Generalized Riesz Potential Operator in the Modified Morrey Spaces

Abstract

In this paper, we prove the boundedness of generalized fractional maximal operator $M_{\rho }$ and generalized Riesz potential operator $I_{\rho }$ in the modified Morrey spaces $\widetilde{L}_{p,\lambda }({\mathbb {R}^n})$. We show that the sufficient conditions for the boundedness of the operator $M_{\rho }$ and the operator $I_{\rho }$ from the modified Morrey spaces $\widetilde{L}_{p,\lambda }({\mathbb {R}^n})$ to another one $\widetilde{L}_{q,\lambda }({\mathbb {R}^n})$, for $1< p<q<\infty$ and from $\widetilde{L}_{1,\lambda }({\mathbb {R}^n})$ to the weak modified Morrey spaces $W\widetilde{L}_{q,\lambda }({\mathbb {R}^n})$, for $p=1, 1<q<\infty$. We get the boundedness of our two-operators $I_{\rho }$ and $M_{\rho }$ in the modified Morrey spaces $\widetilde{L}_{p,\lambda }({\mathbb {R}^n})$ using the local estimate given in the Lemma 2.