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A sharp Schrödinger maximal estimate in \(\mathbb{R}^2\). (English) Zbl 1378.42011
The authors treat the solution of Schrödinger equation, given by \(e^{it\Delta}f(x)=(2\pi)^{-n}\int e^{i(x\dot \xi+t|\xi|^2)}\hat f(\xi)d\xi\). They consider the problem posed by Carleson: determine the optimal \(s\) for which \(\lim_{t\to 0}e^{it\Delta}f(x)=f(x)\) a.e. whenever \(f\) is in the Sobolev space \(H^s(\mathbb R^n)\). Their main result is: For every \(f\in H^s(\mathbb R^2)\) with \(s>1/3\), \(\lim_{t\to 0}e^{it\Delta}f(x)=f(x)\) almost everywhere.
On the other hand, Bourgain showed that the convergence can fail if \(s<n/(2(n+1))\) (in the case \(n=2\): \(s<1/3\)).
So, they solve Carleson’s problem in the case \(n=2\). For the proof, they use polynomial partitioning and decoupling, introduced by L. Guth and N. H. Katz [ibid. 181, No. 1, 155–190 (2015; Zbl 1310.52019)].
In the case \(n=1\) the optimal number is \(1/4\), which was proved by Carleson and Dahlberg and Kenig.

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
42B37 Harmonic analysis and PDEs
Full Text: DOI arXiv
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