Masani and Wiener asked to characterize the regularity of vector stationary stochastic processes. The question easily translates to a harmonic analysis question: for what matrix weights the Hilbert transform is bounded with respect to this weight? We solved this problem with Sergei Treil in 1996 introducing the matrix A_2 condition.
But what is the sharp estimate of the Hilbert transform in terms of matrix A_2 norm? This is still unknown in a striking difference with scalar case.
Convex body valued operators helped to get the estimate via norm raised to the power 3/2. But shouldn't it be power 1?
We construct an example of a rather natural operator for which the estimate in scalar and vector case is indeed different. But it is not the Hilbert transform.