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

TL;DR

This paper establishes the equivalence of various boundedness conditions for maximal singular integrals on the affine group with exponential growth, extending classical harmonic analysis results to this non-commutative setting.

Contribution

It provides new characterizations of boundedness for maximal singular integrals on the affine group, including applications to spectral multipliers of the Laplacian.

Findings

Equivalence of boundedness conditions for $T^*$ on the affine group.

Boundedness of spectral multipliers under Mihlin-H"ormander conditions.

Maximal singular integrals are bounded on $L^p$, from $L^1$ to $L^{1,\, ext{weak}}$, and from $L_c^\infty$ to BMO.

Abstract

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