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

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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 $M_{p,\varphi}(\Gamma)$.