The A_2 theorem and the local oscillation decomposition for Banach space   valued functions
  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. 
Timo S. H\"anninen, Tuomas P. Hyt\"onen
math.CA math.FA