Vector-valued operators, optimal weighted estimates and the $C_p$ condition

TL;DR

This paper establishes sharp weighted bounds for vector-valued operators like the Hardy-Littlewood maximal operator and Calderón-Zygmund operators, using sparse domination and extending the $C_p$ class results to broader contexts.

Contribution

It introduces new sharp weighted estimates for vector-valued operators and extends the $C_p$ class theory to a wider range of operators and norms.

Findings

Sharp weighted estimates for vector-valued operators

Extension of $C_p$ class results to $p > 0$ and weak norms

Application to a broader class of operators and their vector-valued extensions

Abstract

Sharp weighted estimates are obtained for vector-valued extensions of the Hardy-Littlewood maximal operator, Calder\'on-Zygmund operators and Coifman-Rochberg-Weiss commutator. Those estimates will rely upon suitable pointwise estimates in terms of sparse operators. We also prove some new results for the $C_p$ classes introduced by Muckenhoupt and later extended by Sawyer, in particular we extend the result to the full expected range $p > 0$, to the weak norm, to other operators and to the their vector-valued extensions.