Muckenhoupt–Wheeden conjectures for sparse operators

Abstract

We provide an explicit example of a pair of weights and a dyadic sparse operator for which the Hardy–Littlewood maximal function is bounded from \(L^p(v)\) to \(L^p(u)\) and from \(L^{p'}(u^{1-p'})\) to \(L^{p'}(v^{1-p'})\) while the sparse operator is not bounded on the same spaces. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.