On the matrix $A_p$ conjecture

RESUMO

This paper provides a solution to the matrix $A_p$ conjecture. In particular, for every \(1<p<\infty\), we show that the norm of the Hilbert transform on the matrix-weighted space \(L^p(W)\) is bounded below by \(c_p[W]_{A_p}^{1+1/[p(p-1)]}\) for a positive constant \(c_p\) depending only on \(p\). This result, combined with the case $p=2$ due to Domelevo, Petermichl, Treil and Volberg \cite{domelevo-petermichl-treil-volberg-2024}, completes the picture of the matrix $A_p$ conjecture.