Image of L^p(â„ ⁿ) under the Hermite semigroup

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+e)/2) for each ε > 0, where ρ is the Gaussian measure, Vt,p/2r+e/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 Lrℂn, Vt,p/2r/2), then it is shown that there is a function f ∈ Lpℝn, ρ) such that e-tH f = F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,Pp/2 under e-tH, 1 < p < ∞. Further it is shown that if F is a holomorphic function such that F ε L1 (ℂn, Vt,p/21/2 or F ε Lm1,p ℝ2n, then there exists a function f ε Lp (ℝn,p) such that e-tℍf = F, where m(x,y), e-x2/ (p-1)e4t+1e-y2/e4t-1 and 1 < p < ∞.