A Note On $\ell^r$-Valued Calderon-Zygmund Operators

TL;DR

This paper extends Calderon-Zygmund operators to $ell^r$-valued functions on weighted $L^p$ spaces, providing quantitative norm estimates based on $A_p$ weight characteristics, and connects scalar and vector-valued cases.

Contribution

It introduces $ell^r$-valued Calderon-Zygmund operators on weighted spaces and establishes their norm bounds using a decomposition theorem, bridging scalar and vector-valued estimates.

Findings

Quantitative bounds for $ell^r$-valued Calderon-Zygmund operators.

Recovery of scalar estimates via $ell^r$ extensions.

Application of Lerner's decomposition theorem to vector-valued operators.

Abstract

We consider $\ell^r$ extensions of Calderon-Zygmund operators on weighted spaces $L^p(w)$ with $w$ an $A_p$ weight and $1 < p < \infty$. We give quantitative estimates of these operators' norm in terms of a given weight's $A_p$ characteristic. In particular, using a decomposition theorem of A. Lerner, we show that these extensions recover certain scalar estimates. Related results on general Banach spaces have been studied by Hytonen and Hannien.