Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 014, 37 pages      arXiv:2308.16815      https://doi.org/10.3842/SIGMA.2025.014

Strichartz Estimates for the $(k,a)$-Generalized Laguerre Operators

Kouichi Taira a and Hiroyoshi Tamori b
a) Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, Japan
b) Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, 337-8570, Japan

Received June 24, 2024, in final form February 12, 2025; Published online March 02, 2025

Abstract
In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}\bigl(-|x|^{2-a}\Delta_k+|x|^a\bigr)$ which were introduced by Ben Saïd-Kobayashi-Ørsted, and for the operators $|x|^{2-a}\Delta_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $\Delta_k$ and $a$ denotes a positive real number satisfying certain conditions. The cases $a=1,2$ were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.

Key words: Strichartz estimates; oscillatory integrals; representation theory; Schrödinger equations.

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