The exact Sobolev endpoint for Schrödinger convergence
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Proves almost-everywhere convergence e^{it\Delta}f→ f as t\downarrow0 for every f∈ 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 H^(1/3).
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MANUSCRIPT 01
Endpoint convergence for the planar Schrodinger equation
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…
Full source 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< t< 1, and the resulting evolution is continuous at t = 0.
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Endpoint pointwise convergence for the Schrodinger equation in higher dimensions
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…
Full source 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< t< 1.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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