Estimates for bilinear generalized fractional integral operator and its commutator on generalized Morrey spaces over RD-spaces

Let \((X,d,\mu )\) be an RD-space. In this paper, we prove that a bilinear generalized fractional integral \(\widetilde{T}_{\alpha }\) is bounded from the product of generalized Morrey spaces \(\mathcal {L}^{\varphi _{1},p_{1}}(X)\times \mathcal {L}^{\varphi _{2},p_{2}}(X)\) into spaces \(\mathcal {L}^{\varphi ,q}(X)\), and it is also bounded from the product of spaces \(\mathcal {L}^{\varphi _{1},p_{1}}(X)\times \mathcal {L}^{\varphi _{2},p_{2}}(X)\) into generalized weak Morrey spaces \(W\mathcal {L}^{\varphi ,q}(X)\), where the Lebesgue measurable functions \(\varphi _{1}, \varphi _{2}\) and \(\varphi \) satisfy certain conditions and \(\varphi _{1}\varphi _{2}=\varphi \), \(\alpha \in (0,1)\) and \(\frac{1}{q}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-2\alpha \) for \(1<p_{1}, p_{2}<\frac{1}{\alpha }\). Furthermore, we establish the boundedness of the commutator \(\widetilde{T}_{\alpha ,b_{1},b_{2}}\) formed by \(b_{1},b_{2}\in \) \(\textrm{BMO}(X)(\hbox {or }\textrm{Lip}_{\beta }(X))\) and \(\widetilde{T}_{\alpha }\) on spaces \(\mathcal {L}^{\varphi ,q}(X)\) and on spaces \(W\mathcal {L}^{\varphi ,q}(X)\). As applications, we show that the \(\widetilde{T}_{\alpha }\) and its commutator \(\widetilde{T}_{\alpha ,b_{1},b_{2}}\) are bounded on grand generalized Morrey spaces \(\mathcal {L}^{\theta ,\varphi ,p)}(X)\) over \((X,d,\mu )\).