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

Abstract

Denote by γ the Gauss measure on ℝⁿ and by ${\mathcal{L}}$ the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space ${{\mathfrak{h}}^1}{{\rm \gamma}}$ of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator $(r{\mathcal{I}}+{\mathcal{L}})^{iu}$ is unbounded from ${{\mathfrak{h}}^1}{{\rm \gamma}}$ to L ¹γ. This result is in sharp contrast both with the fact that $(r{\mathcal{I}}+{\mathcal{L}})^{iu}$ is bounded from H ¹γ to L ¹γ, where H ¹γ 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 $(r{\mathcal{I}}-\Delta)^{iu}$ is bounded from the Goldberg space ${{\mathfrak{h}}^1}{{\mathbb{R}}^n}$ to L ¹ℝⁿ. We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator ${\mathcal{T}}$, bounded on L ² μ, and with a kernel satisfying certain analytic assumptions, is bounded from H ¹ μ to L ¹ μ if and only if it is bounded from ${{\mathfrak{h}}^1}{\mu}$ to L ¹ μ. Here H ¹ μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and ${{\mathfrak{h}}^1}{\mu}$ 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.