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

Published online by Cambridge University Press:  25 July 2016

Ana Grau De La Herrán
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Finland email [email protected]
Steve Hofmann
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MI 65211, U.S.A. email [email protected]
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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.

Type
Research Article
Copyright
Copyright © University College London 2016 

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