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

TL;DR

This paper investigates the growth bounds of the vector Hilbert transform in matrix weighted spaces, confirming the sharpness of the 3/2 power growth even with an enlarged matrix Poisson $A_2$ characteristic.

Contribution

It demonstrates that the 3/2 power growth bound remains sharp when replacing the classical matrix $A_2$ characteristic with the larger matrix Poisson $A_2$ characteristic.

Findings

The $3/2$ power growth bound for the vector Hilbert transform is sharp.

Replacing the $A_2$ characteristic with the matrix Poisson $A_2$ characteristic does not improve the bound.

The growth of the vector Hilbert transform cannot be bounded linearly in the matrix $A_2$ characteristic.

Abstract

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 $[W]_{\mathbf{A}_2}^{3/2}$. 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 $3/2$ 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.