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Weighted norm inequalities for maximally modulated singular integral operators

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Abstract.

We present a framework that yields a variety of weighted and vector-valued estimates for maximally modulated Calderón-Zygmund singular (and maximal singular) integrals from a single a priori weak type unweighted estimate for the maximal modulations of such operators. We discuss two approaches, one based on the good-λ method of Coifman and Fefferman [CF] and an alternative method employing the sharp maximal operator. As an application we obtain new weighted and vector-valued inequalities for the Carleson operator proving that it is controlled by a natural maximal function associated with the Orlicz space L(log L)(log log log L). This control is in the sense of a good-λ inequality and yields strong and weak type estimates as well as vector-valued and weighted estimates for the operator in question.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    José María Martell & Fernando Soria

Authors
  1. Loukas Grafakos
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  2. José María Martell
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  3. Fernando Soria
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Correspondence to Loukas Grafakos.

Additional information

Mathematics Subject Classification (2000): 42B20, 42B25

The first author is supported by the National Science Foundation under grant DMS 0099881.

Part of this work was carried out while the second author was a Postdoctoral Fellow at University of Missouri-Columbia. The second author would like to thank this department for its support and hospitality.

The second and the third authors are partially supported by MCYT Grant BFM2001-0189.

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Grafakos, L., Martell, J. & Soria, F. Weighted norm inequalities for maximally modulated singular integral operators. Math. Ann. 331, 359–394 (2005). https://doi.org/10.1007/s00208-004-0586-2

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