Norms of certain functions of a distinguished Laplacian on the $ax+b$ groups
The aim of this paper is to find new estimates for the norms of functions of the (minus) Laplace operator $\cal L$ on the `$ax+b$' groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type $ψ(\sqrt{\cal L})\exp(it \sqrt{\cal L})$, with $ψ\in C_0(\mathbb{R})$. We show that for $t\to+\infty$, the convolution kernel $k_t$ of this operator satisfies $$ \|k_t\|_1\asymp t, \qquad \|k_t\|_\infty\asymp 1, $$ so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of $\cal L$ and spend certain time presenting and comparing different approaches to its calculation. Using its explicit form, we estimate uniform norms of several functions of the shifted Laplace-Beltrami operator $\tildeΔ$, closely related to $\cal L$. The functions include in particular $\exp(-t\tildeΔ^γ)$, $t>0,γ>0$, and $(\tildeΔ-z)^s$, with complex $z,s$.