Denote by γ the Gauss measure on ℝ
n
and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space
\mathfrak
h1g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each
u in ℝ ∖ {0} and
r > 0 the operator (
rI+
L)
iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from
\mathfrak
h1g{{\mathfrak{h}}^1}{{\rm \gamma}} to
L
1γ. This result is in sharp contrast both with the fact that (
rI+
L)
iu(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 (
rI-D)
iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space
\mathfrak
h1\mathbb
Rn{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to
L
1ℝ
n
. 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{\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
h1m{{\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
h1m{{\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.
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