A local smoothing estimate for the Schrödinger equation

AbstractWe 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 sup0<t<1|eitΔf| is bounded from Hs(Rn) to Lloc2(Rn) when s>s0 if and only if it is bounded from Hs(Rn) to L2(Rn) when s>2s0. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R2) to L2(R2) when s>3/4