Sharp Matrix-Weighted Lower Bounds for Non-Degenerate Convolution Calderón--Zygmund Operators

Abstract

Let $T$ be a convolution Calderón--Zygmund operator satisfying Hytönen's non-degeneracy condition. We prove that, for every $p\in (1,\infty )$ and $m\ge 3$, 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*} where the supremum is taken over all matrix weights $W:{\mathbb{R}}^n\to {\mathbb{C}}^{m\times m}$ with their weight constants $[W{]}_{{\mathcal{A}}_p}\le t$. If, in addition, the odd part of the kernel is non-degenerate (such as the Hilbert transform and the Riesz transforms), the same conclusion holds for every $m\ge 2$.

arXiv comment21 pages