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In this paper, the boundedness of the Riesz potential generated by generalized shift operator $I_{B_k}^{\alpha }$ from the spaces $a=(L_{p_m,\nu }\left({ℝ}_n^k\right),a_m)$ to the spaces $a^{\text{'}}=(L_{q_m,\nu }\left({ℝ}_n^k\right),a_m^{\text{'}})$ is examined.