Browse › Real and complex analysis › 080 The exact Sobolev endpoint for Schrödinger convergence
Endpoint pointwise convergence for the Schrodinger equation in higher dimensions
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Abstract
We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in $H^{n/(2(n+1))}(\mathbb R^n)$, the free Schrödinger evolution converges almost everywhere to the initial data as $t\downarrow0$. The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval $0\lt t\lt 1$.
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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 Bourgain2016) for the necessary condition $s\ge n/(2(n+1))$ that does not assert failure at 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 local maximal bound with $L^1$ output at $H^{s_n}$ and to its comparison with $L^p$ maximal bounds of Du and Li.
- 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 higher-dimensional paper depends on lemmas from the planar paper and that no Lean development covers it. [family-level]
- Perplexity / Orply, Schrödinger Waves Converge Pointwise at the Sobolev Endpoint in Dimensions Three and Higher (Orply, 9 October 2026). This AI-assisted exposition describes the manuscript's Gaussian representative, packet-geometric proof outline, and time-tree frequency summation. [AI-generated per source]
Elsewhere
- Emergent Mind: family 080
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026,
author = {{OpenAI}},
title = {{Endpoint pointwise convergence for the Schr\"odinger equation in higher dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-September-24-2026/paper.pdf}{OAI:Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-dimensions-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 convergence for the planar Schrodinger equation · September 24, 2026