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Cesàro summability of Hölder functions and Talbot effect on rank one Riemannian symmetric spaces of compact type

Utsav Dewan

Abstract

On rank one Riemannian symmetric spaces of compact type (of dimension $\ge 2$), we first obtain a quantitative characterization of Hölder continuity in terms of Cesàro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the Hölder continuity and the fractal profile of the Schrödinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.

Cesàro summability of Hölder functions and Talbot effect on rank one Riemannian symmetric spaces of compact type

Abstract

On rank one Riemannian symmetric spaces of compact type (of dimension ), we first obtain a quantitative characterization of Hölder continuity in terms of Cesàro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the Hölder continuity and the fractal profile of the Schrödinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.

Paper Structure

This paper contains 10 sections, 22 theorems, 150 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notations
  4. Harmonic analysis on rank one Riemannian symmetric spaces of compact type
  5. Cesàro summability of Hölder functions
  6. Some applications
  7. Upper Minkowski dimension of graphs of Hölder functions
  8. Oscillatory expansions of zonal spherical functions
  9. Talbot effect
  10. Concluding remarks

Key Result

Theorem 1.1

For $d \ge 2$, set Let $\delta \ge \delta_*$ and $\gamma \in (0,1)$. Then $f \in C^\gamma(\mathbb X)$ if and only if $\|C^\delta_Nf-f\|_{L^\infty(\mathbb X)} \lesssim N^{-\gamma}\:.$$\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: $L^p$ scale of Littlewood-Paley projections and the extra regularity parameter $s$

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 30 more