A local $T(b)$ theorem for perfect multilinear Calder\'{o}n-Zygmund operators

TL;DR

This paper establishes a new multilinear local T(b) theorem for perfect dyadic models, using exclusively general testing functions and revealing novel relations among them, advancing the theoretical understanding of Calderón-Zygmund operators.

Contribution

It introduces a multilinear local T(b) theorem that uniquely employs only general testing functions, unlike previous mixed approaches, and uncovers new relations among these functions.

Findings

Proves a multilinear local T(b) theorem for perfect dyadic models.

Identifies new relations between testing functions in the multilinear setting.

Simplifies the testing framework by using only general functions.

Abstract

We prove a multilinear local $T(b)$ theorem that differs from previously considered multilinear local $T(b)$ theorems in using exclusively general testing functions $b$ as opposed to a mix of general testing functions and indicator functions. The main new feature is a set of relations between the various testing functions $b$ that to our knowledge has not been observed in the literature and is necessitated by our approach. For simplicity we restrict attention to the perfect dyadic model.