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

Kouichi Taira, Hiroyoshi Tamori

TL;DR

This work extends Strichartz estimates for Schrödinger equations to the $(k,a)$-generalized Laguerre operators $H_{k,a}=\frac{-|x|^{2-a}\Delta_k+|x|^a}{a}$, covering general positive $a$ (beyond the previously treated cases $a=1,2$). The authors develop a robust analytic framework based on symbol-type estimates for Bessel and Gegenbauer functions and a discrete stationary-phase approach inspired by Ionescu–Jerison, enabling uniform control of the radial–angular kernel sum $\mathscr{I}(b,\nu;w;t)$ across multiple parameters. Using these dispersive bounds, they apply Keel–Tao theory to obtain sharp homogeneous and inhomogeneous Strichartz estimates on finite time intervals for a broad range of $(p,q)$-admissible pairs, under precise dimensional and multiplicity constraints. The results integrate representation-theoretic structure (Dunkl theory) with harmonic-analytic techniques to advance understanding of dispersive dynamics in Dunkl-type settings, with potential extensions to fractional powers and related geometries. All estimates are carried with careful attention to endpoint behavior and parameter uniformity, making the approach adaptable to related radially symmetric dispersive problems.

Abstract

In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}\bigl(-|x|^{2-a}Δ_k+|x|^a\bigr)$ which were introduced by Ben Saïd-Kobayashi-Orsted, and for the operators $|x|^{2-a}Δ_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $Δ_k$ 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.

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

TL;DR

This work extends Strichartz estimates for Schrödinger equations to the -generalized Laguerre operators , covering general positive (beyond the previously treated cases ). The authors develop a robust analytic framework based on symbol-type estimates for Bessel and Gegenbauer functions and a discrete stationary-phase approach inspired by Ionescu–Jerison, enabling uniform control of the radial–angular kernel sum across multiple parameters. Using these dispersive bounds, they apply Keel–Tao theory to obtain sharp homogeneous and inhomogeneous Strichartz estimates on finite time intervals for a broad range of -admissible pairs, under precise dimensional and multiplicity constraints. The results integrate representation-theoretic structure (Dunkl theory) with harmonic-analytic techniques to advance understanding of dispersive dynamics in Dunkl-type settings, with potential extensions to fractional powers and related geometries. All estimates are carried with careful attention to endpoint behavior and parameter uniformity, making the approach adaptable to related radially symmetric dispersive problems.

Abstract

In this paper, we prove Strichartz estimates for the -generalized Laguerre operators which were introduced by Ben Saïd-Kobayashi-Orsted, and for the operators . Here denotes a non-negative multiplicity function for the Dunkl Laplacian and denotes a positive real number satisfying certain conditions. The cases 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.
Paper Structure (34 sections, 26 theorems, 84 equations)

This paper contains 34 sections, 26 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Key theorem
  4. Idea of the proof
  5. Related problems
  6. Notation
  7. Preliminary
  8. A sum of monotonic functions
  9. Discrete oscillatory integrals
  10. A variant of the stationary phase theorem
  11. Asymptotic expansion of special functions and decomposition of the sum
  12. Asymptotic expansion of Bessel functions
  13. Asymptotic expansion of Gegenbauer polynomials
  14. Decomposition of the sum
  15. Properties of the phase functions
  16. ...and 19 more sections

Key Result

Theorem 1.1

We assume one of the following, which implies eq:positive: Let $(p,q), (p_1,q_1), (p_2,q_2)\in [2,\infty]^2$ be $\sigma_{k,a}$-admissible exponents. Then, for $T>0$, there exists ${C>0}$ such that where $r^*$ denotes the Hölder conjugate of $r$: $r^*=r/(r-1)$ and in addition $(p_j,q_j)\neq \bigl(2,\frac{2\sigma_{k,a}}{\sigma_{k,a}-1}\bigr)$ when all of the conditions $n\geq 2$, $1<a<2$ and $k\eq

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • ...and 27 more