Endpoint eigenfunction restriction estimates in codimension two
Abstract
We investigate the optimal endpoint L2 restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold M. For every fixed smooth codimension-two submanifold Σ, we prove a little-o improvement o(λ1/2√(logλ)) on the classical estimate O(λ1/2√(logλ)) of Burq--Gérard--Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein--Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci--Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery--Smith and Carbery--Hofmann. In particular, we establish the log-free estimate O(λ1/2) when (M,g) and Σ are real analytic, whereas this estimate fails in general in the smooth setting.
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BibTeX
@misc{wang2026endpoint,
title = {{Endpoint eigenfunction restriction estimates in codimension two}},
author = {Xing Wang and Cheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18290},
eprint = {2609.18290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18290},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endpoint eigenfunction restriction estimates in codimension two
AU - Xing Wang
AU - Cheng Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.18290
UR - https://arxiv.org/abs/2609.18290
AB - We investigate the optimal endpoint L^2 restriction estimates of Laplace eigenfunctions on submanifolds of codimension 2 in a smooth closed Riemannian manifold M. For every fixed smooth codimension-two submanifold Σ, we prove a little-o improvement o(λ^1/2√(logλ)) on the classical estimate O(λ^1/2√(logλ)) of Burq--Gérard--Tzvetkov and Hu. Our proof uses the Bargmann transform and Tataru's phase-space representation to reduce the problem to Stein--Street's estimate for singular Radon transforms. The three-dimensional case can be handled directly by Ricci--Stein's estimate. Moreover, we construct explicit examples to show that the little-o improvement is optimal in general. These are closely related to earlier counterexamples for endpoint Strichartz estimates by Montgomery--Smith and Carbery--Hofmann. In particular, we establish the log-free estimate O(λ^1/2) when (M,g) and Σ are real analytic, whereas this estimate fails in general in the smooth setting.
ER -