@article{12210,
  abstract     = {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.},
  author       = {Akylzhanov, Rauan and Kuznetsova, Yulia and Ruzhansky, Michael and Zhang, Haonan},
  issn         = {1432-1823},
  journal      = {Mathematische Zeitschrift},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {2327--2352},
  publisher    = {Springer Nature},
  title        = {{Norms of certain functions of a distinguished Laplacian on the ax + b groups}},
  doi          = {10.1007/s00209-022-03143-z},
  volume       = {302},
  year         = {2022},
}