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.
released 2026-09-24 | 4 theorems · 37 lemmas · 47 proofs · 37,897 words |
PLAY LEVEL 1 »(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.
released 2026-09-24 | 6 theorems · 42 lemmas · 60 proofs · 45,428 words |
PLAY LEVEL 2 »(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 $0\lt t\lt 1$.