Higher Order Commutators of Marcinkiewicz Integral Operator on Herz-Morrey Spaces with Variable Exponent

Abstract:

      In the case of $\Omega\in {\rm Lip}_\gamma({{\mathbb S}^{n-1}})~(0<\gamma\leq1)$, we prove the boundedness of the Marcinkiewicz integral operator $\mu_{\Omega}$ on the variable exponent Herz-Morrey spaces. Also, we prove the boundedness of the higher order commutators $\mu^{m}_{\Omega,b}$ with $b\in {{\rm BMO}({\mathbb R}^n)}$ on both variable exponent Herz spaces and Herz-Morrey spaces, and extend some known results.