---
_id: '12210'
abstract:
- lang: eng
  text: "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\r\n∥kt∥1≍t,∥kt∥∞≍1,\r\nso 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."
acknowledgement: "Yu. K. thanks Professor Waldemar Hebisch for valuable discussions
  on the general context of multipliers on Lie groups. This work was started during
  an ICL-CNRS fellowship of the second\r\nnamed author at the Imperial College London.
  Yu. K. is supported by the ANR-19-CE40-0002 grant of the French National Research
  Agency (ANR). H. Z. is supported by the European Union’s Horizon 2020 research and
  innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411
  and the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. R. A. was supported
  by the EPSRC grant EP/R003025. M. R. is supported by the EPSRC grant EP/R003025/2
  and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rauan
  full_name: Akylzhanov, Rauan
  last_name: Akylzhanov
- first_name: Yulia
  full_name: Kuznetsova, Yulia
  last_name: Kuznetsova
- first_name: Michael
  full_name: Ruzhansky, Michael
  last_name: Ruzhansky
- first_name: Haonan
  full_name: Zhang, Haonan
  id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
  last_name: Zhang
citation:
  ama: 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>
  apa: Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., &#38; Zhang, H. (2022). Norms
    of certain functions of a distinguished Laplacian on the ax + b groups. <i>Mathematische
    Zeitschrift</i>. Springer Nature. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a>
  chicago: Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang.
    “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.”
    <i>Mathematische Zeitschrift</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a>.
  ieee: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain
    functions of a distinguished Laplacian on the ax + b groups,” <i>Mathematische
    Zeitschrift</i>, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022.
  ista: Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. 2022. Norms of certain functions
    of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift.
    302(4), 2327–2352.
  mla: Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian
    on the Ax + b Groups.” <i>Mathematische Zeitschrift</i>, vol. 302, no. 4, Springer
    Nature, 2022, pp. 2327–52, doi:<a href="https://doi.org/10.1007/s00209-022-03143-z">10.1007/s00209-022-03143-z</a>.
  short: R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift
    302 (2022) 2327–2352.
date_created: 2023-01-16T09:45:31Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2025-04-14T07:44:00Z
day: '01'
department:
- _id: JaMa
doi: 10.1007/s00209-022-03143-z
ec_funded: 1
external_id:
  arxiv:
  - '2101.00584'
  isi:
  - '000859680700001'
fulldoi: https://doi.org/10.1007/s00209-022-03143-z
intvolume: '       302'
isi: 1
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2101.00584
month: '12'
oa: 1
oa_version: Preprint
page: 2327-2352
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
  grant_number: M03337
  name: Curvature-dimension in noncommutative analysis
publication: Mathematische Zeitschrift
publication_identifier:
  eissn:
  - 1432-1823
  issn:
  - 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Norms of certain functions of a distinguished Laplacian on the ax + b groups
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 302
year: '2022'
...