Lp − Lq Estimates for the Solution of the Dunkl-Klein-Gordon Equation

Résumé

Abstract In Dunkl theory on \(\mathbb{R}^{n}\) which generalizes classical Fourier analysis, we study the boundedeness of the solution of the Klein-Gordon equation given by$$\partial_{t}^{2}u-\Delta_{x}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \ \ \partial_{t}u(x,0)=f(x)$$with \$m > 0 \(\and \$\partial_{t}^{2}u\) \is the second derivative of the solution \(u\) with respect to \(t\) and \(\Delta_{x}u\) is the Laplacian with respect to \(x\) and \(g\) and \(f\) the two functions in \(\mathcal{S}(\mathbb{R}^{n})\) which surround the initial conditions. We gives \$L^{p}-L^{q} \(\estimates satisfied by the solution of the Klein-Gordon equation for cerain values of\) p \(and\) q$ in the Dunkl setting.

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