An Extension of the Adams-type Theorem to the Vanishing Generalized Weighted Morrey Spaces

Abstract

In this paper, we generalize Adams-type theorems given in [1,13] (which are the following Theorem A and Theorem B, respectively) to the vanishing generalized weighted Morrey spaces. We prove the Adams-type boundedness of the generalized fractional maximal operator $M_{\rho}$ from the vanishing generalized weighted Morrey spaces $\mathcal{\mathcal{VM}}_{p,\varphi^{\frac{1}{p}}}(\mathbb{R}^n, w)$ to another one $\mathcal{\mathcal{VM}}_{q,\varphi^{\frac{1}{q}}}(\mathbb{R}^n, w)$ with $w \in A_{p,q}$ for $1$<$p$<$\infty,\ q$>$p$; and from the vanishing generalized weighted Morrey spaces $\mathcal{\mathcal{VM}}_{1,\varphi}(\mathbb{R}^n, w)$ to the vanishing generalized weighted weak Morrey spaces $\mathcal{\mathcal{VWM}}_{q,\varphi^{\frac{1}{q}}}(\mathbb{R}^n, w)$ with $w \in A_{1,q}$ for $p=1,\ 1$<$ q$<$\infty$. The all weight functions belong to Muckenhoupt-Weeden classes $A_{p,q}$.

Keywords

• Generalized fractional maximal operator
• Vanishing generalized weighted Morrey space
• Muckenhoupt-Weeden classes
• Muckenhoupt-Weeden class

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