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GENERALIZED LOCAL $\mathit{Tb}$ THEOREMS FOR SQUARE FUNCTIONS

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  25 July 2016

Ana Grau De La Herrán  and
Steve Hofmann
Ana Grau De La Herrán
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Finland email ana.grau@helsinki.fi
Steve Hofmann
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MI 65211, U.S.A. email hofmanns@missouri.edu
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Abstract

A local $\mathit{Tb}$ theorem is an $L^{2}$ boundedness criterion by which the question of the global behavior of an operator is reduced to its local behavior, acting on a family of test functions $b_{Q}$ indexed by the dyadic cubes. We present two versions of such results, in particular, treating square function operators whose kernels do not satisfy the standard Littlewood–Paley pointwise estimates. As an application of one version of the local $\mathit{Tb}$ theorem, we show how the solvability of the Kato problem (which was implicitly based on local $\mathit{Tb}$ theory) may be deduced from this general criterion.

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B37: Harmonic analysis and PDE
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 1 - 28
Copyright
Copyright © University College London 2016 

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