Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

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 some embeddings into the Morrey space ($D$-Morrey space) $L_{p,\lambda ,\alpha }$, $0\le \lambda <2\alpha +2$ and modified Morrey space (modified $D$-Morrey space) ${\tilde{L}}_{p,\lambda ,\alpha }$ associated with the Dunkl operator on $\mathbb{ℝ}$. As applications we get boundedness of the fractional maximal operator $M_{\beta }$, $0\le \beta <2\alpha +2$, associated with the Dunkl operator (fractional $D$-maximal operator) from the spaces $L_{p,\lambda ,\alpha }$ to $L_{\infty }\left(\mathbb{ℝ}\right)$ for $p=(2\alpha +2-\lambda )/\beta$ and from the spaces ${\tilde{L}}_{p,\lambda ,\alpha }\left(\mathbb{ℝ}\right)$ to $L_{\infty }\left(\mathbb{ℝ}\right)$ for $(2\alpha +2-\lambda )/\beta \le p\le (2\alpha +2)/\beta$.