The sharp two-weight $A_p$–$A_c0$ inequality for truncated Calder3n\adZygmund operators

Let $T$ be an $L^2(\mathbb R^d)$-bounded Calder\f3n\adZygmund operator, let $1<p<\infty$, write $p'=p/(p-1)$, and let $w\in A_p$. Denote by $T_{\natural}$ the truncated maximal Calder\f3n\adZygmund operator, and by $A_p$ and $A_\infty$ the usual Muckenhoupt weight classes.

Sharp $A_p$–$A_\infty$ conjecture. There is a constant $C_{T,p}$ such that

$$\lVert T_{\natural}f\rVert_{L^p(w)}\le C_{T,p}\|w\|_{A_p}^{1/p}\max\left\{\|w\|_{A_\infty}^{1/p'},\left\lVert w^{-p'+1}\right\rVert_{A_\infty}^{1/p}\right\}\lVert f\rVert_{L^p(w)}.$$

This conjecture seeks a sharp weighted bound for truncated Calder\f3n\adZygmund operators. The corresponding estimate is known for the untruncated operator when $p=2$, but the truncated case remains open in the stated generality.