On the boundary convergence of solutions to the Hermite-Schr\uf6dinger equation

In a half-space we consider the Hermite Schr\uf6dinger equation with given boundary values. We prove a formula that links the solution of this problem to that of the classical Schr\uf6dinger equation. It shows that mixed norm estimates for the Hermite- Schr\uf6dinger equation can be obtained immediately fromthose known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link

Functional calculus for the Ornstein-Uhlenbeck operator

AbstractLet γ be the Gauss measure on Rd and L the Ornstein–Uhlenbeck operator, which is self adjoint in L2(γ). For every p in (1, ∞), p≠2, set φ*p=arcsin|2/p−1| and consider the sector Sφ*p={z∈C:|argz|<φ*p}. The main result of this paper is that if M is a bounded holomorphic function on Sφ*p whose boundary values on ∂Sφ*p satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on Lp(γ) and hence on Lq(γ) for all q such that |1/q−1/2|⩽|1/p−1/2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than Sφ*p