Abstract. In this paper we consider the generalized shift operator associated to the Laplace--Bessel differential operator $\Delta _{B}$ and investigate B-maximal commutators, commutators of B-Riesz potentials and commutators of B-singular integral operators associated to the generalized shift operator. The boundedness of the $B$-maximal commutator $M_{b,\gamma }$ and the commutator $[b,A_{\gamma }]$ of the $B$-singular integral operator on the modified $B$-Morrey spaces $\widetilde {L}_{p,\lambda ,\gamma }(\Rnk )$ for all $1 < p < \infty $ when $b \in BMO_\gamma ({\Rnk })$ are proved. In addition, we obtain that the commutator $[b,I_{\alpha ,\gamma }]$ of the $B$-Riesz potential $I_{\alpha ,\gamma }$ is bounded from the modified $B$-Morrey space $\widetilde {L}_{p,\lambda ,\gamma }(\Rnk )$ to $\widetilde {L}_{q,\lambda ,\gamma }(\Rnk )$, $1<p<\frac {n+|\gamma |-\lambda }{\alpha }$, $\frac {\alpha }{n+|\gamma |} \le \frac 1p-\frac 1q \le \frac {\alpha }{n+|\gamma |-\lambda }$ and from the space $\widetilde {L}_{1,\lambda ,\gamma } (\mathbb {R} _{k,+}^{n})$ to $W\widetilde {L}_{q,\lambda ,\gamma } (\Rnk )$, $\frac {\alpha }{n+|\gamma |} \le 1-\frac 1q \le \frac {\alpha }{n+|\gamma |-\lambda }$.

DOI: 10.14232/actasm-020-224-y

AMS Subject Classification (1991): 42B20; 42B25, 42B35

Keyword(s): commutator, generalized shift operator, $B$-maximal function, $B$-Riesz potential, Morrey space, modified Morrey space, $BMO_{\gamma }$ space

received 24.2.2020, revised 4.4.2020, accepted 18.4.2020. (Registered under 224/2020.)