Mixed estimates for singular integrals on weighted Hardy spaces

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).

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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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Correspondence to Estefanía Dalmasso.

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Cejas, M.E., Dalmasso, E. Mixed estimates for singular integrals on weighted Hardy spaces. Rev Mat Complut 33, 745–766 (2020). https://doi.org/10.1007/s13163-019-00331-0

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Keywords

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

Mathematics Subject Classification

  • 42B30
  • 42B20
  • 42B15
  • 42B25