(\mathbb {R}^n)\) and \(BV(\mathbb {R}^n)\), we also prove the affine convex Lorentz–Sobolev inequalities on \(C_{0}^{\infty }(\mathbb {R}^n)\) and \(BV(\mathbb {R}^n)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

The manuscript has no associated data.

References

  1. Alvino, A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. 14, 148–156 (1977)

    MathSciNet  Google Scholar 

  2. Aubin, T.: \(Probl\grave{e}mes\) isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)

    Google Scholar 

  3. Bobkov, S.G., Ledoux, M.: From Brunn–Minkowski to sharp Sobolev inequalities. Ann. Mat. Pura Appl. 187, 369–384 (2008)

    MathSciNet  Google Scholar 

  4. Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)

    MathSciNet  Google Scholar 

  5. Campi, S., Gronchi, P.: The \(L^p\)-Busemann–Petty centroid inequality. Adv. Math. 167, 128–141 (2002)

    MathSciNet  Google Scholar 

  6. Cianchi, A.: A quantitative Sobolev inequality in \(BV\). J. Funct. Anal. 237, 466–481 (2006)

    MathSciNet  Google Scholar 

  7. Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser–Trudinger and Morrey–Sobolev inequalities. Calc. Var. PDE. 36, 419–436 (2009)

    MathSciNet  Google Scholar 

  8. Colesanti, A., Salani, P.: The Brunn–Minkowski inequality for \(p\)-capacity of convex bodies. Math. Ann. 327, 459–479 (2003)

    MathSciNet  Google Scholar 

  9. De Nápoli, P.L., Haddad, J.E., Jiménez, C.H., Montenegro, M.: The sharp affine \(L^2\) Sobolev trace inequality and variants. Math. Ann. 370, 287–308 (2018)

    MathSciNet  Google Scholar 

  10. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, Revised CRC Press, Boca Raton (2015)

    Google Scholar 

  11. Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)

    MathSciNet  Google Scholar 

  12. Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)

    MathSciNet  Google Scholar 

  13. Fleming, W.H., Rischel, R.: An integral formula for total gradient variation. Arch. Math. 11, 218–232 (1960)

    MathSciNet  Google Scholar 

  14. Fang, N., Xu, W., Zhou, J., Zhu, B.: The sharp convex mixed Lorentz–Sobolev inequality. Adv. Appl. Math. 111 (2019). Paper No. 101936

  15. Gagliardo, E.: \(Propriet\grave{a}\) di alcune classi di funzioni in \(pi\grave{u}\) variabili. Ricerche Mat. 7, 102–137 (1958)

    MathSciNet  Google Scholar 

  16. Gardner, R.J.: Geometric Tomography. Encyclopedia of Mathematics and its Applications, vol. 58, 2nd edn. Cambridge University Press, New York (2006)

    Google Scholar 

  17. Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336. Springer, Berlin (2007)

    Google Scholar 

  18. Haddad, J.E., Jiménez, C.H., Montenegro, M.: Sharp affine Sobolev type inequalities via the \(L_{p}\) Busemann–Petty centroid inequality. J. Funct. Anal. 271, 454–473 (2016)

    MathSciNet  Google Scholar 

  19. Haddad, J.E., Jiménez, C.H., Montenegro, M.: Sharp affine weighted \(L^p\) Sobolev type inequalities. Trans. Am. Math. Soc. 372, 2753–2776 (2019)

    MathSciNet  Google Scholar 

  20. Hardy, G.H., Littlewood, J.E., Pólya, G.: Some simple inequalities satisfied by convex functions. Messenger Math. 58, 145–152 (1929)

    Google Scholar 

  21. Haberl, C., Schuster, F.E.: Asymmetric affine \(L_p\) Sobolev inequalities. J. Funct. Anal. 257, 641–658 (2009)

    MathSciNet  Google Scholar 

  22. Leite, E.J.F., Montenegro, M.: Minimization to the Zhang’s energy on \(BV(\Omega )\) and sharp affine Poincaré-Sobolev inequalities. J. Funct. Anal. 283 (2022). Paper No. 109646

  23. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  24. Lutwak, E.: On the affine isoperimetric inequality. Differential Geometry and Its Applications, World Sci. Publ., Teaneck, NJ, 109–118 (1990)

  25. Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    MathSciNet  Google Scholar 

  26. Lutwak, E.: The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    MathSciNet  Google Scholar 

  27. Lutwak, E., Yang, D., Zhang, G.: \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111–132 (2000)

    Google Scholar 

  28. Lutwak, E., Yang, D., Zhang, G.: Sharp affine \(L_{p}\) Sobolev inequalities. J. Differ. Geom. 62, 17–38 (2002)

    Google Scholar 

  29. Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the \(L^p\) Minkowski problem. Int. Math. Res. Not. (2006). Paper No. 62987

