$L^p-L^q$ 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 ${\Delta }_k$, $${\Delta }_k{u}_k(x,t)={\partial }_t^2{u}_k(x,t),{\partial }_t{u}_k(x,0)=f(x),u_k(x,0)=g(x).$$ We extend to Dunkl setting the estimates given by Srichartz in \cite{Sti} for the ordinary wave equation .

17 pages