---
_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: 2023-08-04T09:22:14Z
day: '01'
department:
- _id: JaMa
doi: 10.1007/s00209-022-03143-z
ec_funded: 1
external_id:
  arxiv:
  - '2101.00584'
  isi:
  - '000859680700001'
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'
...
---
_id: '9260'
abstract:
- lang: eng
  text: We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ)
    when Δ is a Q-divisor involving hyperplanes. This allows us to address a question
    of Tanimoto about whether the set of rational points on such an orbifold constitutes
    a thin set. Our approach relies on the Hardy–Littlewood circle method to first
    study an asymptotic version of Waring’s problem for mixed powers. In doing so
    we make crucial use of the recent resolution of the main conjecture in Vinogradov’s
    mean value theorem, due to Bourgain–Demeter–Guth and Wooley.
acknowledgement: While working on this paper the authors were both supported by EPSRC
  grant EP/P026710/1, and the second author received additional support from the NWO
  Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho
  Tanimoto for useful conversations related to this topic, and to the anonymous referee
  for numerous helpful suggestions.
article_processing_charge: No
article_type: original
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Shuntaro
  full_name: Yamagishi, Shuntaro
  last_name: Yamagishi
citation:
  ama: Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a
    mixed Waring problem. <i>Mathematische Zeitschrift</i>. 2021;299:1071–1101. doi:<a
    href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>
  apa: Browning, T. D., &#38; Yamagishi, S. (2021). Arithmetic of higher-dimensional
    orbifolds and a mixed Waring problem. <i>Mathematische Zeitschrift</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>
  chicago: Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1007/s00209-021-02695-w">https://doi.org/10.1007/s00209-021-02695-w</a>.
  ieee: T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem,” <i>Mathematische Zeitschrift</i>, vol. 299. Springer
    Nature, pp. 1071–1101, 2021.
  ista: Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds
    and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.
  mla: Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
    Orbifolds and a Mixed Waring Problem.” <i>Mathematische Zeitschrift</i>, vol.
    299, Springer Nature, 2021, pp. 1071–1101, doi:<a href="https://doi.org/10.1007/s00209-021-02695-w">10.1007/s00209-021-02695-w</a>.
  short: T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.
date_created: 2021-03-21T23:01:21Z
date_published: 2021-03-05T00:00:00Z
date_updated: 2023-08-07T14:20:00Z
day: '05'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00209-021-02695-w
external_id:
  isi:
  - '000625573800002'
file:
- access_level: open_access
  checksum: 8ed9f49568806894744096dbbca0ad7b
  content_type: application/pdf
  creator: dernst
  date_created: 2021-03-22T12:41:26Z
  date_updated: 2021-03-22T12:41:26Z
  file_id: '9279'
  file_name: 2021_MathZeitschrift_Browning.pdf
  file_size: 492685
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  success: 1
file_date_updated: 2021-03-22T12:41:26Z
has_accepted_license: '1'
intvolume: '       299'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 1071–1101
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
  grant_number: EP-P026710-2
  name: Between rational and integral points
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: Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 299
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...