Ayazoglu R.Hasanov J.J.20.04.20192019-04-2020.04.20192019-04-2020161072-947Xhttps://dx.doi.org/10.1515/gmj-2016-0009https://hdl.handle.net/20.500.12403/615We consider the generalized shift operator associated with the Laplace-Bessel differential operator ? B = i=1 n 2 x i 2 +i=1 k ? i x ix 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 operator M ? ${M-{\gamma }}$ (B-maximal operator) and the Riesz potential I ?,? ${I-{\alpha ,\gamma }}$ (B-Riesz potential), associated with the generalized shift operator are investigated. We prove that the B-maximal operator M ? ${M-{\gamma }}$ and the B-singular integral operator are bounded from the generalized weighted B-Morrey space p,? 1 ,?,? (k,+ n )${{\cal M}-{p,\omega -1,\varphi ,\gamma }(\mathbb {R}-{k,+}^{n})}$ to p,? 2 ,?,? (k,+ n )${{\cal M}-{p,\omega -2,\varphi ,\gamma }(\mathbb {R}-{k,+}^{n})}$ for all 1<p<${1 &lt; p &lt; \infty }$ , ?A p,? (k,+ n )${\varphi \in A-{p,\gamma }(\mathbb {R}-{k,+}^{n})}$ . Furthermore, we prove that the B-Riesz potential I ?,? ${I-{\alpha ,\gamma }}$ , 0<?<n+|?|${0&lt;\alpha &lt;n+|\gamma |}$ , is bounded from the generalized weighted B-Morrey space p,? 1 ,?,? (k,+ n )${{\cal M}-{p,\omega -1,\varphi ,\gamma }(\mathbb {R}-{k,+}^{n})}$ to q,? 2 ,?,? (k,+ n )${{\cal M}-{q,\omega -2,\varphi ,\gamma }(\mathbb {R}-{k,+}^{n})}$ , where ?/(n+|?|)=1/p-1/q${{\alpha }/{(n+|\gamma |)}=1/p-1/q}$ , 1<p<(n+|?|)/?${1&lt;p&lt;(n+|\gamma |)/{\alpha }}$ , ?A 1+q/p ' ,? (k,+ n )${\varphi \in A-{1+{q/p^{\prime }},\gamma }(\mathbb {R}-{k,+}^{n})}$ and 1/p+1/p ' =1${{1/p}+{1/p^{\prime }}=1}$ . © 2016 by De Gruyter.eninfo:eu-repo/semantics/closedAccessB-maximal operatorB-Riesz potentialgeneralized B-Morrey spaceB-maximal operatorB-Riesz potentialgeneralized B-Morrey spaceArticle23214315510.1515/gmj-2016-00092-s2.0-84973926512Q3WOS:000377453300001Q4