In this paper, we prove Hörmander’s \(L^p\) - \(L^q\) multiplier theorem for the hypergeometric (or Heckman-Opdam) Fourier multipliers associated with root systems for the range \(1<p\le 2 \le q<\infty \) . We also establish the \(L^p\) - \(L^q\) boundedness of spectral multipliers of the Heckman-Opdam Laplacian and consequently we obtain time asymptotics for the \(L^p\) - \(L^q\) norms of the hypergeometric heat kernels and the Sobolev-type embedding for the Heckman-Opdam Laplacian. The proof hinges on the Paley and Hausdorff-Young-Paley inequalities for the hypergeometric Fourier transform (also known as the Heckman-Opdam transform) associated with root systems. Moreover, we also study the aforementioned spectral multiplier results in the compact Heckman-Opdam setting.
