Generalized Ornstein–Uhlenbeck Semigroups: Littlewood–Paley–Stein Inequalities and the P. A. Meyer Equivalence of Norms

Let μ be a nondegenerate Gaussian measure on a Hilbert space H. For an arbitrary selfadjoint nonnegative operator A we consider the semigroup etL=Γ(e−tA) on Lp(μ), where Γ stands for the second quantization operator. We provide an explicit characterization of the domains of (I−L)m/2 in Lp(μ) in terms of Gaussian Sobolev spaces thus extending the P. A. Meyer result on equivalence of norms. The main tools are the Littlewood–Paley–Stein inequalities which are proved under minimal assumptions by a purely analytic method following E. Stein (1970, “Topics in Harmonic Analysis,” Princeton Univ. Press, Princeton, NJ).