Potential operators in modified Morrey spaces defined on Carleson curves

In this paper we study the fractional maximal operator $\mathcal{M}^{\alpha}$ in the local generalized Morrey space $LM_{p,\varphi}^{\{t_0\}}(\Gamma)$ and the generalized Morrey space $M_{p,\varphi}(\Gamma)$ defined on Carleson curves $\Gamma$, respectively. For the operator $\mathcal{M}^{\alpha}$ we shall give a characterization the strong and weak Spanne-Guliyev type boundedness on $LM_{p,\varphi}^{\{t_0\}}(\Gamma)$ and the strong and weak Adams-Guliyev type boundedness on $M_{p,\varphi}(\Gamma)$.

Characterizations for the fractional maximal operators on Carleson curves in local generalized Morrey spaces

Abstract In this paper, we give a boundedness criterion for the potential operator ℐ α { {\mathcal I} }^{\alpha } in the local generalized Morrey space L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the generalized Morrey space M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) defined on Carleson curves Γ \text{Γ} , respectively. For the operator ℐ α { {\mathcal I} }^{\alpha } , we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the strong and weak Adams-type boundedness on M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) .

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Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces

In this paper we study the potential operator Iα in the Morrey space Lp,λ and the spaces BMO defined on Carleson curves Γ. We prove that for 0 < α < 1, Iα is bounded from the Morrey space Lp,λ(Γ) to Lq,λ(Γ) on simple Carleson curves if (and only if in the infinite simple Carleson curve Γ) 1/p − 1/q = α/(1 − λ), 1 < p < (1 − λ)/α, and from the spaces L1,λ(Γ) to WLq,λ(Γ) if (and only if in the infinite case) 1− 1q = α 1−λ .

Potential operators on carleson curves in morrey spaces

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Potential operators on carleson curves in morrey spaces