Potential Operators on Cones of Nonincreasing Functions

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (∫)()=∞0(()/|？|1？) and (？1,2∫)(,)=∞0∫∞0((,)/|？|1？1|？|1？2) on the cone of nonincreasing functions are derived. In the case of ？1,2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.