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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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 42B37 Harmonic analysis and PDEs
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##### References:
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