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 $\alpha \in [0,n),m⩾1,{\sum }_{i=1}^m\frac{n}{q_i}=n-\alpha$, $\{A_i{\}}_{i=1}^m$ 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 $L^p$ 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 $A=I$.