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

Abstract

The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator \({\mathcal {L}}\) on the ‘\(ax+b\)’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type \(\psi (\sqrt{{\mathcal {L}}})\exp (it \sqrt{{\mathcal {L}}})\), with \(\psi \in C_0({\mathbb {R}})\). We show that for \(t\rightarrow +\infty \), the convolution kernel \(k_t\) of this operator satisfies $$\begin{aligned} \Vert k_t\Vert _1\asymp t, \qquad \Vert k_t\Vert _\infty \asymp 1, \end{aligned}$$so that the upper estimates of D. Müller and C. Thiele (Studia Math., 2007) are sharp. As a necessary component, we recall the Plancherel density of \({\mathcal {L}}\) 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 \({\mathcal {L}}\). The functions include in particular \(\exp (-t{\tilde{\Delta }}^\gamma )\), \(t>0,\gamma >0\), and \(({\tilde{\Delta }}-z)^s\), with complex z, s.