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

Abstract

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