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Abstract
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We represent a bilinear Calderón–Zygmund operator at a given smoothness level as
a finite sum of cancellative, complexity-zero operators, involving smooth wavelet
forms, and continuous paraproduct forms. This representation results in a sparse
-type
bound, which in turn yields directly new sharp weighted bilinear estimates on
Lebesgue and Sobolev spaces. Moreover, we apply the representation theorem to
study fractional differentiation of bilinear operators, establishing Leibniz-type rules in
weighted Sobolev spaces which are new even in the simplest case of the pointwise
product.
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Keywords
wavelet representation theorem, bilinear singular
integrals, $T(1)$-theorems, sharp weighted bounds, Leibniz
rules, fractional differentiation, sparse domination
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Mathematical Subject Classification
Primary: 42B20
Secondary: 42B25
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Milestones
Received: 10 June 2021
Revised: 13 July 2022
Accepted: 26 August 2022
Published: 24 April 2023
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