# Journal of Operator Theory

Volume 72, Issue 1, Summer 2014 pp. 193-218.

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

**Authors**:
Timo S. Hanninen (1)
and Tuomas P. Hytonen (2)

**Author institution:** (1) Department of Mathematics and Statistics,
University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland

(2) Department of Mathematics and Statistics, University of Helsinki, P.O.
Box 68, FI-00014 Helsinki, Finland

**Summary: **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 the real-valued case to the
Banach
space valued case. This extension is based on a Banach space valued
generalization of the notion of median.

**DOI: **http://dx.doi.org/10.7900/jot.2012nov21.1972

**Keywords: **Banach space, vector-valued, Calderon-Zygmund
operator, Bochner space,
local oscillation decomposition, Lerner's formula, Muckenhoupt weight,
median,
dyadic domination, $A_2$

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