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In this paper, we give a boundedness criterion for the potential operatorℐα{ {\mathcal I} }^{\alpha }in the local generalized Morrey spaceLMp,φ{t0}(Γ)L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{&amp;#x0393;})and the generalized Morrey spaceMp,φ(Γ){M}_{p,\varphi }(\text{&amp;#x0393;})defined on Carleson curvesΓ\text{&amp;#x0393;}, respectively. For the operatorℐα{ {\mathcal I} }^{\alpha }, we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness onLMp,φ{t0}(Γ)L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{&amp;#x0393;})and the strong and weak Adams-type boundedness onMp,φ(Γ){M}_{p,\varphi }(\text{&amp;#x0393;}).

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