# 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, $A_2$