For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and A_∞-characteristics. These bounds are known to be sharp in many cases; as a new result, we prove their sharpness for fractional square functions.

The first author was supported by the DFG grant AR 134/4-1 "Regularität evolutionärer Probleme mittels Harmonischer Analyse und Operatortheorie''. The second author was supported by the ERC Starting Grant "Analytic-probabilistic methods for borderline singular integrals'' (grant agreement No. 278558), and by the Academy of Finland via the Finnish Centre of Excellence in Analysis and Dynamics Research (project Nos. 271983 and 307333). Parts of the work were done during a research stay of the first author at the University of Helsinki. He thanks the members of the harmonic analysis group for their hospitality.