### Abstract

We consider the generalized shift operator associated with the Laplace–Bessel differential operator ΔB=∑i=1n∂2∂xi2+∑i=1kγixi∂∂xi.$ \Delta _{B}=\sum _{i=1}^{n}\frac{\partial ^2 }{\partial x_i^2} +\sum _{i=1}^{k} \frac{\gamma _i }{x_i}\frac{\partial }{\partial x_i}. $ The maximal operator Mγ${M_{\gamma }}$ ( B -maximal operator) and the Riesz potential Iα,γ${I_{\alpha ,\gamma }}$ ( B -Riesz potential), associated with the generalized shift operator are investigated. We prove that the B -maximal operator Mγ${M_{\gamma }}$ and the B -singular integral operator are bounded from the generalized weighted B -Morrey space ℳp,ω1,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ to ℳp,ω2,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ for all 1<p<∞${1 < p < \infty }$, φ∈Ap,γ(ℝk,+n)${\varphi \in A_{p,\gamma }(\mathbb {R}_{k,+}^{n})}$. Furthermore, we prove that the B -Riesz potential Iα,γ${I_{\alpha ,\gamma }}$, 0<α<n+|γ|${0<\alpha <n+|\gamma |}$, is bounded from the generalized weighted B -Morrey space ℳp,ω1,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ to ℳq,ω2,φ,γ(ℝk,+n)${{\cal M}_{q,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$, where α/(n+|γ|)=1/p-1/q${{\alpha }/{(n+|\gamma |)}=1/p-1/q}$, 1<p<(n+|γ|)/α${1<p<(n+|\gamma |)/{\alpha }}$, φ∈A1+q/p',γ(ℝk,+n)${\varphi \in A_{1+{q/p^{\prime }},\gamma }(\mathbb {R}_{k,+}^{n})}$ and 1/p+1/p'=1${{1/p}+{1/p^{\prime }}=1}$.