Sharp L² estimate of Schrödinger maximal function in higher dimensions

2018/05/07 by Du, Xiumin, Zhang, Ruixiang · 2 citations
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.02775

Abstract

We show that, for n≥ 3, lim_{t → 0} e^itΔf(x) = f(x) holds almost everywhere for all f ∈ H^s (ℝⁿ) provided that s>(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 L² restriction estimate, which also gives improved results on the size of divergence set of Schrödinger 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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• On Falconer type functions and the distance set problem
• Maximal estimates for orthonormal systems of wave equations

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