A Note on the Weighted Modified Morrey Spaces

Abstract

In this study, firstly we define the weighted modified Morrey spaces $\widetilde{L}_{p,\lambda}(\Rn, \mu)$ for the weight function $\mu$ in the class $A_{p}(\Rn)$ and we prove the boundedness of some classical operators as the generalized fractional maximal operator $M_{\rho}$ and the generalized fractional integral operator $I_{\rho}$ from the weighted modified Morrey spaces $\widetilde{L}_{p,\lambda}(\Rn, \mu^{p})$ to $\widetilde{L}_{q,\lambda}(\Rn, \mu^{q})$ with $\mu^{q} \in A_{1+\frac{q}{p'}}(\Rn)$ and from $\widetilde{L}_{1,\lambda}(\Rn, \mu)$ to weighted weak modified Morrey spaces  $W\widetilde{L}_{q,\lambda}(\Rn, \mu^{q})$,  with $\mu \in A_{1,q}(\Rn)$ by proving the appropriate weighted norm  inequalities.