R. Radha and D. Venku Naidu

Copyright © 2008 R. Radha and D. Venku Naidu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is shown that the Hermite (polynomial) semigroup {e−tℍ:t>0} maps Lp(ℝn,ρ) into the space of holomorphic functions in Lr(ℂn,Vt,p/2(r+ϵ)/2) for each ϵ>0, where ρ is the Gaussian measure, Vt,p/2(r+ϵ)/2 is a scaled version of Gaussian measure with r=p if 1<p<2 and r=p′ if 2<p<∞ with 1/p+1/p′=1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely Lr(ℂn,Vt,p/2r/2), then it is shown that there is a function f∈Lp(ℝn,ρ) such that e−tℍf=F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,ρp/2) under e−tℍ, 1<p<∞. Further it is shown that if F is a holomorphic function such that F∈L1(ℂn,Vt,p/21/2) or F∈Lm1,p(ℝ2n), then there exists a function f∈Lp(ℝn,ρ) such that e−tℍf=F, where m(x,y)=e−x2/(p−1)e4t+1e−y2/e4t−1 and 1<p<∞.