Estimates for the Maximal Operator of the Ornstein-uhlenbeck Semigroup

We show pointwise estimates for the maximal operator of the Ornstein-Uhlenbeck semigroup for functions that are integrable with respect to the Gaussian measure. The estimates are used to prove pointwise convergence. The Ornstein-Uhlenbeck semigroup is defined by Ttf(x)= [ k(t,x,y)f(y)dy, Jr" where ,1. \ -"/2m -2f,-«/2 / \e~'x-y\ \ 4 ^ _ ^ D« k(t, x, y) n (1 -e ) exp -■-Z^L, t > 0, x £ R . This family of operators was considered by L. S. Ornstein and G. E. Uhlenbeck to construct a theory of Brownian motion [N]. The infinitesimal generator of this semigroup is L = ^A x ■ V, and the eigenfunctions are the Hermite polynomials. If 0 < r < 1 then the Poisson-Hermite integral of the function / is given by PrfM = TXogif(x). For x £ R" we set y(x) = n~"/2e~'x' , and given p > 1 , by Lp(Rn, y) we denote the class of functions that are integrable to the pth power over Rn with respect to the measure y(x) dx . Also, || • || denotes the usual norm. Let P*f(x) = sup \Prf(x)\. 0 1 and is of weak-type (1,1) with respect to y. C. P. Calderón [C], by adapting the well-known results of Abel and Cesàro summability of the multiple Fourier series to the case of Abel Received by the editors March 5, 1990 and, in revised form, July 12, 1990. 1991 Mathematics Subject Classification. Primary 42B25; Secondary 47D06, 60J60. The first author was supported in part by NSF Grant #DMS-9003095. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page