# Sarason Conjecture and the Composition of Paraproducts (12rit186)

## Organizers

Eric Sawyer (McMaster University)

(Georgia Tech)

## Description

The Banff International Research Station will host the "Sarason Conjecture and the Composition of Paraproducts" workshop from November 4th to November 11th, 2012.

Given a function $b$, and choices $\epsilon,\delta\in\{0,1\}$ define the
paraproduct
$$\label{e.P} f\in L^2(\mathbb{R})\mapsto \operatorname P ^{\epsilon ,\delta } _{b}f(x) := \sum _{I\in \mathcal D} \frac {\langle b, h_I\rangle_{L^2}} {\sqrt {\left\vert I\right\vert}} \left\langle h ^{\delta }_I, f\right\rangle_{L^2} h ^{\epsilon }_I(x).$$
These discrete paraproduct operators are fundamental in harmonic analysis
since they serve as dyadic examples of Calder\'on-Zygmund operators.
Additionally, they are connected to questions in analytic function theory
through Sarason's Conjecture about Toeplitz operators on the Hardy Space.

This Research in Teams would be studying the question of the boundedness of
the composition
$$\operatorname P ^{\epsilon ,\delta } _{b} \operatorname P ^{\epsilon' ,\delta' } _{\beta}:L^2(\mathbb{R})\to L^2(\mathbb{R})$$
and what are the conditions on the symbols $b$ and $\beta$ that
characterize the boundedness of this operator. It will bring together Eric
T. Sawyer, Maria Cristina Pereyra, Maria Carmen Reguera and Brett D.
Wick to address the question about boundedness of the composition of
paraproducts. Each of these mathematicians brings unique expertise,
background, and motivations for this problem. It is hoped that through a
team effort a resolution to this question will be obtained using recent
advances in harmonic analysis and operator theory.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).