Generalized Fractional Integrals in the Vanishing Generalized Weighted Local and Global Morrey Spaces

Abstract

In this paper, we prove the boundedness of generalized fractional integral operators $I_{\rho}$ in the vanishing generalized weighted Morrey-type spaces, such as vanishing generalized weighted local Morrey spaces $\mathcal{\mathcal{\mathcal{VM}}}_{p,\varphi}^{\{x_0\}}(\Rn, w^{p})$ and vanishing generalized weighted global Morrey spaces $\mathcal{\mathcal{VM}}_{p,\varphi^{\frac{1}{p}}}(\Rn, w)$ by using weighted $L_{p}$ estimates over balls.

In more detail, we obtain the Spanne-type boundedness of the generalized fractional integral operators $I_{\rho}$ from the vanishing generalized weighted local Morrey spaces $\mathcal{\mathcal{\mathcal{VM}}}_{p,\varphi_1}^{\{x_0\}}(\Rn, w^{p})$ to another one
$\mathcal{\mathcal{VM}}_{q,\varphi_2}^{\{x_0\}}(\Rn, w^{q})$ with $w^{q} \in A_{1+\frac{q}{p'}}$ for $1<p< q<\infty$, and from the vanishing generalized weighted local Morrey spaces
$\mathcal{VM}_{1,\varphi_1}^{\{x_0\}}(\Rn, w)$ to the vanishing generalized weighted weak local Morrey spaces \,\,\,
$\mathcal{VWM}_{q,\varphi_2}^{\{x_0\}}(\Rn, w^{q})$ with $w \in A_{1,q}$ for $p=1,1< q<\infty$, where $\varphi_1, \varphi_2 \in \mathfrak{M}_{\rm loc}$ class. We also prove the Adams-type boundedness of the generalized fractional integral operators $I_{\rho}$ from the vanishing generalized weighted global Morrey spaces $\mathcal{\mathcal{VM}}_{p,\varphi^{\frac{1}{p}}}(\Rn, w)$ to $\mathcal{\mathcal{VM}}_{q,\varphi^{\frac{1}{q}}}(\Rn, w)$ with $ w \in A_{p,q}$ for $1<p<q<\infty$ and from the vanishing generalized weighted global Morrey spaces $\mathcal{\mathcal{VM}}_{1,\varphi}(\Rn, w)$ to the vanishing generalized weighted weak global Morrey spaces $\mathcal{WM}_{q,\varphi^{\frac{1}{q}}}(\Rn, w)$ with $w \in A_{1,q}$ for $p=1, 1<q<\infty$, where $\varphi \in \mathfrak{M}_{\rm glob}$ class. The our all weight functions belong to Muckenhoupt-Weeden classes $A_{p,q}$.