Vol. 15, No. 4, 2022

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Five-linear singular integral estimates of Brascamp–Lieb-type

Camil Muscalu and Yujia Zhai

Vol. 15 (2022), No. 4, 1011–1095
Abstract

We prove the full range of estimates for a five-linear singular integral of Brascamp–Lieb type. The study is methodology-oriented with the goal of developing a sufficiently general technique to estimate singular integral variants of Brascamp–Lieb inequalities that do not obey Hölder scaling. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule which captures nonlinear interactions of waves from transversal directions.

Keywords
multilinear and multiparameter operators, Brascamp–Lieb singular integrals, non-Hölder scaling, Leibniz rule, tensor-type stopping-time decompositions
Mathematical Subject Classification 2010
Primary: 42A45, 42B15, 42B20, 42B25, 42B37
Milestones
Received: 25 January 2020
Revised: 10 October 2020
Accepted: 31 December 2020
Published: 3 September 2022
Authors
Camil Muscalu
Department of Mathematics
Cornell University
Ithaca, NY
United States
Yujia Zhai
Laboratoire de Mathématiques
Université de Nantes
Nantes
France