Abstract

The aim of the paper is to prove weighted John-Nirenberg and sharp Trudinger type inequalities for measure-valued (α, p) - Lagrangeans.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Embedding theorems
  • metric spaces
  • doubling measure
  • Lagrangeans
  • weight

Mathematics Subject Classification

35J60, 35J65, 35B30, 35J87

References

  1. M. T. Barlow, B. M. Hambly: Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets, Ann. Inst. H. Poincaré Probab. Statist. 33(5) (1997) 531–556.https://doi.org/10.1016/S0246-0203(97)80104-5Google Scholar
  2. H. Brézis, S. Wainger: A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Diff. Equations 5(7) (1980) 773–789.https://doi.org/10.1080/03605308008820154Google Scholar
  3. R. Capitanelli: Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Analysis 3(1) (2002) 67–80.Google Scholar
  4. R. Capitanelli: Functional inequalities for measure-valued Lagrangeans on homogeneous spaces, Adv. Math. Sci. Appl. 13(1) (2003) 301–313.Google Scholar
  5. R. Capitanelli: Harnack inequalities for p-Laplacians associated to homogeneous p-Lagrangians, Nonlinear Analysis, Ser. A: Theory, Methods Appl. 66(6) (2007) 1302–1317.https://doi.org/10.1016/j.na.2006.01.017Google Scholar
  6. M. J. Carro, J. Soria: Weighted Lorentz spaces and the Hardy operator, J. Funct. Analysis 112 (1993) 480–494.https://doi.org/10.1006/jfan.1993.1042Google Scholar
  7. M. Cwikel, E. Pustylnik: Sobolev type embeddings in the limiting case, J. Fourier Analysis Appl. 4(4-5) (1998) 433–446.https://doi.org/10.1007/BF02498218Google Scholar
  8. B. Franchi, C. Pérez, R. L. Wheeden: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Analysis 153 (1998) 108–146.https://doi.org/10.1006/jfan.1997.3175Google Scholar
  9. J. Garcia-Cuerva, J. L. Rubio De Francia: Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam (1985).Google Scholar
  10. P. Hajłasz, P. Koskela: Sobolev met Poincaré, Memoirs of the AMS, vol. 145, no. 688, American Mathematical Society, Providence (2000).https://doi.org/10.1090/memo/0688Google Scholar
  11. K. Hansson: Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979) 77–102.https://doi.org/10.7146/math.scand.a-11827Google Scholar
  12. P. E. Herman, R. Peirone, R. S. Strichartz: p-Energy and p-harmonic functions on Sierpinski gasket type fractals, Potential Analysis 20(2) (2004) 125–148.https://doi.org/10.1023/A:1026377524793Google Scholar
  13. P. Koskela, J. Onninen: Sharp inequalities via truncation, J. Math. Analysis Appl. 278(2) (2003) 324–334.https://doi.org/10.1016/S0022-247X(02)00465-1Google Scholar
  14. R. A. Macías, C. Segovia: Lipschitz functions on spaces of homogeneous type, Advances Math. 33(3) (1979) 257–270.https://doi.org/10.1016/0001-8708(79)90012-4Google Scholar
  15. P. MacManus, C. Pérez: Generalized Poincaré inequalities: Sharp self-improving properties, Int. Math. Res. Notices 2 (1998) 101–116.Google Scholar
  16. P. MacManus, C. Pérez: Trudinger inequalities without derivatives, Trans. Amer. Math. Soc. 354(5) (2002) 1997–2012.https://doi.org/10.1090/S0002-9947-02-02918-5Google Scholar
  17. J. Malý, U. Mosco: Remarks on measure-valued Lagrangeans on homogeneous spaces, in: Papers in Memory of Ennio De Giorgi, Ricerche Mat. 48 (1999), supplement, 217–231.Google Scholar
  18. J. Malý, L. Pick: An elementary proof of sharp Sobolev embeddings, Proc. Amer. Math. Soc. 130 (2002) 555–563.https://doi.org/10.1090/S0002-9939-01-06060-9Google Scholar
  19. J. Malý, L. Pick: The sharp Riesz potential estimates in metric spaces, Indiana Univ. Math. J. 51(2) (2002) 251–268.https://doi.org/10.1512/iumj.2002.51.2116Google Scholar
  20. V. G. Maz’ya: Sobolev Spaces, Springer, Berlin (1975).Google Scholar
  21. E. Mbakop: Trudinger’s inequality on spaces with measure valued Lagrangeans, Summary WPI (2011).Google Scholar
  22. E. Mbakop, U. Mosco: Gauge functions and Sobolev inequalities on fluctuating domains, in: Around the Research of Vladimir Maz’ya, A. Laptev (ed.), International Mathematical Series 11, Springer, New York (2010) 299–320.https://doi.org/10.1007/978-1-4419-1341-8_14Google Scholar
  23. U. Mosco: Variational fractals, Ann. Scuola Norm. Sup. Pisa, Cl. Sci.(4) 25 (1997) 683–712.Google Scholar
  24. U. Mosco: Harnack inequalities on scale irregular Sierpinski gaskets, in: Nonlinear Problems in Mathematical Physics and Related Topics II, M. S. Birman et al. (eds.), International Mathematical Series 2, Springer, New York (2002) 305–328.https://doi.org/10.1007/978-1-4615-0701-7_17Google Scholar
  25. U. Mosco: Gauged Sobolev inequalities, Appl. Analysis 86(3) (2007) 367–402.https://doi.org/10.1080/00036810701206617Google Scholar
  26. S. I. Pokhozhaev: On eigenfunctions of the equation ∆u + λf (u) = 0, Dokl. Akad. Nauk SSR 165 (1965) 36–39.Google Scholar
  27. R. S. Strichartz: A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Univ. Math. J. 21 (1971/72) 841–842.https://doi.org/10.1512/iumj.1972.21.21066Google Scholar
  28. L. Tartar: Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Ric. Mat. (8) 1(3) (1998) 479–500.Google Scholar
  29. N. S. Trudinger: On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473–483.https://doi.org/10.1512/iumj.1968.17.17028Google Scholar
  30. V. I. Yudovich: Some estimates connected with integral operators and with solutions of elliptic equations, Soviet Math. Doklady 2 (1961) 746–749.Google Scholar
  31. W. P. Ziemer: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variations, Springer, Berlin (1993).Google Scholar

Suggested citation

R. Capitanelli, M. A. Vivaldi. “Sharp Trudinger Type Inequalities for Measure-Valued Lagrangeans.” Journal of Convex Analysis 28 (2021), No. 2, 471–494. https://doi.org/10.68381/jca28027

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.