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arXiv · 1606.04309

Non-homogeneous square functions on general sets: suppression and big pieces methods

Henri Martikainen·Mihalis Mourgoglou·Emil Vuorinen
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Abstract

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 $\{x \in \mathbb{R}^n \setminus E: |x-y| < 2 \operatorname{dist}(x,E)\}$, $y \in E$, defined on general closed subsets $E \subset \mathbb{R}^n$ supporting a non-homogeneous measure $\mu$. We obtain boundedness criteria in this generality in terms of weak type testing of measures on regular balls $B \subset E$, 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.

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Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen. 2016-06-14. Non-homogeneous square functions on general sets: suppression and big pieces methods. https://doi.org/10.1007/s12220-017-9801-8

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