Endpoint eigenfunction restriction estimates in codimension two

Technical Summary: Endpoint Eigenfunction Restriction Estimates in Codimension Two

Problem Statement
The paper investigates the optimal growth rate of $L^2$ norms of Laplace eigenfunctions restricted to smooth, closed submanifolds of codimension two in a smooth, closed Riemannian manifold $M$ of dimension $n \ge 3$. Let $e_\lambda$ be an $L^2$-normalized eigenfunction satisfying $-\Delta_g e_\lambda = \lambda^2 e_\lambda$. The classical endpoint restriction estimate established by Burq–Gérard–Tzvetkov [7] and Hu [15] states:
$$\|e_\lambda\|_{L^2(\Sigma)} \le C \lambda^{1/2} \sqrt{\log \lambda} \|e_\lambda\|_{L^2(M)}.$$
While examples on the standard sphere $S^n$ exhibit growth of order $\lambda^{1/2}$ without the logarithmic factor, it has been a longstanding open problem whether the $\sqrt{\log \lambda}$ factor is necessary for general smooth codimension-two submanifolds. Specifically, it was expected that the estimate could be improved to the "log-free" bound $O(\lambda^{1/2})$. This paper addresses the validity of this log-free conjecture and the precise nature of the optimal growth rate.

Methodology
The authors employ a combination of microlocal analysis, oscillatory integral theory, and singular Radon transform estimates.

1. $TT^*$ Reduction and Spectral Kernel Analysis:
   The proof of the upper bound (Theorem 1) utilizes a localized $TT^*$ argument. The operator $T_\lambda = R_\Sigma \varrho(\lambda - \sqrt{-\Delta_g})$ is analyzed, where $R_\Sigma$ is the restriction to $\Sigma$. The kernel of $T_\lambda T_\lambda^*$ is expanded using the Hadamard parametrix. By applying stationary phase in the $n-3$ transverse variables, the problem is reduced to analyzing oscillatory Hilbert-type operators.

2. Kernel Decomposition:
   The resulting kernel is split into "near" and "far" components based on the distance $|t-s|$ between points on the submanifold, with a cutoff at $|t-s| \approx \lambda^{-1/m}$.

   • Far Part: Bounded using uniform transverse $L^2$ estimates and Schur's test, yielding a bound proportional to $\log \lambda$.
   • Near Part: This is the critical component. The authors use a Gaussian transform and Taylor approximation to reduce the smooth phase and amplitude to polynomial models. This reduction allows the application of Stein–Street's estimate [32] on singular Radon transforms, which provides a bound independent of the logarithmic factor for these polynomial models.

3. Counterexample Construction:
   To prove the optimality of the improvement (Theorem 2), the authors construct explicit counterexamples on the product manifold $M = S^3 \times Y$.

   • They define curves on the round sphere $S^3$ using lacunary trigonometric profiles and quadratic-increment phases, analogous to Montgomery–Smith's counterexamples for endpoint Strichartz estimates.
   • By gluing local arcs with specific scales and frequencies, they construct closed embedded curves of varying regularity ($C^K$ and $C^\infty$).
   • The construction relies on the exact spherical projector and the orthogonality of trigonometric profiles to ensure that each dyadic frequency contributes a fixed amount, leading to logarithmic growth in the restriction norm.

Key Contributions and Results

• Theorem 1 (Little-oh Improvement): For any fixed smooth codimension-two submanifold $\Sigma$, the authors prove that the classical bound admits a "little-oh" improvement. Specifically, for any integer $m \ge 2$, there exist constants such that:
  $$\|e_\lambda\|_{L^2(\Sigma)}^2 \le \left( A_m \lambda + \frac{B}{m \lambda \log \lambda} \right) \|e_\lambda\|_{L^2(M)}^2.$$
  Consequently, as $\lambda \to \infty$:
  $$\|e_\lambda\|_{L^2(\Sigma)} = o\left( \lambda^{1/2} \sqrt{\log \lambda} \right) \|e_\lambda\|_{L^2(M)}.$$
  This establishes that the classical bound is not sharp, but the improvement depends on the parameter $m$ and vanishes as $m \to \infty$.

• Theorem 2 (Optimality and Log-Removal Failure): The paper demonstrates that no uniform quantitative improvement over the $\sqrt{\log \lambda}$ factor is possible for all smooth submanifolds.

  • Part (i): For every integer $K \ge 1$, there exists a $C^K$ submanifold and a sequence of eigenfunctions where the restriction norm grows as $c_K \lambda^{1/2} \sqrt{\log \lambda}$.
  • Part (ii): For any function $L(\lambda) = o(\sqrt{\log \lambda})$, there exists a smooth ($C^\infty$) submanifold and a sequence of eigenfunctions such that the ratio $\|e_\lambda\|_{L^2(\Sigma)} / (\lambda^{1/2} L(\lambda))$ tends to infinity.
  • This result provides a negative answer to the "log-removal question," proving that the estimate $O(\lambda^{1/2})$ cannot hold in general for smooth codimension-two submanifolds.

Significance
The paper resolves the long-standing problem of the optimal growth rate for codimension-two endpoint restriction estimates. It clarifies that while the logarithmic loss in the classical Burq–Gérard–Tzvetkov bound is not sharp for any fixed smooth submanifold (allowing for a little-oh improvement), the logarithmic factor cannot be removed uniformly across the class of all smooth submanifolds.

The work draws a parallel between eigenfunction restriction estimates and endpoint Strichartz estimates, as well as oscillatory singular integrals. It highlights a distinction similar to that found by Stein–Wainger and Zhang–Zhu: while uniform bounds may fail or require logarithmic losses for specific regularity classes, smooth phases allow for slightly better behavior (little-oh) that is nonetheless insufficient to eliminate the logarithmic growth entirely in the general case. The results settle the question of whether the log-free estimate $O(\lambda^{1/2})$ is generally valid, confirming it is not.