Norms of certain functions of a distinguished Laplacian on the ax + b groups Akylzhanov, Rauan Kuznetsova, Yulia Ruzhansky, Michael Zhang, Haonan General Mathematics The aim of this paper is to find new estimates for the norms of functions of a (minus) distinguished Laplace operator L on the ‘ax+b’ groups. The central part is devoted to spectrally localized wave propagators, that is, functions of the type ψ(L−−√)exp(itL−−√), with ψ∈C0(R). We show that for t→+∞, the convolution kernel kt of this operator satisfies ∥kt∥1≍t,∥kt∥∞≍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 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 Δ~, closely related to L. The functions include in particular exp(−tΔ~γ), t>0,γ>0, and (Δ~−z)s, with complex z, s. Springer Nature 2022 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/12210 Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische Zeitschrift</i>. 2022;302(4):2327-2352. doi:<a href="https://doi.org/10.1007/s00209-022-03143-z">10.1007/s00209-022-03143-z</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00209-022-03143-z info:eu-repo/semantics/altIdentifier/issn/0025-5874 info:eu-repo/semantics/altIdentifier/e-issn/1432-1823 info:eu-repo/semantics/altIdentifier/wos/000859680700001 info:eu-repo/semantics/altIdentifier/arxiv/2101.00584 info:eu-repo/semantics/openAccess