International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 287218, 13 pages
Research Article

Image of Lp(n) under the Hermite Semigroup

R. Radha and D. Venku Naidu

Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Received 9 June 2008; Revised 8 October 2008; Accepted 9 December 2008

Academic Editor: Misha Rudnev

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.


It is shown that the Hermite (polynomial) semigroup {et:t>0} maps Lp(n,ρ) into the space of holomorphic functions in Lr(n,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(n,Vt,p/2r/2), then it is shown that there is a function fLp(n,ρ) such that etf=F. However, a single necessary and sufficient condition is obtained for the image of L2(n,ρp/2) under et, 1<p<. Further it is shown that if F is a holomorphic function such that FL1(n,Vt,p/21/2) or FLm1,p(2n), then there exists a function fLp(n,ρ) such that etf=F, where m(x,y)=ex2/(p1)e4t+1ey2/e4t1 and 1<p<.