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Bilinear wavelet representation of Calderón–Zygmund forms

Francesco Di Plinio, Walton Green and Brett D. Wick

Vol. 5 (2023), No. 1, 47–83

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 T(1)-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.

wavelet representation theorem, bilinear singular integrals, $T(1)$-theorems, sharp weighted bounds, Leibniz rules, fractional differentiation, sparse domination
Mathematical Subject Classification
Primary: 42B20
Secondary: 42B25
Received: 10 June 2021
Revised: 13 July 2022
Accepted: 26 August 2022
Published: 24 April 2023
Francesco Di Plinio
Dipartimento di Matematica e Applicazioni
Università di Napoli “Federico II”
Walton Green
Department of Mathematics and Statistics
Washington University in St. Louis
St. Louis, MO
United States
Brett D. Wick
Department of Mathematics and Statistics
Washington University in St. Louis
St. Louis, MO
United States