Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

Abstract

We consider the generalized shift operator, associated with the Dunkl operator ${\Lambda }_{\alpha }\left(f\right)\left(x\right)=(d/dx)f\left(x\right)+\left(\right(2\alpha +1)/x)\left(\right(f\left(x\right)-f(-x))/2)$, $\alpha >-1/2$. We study the boundedness of the Dunkl-type fractional maximal operator $M_{\beta }$ in the Dunkl-type Morrey space $L_{p,\lambda ,\alpha }\left(ℝ\right)$, $0\le \lambda <2\alpha +2$. We obtain necessary and sufficient conditions on the parameters for the boundedness $M_{\beta }$, $0\le \beta <2\alpha +2$ from the spaces $L_{p,\lambda ,\alpha }\left(ℝ\right)$ to the spaces $L_{q,\lambda ,\alpha }\left(ℝ\right)$, $1<p\le q<\infty$, and from the spaces $L_{1,\lambda ,\alpha }\left(ℝ\right)$ to the weak spaces $W{L}_{q,\lambda ,\alpha }\left(ℝ\right)$, $1<q<\infty$. As an application of this result, we get the boundedness of $M_{\beta }$ from the Dunkl-type Besov-Morrey spaces $B_{p\theta ,\lambda ,\alpha }^s\left(ℝ\right)$ to the spaces $B_{q\theta ,\lambda ,\alpha }^s\left(ℝ\right)$, $1<p\le q<\infty$, $0\le \lambda <2\alpha +2$, $1/p-1/q=\beta /(2\alpha +2-\lambda )$, $1\le \theta \le \infty$, and $0<s<1$.