Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications

We prove that the spherical mean value of the Dunkl-type generalized translation operator ${\tau }^y$ is a positive $L^p$-bounded generalized translation operator $T^t$. As application, we prove the Young inequality for a convolution defined by $T^t$, the $L^p$-boundedness of ${\tau }^y$ on a radial functions for $p>2$, the $L^p$-boundedness of the Riesz potential for the Dunkl transform and direct and inverse theorems of approximation theory in $L^p$-spaces with the Dunkl weight.