Bilinear singular integral operators with generalized kernels and their commutators on product of (grand) generalized fractional (weighted) Morrey spaces

The purpose of this article is to establish the boundedness of a bilinear singular integral operator $\tilde{T}$ with generalized kernels and its commutator ${\tilde{T}}_{b_1,b_2}$ formed by $b_1,b_2\in \mathrm{BMO}\left(R^n\right)$ and the $\tilde{T}$ on product of generalized fractional Morrey spaces $L^{p_1,{\eta }_1,\phi }\left(R^n\right)\times L^{p_2,{\eta }_2,\phi }\left(R^n\right)$, product of generalized fractional weighted Morrey spaces $L_{{\omega }_1}^{p_1,{\eta }_1,\phi }\left(R^n\right)\times L_{{\omega }_2}^{p_2,{\eta }_2,\phi }\left(R^n\right)$ and product of grand generalized fractional weighted Morrey spaces $L_{{\omega }_1}^{p_1),{\eta }_1,\phi ,\theta ,\sigma }\left(R^n\right)\times L_{{\omega }_2}^{p_2),{\eta }_2,\phi ,\theta ,\sigma }\left(R^n\right)$, where $\phi (\cdot )$ is a positive, increasing function defined on $[0,\infty )$ and satisfies doubling conditions, ω is an $A_p\left(R^n\right)$ weight, $\theta >0$, $\sigma <p-1$ and $p\in (1,\frac{n}{\eta })$ with $\eta \in (0,n)$. Via some known results, the authors show that the $\tilde{T}$ and ${\tilde{T}}_{b_1,b_2}$ associated with BMO functions are bounded from product of spaces $L^{p_1,{\eta }_1,\phi }\left(R^n\right)\times L^{p_2,{\eta }_2,\phi }\left(R^n\right)$ into spaces $L^{p,\eta ,\phi }\left(R^n\right)$, and they are also bounded from product of spaces $L^{p_1,{\eta }_1,\phi }\left(R^n\right)\times L^{p_2,{\eta }_2,\phi }\left(R^n\right)$ into generalized fractional weak Morrey spaces $W{L}^{p,\eta ,\phi }\left(R^n\right)$, where $\eta ={\eta }_1+{\eta }_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ for $p_i\in (1,\frac{n}{{\eta }_i})(i=1,2)$. Furthermore, the boundedness of the $\tilde{T}$ and ${\tilde{T}}_{b_1,b_2}$ on product of spaces $L_{{\omega }_1}^{p_1,{\eta }_1,\phi }\left(R^n\right)\times L_{{\omega }_2}^{p_2,{\eta }_2,\phi }\left(R^n\right)$ and product of spaces $L_{{\omega }_1}^{p_1),{\eta }_1,\phi ,\theta ,\sigma }\left(R^n\right)\times L_{{\omega }_2}^{p_2),{\eta }_2,\phi ,\theta ,\sigma }\left(R^n\right)$ is obtained, respectively.