PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 4, April 1988 SCHRODINGER EQUATIONS: POINTWISE CONVERGENCE TO THE INITIAL DATA LUIS VEGA (Communicated by Richard R. Goldberg) ABSTRACT. Let u(x,i) be the solution of the Schrödinger equation with initial data / in the Sobolev space ifs(Rn) with a > t¡. The a.e. convergence of u(x, i) to f(x) follows from a weighted estimate of the maximal function u*(x,t) = supi>0 |u(x,t)|. 0. Let / be in the Schwartz space S(R") and for x e R", í € R set u(x,t)= í /(e^WV^dí. ÍR" It is well known that u is the solution of the Schrödinger equation with initial data /. Au = i-u, t > 0, u(x,0) = f(x). For s € R we denote by Hs(Rn) the Sobolev space H9(Rn)= if € S'(R") s.t. H/IU^r.) = (|(1 + |e|3)*l/(0l2) < +001 . We obtain the following result. THEOREM 1. Let f be in Hs{Rn) with s> \. Then lim u(x,t) = f(x) a.e. x. This result is a consequence of the boundedness of the maximal operator u* (x) = sup0<lti\u(x,t)\. THEOREM 2. Let f be in Hs{Rn) with s > a/2 and a > 1. Then Since HT(R) with r > | is embedded in L°°(R) by the classical Sobolev inequal- ities, Theorem 2 is an immediate consequence of (/' THEOREM 3. IfO<a anda > 1, then f r roc ,JR" J -ao -—u(x,t) dta y ' 1/2 (i'X dt(\ + \x\)a 1 ^Cll/H//^-' + »/2(R")- Received by the editors December 4, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 42A45; Secondary 42B25. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 874 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use