transactions of the american mathematical society Volume A STRONG TYPE (2,2) ESTIMATE FOR A MAXIMAL OPERATORASSOCIATED TO THE SCHRÖDINGEREQUATION BY CARLOS E. KENIG1 AND ALBERTO RUIZ Abstract. Let T*f(x) = sup/>0|7;/(x)|, where (T,f)~(t) = e',|í|2/(í)/|í|'/4- We show that, given any finite interval /, j,\ T*f\2 dx < C, fR\f(x)\2 dx, and that the above inequality is false with 2 replaced by any p < 2. This maximal operator is related to solutions of the Schrödinger equation. Introduction. Let u(x, t) be the solution of the initial value problem /' du(x, t)/èt — d2u(x, t)/dx2 in t > 0, limr^0 u(x, t) = f, for/say in L2(R), and where the limit is to be understood in the L2-sense. By taking Fourier transforms, it is easy to see that u(x, t) = SJ(x) = jf™ eixieHij(i) d& In [C], L. Carleson raised the question: Under what conditions does u(x, t) ->r^0/pointwise a.e.? Carleson showed in [C] that if/belongs to the Sobolev space HS(R) = {/G L2(R): /|/(£)|2(1 + \£\2)sd$ < oo}, ä > 1/4, u(x,t)->f a.e. as t -* 0. He also constructed an example of an / G H]/\R) such that u(x, t) does not converge to / a.e. In [DK], the authors showed that, in fact, given s < 1/4, there exists / G HS(R) so that u(x, t) does not converge a.e. to /. Carleson obtained the positive result by showing (using the Kolmogorov-Seliverstov-Plessner method (see [KS])) that if t(x) is an arbitrary measurable function of x, then the operator Sl(x)f(x) = /J^00eix*e''(x)*2f(i-)d£ satis- fies the inequality | /_', St(x)f(x) dx\< C||/||Hi/4, for/ G C™, with C independent of t(x) and /. This inequality is enough to guarantee almost everywhere convergence, but does not answer the question: What are the mapping properties of the maximal operator 5/ = sup0<r<, | S,f(x) | ? In this note we show that given any finite interval / C R, we have the 'strong type' inequality f,\ Sf\2 < C/||/||2ïi/4, where C, depends only on /. Moreover, the example in [DK] shows that the above inequality fails if ||/||Hi/4 is replaced by ||/|[ff« for any s < 1/4. Instead of treating the operator 5 for /G Hl/4, we study the maximal operator Tf= sup0<t^x\Tlf(x)\, where TJ(x) = /_+0000e'JCVíj?(¿)^/|l|1/4, and / G L2(R) now, and we show that J\Tf\2^ C,\\f \\l^. Moreover, the example in [DK] shows that the inequality ¡\mp<Cj\\f wi,^ Received by the editors March 15, 1982 and, in revised form, August 31, 1982. 1980Mathematics Subject Classification. Primary 42A45; Secondary 42B25. 1 Alfred P. Sloan Fellow 1981-83. Research supported in part by the NSF. ©1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page 239 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use