Comptes Rendus
no. 17-18
Harmonic Analysis
Boundedness of the square function and rectifiability
[Une fonction carrée de la transformation de Riesz et rectifiabilité]

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. 17-18, pp. 1051-1056.

On peut modifier l'article récent [X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal. 254 (7) (2008) 1811–1863] pour démontrer que la convergence de la fonction carrée associée aux transformations de Riesz de mesure de Hausdorff Hn (n est un nombre entier) sur un compact E implique que E est rectifiable.

Following a recent paper [X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal. 254 (7) (2008) 1811–1863] we show that the finiteness of the square function associated with the Riesz transforms with respect to Hausdorff measure Hn (n is an integer) on a set E, implies that E is rectifiable.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.07.007
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. Boundedness of the square function and rectifiability. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1051-1056. doi : 10.1016/j.crma.2009.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.007/

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[3] T. Hytönen The vector-valued non-homogeneous Tb theorem | arXiv

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[5] S. Mayboroda, A. Volberg, Square function and Riesz transform in non-integer dimensions, preprint

[6] M. Melnikov; J. Verdera A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices, Volume 7 (1995), pp. 325-331

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

[8] L. Prat, Principal values for the signed Riesz kernels of non-integer dimensions, preprint

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

[10] X. Tolsa Principal values for Riesz transforms and rectifiability, J. Funct. Anal., Volume 254 (2008) no. 7, pp. 1811-1863

[11] X. Tolsa Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality, Proc. London Math. Soc., Volume 98 (2009) no. 2, pp. 393-426

[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)

Cité par Sources :

☆ The first author was partially supported by the NSF grant 0929382. The second author was partially supported by the NSF grant 0758552.

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