Weighted inequalities involving Riemann-Liouville and Hardy-type operators

Abstract: Necessary and sufficient conditions for the weighted boundedness and compactness of the Riemann-Liouville operators are obtained. Applications to the solvability of the Abel nonlinear integral equations and to embedding theorems of some Besov-type spaces into weighted Lebesgue spaces on the semiaxis are given. A criterion for the boundedness of the Riemann-Liouville type operator with variable limits between Lebesgue spaces on the semiaxis is given. Some Sobolev-type spaces are considered and a necessary and sufficient condition for their embedding into a Lebesgue space on the real axis is given. A new geometric mean integral operator is introduced and a necessary and sufficient condition for its mapping between Lebesgue spaces on the semiaxis is proved. The key point of the proof is to first derive a similar result for the corresponding Hardy operator. A precise characterization of Hardy type inequalities with weights for the negative indices and the indices between 0 and 1 are obtained and a duality between these cases is established.