A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q⩾2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0s0 if and only if it is bounded from Hs(Rn) to L2(Rn) when s>2s0. A corollary is that sup03/4.