Alexander Volberg : A_2 conjecture fails for matrix weights because 3/2 >1.
Abstract: How to estimate the norm of the Hilbert transform (or other singular operators) in space L^2(w) via weight w This classical question appeared from probability (i.e. from the regularity theory of stationary stochastic processes). Initial answer was given by works of Hunt—Muckenhoupt—Wheeden and Helson—Szeg\”o in the 1970’s.
But the sharp estimates are from 2000’s due to Petermichl, Volberg, Hytönen.
The answer (obtained first for Ahlfors—Beurling transform, then for the Hilbert transform, and then for all Calder\’on—Zygmund operators) is that the bound is linear in A_2 Hunt—Muckenhoupt—Wheeden characteristic of weight w . This was a solution of the so-called A_2 conjecture.
But what if the stationary process in question is a vector one (as Wiener and Masani asked)?
Then the weight W is a matrix weight. It was a long standing problem to prove that the same linear estimate holds.
However, recently we, Domelevo—Petermichl—Treil—Volberg, showed that this is not the case, A_2 conjecture fails, and the right sharp estimate is [W]_{A_2}^{3/2} .