On the Marcinkiewicz integral with variable kernels

Yong Ding, Chin Cheng Lin, Shuanglin Shao

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32 Scopus citations

Abstract

In this paper we prove that the Marcinkiewicz integral μΩ with variable kernels is an operator of type (2, 2), where the kernel function Ω does not have any smoothness on the unit sphere in ℝn. We prove further that, when the variable kernel Ω satisfies a class of integral Dini condition, μΩ is a bounded operator from the Hardy space H1 (ℝn) to L1 (ℝn) and from the weak Hardy space H 1,∞ (ℝn) to the weak L1 space L 1-∞(ℝn), respectively. As corollaries of the above conclusions, we show that μΩ is also of the weak type (1, 1) and of type (p, p) for 1 < p < 2. Moreover, the L2 boundedness of a class of the Littlewood-Paley type operators with variable kernels also are obtained, which are related to the Littlewood-Paley g*λ-function and Lusin area integral, respectively.

Original languageEnglish
Pages (from-to)805-821
Number of pages17
JournalIndiana University Mathematics Journal
Volume53
Issue number3
DOIs
StatePublished - 2004

Keywords

  • Marcinkiewicz integral
  • Rough kernel
  • Variable kernel

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