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 the (minus) 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 $t\to +\infty$, the convolution kernel $k_t$ of this operator satisfies $$\Vert k_t{\Vert }_1\asymp t,\Vert k_t{\Vert }_{\infty }\asymp 1,$$ 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 $z,s$.

To appear in Math.Z