# 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.