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

Abstract

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.