Potential Operators on Cones of Nonincreasing Functions

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (I(alpha)f) (x) = integral(infinity)(0)(f(t)/vertical bar x-t vertical bar(1-alpha))dt and (O(alpha 1,alpha 2)f)(x, y) = integral(infinity)(0) integral(infinity)(0) (f(t,tau)/ vertical bar x-t vertical bar(1-alpha 1)vertical bar y-tau vertical bar(1-alpha 2)) dtd tau on the cone of nonincreasing functions are derived. In the case of O-alpha 1,O-alpha 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 infinity.