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Electronic Research Announcements in Mathematical Sciences (ERA-MS)
 

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

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

 
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Steve Hofmann - University of Missouri, Columbia, MO 65211, United States (email)
Dorina Mitrea - University of Missouri, Columbia, MO 65211, United States (email)
Marius Mitrea - University of Missouri, Columbia, MO 65211, United States (email)
Andrew J. Morris - University of Missouri, Columbia, MO 65211, United States (email)

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.

Received: November 2013;      Available Online: February 2014.

 References