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

Abstract

Let $(S,d,\rho )$ be the affine group ${\mathrm{R}}^n⋉{\mathrm{R}}^{+}$ 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 Hörmander'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-Hörmander 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 $p\in (1,\infty )$.

34 pages, J. Fourier Anal. Appl. (to appear)