Generalized local Tb Theorems for Square Functions, and applications

A local 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 several versions of such results, in particular, treating square function...

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Bibliographic Details
Published in:arXiv.org 2012-12
Main Authors: Ana Grau de la Herran, Hofmann, Steve
Format: Article
Language:English
Subjects:
Criteria
Cubes
Divergence
Operators (mathematics)
Theorems
Online Access:Get full text
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Summary:A local 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 several 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 Tb theorem, we show how the solvability of the Kato problem (which was implicitly based on local Tb theory) may be deduced from this general criterion. We also present another version, from which we deduce boundedness of layer potentials associated to certain complex elliptic operators in divergence form.
ISSN:2331-8422
DOI:10.48550/arxiv.1212.5870