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$ L^p$ theory for outer measures and two themes of Lennart Carleson united

Authors: Yen Do and Christoph Thiele
Journal: Bull. Amer. Math. Soc. 52 (2015), 249-296
MSC (2010): Primary 42B20
Published electronically: December 29, 2014
MathSciNet review: 3312633
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Abstract: We develop a theory of $ L^p$ spaces based on outer measures generated through coverings by distinguished sets. The theory includes as a special case the classical $ L^p$ theory on Euclidean spaces as well as some previously considered generalizations. The theory is a framework to describe aspects of singular integral theory, such as Carleson embedding theorems, paraproduct estimates, and $ T(1)$ theorems. It is particularly useful for generalizations of singular integral theory in time-frequency analysis, the latter originating in Carleson's investigation of convergence of Fourier series. We formulate and prove a generalized Carleson embedding theorem and give a relatively short reduction of the most basic $ L^p$ estimates for the bilinear Hilbert transform to this new Carleson embedding theorem.

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Additional Information

Yen Do
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511

Christoph Thiele
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Alle 60, D-53115 Bonn, and Department of Mathematics, UCLA, Los Angeles, California 90095

Received by editor(s): September 4, 2013
Published electronically: December 29, 2014
Additional Notes: The first author was partially supported by NSF grant DMS 1201456
The second author was partially supported by NSF grant DMS 1001535.
Dedicated: Dedicated to Lennart Carleson
Article copyright: © Copyright 2014 American Mathematical Society

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