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$L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets

About this Title

Steve Hofmann, University of Missouri, Columbia, Dorina Mitrea, University of Missouri, Columbia, Marius Mitrea, University of Missouri, Columbia and Andrew J. Morris, University of Missouri, Columbia

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 245, Number 1159
ISBNs: 978-1-4704-2260-8 (print); 978-1-4704-3607-0 (online)
Published electronically: July 25, 2016
Keywords: Square function, quasi-metric space, space of homogeneous type, Ahlfors-David regularity, singular integral operators, area function, Carleson operator, $T(1)$ theorem for the square function, local $T(b)$ theorem for the square function, uniformly rectifiable sets, tent spaces, variable coefficient kernels
MSC: Primary 28A75, 42B20; Secondary 28A78, 42B25, 42B30

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Table of Contents


  • 1. Introduction
  • 2. Analysis and Geometry on Quasi-Metric Spaces
  • 3. $T(1)$ and local $T(b)$ Theorems for Square Functions
  • 4. An Inductive Scheme for Square Function Estimates
  • 5. Square Function Estimates on Uniformly Rectifiable Sets
  • 6. $L^p$ Square Function Estimates
  • 7. Conclusion


We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for $L^p$ and Hardy space versions of these estimates are also established. Moreover, we prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

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