Commutators for certain fractional type operators on weighted spaces and Orlicz–Morrey spaces

In this paper, we focus on a class of fractional type integral operators that can be served as extensions of Riesz potential with kernels \(\begin{aligned} K(x,y)=\frac{\Omega _1(x-A_1 y)}{|x-A_1 y |^{{n}/{q_1}}} \cdots \frac{\Omega _m(x-A_m y)}{|x-A_m y |^{{n}/{q_m}}}, \end{aligned}\) $$\begin{array}{c}\hfill K(x,y)=\frac{{\mathrm{\Omega }}_1(x-A_{1}y)}{|x-A_1{y|}^{n/q_1}}\cdots \frac{{\mathrm{\Omega }}_m(x-A_{m}y)}{|x-A_m{y|}^{n/q_m}},\end{array}$$ where \(\alpha \in [0,n)\) $\alpha \in [0,n)$ , \( m\geqslant 1\) $m⩾1$ , \(\sum \limits _{i=1}^m\frac{n}{q_i}=n-\alpha \) $\sum _{i=1}^m\frac{n}{q_i}=n-\alpha$ , \(\{A_i\}^m_{i=1}\) ${\{A_i\}}_{i=1}^m$ are invertible matrixes, \(\Omega _i\) ${\mathrm{\Omega }}_i$ is homogeneous of degree 0 on \(\mathbb R^n\) ${\mathbb{R}}^n$ and \(\Omega _i\in L^{p_i}(S^{n-1})\) ${\mathrm{\Omega }}_i\in L^{p_i}(S^{n-1})$ for some \(p_i\in [1,\infty )\) $p_i\in [1,\infty )$ . Under appropriate assumptions, we obtain the weighted \(L^p(\mathbb R^n)-L^q(\mathbb R^n)\) $L^p({\mathbb{R}}^n)-L^q({\mathbb{R}}^n)$ estimates as well as weighted \(H^p(\mathbb R^n)-L^q(\mathbb R^n)\) $H^p({\mathbb{R}}^n)-L^q({\mathbb{R}}^n)$ estimates of the commutators for such operators with BMO-type function when \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}\) $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ . In addition, we acquire the boundedness of these operators and their commutators with a function in Campanato spaces on Orcliz–Morrey spaces as well as the compactness for such commutators in a special case: \(m=1\) $m=1$ and \(A=I\) $A=I$ .