Norms of certain functions of a distinguished Laplacian on the ax + b groups

article Norms of certain functions of a distinguished Laplacian on the ax + b groups published yes Rauan Akylzhanov author Yulia Kuznetsova author Michael Ruzhansky author Haonan Zhang author D8F41E38-9E66-11E9-A9E2-65C2E5697425 JaMa department ISTplus - Postdoctoral Fellowships project Curvature-dimension in noncommutative analysis project 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 Nature2022 eng General Mathematics Mathematische Zeitschrift 0025-5874 1432-1823 2101.00584 00085968070000110.1007/s00209-022-03143-z 30242327-2352 R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, and H. Zhang, “Norms of certain functions of a distinguished Laplacian on the ax + b groups,” Mathematische Zeitschrift, vol. 302, no. 4. Springer Nature, pp. 2327–2352, 2022. Akylzhanov R, Kuznetsova Y, Ruzhansky M, Zhang H. Norms of certain functions of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift. 2022;302(4):2327-2352. doi:<a href="https://doi.org/10.1007/s00209-022-03143-z">10.1007/s00209-022-03143-z</a> R. Akylzhanov, Y. Kuznetsova, M. Ruzhansky, H. Zhang, Mathematische Zeitschrift 302 (2022) 2327–2352. Akylzhanov, R., Kuznetsova, Y., Ruzhansky, M., & Zhang, H. (2022). Norms of certain functions of a distinguished Laplacian on the ax + b groups. Mathematische Zeitschrift. Springer Nature. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a> 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. Akylzhanov, Rauan, et al. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” Mathematische Zeitschrift, 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>. Akylzhanov, Rauan, Yulia Kuznetsova, Michael Ruzhansky, and Haonan Zhang. “Norms of Certain Functions of a Distinguished Laplacian on the Ax + b Groups.” Mathematische Zeitschrift. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00209-022-03143-z">https://doi.org/10.1007/s00209-022-03143-z</a>. 122102023-01-16T09:45:31Z2025-04-14T07:44:00Z