Sharp L² estimates of the Schrödinger maximal function in higher dimensions

We show that, for n≥3, lim_t→0e^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 the divergence set of the 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.