Image of Lp(ℝn) under the Hermite Semigroup - R. Radha; D. Venku Naidu (International Journal of Mathematics and Mathematical Sciences)
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<∞.