Image ofLp(ℝn)under the Hermite Semigroup

Abstract

It is shown that the Hermite (polynomial) semigroup{e−tℍ:t>0}mapsLp(ℝn,ρ)into the space of holomorphic functions inLr(ℂn,Vt,p/2(r+ϵ)/2)for eachϵ>0, whereρis the Gaussian measure,Vt,p/2(r+ϵ)/2is a scaled version of Gaussian measure withr=pif1<p<2andr=p′if2<p<∞with1/p+1/p′=1. Conversely ifFis a holomorphic function which is in a “slightly” smaller space, namelyLr(ℂn,Vt,p/2r/2), then it is shown that there is a functionf∈Lp(ℝn,ρ)such thate−tℍf=F. However, a single necessary and sufficient condition is obtained for the image ofL2(ℝn,ρp/2)undere−tℍ,1<p<∞. Further it is shown that ifFis a holomorphic function such thatF∈L1(ℂn,Vt,p/21/2)orF∈Lm1,p(ℝ2n), then there exists a functionf∈Lp(ℝn,ρ)such thate−tℍf=F, wherem(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1and1<p<∞.