Comptes Rendus
no. 21-22
Harmonic Analysis/Functional Analysis
Finite square function implies integer dimension
[Pour que la fonction carrée de mesure de Hausdorff soit finie il faut que la dimension de mesure soit entier]

Présenté par : Gilles Pisier

Svitlana Mayboroda1 ; Alexander Volberg2
1 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1271-1276.

On peut modifier l'article recent (Tolsa and Ruiz de Villa, 2008) pour démontrer que la convergence de la fonction carrée associée aux transformations de Riesz de mesure de Hausdorff Hs implique que s est un nombre entier.

Following a recent paper (Tolsa and Ruiz de Villa, 2008) we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure Hs implies that s is integer.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.020
Svitlana Mayboroda 1 ; Alexander Volberg 2

1 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
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Svitlana Mayboroda; Alexander Volberg. Finite square function implies integer dimension. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1271-1276. doi : 10.1016/j.crma.2009.09.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.020/

[1] G. David; S. Semmes Singular integrals and rectifiable sets in Rn: Beyond Lipschitz graphs, Astérisque, Volume 193 (1991), p. 152

[2] G. David; S. Semmes Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993

[3] P. Mattila; M. Melnikov; J. Verdera The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2), Volume 144 (1996) no. 1, pp. 127-136

[4] F. Nazarov; S. Treil; A. Volberg Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices, Volume 9 (1998), pp. 463-487

[5] F. Nazarov; S. Treil; A. Volberg The Tb-theorem on non-homogeneous spaces, Acta Math., Volume 190 (2003) no. 2, pp. 151-239

[6] L. Prat Potential theory of signed Riesz kernels: Capacity and Hausdorff measure, Internat. Math. Res. Notices, Volume 19 (2004), pp. 937-981

[7] L. Prat, Principal values for the signed Riesz kernels of non-integer dimensions, preprint, 2006, as cited in [10]

[8] X. Tolsa Principal values for the Cauchy integral and rectifiability, Proc. Amer. Math. Soc., Volume 128 (2000) no. 7, pp. 2111-2119

[9] X. Tolsa Xavier growth estimates for Cauchy integrals of measures and rectifiability, Geom. Funct. Anal., Volume 17 (2007) no. 2, pp. 605-643

[10] X. Tolsa, A. Ruiz de Villa, Non existence of principal values of signed Riesz transforms of non integer dimension, preprint, 2008

[11] M. Vihtilä The boundedness of Riesz s-transforms of measures in Rn, Proc. Amer. Math. Soc., Volume 124 (1996) no. 12, pp. 3797-3804

[12] A. Volberg Calderón–Zygmund Capacities and Operators on Nonhomogeneous Spaces, CBMS Regional Conference Series in Mathematics, vol. 100, American Mathematical Society, Providence, RI, 2003 (published for the Conference Board of the Mathematical Sciences, Washington, DC)

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