On the Calderón–Zygmund structure of Petermichl's kernel

Hugo Aimar; Ivana Gómez

doi:10.1016/j.crma.2018.04.002

Harmonic analysis

On the Calderón–Zygmund structure of Petermichl's kernel

Presented by: Yves Meyer

Hugo Aimar ¹; Ivana Gómez ¹

¹ Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ, CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac. 168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina

Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 509-516.

Abstract
Résumé

We show that Petermichl's dyadic operator $P$ (Petermichl (2000) [8]) is a Calderón–Zygmund-type operator on an adequate metric normal space of homogeneous type. We also compare the maximal operators associated with truncations of the kernel and to the summability of the Haar series.

Nous démontrons que l'opérateur dyadique de Petermichl $P$ est un opérateur de type Calderón–Zygmund sur un espace normal métrique de type homogène. Nous comparons les opérateurs maximaux associés aux troncatures du noyau et à la sommabilité de la série de Haar.

Received: 2017-07-11
Accepted: 2018-04-05
Published online: 2018-04-10

DOI: 10.1016/j.crma.2018.04.002

Author's affiliations:

Hugo Aimar ¹; Ivana Gómez ¹

¹ Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ, CCT CONICET Santa Fe, Predio “Alberto Cassano”, Colectora Ruta Nac. 168 km 0, Paraje El Pozo, S3007ABA Santa Fe, Argentina

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     author = {Hugo Aimar and Ivana G\'omez},
     title = {On the {Calder\'on{\textendash}Zygmund} structure of {Petermichl's} kernel},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {509--516},
     publisher = {Elsevier},
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     year = {2018},
     doi = {10.1016/j.crma.2018.04.002},
     language = {en},
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Hugo Aimar; Ivana Gómez. On the Calderón–Zygmund structure of Petermichl's kernel. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 509-516. doi : 10.1016/j.crma.2018.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.002/

References
Cited by

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[2] H. Aimar Singular integrals and approximate identities on spaces of homogeneous type, Transl. Amer. Math. Soc., Volume 292 (1985) no. 1, pp. 135-153 (MR805957)

[3] H. Aimar; R.A. Macías Weighted norm inequalities for the Hardy–Littlewood maximal operator on spaces of homogeneous type, Proc. Amer. Math. Soc., Volume 91 (1984) no. 2, pp. 213-216 (MR740173)

[4] R.R. Coifman; G. Weiss Analyse harmonique non-commutative sur certains espaces homogènes : Étude de certaines intégrales singulières, Lecture Notes in Mathematics, vol. 242, Springer-Verlag, Berlin–New York, 1971 (MR0499948)

[5] R.A. Macías; C. Segovia A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math., Volume 33 (1979) no. 3, pp. 271-309 (MR546296)

[6] R.A. Macías; C. Segovia Lipschitz functions on spaces of homogeneous type, Adv. Math., Volume 33 (1979) no. 3, pp. 257-270 (MR546295)

[7] Y. Meyer; R. Coifman Wavelets – Calderón – Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge, UK, 1997 (translated from the 1990 and 1991 French originals by David Salinger, MR1456993)

[8] S. Petermichl Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris, Ser. I, Volume 330 (2000) no. 6, pp. 455-460 (MR1756958)

Cited by Sources:

^☆ This work was supported by CONICET (grant PIP-112-2011010-0877, 2012); ANPCyT-MINCyT (grants PICT-2568, 2012; PICT-3631, 2015); and UNL (grant CAID-50120110100371LI, 2013).

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