1 IG of Poincaré inequalities …
5
Let us denote by C
k
poly (R
n
), k = 0, 1, . . . , the vector space of functions which
are differentiable up to order k and which are bounded, together with all derivatives,
by a polynomial. This class of functions is dense in L
2
(γ ). For each couple f, g ∈
C
1
poly (R
n
), we have
f (x) ∂ i g(x) γ (x) dx =
δ i f (x) g(x) γ (x) dx ,
where the divergence operator δ i is defined by δ i f (x) = x i f (x) − ∂ i f (x). Multidimensional notations will be used, for example,
∇ f (x) · ∇g(x) γ (x) dx =
f (x) δ · ∇g(x) γ (x) dx , f, g ∈ C
2
poly
R
n
,
with δ · ∇g(x) = x · ∇g(x) − Δg(x).
For example, in this notation, the divergence of Eq. (1.1) with P = p · γ , Q =
q · γ , and p, q ∈ C
2
poly (R
n
), becomes
1
2
∇ log
p(x)
q(x)
· ∇ log
p(x)
q(x)
p(x) γ (x) dx =
1
2
log
p(x)
q(x)
δ ·
∇ log
p(x)
q(x)
p(x)
γ (x) dx .
The inner product Eq. (1.2) becomes, with P = p · γ and f, g, p ∈ C
2
poly (R
n
),
∇ f (x) · ∇g(x) p(x) γ (x) dx =
f (x)δ · ∇(g(x) p(x)) γ (x) dx .
Hermite polynomials H α = δ
α 1 provide an orthogonal basis for L
2
(γ ) such that
∂ i H α = α i H α−e i , e 1 the i-th element of the standard basis of R
n . In turn, this provides
a way to prove that there is a closure of both operator ∂ i and δ i on a domain which is
an Hilbert subspace of L
2
(γ ). Such a space is denoted by D
2 in [15]. Moreover, the
closure of ∂ i is the infinitesimal generator of the translation operator, [3, 16]. The
space D
2 is a Sobolev Space with Gaussian weight based on the L
2 norm, [1]. By
replacing that norm with a (cosh −1) Orlicz norm, one derives the applications to
IG that are presented in [13, 27].
1.2.3 Exponential Statistical Bundle
We refer to [25, 27] for the definition of maximal exponential manifold E (γ ), and
of statistical bundle SE (γ ). Below we report the results that are necessary in the
context of the present paper.
5
Let us denote by C
k
poly (R
n
), k = 0, 1, . . . , the vector space of functions which
are differentiable up to order k and which are bounded, together with all derivatives,
by a polynomial. This class of functions is dense in L
2
(γ ). For each couple f, g ∈
C
1
poly (R
n
), we have
f (x) ∂ i g(x) γ (x) dx =
δ i f (x) g(x) γ (x) dx ,
where the divergence operator δ i is defined by δ i f (x) = x i f (x) − ∂ i f (x). