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 $$K(x,y)=\frac{\Omega_1(x-A_1 y)}{|x-A_1 y |^{\frac{n}{q_1}}} \cdots \frac{\Omega_m(x-A_m y)}{|x-A_m y |^{\frac{n}{q_m}}},$$ where $\alpha\in [0,n), m\geqslant1, \sum_{i=1}^m\frac{n}{q_i}=n-\alpha$, $\{A_i\}^m_{i=1}$ are invertible matrixes, $\Omega_i$ is homogeneous of degree 0 on $\R^n$ and $\Omega_i\in L^{p_i}(S^{n-1})$ for some $p_i\in [1,\infty)$. Under appropriate assumptions, we obtain the weighted $L^p$ estimates as well as weighted Hardy estimates of the commutator for such operators with $BMO$-type function. In addition, we acquire the boundedness of these operators and their commutators with a function in Campanato space on Orcliz-Morrey spaces as well as the compactness for such commutators in a special case: $m=1$ and $A=I$.