Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel

Abstract

The authors consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: where $\mathrm{\Omega }\in L^r({𝕊}^{n-\mathrm{1}})$, , and with $m\in Z^{+}$. The authors prove that ${\mathscr{H}}_{\mathrm{\Omega },A,\beta }^m$ is bounded on Herz type space and $\lambda$-Central Morrey space with $m\ge \mathrm{1}$, which is an open problem for $m>2$.