| Summary: | In this paper we study the potential operator I Γ α , 0 < 1 in the modified Morrey space L ˜ p , λ ( Γ ) and the spaces B M O ( Γ ) defined on Carleson curves Γ . We prove that for 1 < p < ( 1 − λ ) ∕ α the potential operator I Γ α 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 W L ˜ q , λ ( Γ ) if and in the case of infinite curve only if α ≤ 1 − 1 q ≤ α 1 − 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 ˜ Γ α is bounded from L ˜ p , λ ( Γ ) to B M O ( Γ ) , and if Γ is a finite Carleson curve, then the operator I Γ α is bounded from L ˜ p , λ ( Γ ) to B M O ( Γ ) . Keywords: Carleson curves, Modified Morrey spaces, B M O spaces, Potential operators, Sobolev–Morrey inequality
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