Comptes Rendus
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]
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/

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[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

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[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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