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On certain estimates for Marcinkiewicz integrals and extrapolation

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Abstract

We obtainL p estimates for parametric Marcinkiewicz integrals associated to polynomial mappings and with rough kernels on the unit sphere as well as on the radial direction. These estimates will allow us to use an extrapolation argument to obtain some new and improved results on Marcinkiewicz integrals. Also, such estimates provide us with a unifying approach in dealing with Marcinkiewicz integrals when the kernel function ω belongs to the class of block spaces (0,α) B q (S n-1) as well as when ω belongs to the classL(logL)α (S n-1). Our conditions on the kernels are known to be the best possible in their respective classes.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Qatar University, Doha-Qatar

    Hussain Al-Qassem

  2. Department of Mathematics, University of Pittsburgh, 15260, PA, USA

    Yibiao Pan

Authors
  1. Hussain Al-Qassem
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  2. Yibiao Pan
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Correspondence to Hussain Al-Qassem.

Additional information

The work of this paper was done while the first author was on sabbatical leave from Yormouk University

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Al-Qassem, H., Pan, Y. On certain estimates for Marcinkiewicz integrals and extrapolation. Collect. Math. 60, 123–145 (2009). https://doi.org/10.1007/BF03191206

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  • DOI: https://doi.org/10.1007/BF03191206

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