The Poisson Matrix ${\mathbf{A}}_2$ characteristic and the 3/2 blow up of the Hilbert transform

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W{]}_{{\mathbf{A}}_2}^{3\mathrm{/}2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3\mathrm{/}2$ power persists if we replace the classical matrix $A_2$ characteristic by the “fattened”, larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.