{"abstract":[{"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.","lang":"eng"}],"publisher":"Springer Nature","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"external_id":{"arxiv":["2101.00584"],"isi":["000859680700001"]},"oa_version":"Preprint","page":"2327-2352","keyword":["General Mathematics"],"isi":1,"ec_funded":1,"publication":"Mathematische Zeitschrift","doi":"10.1007/s00209-022-03143-z","language":[{"iso":"eng"}],"year":"2022","date_updated":"2023-08-04T09:22:14Z","day":"01","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"article_type":"original","intvolume":"       302","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2101.00584"}],"citation":{"short":"R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift 302 (2022) 2327–2352.","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.","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>","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>.","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>.","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>"},"month":"12","author":[{"full_name":"Akylzhanov, Rauan","first_name":"Rauan","last_name":"Akylzhanov"},{"full_name":"Kuznetsova, Yulia","first_name":"Yulia","last_name":"Kuznetsova"},{"full_name":"Ruzhansky, Michael","first_name":"Michael","last_name":"Ruzhansky"},{"full_name":"Zhang, Haonan","first_name":"Haonan","last_name":"Zhang","id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425"}],"date_published":"2022-12-01T00:00:00Z","title":"Norms of certain functions of a distinguished Laplacian on the ax + b groups","project":[{"call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"},{"grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis","_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6"}],"scopus_import":"1","date_created":"2023-01-16T09:45:31Z","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.","issue":"4","department":[{"_id":"JaMa"}],"article_processing_charge":"No","type":"journal_article","_id":"12210","status":"public","publication_status":"published","quality_controlled":"1","volume":302}