Restriction Theorem for the Fourier–Dunkl Transform and Its Applications to Strichartz Inequalities

In this article, we address the following Strichartz’s restriction problem: For a given surface S embedded in \(\mathbb {R}^{n}\times \mathbb {R}^{d}\) ${\mathbb{R}}^n\times {\mathbb{R}}^d$ with \(n+d\ge 2\) $n+d\ge 2$ , for what values of \(1\le p < 2,\) $1\le p<2,$ do we have \(\begin{aligned} \left( \int _S| {\widehat{f}}(\xi , \zeta ) |^2 h^2_\kappa (\zeta )d\sigma (\xi ,\zeta )\right) ^\frac{1}{2}\le C \Vert f\Vert _{L^p_\kappa (\mathbb R^n\times \mathbb R^d)}? \end{aligned}\) $$\begin{array}{c}\hfill {\left({\int }_S,{|\widehat{f}(\xi ,\zeta )|}^2,h_{\kappa }^2,(\zeta ),d,\sigma ,(\xi ,\zeta )\right)}^{\frac{1}{2}}\le C{\Vert f\Vert }_{L_{\kappa }^p({\mathbb{R}}^n\times {\mathbb{R}}^d)}?\end{array}$$ Here \({\widehat{f}}\) $\widehat{f}$ is the Fourier–Dunkl transform of f (defined in (1.2)). In particular, we prove Strichartz’s restriction theorem for the Fourier–Dunkl transform for certain surfaces, namely, paraboloid, sphere, and hyperboloid, and its generalization to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish respected versions of Strichartz estimates for Schrödinger’s propagator and Klein–Gordon propagator associated with the Dunkl Laplacian. This restriction theorem generalizes the Stein-Tomas and Strichartz’s restrictions theorems in the special cases.