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arXiv · 1108.5119

Representation of singular integrals by dyadic operators, and the A_2 theorem

Tuomas P. Hytönen
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Abstract

This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journ\'e $T(1)$ theorem, whose proof is obtained as a byproduct. Only very basic Probability is used; in particular, the conditional probabilities of the original proof are completely avoided.

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Tuomas P. Hytönen. 2011-08-25. Representation of singular integrals by dyadic operators, and the A_2 theorem. https://doi.org/10.1016/j.exmath.2016.09.003

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