Equivalent Characterizations for Boundedness of Maximal Singular Integrals on $ax+b$\,--Groups

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 $p\in(1, \infty)$.