The A_2 theorem and the local oscillation decomposition for Banach space valued functions

TL;DR

This paper extends the local oscillation decomposition and dyadic domination techniques to Banach space valued functions, proving linear dependence of Calderon-Zygmund operator norms on Muckenhoupt A_2 weights.

Contribution

It generalizes Lerner's local oscillation decomposition and median concepts from real-valued to Banach space valued functions, enabling new weighted norm estimates.

Findings

Operator norm depends linearly on A_2 characteristic.

Banach space valued Calderon-Zygmund operators are dominated by positive dyadic shifts.

Extension of local oscillation decomposition to Banach space valued functions.

Abstract

We prove that the operator norm of every Banach space valued Calderon-Zygmund operator T on the weighted Lebesgue-Bochner space depends linearly on the Muckenhoupt A_2 characteristic of the weight. In parallel with the proof of the real-valued case, the proof is based on pointwise dominating every Banach space valued Calderon-Zygmund operator by a series of positive dyadic shifts. In common with the real-valued case, the pointwise dyadic domination relies on Lerner's local oscillation decomposition formula, which we extend from real-valued functions to Banach space valued functions. The extension of Lerner's local oscillation decomposition formula is based on a Banach space valued generalization of the notion of median.