The dual conjecture of Muckenhoupt and Wheeden (Q1982551)

Summary: 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 \[w\left(\{x\in \mathbb{R}^d:|Tf(x)| > w(x)\}\right) \leq C_{T,d}[w]_{A_1}\int_{\mathbb{R}^d}f \ \mathrm{d}x.\] 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.