Browse › Real and complex analysis › 080 The exact Sobolev endpoint for Schrödinger convergence
Endpoint convergence for the planar Schrodinger equation
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Abstract
We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in $H^{1/3}(\mathbb R^2)$, the Schrödinger evolution converges almost everywhere to the initial data as $t\downarrow0$. The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every $0\lt t\lt 1$, and the resulting evolution is continuous at t = 0.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, lean/OAI/Analysis/PlanarEndpoint (Lean development in the release, not catalogued in a
lean/docs/080.mdscope note). ItsRepresentative.leanderives the paper's main conclusion from a uniform local $L^1$ bound for time paths that agree with the $L^2$ evolution; the AI Proved What? episode page describes it as checking "the planar paper's final step, assuming its main estimate".
Possibly relevant literature (found by the agent, not checked)
- L. B. Pierce, On Bourgain's counterexample for the Schrödinger maximal function (arXiv:1912.10574, 23 Dec 2019; v4 30 Oct 2020). A rigorous derivation of Bourgain's counterexample, which the paper cites (as Bourgain16) for the necessary condition $s\ge n/(2(n+1))$ that does not rule out equality.
- X. Du, J. Kim, H. Wang, R. Zhang, Lower bounds for estimates of the Schrödinger maximal function (arXiv:1902.01430, Math. Res. Lett. 27 (2020)). Gives lower bounds for $L^p$ Schrödinger maximal estimates by generalizing Bourgain's example; relevant to the paper's remark that an endpoint local $L^3$ maximal estimate is not asserted.
- X. Cen, X. Gao, J. Zhang, A sharp regularity threshold for Schrödinger maximal estimates on standard tori (arXiv:2609.29047, 24 Sep 2026). A periodic-setting paper with the same date as this paper, which states "The endpoint case s=d/[2(d+1)] remains open for d≥2" for the Euclidean problem this paper addresses.
Digests and expositions
- AI Proved What?, Result 080: The exact Sobolev endpoint for Schrödinger convergence (video explainer with transcript, October 2026). Summarizes both preprints and their status, noting that the planar paper does not claim the endpoint local $L^3$ estimate and that the page has "seen no expert review". [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 080
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026,
author = {{OpenAI}},
title = {{Endpoint convergence for the planar Schr\"odinger equation}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf}{OAI:Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026}},
year = {2026}
}Family 080: The exact Sobolev endpoint for Schrödinger convergence
Proves almost-everywhere convergence $e^{it\Delta}f\to f$ as $t\downarrow0$ for every $f\in H^{n/(2(n+1))}(\mathbb R^n)$ and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.
Other papers in this family:
- Endpoint pointwise convergence for the Schrodinger equation in higher dimensions · September 24, 2026