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

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We prove that the spherical mean value of the Dunkl-type generalized translation operator τ 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 τ 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.