LAUSR

In this paper, the authors prove that the following form of bilinear fractional integral operator $$\begin{aligned} B_{\alpha }(f,g)(x):=\int _{\mathbb {R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha }}\mathrm {d}y \end{aligned}$$ is bounded from the product of generalized fractional Morrey spaces $${\mathcal {L}}^{p_{1},\eta _{1},\varphi }(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\eta _{2},\varphi }(\mathbb {R}^{n})$$ to space $${\mathcal {L}}^{q,\eta -\alpha ,\varphi }(\mathbb {R}^{n})$$ , and also bounded from the product space $${\mathcal {L}}^{p_{1},\varphi }$$ $$(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\varphi }(\mathbb {R}^{n})$$ to $${\mathcal {L}}^{q,\varphi ^{\frac{q}{p}}}(\mathbb {R}^{n})$$ . Moreover, the boundedness of commutator $$[b_{1},b_{2},{\mathcal {I}}_{\alpha }]$$ generated by $$b_{1}, b_{2}\in \mathrm {BMO}(\mathbb {R}^{n})$$ and bilinear fractional integral operator $${\mathcal {I}}_{\alpha }$$ defined as in (1.1) is bounded from the product space $${\mathcal {L}}^{p_{1},\eta _{1},\varphi }(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\eta _{2},\varphi }(\mathbb {R}^{n})$$ to space $${\mathcal {L}}^{q,\eta -\alpha ,\varphi }(\mathbb {R}^{n})$$ ; as a corollary, the boundedness of the commutator $$[b_{1},b_{2},{\mathcal {I}}_{\alpha }]$$ on generalized Morrey space is also obtained.