Rough fractional integral operators and beyond Adams inequalities

Abstract

We consider the boundedness of fractional integral operators with rough kernel from Morrey spaces L p,λ to L q,μ . Our main concern is proving the boundedness property for μ < λ as an extension of Adams inequality on some special subsets of the operator's domain namely classes of A p , simple function, and radial function respectively. For radial function, we prove the boundedness on local Morrey spaces. We also prove the boundedness property for μ ≥ λ as well as the special case of q ≦ p. It is interesting on its own term since the operator is not bounded from Lp to L q if q ≦ p. We also establish necessary conditions for boundedness. Our proposed condition for boundedness includes the sufficient conditions for both Adams inequality and Spanne inequality.