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The exact Sobolev endpoint for Schrödinger convergence

Proves almost-everywhere convergence

e^{it\Delta}f\to f

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 H

^{1/3}

Manuscripts in this family · 2

Endpoint convergence for the planar Schrodinger equation

Sep 24, 2026·paper.pdf

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.

Full abstract

PDF on GitHub LaTeX source

Endpoint pointwise convergence for the Schrodinger equation in higher dimensions

Sep 24, 2026·paper.pdf

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

Full abstract

PDF on GitHub LaTeX source