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 (n is an integer) on a set E, implies that E is rectifiable.
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 (n est un nombre entier) sur un compact E implique que E est rectifiable.
Accepted:
Published online:
@article{CRMATH_2009__347_17-18_1051_0, author = {Mayboroda, Svitlana and Volberg, Alexander}, title = {Boundedness of the square function and rectifiability}, journal = {Comptes Rendus. Math\'ematique}, pages = {1051--1056}, publisher = {Elsevier}, volume = {347}, number = {17-18}, year = {2009}, doi = {10.1016/j.crma.2009.07.007}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.07.007/} }
TY - JOUR AU - Mayboroda, Svitlana AU - Volberg, Alexander TI - Boundedness of the square function and rectifiability JO - Comptes Rendus. Mathématique PY - 2009 SP - 1051 EP - 1056 VL - 347 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.07.007/ DO - 10.1016/j.crma.2009.07.007 LA - en ID - CRMATH_2009__347_17-18_1051_0 ER -
%0 Journal Article %A Mayboroda, Svitlana %A Volberg, Alexander %T Boundedness of the square function and rectifiability %J Comptes Rendus. Mathématique %D 2009 %P 1051-1056 %V 347 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.07.007/ %R 10.1016/j.crma.2009.07.007 %G en %F CRMATH_2009__347_17-18_1051_0
Mayboroda, Svitlana; Volberg, Alexander. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.07.007/
[1] Singular integrals and rectifiable sets in : Beyond Lipschitz graphs, Astérisque, Volume 193 (1991)
[2] Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993
[3] The vector-valued non-homogeneous Tb theorem | arXiv
[4] Menger curvature and rectifiability, Ann. of Math. (2), Volume 149 (1999) no. 3, pp. 831-869
[5] S. Mayboroda, A. Volberg, Square function and Riesz transform in non-integer dimensions, preprint
[6] A geometric proof of the boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices, Volume 7 (1995), pp. 325-331
[7] 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] Principal values for the Cauchy integral and rectifiability, Proc. Amer. Math. Soc., Volume 128 (2000) no. 7, pp. 2111-2119
[10] Principal values for Riesz transforms and rectifiability, J. Funct. Anal., Volume 254 (2008) no. 7, pp. 1811-1863
[11] Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality, Proc. London Math. Soc., Volume 98 (2009) no. 2, pp. 393-426
[12] 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)
Cited by Sources:
☆ The first author was partially supported by the NSF grant 0929382. The second author was partially supported by the NSF grant 0758552.