The dual conjecture of Muckenhoupt and Wheeden

Abstract

Let $T$ be a Calderón–Zygmund operator on ${\mathbb{R}}^d$. We prove the existence of a constant $C_{T,d}<\infty$ such that for any weight $w$ on ${\mathbb{R}}^d$ satisfying Muckenhoupt's condition $A_1$, we have

The linear dependence on $[w{]}_{A_1}$, the $A_1$ characteristic of $w$, is optimal. The proof exploits the associated dimension-free inequalities for dyadic shifts.