Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces

Abstract

The goal of this paper is to characterize the local sharp estimate $$(I_{\rho } f)^{\#}(x) \le C \, M_{\rho } f(x)$$ and by using this inequality to get necessary and sufficient conditions on the triple functions $$(\varphi , \rho , \omega )$$ which satisfy the equivalence of norms of the generalized fractional integral operator $$I_{\rho }$$ and the generalized fractional maximal operator $$M_{\rho }$$ on the generalized weighted Morrey spaces $${\mathcal {M}}_{p,\varphi }({\mathbb {R}}^{n},\omega )$$ and generalized weighted central Morrey spaces $$\dot{{\mathcal {M}}}_{p,\varphi }({{\mathbb {R}}}^n,\omega )$$ , when $$\omega \in A_{\infty }$$ -Muckenhoupt class.