Norms of certain functions of a distinguished Laplacian on the $ax+b$ groups

TL;DR

This paper derives sharp estimates for the norms of functions of the Laplacian on $ax+b$ groups, focusing on spectrally localized wave propagators and their convolution kernels, with implications for harmonic analysis on these groups.

Contribution

It provides explicit norm estimates for functions of the Laplacian on $ax+b$ groups, confirming sharpness of previous bounds and comparing approaches to spectral density calculation.

Findings

Convolution kernels of wave operators satisfy  asymptotic bounds.

Upper estimates of D. Mller and C. Thiele are sharp.

Explicit spectral density and norm estimates for functions of the shifted Laplace-Beltrami operator.

Abstract

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 $z,s$.