The Poisson–Hermite Semigroup

In this chapter, we consider the Poisson–Hermite semigroup, which is the semigroup subordinated to the Ornstein–Uhlenbeck semigroup. This is analogous to the classical case in which the Poisson semigroup is obtained by subordination of the heat semigroup (for more details see the Appendix). Then, we study the characterization of the \(\frac {\partial ^2}{\partial t^2} + L\)-harmonic functions, the generalized Poisson–Hermite semigroups, and the conjugated Poisson–Hermite semigroup which, as in the classical case, is closely related to the notion of singular integrals.