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Some inequalities related to Sobolev norms

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Abstract

In this paper, we study some properties related to the new characterizations of Sobolev spaces introduced in Bourgain and Nguyen (C R Acad Sci, 343:75–80, [2006]), Nguyen (J Funct Anal 237: 689–720, [2006]; J Eur Math Soc 10:191–229, [2008]). More precisely, we establish variants of the Poincaré inequality, the Sobolev inequality, and the Rellich–Kondrachov compactness theorem, where \({\int_{\mathbb{R}^N} |\nabla g|^p \;dx}\) is replaced by some quantity of the type

$$I_{\delta} (g) =\mathop{\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N}}_{|g(x) - g(y)| > \delta}\frac{\delta^p}{|x-y|^{N+p}}\, dx \, dy.$$

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References

  1. Ambrosio L.: Metric space valued functions of bounded variation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17, 493–498 (1990)

    MathSciNet  Google Scholar 

  2. Bourgain J., Nguyen H.-M.: A new characterization of Sobolev spaces. C. R. Acad. Sci. Paris 343, 75–80 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations. A Volume in Honour of A. Bensoussan’s 60th Birthday, pp. 439–455. IOS Press, Amsterdam (2001)

  4. Bourgain, J., Brezis, H., Mironescu, P.: Complements to the paper: Lifting, Degree, and Distributional Jacobian Revisited, personal communication (2004)

  5. Brezis, H.: Analyse Fonctionnelle. Théorie et applications, Mathématiques appliquées pour la maîtrise. Dunod, Paris (2002)

  6. Brezis, H., Nguyen, H.-M.: On the distributional Jacobian of maps from \({\mathbb{S}^N}\) into \({\mathbb{S}^N}\) in fractional Sobolev and Hölder spaces. Ann. Math. to appear (2010)

  7. Brezis, H., Nguyen, H.-M.: On a new class of functions related to VMO, C. R. Acad. Sci. Paris, to appear (2010)

  8. Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)

    Google Scholar 

  9. Fefferman C., Stein E.: H p spaces of several variables. Acta. Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Franchi B., Lu G., Wheeden R.: A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type. Int. Math. Res. Not. 1, 1–14 (1996)

    Article  MathSciNet  Google Scholar 

  11. Garofalo N., Nhieu D.-M.: Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hajlasz P., Koskela P.: Sobolev met Poincaré. Memoirs Am. Math. Soc. 145, 1–101 (2000)

    MathSciNet  Google Scholar 

  13. John F., Nirenberg L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  14. Korevaar N., Schoen R.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1, 561–659 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Mazya V.G.: Sobolev Spaces. Springer, Berlin (1985)

    Google Scholar 

  16. Nguyen H.-M.: Some new characterizations of Sobolev spaces. J. Funct. Anal. 237, 689–720 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nguyen H.-M.: Γ-convergence and Sobolev norms. C. R. Acad. Sci. Paris 345, 679–684 (2007)

    MATH  Google Scholar 

  18. Nguyen H.-M.: Further characterizations of Sobolev spaces. J. Eur. Math. Soc. 10, 191–229 (2008)

    MATH  Google Scholar 

  19. Nguyen, H.-M.: Γ-convergence, Sobolev norms, and BV functions, Duke Math. J., to appear (2010)

  20. Ponce A.C.: An estimate in the spirit of Poincaré’s inequality. J. Eur. Math. Soc. 6, 1–15 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reshetnyak Y.: Sobolev classes of functions with values in a metric space. Sib. Math. J. 38, 567–583 (1997)

    Article  MathSciNet  Google Scholar 

  22. Saloff-Coste L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992)

    Article  MathSciNet  Google Scholar 

  23. Stein E.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

    Google Scholar 

  24. Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43. Princeton University Press, Princeton (1993)

    Google Scholar 

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Authors and Affiliations

  1. Courant Institute, NYU, 251 Mercer Street, New York, NY, 10012, USA

    Hoai-Minh Nguyen

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  1. Hoai-Minh Nguyen
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Correspondence to Hoai-Minh Nguyen.

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Communicated by H. Brezis.

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Nguyen, HM. Some inequalities related to Sobolev norms. Calc. Var. 41, 483–509 (2011). https://doi.org/10.1007/s00526-010-0373-8

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  • DOI: https://doi.org/10.1007/s00526-010-0373-8

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