Off-diagonal sharp two-weight estimates for sparse operators

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 $A_{\mathrm{\infty }}$-characteristics. These bounds are known to be sharp in many cases; as a new result, we prove their sharpness for fractional square functions.