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arXiv · 2605.19637

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

Komla Domelevo·Spyridon Kakaroumpas·Stefanie Petermichl·Sergei Treil·Alexander Volberg
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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.

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Komla Domelevo, Spyridon Kakaroumpas, Stefanie Petermichl, Sergei Treil, Alexander Volberg. 2026-05-19. The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform. https://arxiv.org/abs/2605.19637

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