Potential operators in modified Morrey spaces defined on Carleson curves

Abstract

In this paper we study the potential operator IΓα, 0<1 in the modified Morrey space L˜p,λ(Γ) and the spaces BMO(Γ) defined on Carleson curves Γ. We prove that for 1<p<(1−λ)∕α the potential operator IΓα is bounded from the modified Morrey space L˜p,λ(Γ) to L˜q,λ(Γ) if and in the case of infinite curve only if α≤1∕p−1∕q≤α∕(1−λ), and from the spaces L˜1,λ(Γ) to WL˜q,λ(Γ) if and in the case of infinite curve only if α≤1−1q≤α1−λ. Furthermore, for the limiting case (1−λ)∕α≤p≤1∕α we show that if Γ is an infinite Carleson curve, then the modified potential operator I˜Γα is bounded from L˜p,λ(Γ) to BMO(Γ), and if Γ is a finite Carleson curve, then the operator IΓα is bounded from L˜p,λ(Γ) to BMO(Γ).