Singular integral operators with non-smooth kernels on irregular domains

Xuan Thinh Duong, Alan MacIntosh

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    166 Citations (Scopus)

    Abstract

    Let χ be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(χ) for 1 < p ≤ 2; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable subset of χ, we give a sufficient condition on the kernel of T so that T is of weak type (1,1), hence bounded on Lp(Ω) for 1 < p ≤ 2. iii) We establish sufficient conditions for the maximal truncated operator T*, which is defined by T*u(Greek cursive chi) = supε>0|Tεu(Greek curve chi)|, to be Lp bounded, 1 < p < ∞. Applications include weak (1, 1) estimates of certain Riesz transforms, and Lp boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

    Original languageEnglish
    Pages (from-to)233-265
    Number of pages33
    JournalRevista Matematica Iberoamericana
    Volume15
    Issue number2
    DOIs
    Publication statusPublished - 1999

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