Comptes Rendus
no. 5
Harmonic analysis
On the Calderón–Zygmund structure of Petermichl's kernel
[Sur la structure Calderón–Zygmund du noyau de Petermichl]

Présenté par : Yves Meyer

Hugo Aimar1 ; Ivana Gómez1
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.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.002
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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     pages = {509--516},
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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/

[1] H. Aimar Integrales singulares y aproximaciones de la identidad en espacios de tipo homogéneo, Universidad Nacional de Buenos Aires, PEMA–INTEC, 1983 http://www.imal.santafe-conicet.gov.ar/TesisIMAL/tesisAimarH.pdf (Doctoral thesis Available in:)

[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)

Cité par 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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