Positive $$L^p$$ L p -Bounded Dunkl-Type Generalized Translation Operator and Its Applications

Abstract

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 applications, we prove the Young inequality for a convolution defined by $$T^t$$ , the $$L^p$$ -boundedness of $$\tau ^y$$ on 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.