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