Sharp Matrix $\mathcal A_p$ lower bounds for general non-degenerate convolution Calder\'on--Zygmund operators

Abstract

Let $T$ be a convolution Calder\'on--Zygmund operator satisfying Hyt\"onen's non-degeneracy condition. We prove that, for every $p\in(1,\infty)$ and every matrix dimension $m\ge3$, there exists a positive constant $C$ such that, for any $t\in[1,\infty)$, \begin{equation*} \sup_{[W]_{\mathcal A_p}\le t} \|T\|_{L^p(W)\to L^p(W)} \ge C t^{1+\frac{1}{p(p-1)}}. \end{equation*} If, in addition, the odd part of the kernel is non-degenerate (as for the Hilbert transform and the Riesz transforms), the same conclusion holds for every $m\ge2$.