  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 $\psi(\sqrt{\cal L})\exp(it \sqrt{\cal L})$, with
$\psi\in C_0(\mathbb{R})$. 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. M\"uller 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 $\tilde\Delta$, closely
related to $\cal L$. The functions include in particular
$\exp(-t\tilde\Delta^\gamma)$, $t>0,\gamma>0$, and $(\tilde\Delta-z)^s$, with
complex $z,s$.
