$L^p-L^q$ estimates for the solution of the Dunkl wave equation

Abstract

In this paper, our main aim is to derive $L^p-L^q$ estimates of the solution $u_k(x,t)$ ( t fixed) of the Cauchy problem for the homogeneous linear wave equation associated to the Dunkl Laplacian $\Delta_k$, $$\Delta_ku_k(x,t)= \partial_t^2u_k (x,t),\quad \partial_tu_k(x,0)= f(x),\quad u_k(x,0)= g(x).$$ We extend to Dunkl setting the estimates given by Srichartz in \cite{Sti} for the ordinary wave equation .