Some embeddings into the Morrey and modified Morrey spaces associated with the Dunkl operator (Q963179)

Summary: We consider the generalized shift operator associated with the Dunkl operator \(\Lambda_\alpha(f)(x)=(d/dx)f(x)+((2\alpha+1)/x)((f(x)-f(-x))/2)\), \(\alpha>-1/2\). We study some embeddings into the Morrey space (\(D\)-Morrey space) \(L_{p,\lambda,\alpha}\), \(0\leq\lambda<2\alpha +2\), and modified Morrey space (modified \(D\)-Morrey space) \(\widetilde L_{p,\lambda,\alpha}\) associated with the Dunkl operator on \(\mathbb R\). As applications, we get boundedness of the fractional maximal operator \(M_\beta\), \(0\leq\beta<2\alpha+2\), associated with the Dunkl operator (fractional \(D\)-maximal operator) from the spaces \(L_{p,\lambda,\alpha}\) to \(L_\infty(\mathbb R)\) for \(p=(2\alpha+2-\lambda)/\beta\) and from the spaces \(\widetilde L_{p,\lambda,\alpha}(\mathbb R)\) to \(L_\infty(\mathbb R)\) for \((2\alpha+2-\lambda)/\beta\leq p\leq (2\alpha+2)/\beta\).