  This paper gives new two-weight bump conditions for the sparse operators
related to iterated commutators of fractional integrals. As applications, the
two-weight bounds for iterated commutators of fractional integrals under more
general bump conditions are obtained. Meanwhile, the necessity of two-weight
bump conditions as well as the converse of Bloom type estimates for iterated
commutators of fractional integrals are also given.
