Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp $L^2$ estimate of Schr\"odinger maximal function in higher
  dimensions   We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds
almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that
$s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint,
this result is sharp and fully resolves a problem raised by Carleson. Our main
theorem is a fractal $L^2$ restriction estimate, which also gives improved
results on the size of divergence set of Schr\"odinger solutions, the Falconer
distance set problem and the spherical average Fourier decay rates of fractal
measures. The key ingredients of the proof include multilinear Kakeya
estimates, decoupling and induction on scales.

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The categories of this paper are: quant-ph
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn