An endpoint space-time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in Lxq(Rn,Ltr(R)) with initial data f in the homogeneous Sobolev space H˙s(Rn). The exponents and regularity index satisfy n+1q+1r=n2 and s=n2−nq−2r. For n=2 we prove the estimates in the range q>16/5, and for n⩾3 in the range q>2+4/(n+1).