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

TL;DR

This paper establishes $L^p-L^q$ estimates for solutions to the Dunkl wave equation, extending classical wave equation estimates to the Dunkl setting, which involves a generalized Laplacian operator.

Contribution

It derives $L^p-L^q$ estimates for the Dunkl wave equation, extending known results from the classical wave equation to the Dunkl framework.

Findings

Established $L^p-L^q$ estimates for Dunkl wave solutions.

Extended classical wave equation estimates to Dunkl operators.

Provides a foundation for further analysis of PDEs in Dunkl setting.

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 .