Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type

Technical Summary: Regularity and Pointwise Convergence for Dispersive Equations on Riemannian Symmetric Spaces of Compact Type

Problem Statement
This article addresses the Carleson problem in the setting of Riemannian symmetric spaces of compact type $X = U/K$. The central question is to determine the optimal Sobolev regularity threshold $\alpha$ such that for initial data $f \in H^\alpha(X)$, the solution $u(x,t)$ to general dispersive equations converges pointwise to $f(x)$ almost everywhere as $t \to 0^+$. Specifically, the authors investigate the equation:
$$i \frac{\partial u}{\partial t} - \psi(\sqrt{-\Delta})u = 0, \quad u(\cdot, 0) = f,$$
where $\Delta$ is the Laplace-Beltrami operator and $\psi$ is a phase function. The study focuses on spaces of rank 1 and 2, including spheres, real/complex/quaternionic projective spaces, and the Cayley plane.

Methodology
The authors employ a synthesis of harmonic analysis on compact semi-simple Lie groups, spectral theory of the Laplacian, and analytic number theory.

1. Harmonic Analysis and Spectral Decomposition:

   • The $L^2(X)$ space is decomposed using the Peter-Weyl theorem into irreducible unitary representations of $U$. For $K$-biinvariant functions (radial functions), this reduces to a series of zonal spherical functions (Jacobi polynomials).
   • The authors utilize the specific growth rates of eigenvalues $\lambda_\mu^2$ and the dimensions of eigenspaces $d_\mu$ for rank 1 and rank 2 spaces. For rank 1, eigenvalues behave quadratically ($\lambda_n^2 \sim n^2$), while for rank 2, they are analyzed via lattice point counting in annuli (Gauss circle problem).
   • Maximal function estimates are reduced to bounding the series $\sum (1+\lambda_\mu^2)^{-\alpha}$.

2. Transference Principle:

   • A novel transference principle (Theorem 1.10) is established. It asserts that if two dispersive equations have phase functions $\psi_1$ and $\psi_2$ of "comparable oscillation" (i.e., $|\psi_1(r) - \psi_2(r)| \leq C$ for large $r$), then the validity of maximal estimates for one implies the validity for the other. This allows results derived for the Schrödinger equation ($\psi(r)=r^2$) to be transferred to the Boussinesq and Beam equations.

3. Number Theoretic Techniques:

   • For the sufficiency of $\alpha > 1/3$ in the $K$-biinvariant case, the authors adapt Strichartz estimates on the circle $\mathbb{T}$ (using decoupling and polynomial partitioning ideas from Bourgain and Moyua-Vega) to the symmetric space setting. This involves expanding Jacobi polynomials and reducing the problem to maximal estimates on $\mathbb{T}$.
   • For the failure of $\alpha < 1/4$, the authors construct counterexamples using quadratic Gauss sums. They identify specific sets of times and spatial points where the solution diverges, relying on the periodicity of the phase functions and the arithmetic properties of the eigenvalues.

Key Contributions and Results

1. General Regularity Thresholds (Theorem 1.1):

   • For general initial data on rank 1 spaces, the authors prove that $\alpha > 1/2$ is sufficient for pointwise convergence. This improves upon the previous best-known bound of $\alpha > 1$ (Cowling) and matches the bound for lower-dimensional Zoll manifolds.
   • For rank 2 spaces, they prove $\alpha > 1$ is sufficient, improving the previous bound of $\alpha > a/2$ (for fractional Schrödinger equations with phase $r^a$).

2. Improved Thresholds for Symmetric Data (Theorem 1.5):

   • Focusing on $K$-biinvariant (radial) initial data on specific rank 1 spaces ($S^d$, $\mathbb{P}^d(\mathbb{R})$, $\mathbb{P}^4(\mathbb{C})$, $\mathbb{P}^8(\mathbb{H})$, $\mathbb{P}^{16}(\text{Cay})$), the authors prove that the threshold improves to $\alpha > 1/3$.
   • This result generalizes recent work by Chen et al. on the sphere to other projective spaces and extends the result to the Boussinesq and Beam equations via the transference principle.

3. Sharpness and Failure Results (Theorem 1.6):

   • The authors demonstrate that for the same class of $K$-biinvariant functions, the maximal estimate fails if $\alpha < 1/4$. This establishes a gap between the sufficient condition ($\alpha > 1/3$) and the necessary condition ($\alpha \geq 1/4$), mirroring the situation on the torus $\mathbb{T}^d$.

4. Transference Principle (Theorem 1.10):

   • The paper establishes that equations with comparable oscillation phases share the same regularity thresholds for pointwise convergence. This is presented as a new tool, even for the circle $\mathbb{T} \cong SO(2)$.

Significance and Claims
The paper claims to resolve the Carleson problem for general dispersive equations on compact symmetric spaces of rank 1 and 2, providing the first results for rank 2 spaces. By introducing a transference principle, the authors unify the analysis of the Schrödinger, Boussinesq, and Beam equations, showing that their pointwise convergence behaviors are governed by the same geometric and spectral constraints.

The authors note that their approach for the improved $\alpha > 1/3$ bound relies on the specific geometry of the antipodal manifold. They explicitly state that their method fails for $\mathbb{P}^d(\mathbb{C})$ and $\mathbb{P}^d(\mathbb{H})$ when $d$ is even (specifically $d=6, 8, \dots$ and $d=12, 16, \dots$) due to the concentration of the antipodal manifold, leaving the optimal threshold for these specific cases as an open problem. Similarly, the failure result for $\alpha < 1/4$ is currently limited to specific projective spaces, with the even-dimensional real projective spaces remaining an open question.

The work relies heavily on the interplay between the discrete spectrum of the Laplacian on symmetric spaces and number-theoretic properties of Gauss sums, highlighting the deep connection between harmonic analysis on Lie groups and arithmetic.