Quantitative weighted estimates for rough homogeneous singular integrals

  1. Hytönen, T.P. 3
  2. Roncal, L. 2
  3. Tapiola, O. 1
  1. 1 Basque Center for Applied Mathematics
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    Basque Center for Applied Mathematics

    Bilbao, España

    ROR 03b21sh32

  2. 2 Universidad de La Rioja
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    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  3. 3 University of Helsinki
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    University of Helsinki

    Helsinki, Finlandia

    ROR https://ror.org/040af2s02

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Revista:
Israel Journal of Mathematics

ISSN: 0021-2172

Año de publicación: 2017

Volumen: 218

Número: 1

Páginas: 133-164

Tipo: Artículo

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DOI: 10.1007/S11856-017-1462-6 SCOPUS: 2-s2.0-85015747736 WoS: WOS:000398070100007 GOOGLE SCHOLAR

Otras publicaciones en: Israel Journal of Mathematics

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Resumen

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L2(w), we obtain a bound that is quadratic in A2 constant [w]A2. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform. © 2017, Hebrew University of Jerusalem.