Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

Denote by γ the Gauss measure on ℝn and by L the Ornstein-Uhlenbeck operator. In this paper we introduce a Hardy space h1(γ) of Goldberg type and show that for each u in ℝ\{0} and r > 0 the operator (rI + L)iu is unbounded from h1(γ) to L1(γ). This result is in sharp contrast both with the fact that (rI + L)iu is bounded from H1(γ) to L1(γ), where H1(γ) denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278-313, 2007), and with the fact that in the Euclidean case (rI -Δ)iu is bounded from the Goldberg space h1(Rn) to L1(ℝn). We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T, bounded on L2(μ), and with a kernel satisfying certain analytic assumptions, is bounded from H1(μ) to L1(μ) if and only if it is bounded from h1(μ) to L1(μ). Here H1(μ) denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and h1(μ) is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137-190, 2009). The case of translation invariant operators on homogeneous trees is also considered. © 2009 Springer Science+Business Media B.V.