  Let $(S, d, \rho)$ be the affine group $\mathrm{R}^n \ltimes \mathrm{R}^+$
endowed with the left-invariant Riemannian metric $d$ and the right Haar
measure $\rho$, which is of exponential growth at infinity. In this paper, for
any linear operator $T$ on $(S, d, \rho)$ associated with a kernel $K$
satisfying certain integral size condition and H\"ormander's condition, the
authors prove that the following four statements regarding the corresponding
maximal singular integral $T^\ast$ are equivalent: $T^\ast$ is bounded from
$L_c^\infty$ to $\mathrm{BMO}$, $T^\ast$ is bounded on $L^p$ for all $p\in(1,
\infty)$, $T^\ast$ is bounded on $L^p$ for certain $p\in(1, \infty)$ and
$T^\ast$ is bounded from $L^1$ to $L^{1,\,\infty}$. As applications of these
results, for spectral multipliers of a distinguished Laplacian on $(S, d,
\rho)$ satisfying certain Mihlin-H\"ormander type condition, the authors obtain
that their maximal singular integrals are bounded from $L_c^\infty$ to
$\mathrm{BMO}$, from $L^1$ to $L^{1,\,\infty}$, and on $L^p$ for all $p\in(1,
\infty)$.
