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Mixed estimates for singular integrals on weighted Hardy spaces

  • María Eugenia Cejas
  • Estefanía DalmassoEmail author
Article
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Abstract

In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces \(H_w^p\), where \(0<p\le 1\) and w is a weight in the Muckenhoupt \(A_{\infty }\) class. We deal with Fourier multiplier operators, Calderón–Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood–Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the \(H^p_w\)-norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound \(A_q-A_\infty \) result for the vector-valued extension of the Hardy–Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993).

Keywords

Weighted Hardy spaces Singular integrals Mixed estimates Calderón–Zygmund operators Fourier multipliers 

Mathematics Subject Classification

42B30 42B20 42B15 42B25 

Notes

Acknowledgements

María Eugenia wants to mention that the study of these estimates was pointed out by Sheldy Ombrosi, her supervisor during postdoctoral research, which was done under a fellowship granted by CONICET, Argentina. Also, she would like to thank Carlos Pérez, Cristina Pereyra and Michael Wilson for their help and interest when contacting them.

Funding

First author was supported by Universidad de Buenos Aires (Grant No. 20020120100050), by Agencia Nacional de Promoción Científica y Tecnológica (Grant No. PICT 2014-1771) and by Universidad Nacional de La Plata (Grant No. UNLP 11/X752 and UNLP 11/X805). Second author was supported by Universidad Nacional del Litoral (Grants No. CAI+D 2015-026 and CAI+D 2015-066).

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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de Ciencias ExactasUniversidad Nacional de La Plata, CONICETLa PlataArgentina
  2. 2.Instituto de Matemática Aplicada del LitoralUNL, CONICET, FIQ.Santa FeArgentina

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