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 ${\Lambda }_{\alpha }(f)(x)=(d/dx)f(x)+((2\alpha +1)/x)((f(x)-f(-x))/2)$, $\alpha\gt -1/2$. We study the boundedness of the Dunkl-type fractional maximal operator ${M}_{\beta }$ in the Dunkl-type Morrey space ${L}_{p,\lambda ,\alpha }(\mathbb{R})$, $0\leq \lambda \lt 2\alpha +2$. We obtain necessary and sufficient conditions on the parameters for the boundedness ${M}_{\beta}$, $0\leq \beta \lt 2\alpha +2$ from the spaces ${L}_{p,\lambda ,\alpha}(\mathbb{R})$ to the spaces ${L}_{q,\lambda ,\alpha}(\mathbb{R})$, $1\lt p\leq q\lt \infty$, and from the spaces ${L}_{1,\lambda ,\alpha }(\mathbb{R})$ to the weak spaces $W{L}_{q,\lambda ,\alpha }(\mathbb{R})$, $1\lt q\lt \infty$. As an application of this result, we get the boundedness of ${M}_{\beta}$ from the Dunkl-type Besov-Morrey spaces ${B}_{p\theta ,\lambda,\alpha }^{s}(\mathbb{R})$ to the spaces ${B}_{q\theta ,\lambda,\alpha }^{s}(\mathbb{R})$, $1\lt p\leq q\lt \infty$, $0\leq \lambda\lt 2\alpha +2$, $1/p-1/q=\beta /(2\alpha +2-\lambda)$, $1\leq \theta \leq \infty$, and $0\lt s\lt 1$.