Archiv der Mathematik

, Volume 103, Issue 2, pp 167–175 | Cite as

Asymmetric anisotropic fractional Sobolev norms



Bourgain, Brezis, and Mironescu showed that (with suitable scaling) the fractional Sobolev s-seminorm of a function \({f \in W^{1,p}(\mathbb{R}^n)}\) converges to the Sobolev seminorm of f as \({s\rightarrow1^-}\) . Ludwig introduced the anisotropic fractional Sobolev s-seminorms of f defined by a norm on \({\mathbb{R}^n}\) with unit ball K and showed that they converge to the anisotropic Sobolev seminorm of f defined by the norm whose unit ball is the polar L p moment body of K, as \({s \rightarrow 1^-}\) . The asymmetric anisotropic s-seminorms are shown to converge to the anisotropic Sobolev seminorm of f defined by the Minkowski functional of the polar asymmetric L p moment body of K.

Mathematics Subject Classification

46E35 52A20 


Fractional sobolev norm Lp moment body 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Alvino et al., Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 275–293.Google Scholar
  2. 2.
    J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, A volume in honor of A. Bensoussans’s 60th birthday (Amsterdam) (J. L. Menaldi, E. Rofman, and A. Sulemn, eds.), IOS Press, 2001, pp. 439–455.Google Scholar
  3. 3.
    J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for W s.p when \({s \uparrow 1}\) and applications, J. Anal. Math. 87 (2002), 77–101, Dedicated to the memory of Thomas H. Wolff.Google Scholar
  4. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Di Nezza E., Palatucci G., Valdinoci E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    L. Evans and R. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.Google Scholar
  7. 7.
    Figalli A., Maggi F., Pratelli A.: Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation. Adv. Math. 242, 80–101 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gagliardo E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)zbMATHMathSciNetGoogle Scholar
  9. 9.
    R. J. Gardner, Geometric tomography, 2nd ed., Cambridge Univ. Press, New York, 2006.Google Scholar
  10. 10.
    M. Gromov, Isoperimetric inequalities in Riemannian manifolds, Asymptotic Theory of Finite-dimensional Normed Spaces (V. D. Milman and G. Schechtman, eds.), Springer-Verlag, Berlin Heidelberg, 1986, pp. 114–129.Google Scholar
  11. 11.
    Haberl C.: Minkowski valuations intertwining the special linear group. J. Eur. Math. Soc. 14, 1565–1597 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Haberl C., Schuster F.: General L p affine isoperimetric inequalities. J. Differential Geom. 83, 1–26 (2009)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Ludwig M.: Ellipsoids and matrix valued valuations. Duke Math. J. 119, 159–188 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ludwig M.: Minkowski valuations. Trans. Amer. Math. Soc. 357, 4191–4213 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ludwig M.: Minkowski areas and valuations. J. Differential Geom. 86, 133–161 (2010)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Ludwig M.: Anisotropic fractional perimeters. J. Differential Geom. 96, 77–93 (2014)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Ludwig M.: Anisotropic fractional Sobolev norms. Adv. Math. 252, 150–157 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lutwak E.: Centroid bodies and dual mixed volumes. Proc. London Math. Soc. 60, 365–391 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    E. Lutwak, D. Yang, and G. Zhang, L p affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111–132.Google Scholar
  20. 20.
    E. Lutwak, D. Yang, and G. Zhang, A new ellipsoid associated with convex bodes, Duke. Math. J. 104 (2000), 375–390.Google Scholar
  21. 21.
    E. Lutwak, D. Yang, and G. Zhang, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), 59–81.Google Scholar
  22. 22.
    E. Lutwak, D. Yang, and G. Zhang, Moment-entropy inequalities, Ann. Probab. 32 (2004), 757–774.Google Scholar
  23. 23.
    E. Lutwak, D. Yang, and G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), 365–387.Google Scholar
  24. 24.
    V. G. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, augmented ed., Grundlehren der Mathematischen Wissenschaften, vol. 342, Springer-Verlag, Berlin Heidelberg, 2011.Google Scholar
  25. 25.
    Paouris G. A.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16, 1021–1049 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    G. A. Paouris and E. Werner, Relative entropy of cone measures and L p centroid bodies. Proc. Lond. Math. Soc. 104 (2012), no. 2, 253–286.Google Scholar
  27. 27.
    L. Parapatits, SL(n)-contravariant L p-Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), 1195–1211.Google Scholar
  28. 28.
    Parapatits L.: SL(n)-covariant L p-Minkowski valuations. J. London Math. Soc. 89, 397–414 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Ponce A.: A new approach to Sobolev spaces and connections to Γ-convergence. Calc. Var. Partial Differential Equations 19, 229–255 (2004)CrossRefMathSciNetGoogle Scholar
  30. 30.
    D. Spector, Characterization of Sobolev and BV spaces, Ph.D. thesis, Carnegie Mellon University, 2011.Google Scholar
  31. 31.
    Wannerer T.: GL(n) equivariant Minkowski valuations. Indiana Univ. Math. J. 60, 1655–1672 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

Personalised recommendations