Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets

Steve Hofmann; Dorina Mitrea; Marius Mitrea; Andrew J. Morris

doi:10.3934/era.2014.21.8

Article Contents

2014, Volume 21: 8-18. Doi: 10.3934/era.2014.21.8

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Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets

1.
University of Missouri, Columbia, MO 65211

Received: October 31, 2013

Published: January 2014

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Abstract

We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.

Keywords:

Square function,
quasi-metric space,
space of homogeneous type,
Ahlfors-David regular,
singular integral operator,
local $T(b)$ theorem for the square function,
uniformly rectifiable set,
tent space,
variable coefficient kernel.

Mathematics Subject Classification: Primary: 28A75, 42B20; Secondary: 28A78, 42B25, 42B30.

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