The concept of k-step full backward extension for subnormal operators is adapted to the context of completely hyperexpansive operators. The question of existence of k-step full backward extension is solved within this class of operators with the help of an operator version of the Levy-Khinchin formula. Some new phenomena in comparison with subnormal operators are found and related classes of operators are discussed as well.

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.