An Extrapolation of Operator Valued Dyadic Paraproducts

TL;DR

This paper extends the boundedness results of operator-valued dyadic paraproducts on the unit circle, showing that boundedness in a specific BMO space implies boundedness across all p-values in the range, with modifications considered.

Contribution

It proves that boundedness of operator-valued dyadic paraproducts on L^p spaces for some p implies boundedness for all p, under BMO conditions, and explores modified versions.

Findings

Boundedness on one L^p implies boundedness on all L^p for operator-valued dyadic paraproducts.

Boundedness results hold under operator-valued BMO space assumptions.

Analysis of modified dyadic paraproducts and their boundedness.

Abstract

We consider the dyadic paraproducts $\pi_\f$ on $\T$ associated with an $\M$-valued function $\f.$ Here $\T$ is the unit circle and $\M$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\T,L^p(\M))$ for some $1<p<\infty $ implies their boundedness on $L^p(\T,L^p(\M))$ for all $1<p<\infty$ provided $\f$ is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M)).