The Muckenhoupt–Wheeden two-weight conjecture for Calderón–Zygmund operators

Let $1<p<\infty$, let $p'=p/(p-1)$, let $u$ and $v$ be weights, and let $T$ be a Calderón–Zygmund operator. Write $M:L^p(u)\to L^p(v)$ for boundedness of the maximal operator between the indicated weighted spaces. Muckenhoupt–Wheeden two-weight conjecture. If

$$M:L^p(u)\to L^p(v)$$

and

$$M:L^{p'}(v^{1-p'})\to L^{p'}(u^{1-p'}),$$

then

$$T:L^p(u)\to L^p(v).$$

This conjecture sought to characterize weighted boundedness of Calderón–Zygmund operators through the corresponding two maximal-operator inequalities. The conjecture is refuted by counterexamples in the paper, so the two maximal bounds do not suffice for the asserted strong-type bound.