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We consider the generalized shift operator associated with the Laplace–Bessel differential operatorΔ  B   =   ∑   i  =  1   n∂  2    ∂   x  i  2+   ∑   i  =  1   kγ  i    x  i∂   ∂   x  i.    $ \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 operatorM  γ    ${M_{\gamma }}$    ( B -maximal operator) and the Riesz potentialI   α  ,  γ${I_{\alpha ,\gamma }}$    ( B -Riesz potential), associated with the generalized shift operator are investigated. We prove that the  B -maximal operatorM  γ    ${M_{\gamma }}$    and the  B -singular integral operator are bounded from the generalized weighted  B -Morrey spaceℳ   p  ,   ω  1   ,  φ  ,  γ(   ℝ   k  ,  +   n   )${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$    toℳ   p  ,   ω  2   ,  φ  ,  γ(   ℝ   k  ,  +   n   )${{\cal M}_{p,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$    for all1  &lt;  p  &lt;  ∞    ${1 &amp;lt; p &amp;lt; \infty }$   ,φ  ∈   A   p  ,  γ(   ℝ   k  ,  +   n   )${\varphi \in A_{p,\gamma }(\mathbb {R}_{k,+}^{n})}$   . Furthermore, we prove that the  B -Riesz potentialI   α  ,  γ${I_{\alpha ,\gamma }}$   ,0  &lt;  α  &lt;  n  +  |  γ  |    ${0&amp;lt;\alpha &amp;lt;n+|\gamma |}$   , is bounded from the generalized weighted  B -Morrey spaceℳ   p  ,   ω  1   ,  φ  ,  γ(   ℝ   k  ,  +   n   )${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$    toℳ   q  ,   ω  2   ,  φ  ,  γ(   ℝ   k  ,  +   n   )${{\cal M}_{q,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$   , whereα  /   (  n  +  |  γ  |  )   =  1  /  p  -  1  /  q    ${{\alpha }/{(n+|\gamma |)}=1/p-1/q}$   ,1  &lt;  p  &lt;  (  n  +  |  γ  |  )  /  α    ${1&amp;lt;p&amp;lt;(n+|\gamma |)/{\alpha }}$   ,φ  ∈   A   1  +   q  /   p  '    ,  γ(   ℝ   k  ,  +   n   )${\varphi \in A_{1+{q/p^{\prime }},\gamma }(\mathbb {R}_{k,+}^{n})}$    and1  /  p   +   1  /   p  '    =  1    ${{1/p}+{1/p^{\prime }}=1}$   .

On the boundedness of a B -Riesz potential in the generalized weighted B -Morrey spaces