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

1. Problem Statement and Context

The paper addresses two interconnected problems in harmonic analysis and geometric measure theory on rank one Riemannian symmetric spaces of compact type ($X$), specifically for dimensions $d \geq 2$. These spaces include spheres ($S^d$), real/complex/quaternionic projective spaces ($P^d(\mathbb{R}), P^d(\mathbb{C}), P^d(\mathbb{H})$), and the Cayley plane ($P^{16}(\text{Cay})$).

The core problems are:

1. Quantitative Characterization of Hölder Continuity: While it is known that Cesàro means ($C_N^\delta$) converge uniformly for continuous functions if the index $\delta$ is sufficiently large, the paper seeks a precise quantitative characterization: specifically, determining the decay rate of $\|C_N^\delta f - f\|_\infty$ that is equivalent to the function $f$ belonging to the Hölder space $C^\gamma(X)$.
2. The Talbot Effect (Fractal Profiles): The Talbot effect describes the self-imaging of diffraction patterns at rational distances (or times in the Schrödinger evolution). At irrational times, the solution to the Schrödinger equation with rough initial data (e.g., step functions or bounded variation) develops a fractal profile. The paper aims to generalize the known results for spheres ($S^d$) and flat tori ($T^d$) to all rank one symmetric spaces, quantifying the Upper Minkowski dimension of the graph of the solution for almost every time instance.

2. Methodology

The author employs a synthesis of harmonic analysis, approximation theory, and geometric measure theory. The methodology is structured as follows:

• Harmonic Analysis on Symmetric Spaces:

  • Utilizes the spectral decomposition of the Laplace-Beltrami operator $\Delta$ on $X$. The space $L^2(X)$ decomposes into eigenspaces $H_n(X)$ associated with eigenvalues $\lambda_n^2$.
  • Uses Zonal Spherical Functions ($Z_n$) as kernels for projections. The paper derives new oscillatory expansions for these functions near the origin and near the cut locus (antipodal manifold). This is crucial because, unlike spheres where the cut locus is a single point, in projective spaces, the cut locus has higher dimension, leading to different asymptotic behaviors.
  • Employs Jacobi polynomials to represent $Z_n$ and utilizes asymptotic expansions involving Bessel functions (via Frenzen and Wong's results) to handle the oscillatory nature of the Schrödinger propagator.

• Approximation Theory:

  • Establishes a Bernstein-type inequality for spherical polynomials on $X$, bounding the gradient of a polynomial by its degree and $L^\infty$ norm.
  • Connects the decay of Cesàro means and Riesz means to Hölder continuity.
  • Uses Littlewood-Paley projections ($P_N$) to define Besov spaces $B^\gamma_{p,\infty}(X)$ and proves embedding theorems ($B^\gamma_{\infty,\infty} \subset C^\gamma$).

• Geometric Measure Theory:

  • Analyzes the Upper Minkowski dimension of the graph of a function.
  • Proves that for a compact Riemannian manifold, the graph of a $C^\gamma$ function has an upper Minkowski dimension bounded by $(d+1) - \gamma$. This is achieved by utilizing the local bi-Lipschitz structure of manifolds and gluing local estimates.

• Weyl Sum Estimates:

  • To handle the Schrödinger evolution at irrational times, the author uses estimates on exponential sums (Weyl sums) involving the eigenvalues and the zonal spherical functions. This allows bounding the $L^\infty$ norm of the Littlewood-Paley projections of the solution.

3. Key Contributions and Results

A. Quantitative Characterization of Hölder Spaces (Theorem 1.1)

The paper provides a necessary and sufficient condition for a function $f \in C(X)$ to be in the Hölder space $C^\gamma(X)$ ($0 < \gamma < 1$) in terms of Cesàro means.

• Result: For a specific threshold index $\delta^*$ (dependent on the type of space $X$), $f \in C^\gamma(X)$ if and only if:
  $$\|C_N^\delta f - f\|_{L^\infty(X)} \lesssim N^{-\gamma}$$
• Significance: This generalizes previous results known only for spheres ($S^d$) to the broader class of rank one symmetric spaces. It also yields quantitative characterizations via Riesz means (Corollary 1.3) and spherical means (Corollary 1.2).

B. Oscillatory Expansions of Zonal Spherical Functions (Theorem 6.1)

A significant technical contribution is the derivation of uniform oscillatory expansions for zonal spherical functions $Z_n(\theta)$ in two distinct regions:

1. Near the origin: Uniform expansion valid for $\theta \in [0, \pi/2]$.
2. Near the cut locus: A distinct expansion valid for $\theta \in [\pi/2, \pi]$.

• Significance: The expansions differ significantly between spheres and projective spaces due to the geometry of the cut locus (a single point vs. a submanifold). These expansions are essential for estimating the kernels of the Schrödinger propagator and are noted as being of independent interest.

C. The Talbot Effect and Fractal Dimension (Theorem 1.6)

The paper analyzes the Schrödinger evolution $u(\cdot, t)$ with initial data $f$. Assuming the initial data satisfies a decay condition on its Littlewood-Paley projections ($\|P_N f\|_{L^p} \lesssim N^{-(d/2 + s)}$), the main result characterizes the regularity and fractal dimension of the solution at almost every time $t$.

• Regularity: The solution $u(\cdot, t)$ belongs to the Hölder space $C^{\gamma'}(X)$ for almost all $t$, where $\gamma'$ depends on $p, s,$ and $d$.
• Fractal Dimension: Consequently, the upper Minkowski dimension of the graph of the real and imaginary parts of the solution is bounded by:
  $$\dim_M(\text{Graph}(u)) \leq (d+1) - \min\{\gamma', 1\}$$
• Corollary 1.8: For initial data analogous to "bounded variation" on $S^1$ (specifically $p = \frac{2d}{d+1}$ and $s=1/2$), the fractal dimension is bounded by $\frac{d+1}{2}$. This generalizes the famous $3/2$ dimension result for $S^1$ to higher-dimensional symmetric spaces.

4. Significance and Impact

1. Generalization of Classical Results: The paper successfully extends deep results regarding the Talbot effect and Hölder summability from the sphere ($S^d$) and flat tori to the entire class of rank one compact symmetric spaces. This is non-trivial due to the varying geometric structures (e.g., the dimension of the cut locus).
2. Bridging Analysis and Physics: It provides a rigorous mathematical framework for the "Talbot effect" in curved spaces, linking diffraction theory, quantum mechanics (Schrödinger propagation), and fractal geometry.
3. Technical Innovation: The derivation of uniform oscillatory expansions for zonal spherical functions near the cut locus is a novel analytical tool. It overcomes the lack of symmetry in projective spaces compared to spheres, allowing for precise kernel estimates required for the Schrödinger propagator.
4. Geometric Measure Theory: The proof of the Minkowski dimension bound for graphs on general compact manifolds (via bi-Lipschitz charts) provides a robust method applicable beyond the specific context of the Talbot effect.

In summary, Dewan's work establishes a comprehensive quantitative link between the spectral decay of functions on symmetric spaces, their Hölder regularity, and the fractal nature of their time-evolution under the Schrödinger equation, resolving the rational-irrational dichotomy for a broad class of curved manifolds.