Non-homogeneous Square Functions on General Sets: Suppression and Big Pieces Methods

Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    © 2017, Mathematica Josephina, Inc. We aim to showcase the wide applicability and power of the big pieces and suppression methods in the theory of local Tb theorems. The setting is new: we consider conical square functions with cones { x∈ Rn\ E: | x- y| < 2 dist (x, E) } , y∈ E, defined on general closed subsets E⊂ Rn supporting a non-homogeneous measure μ. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls B⊂ E, which are doubling and of small boundary. Due to the general set E we use metric space methods. Therefore, we also demonstrate the recent techniques from the metric space point of view, and show that they yield the most general known local Tb theorems even with assumptions formulated using balls rather than the abstract dyadic metric cubes.
    Original languageEnglish
    Pages (from-to)3176-3227
    JournalJournal of Geometric Analysis
    Volume27
    Issue number4
    DOIs
    Publication statusPublished - 1 Oct 2017

    Keywords

    • Big pieces
    • Conical square functions
    • Good lambda method
    • Local Tb theorems

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