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

Denote by γ the Gauss measure on R and by L the Ornstein– Uhlenbeck operator. In this paper we introduce a Hardy space h(γ) of Goldberg type and show that for each u in R\{0} and r > 0 the operator (rI+L) is unbounded from h(γ) to L(γ). This result is in sharp contrast both with the fact that (rI+L) is bounded from H(γ) to L(γ), where H(γ) denotes the Hardy type space introduced in [MM], and with the fact that in the Euclidean case (rI−∆) is bounded from the Goldberg space h(R) to L(R). 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 L(μ), and with a kernel satisfying certain analytic assumptions, is bounded from H(μ) to L(μ) if and only if it is bounded from h(μ) to L(μ). Here H(μ) denotes the Hardy space introduced in [CMM1], and h(μ) is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor [T]. The case of translation invariant operators on homogeneous trees is also considered.