Bump conditions and two-weight inequalities for commutators of fractional integrals

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