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Maximal estimates and pointwise convergence for solutions of certain dispersive equations with radial initial data on Damek-Ricci spaces

Utsav Dewan

TL;DR

This work extends Carleson's pointwise convergence problem to dispersive equations on Damek-Ricci spaces with radial data, focusing on Fractional Schrödinger, Boussinesq, and Beam-type evolutions. By developing a local maximal operator framework and leveraging spherical Fourier analysis, Bessel-type expansions, and a transference principle, the authors obtain a complete description of local $L^p$-Sobolev mapping properties for the maximal functions and prove sharp almost-everywhere convergence for initial data in $H^\beta(S)$ with $\beta \ge 1/4$, establishing the endpoint $\beta=1/4$ as sharp. The analysis combines Euclidean-model reductions with non-Euclidean oscillatory integral estimates, and introduces a transference mechanism that preserves maximal-bound properties across dispersive multipliers with comparable oscillation. These results generalize Euclidean Carleson-type bounds to rank-one Damek-Ricci spaces and set a blueprint for similar investigations on higher-rank symmetric spaces. Overall, the paper provides a robust toolkit for maximal estimates in noncompact harmonic analysis with radial symmetry, connecting deep harmonic-analytic structures to dispersive PDE behavior.

Abstract

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} -Δ_{\mathbb{R}^n} u=0\:,\:\:\: (x,t) \in \mathbb{R}^n \times \mathbb{R}\:, \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index $β$ such that $f$ belongs to the inhomogeneous Sobolev space $H^β(\mathbb{R}^n)$ , so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, $\displaystyle\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we address the Carleson's problem for the fractional Schrödinger equation, the Boussinesq equation and the Beam equation corresponding to both the Laplace-Beltrami operator $Δ$ and the shifted Laplace-Beltrami operator $\tildeΔ$, with radial initial data on Damek-Ricci spaces, by obtaining a complete description of the local (in space) mapping properties for the corresponding local (in time) maximal functions. Consequently, we obtain the sharp bound up to the endpoint $β\ge 1/4$, for (almost everywhere) pointwise convergence. We also establish an abstract transference principle for dispersive equations whose corresponding multipliers have comparable oscillation and also apply it in the proof of our main result.

Maximal estimates and pointwise convergence for solutions of certain dispersive equations with radial initial data on Damek-Ricci spaces

TL;DR

This work extends Carleson's pointwise convergence problem to dispersive equations on Damek-Ricci spaces with radial data, focusing on Fractional Schrödinger, Boussinesq, and Beam-type evolutions. By developing a local maximal operator framework and leveraging spherical Fourier analysis, Bessel-type expansions, and a transference principle, the authors obtain a complete description of local -Sobolev mapping properties for the maximal functions and prove sharp almost-everywhere convergence for initial data in with , establishing the endpoint as sharp. The analysis combines Euclidean-model reductions with non-Euclidean oscillatory integral estimates, and introduces a transference mechanism that preserves maximal-bound properties across dispersive multipliers with comparable oscillation. These results generalize Euclidean Carleson-type bounds to rank-one Damek-Ricci spaces and set a blueprint for similar investigations on higher-rank symmetric spaces. Overall, the paper provides a robust toolkit for maximal estimates in noncompact harmonic analysis with radial symmetry, connecting deep harmonic-analytic structures to dispersive PDE behavior.

Abstract

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition of the Schrödinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} -Δ_{\mathbb{R}^n} u=0\:,\:\:\: (x,t) \in \mathbb{R}^n \times \mathbb{R}\:, \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index such that belongs to the inhomogeneous Sobolev space , so that the solution of the Schrödinger operator converges pointwise to , , almost everywhere. In this article, we address the Carleson's problem for the fractional Schrödinger equation, the Boussinesq equation and the Beam equation corresponding to both the Laplace-Beltrami operator and the shifted Laplace-Beltrami operator , with radial initial data on Damek-Ricci spaces, by obtaining a complete description of the local (in space) mapping properties for the corresponding local (in time) maximal functions. Consequently, we obtain the sharp bound up to the endpoint , for (almost everywhere) pointwise convergence. We also establish an abstract transference principle for dispersive equations whose corresponding multipliers have comparable oscillation and also apply it in the proof of our main result.
Paper Structure (16 sections, 18 theorems, 240 equations, 1 table)

This paper contains 16 sections, 18 theorems, 240 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notations
  4. Fourier Analysis on $\mathbb R^n$:
  5. Preliminaries on oscillatory integral estimates:
  6. Damek-Ricci spaces and spherical Fourier Analysis thereon:
  7. Proof of the sufficient conditions of Theorem \ref{['theorem']}
  8. Estimating $T_{\psi,1}$:
  9. Estimating $T_{\psi,2}$:
  10. Estimating $T_{\psi,2}$ on $\overline{B_{R_0}}$ :
  11. Estimating $T_{\psi,2}$ on $A(R_0,R)$ :
  12. The transference principle
  13. Proof of the necessary conditions of Theorem \ref{['theorem']}
  14. Concluding remarks
  15. Results for general dispersive equations:
  16. ...and 1 more sections

Key Result

Theorem 1.1

For the maximal functions corresponding to the Fractional Schrödinger equation (with $a>1$), the Boussinesq equation and the Beam equation with respect to both $\Delta$ and $\tilde{\Delta}$, we have:

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 23 more