A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Xuan Thinh Duong, Ji Li*, Eric T. Sawyer, Manasa N. Vempati, Brett D. Wick, Dongyong Yang

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B

    ‖T(u1B)‖L2(B,v)≤T‖1BL2(u),

    ‖T(v1B)‖L2(B,u)≤T‖1BL2(v).

    The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

    Original languageEnglish
    Article number109190
    Pages (from-to)1-65
    Number of pages65
    JournalJournal of Functional Analysis
    Volume281
    Issue number9
    DOIs
    Publication statusPublished - 1 Nov 2021

    Keywords

    • Two weight inequality
    • Testing conditions
    • Space of homogeneous type
    • Calderón–Zygmund operator
    • Haar basis

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