Fourier multiplier norms of spherical functions on the generalized Lorentz groups (Q1679458)

Let \(G\) be a locally compact group and \(K\) be a compact subgroup of \(G\). The Gelfand pairs \((G,K)\) are considered, mainly for the cases, when \(G\) is isomorphic to \(\mathrm{SO}_0(1,n)\) (the generalized Lorentz group), \(\mathrm{SU}(1,n)\), \(\mathrm{Sp}(1,n)\) or the exceptional Lie group \(F_{4(-20)}\) (all they are real rank one simple Lie groups) and \(K\) is the standard maximal compact Lie subgroup in \(G\). For these Gelfand pairs the spherical functions are investigated. The completely bounded Fourier norms of the spherical functions on \(\mathrm{SO}_0(1,n)\) are calculated. Also it is proved that there is no uniform bound for the completely bounded Fourier multiplier norm of the spherical functions for aforementioned simple Lie groups. Some results about completely bounded Fourier multipliers for discrete groups \(G\) are also proved.