Some properties of the potential operators in Morrey spaces defined on Carleson curves

Abstract

In this article, we study the potential operator ℐα and fractional maximal operator ℳα in the Morrey space L p,λ(Γ) on Carleson curves Γ in the limiting case p = (1 − λ)/α. We show that if Γ is an infinite Carleson curve, then the modified potential operator is bounded from L (1−λ)/α,λ(Γ) to BMO(Γ), and if Γ is a finite Carleson curve, then the operator ℐα is bounded from L (1−λ)/α,λ(Γ) to BMO(Γ). Also we show that for any Carleson curve the operator ℳα is bounded from L (1−λ)/α,λ(Γ) to L ∞(Γ).