Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator

Technical Summary: Sharp Endpoint Eigenfunction Estimates for the Two-Dimensional Hermite Operator

Problem Statement
The paper addresses the problem of determining the optimal growth or decay rates of the spectral projection operator $\Pi_\lambda$ associated with the Hermite operator $H = -\Delta + |x|^2$ on $\mathbb{R}^2$. Specifically, the authors investigate the operator norm $\|\Pi_\lambda\|_{L^2(\mathbb{R}^2) \to L^p(\mathbb{R}^2)}$ at the critical endpoint exponent $p_e(2) = 10/3$.

Previous work by Koch and Tataru established sharp bounds away from this critical exponent but yielded a logarithmic loss factor of $(\log \lambda)^{1/p_e(d)}$ at the endpoint. While this logarithmic loss was proven unavoidable in dimension $d=1$, its necessity in dimensions $d \ge 2$ remained an open question. Recent work by Jeong, Lee, and Ryu removed this loss for $d \ge 3$ and, independently, for $d=2$ using asymmetric localization and recursive multiscale decomposition. This paper aims to provide an independent proof for the two-dimensional case that removes the logarithmic factor without relying on recursive space-time localization or iterative almost-orthogonality arguments.

Methodology
The proof proceeds through a spectral decomposition in polar coordinates, analyzing the radial modes of the eigenfunctions. The domain is partitioned based on the behavior of the radial variable relative to the turning point, defined by the parameter $u = 1 - |x|^2/\lambda$.

1. Spectral Decomposition and Polar Basis:
   The authors utilize a polar Laguerre–Hermite basis $\{\Phi_{N,m}\}$ to expand the spectral projection. This reduces the problem to estimating radial modes $R_{N,m}(r)$ indexed by angular momentum $m$. A key parameter $\nu_m$ is introduced to identify the outer radial turning point for each mode.

2. Regional Analysis:
   The analysis divides the radial domain into distinct regions:

   • Turning-point core and deep interior/exterior: Controlled using existing localized estimates by Koch and Tataru.
   • Interior collar ($A_\lambda$): The region where $u$ is small but positive ($\lambda^{-2/3} \ll u \ll 1$). This is the critical region requiring new estimates.

3. Mode Classification:
   Within the interior collar, modes are classified based on their proximity to the turning point:

   • Non-glancing modes ($u \gg \nu_m$): These modes stay a fixed relative distance from the turning point.
   • Glancing/Forbidden modes ($u \lesssim \nu_m$): These include modes near the turning point and those deep in the forbidden region ($u < \nu_m$).

4. Key Analytical Tools:

   • Liouville–Green Representation: For non-glancing modes, a uniform Liouville–Green (WKB) representation is derived. This expresses the radial functions as oscillatory exponential sums with specific phase functions.
   • Exponential Sum Estimates: The authors apply van der Corput estimates to these exponential sums. Crucially, they establish a curvature bound for the phase difference between different radial scales (Lemma 4.2), showing that the second derivative of the phase difference is proportional to the distance between scales.
   • Weighted $TT^*$ Argument: To couple all relevant radial scales and avoid the accumulation of logarithmic factors, a weighted $TT^*$ argument is employed (Proposition 5.2). This involves a discrete van der Corput estimate combined with Abel summation and a weighted convolution kernel, allowing the summation over dyadic annuli to be controlled without a $(\log \lambda)$ loss.
   • Rapid Decay: For modes deep in the forbidden region ($u \ll \nu_m$), Lemma 4.3 establishes rapid decay, rendering their contribution negligible.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes the sharp, logarithm-free endpoint estimate:
$$\|\Pi_\lambda\|_{L^2(\mathbb{R}^2) \to L^{10/3}(\mathbb{R}^2)} \lesssim \lambda^{-1/10}.$$
The authors demonstrate that this estimate is sharp by constructing a specific eigenfunction (the highest-angular-momentum state) that achieves this decay rate.

The proof strategy distinguishes itself from previous approaches (specifically the recent work of Jeong–Lee–Ryu) by:

• Avoiding recursive multiscale decompositions in space and time.
• Avoiding iterative almost-orthogonality arguments.
• Instead, utilizing a uniform Liouville–Green representation to reduce the problem to exponential sums, which are then coupled across scales via a single weighted $TT^*$ argument.

Significance
The paper claims to resolve the question of whether the logarithmic loss is necessary in dimension two. By providing an independent proof that removes the logarithmic factor, it confirms that the sharp power law $\lambda^{-1/10}$ holds for the two-dimensional Hermite operator at the critical exponent. The methodology offers a distinct technical pathway to endpoint estimates, relying on the interplay between polar spectral decomposition, uniform asymptotic representations, and weighted harmonic analysis techniques, rather than the recursive localization methods used in higher dimensions or recent two-dimensional preprints.