  30. Ludwig, M., Xiao, J., Zhang, G: Sharp convex Lorentz–Sobolev inequalities. Math. Ann. 350, 169–197 (2011)

  31. Maz’ya, V.G.: Classes of domains and imbedding theorems for function spaces. Soviet Math. Dokl. 1, 882–885 (1960)

    MathSciNet  Google Scholar 

  32. Maz’ya, V.G.: Certain integral inequalities for functions of many variables. Problems in Mathematical Analysis, Leningrad: LGU, No. 33–68 (1972). (Russian). English translation: J. Soviet Math. 1, 205–234 (1973)

  33. Mazya, V.G.: Sobolev Spaces. Springer Series in Soviet Mathematics, Springer, Berlin (1985)

    Google Scholar 

  34. Maz’ya, V.G.: Conductor and capacitary inequalities for functions on topological spaces and their applications to Sobolev-type imbeddings. J. Funct. Anal. 224, 408–430 (2005)

    MathSciNet  Google Scholar 

  35. Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second, Revised and Augmented Edition. Grundlehren der mathematischen Wiassenschaften [Fundamental Principles of Mathematical Sciences], vol. 342. Springer, Heidelberg (2011)

  36. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13, 115–162 (1959)

    MathSciNet  Google Scholar 

  37. Nguyen, V.H.: Sharp Gagliardo–Nirenberg trace inequalities via mass transportation method and their affine versions. J. Geom. Anal. 30, 2132–2156 (2020)

    MathSciNet  Google Scholar 

  38. Ober, M.: Asymmetric \(L_{p}\) convexification and the convex Lorentz–Sobolev inequality. Monatsh. Math. 179, 113–127 (2016)

    MathSciNet  Google Scholar 

  39. Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry, University of Oklahoma, Norman, Okla., pp. 26–41 (1971)

  40. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia Mathematcal Applications, 2nd edn. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  41. Sobolev, S.L.: On a theorem of functional analysis. Mat. Sb. 471–479 (1938). A.M. S. Transl. Ser. 34, 39–68 (1963)

  42. Schindler, I., Tintarev, C.: Compactness properties and ground states for the affine Laplacian. Calc. Var. PDE. 57 (2018). Paper No. 48

  43. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    MathSciNet  Google Scholar 

  44. Tertikas, A., Zographopoulos, N.B.: Best constants in the Hardy–Rellich inequalities and related improvements. Adv. Math. 209, 407–459 (2007)

    MathSciNet  Google Scholar 

  45. Wang, T.: The affine Sobolev–Zhang inequality on \(BV(\mathbb{R} ^n)\). Adv. Math. 230, 2457–2473 (2012)

    MathSciNet  Google Scholar 

  46. Wang, T., Xiao, J.: A new capacity for the affine bounded variation. Adv. Appl. Math. 121 (2020). Paper No. 102084

  47. Xiao, J.: The sharp Sobolev and isoperimetric inequalities split twice. Adv. Math. 211, 417–435 (2007). Corrigendum: Adv. Math. 268, 906–914 (2015)

  48. Xiao, J.: The p-affine capacity. J. Geom. Anal. 26, 947–966 (2016)

    MathSciNet  Google Scholar 

  49. Xiao, J., Zhang, N.: The relative p-affine capacity. Proc. Am. Math. Soc. 144, 3537–3554 (2016)

    MathSciNet  Google Scholar 

  50. Xiao, J.: The p-affine capacity redux. J. Geom. Anal. 27, 2872–2888 (2017)

    MathSciNet  Google Scholar 

  51. Zhu, G.: Continuity of the solution to the \(L_p\) Minkowski problem. Proc. Am. Math. Soc. 145, 379–386 (2017)

    MathSciNet  Google Scholar 

  52. Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)

    MathSciNet  Google Scholar 

  53. Zhang, G.: New affine isoperimetric inequalities. ICCM. 2, 239–267 (2007)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Ning Zhang for many useful discussion. This paper is supported in part by the National Natural Science Foundation of China (No. 11971005), the Fundamental Research Funds for the Central Universities (Nos. GK202202007, GK202102012) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSZ012).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China

    Wan Li & Baocheng Zhu

Authors
  1. Wan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Baocheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Baocheng Zhu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Zhu, B. The Affine Convex Lorentz–Sobolev Inequalities. J Geom Anal 34, 30 (2024). https://doi.org/10.1007/s12220-023-01471-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01471-y

Keywords

Mathematics Subject Classification

Search

Navigation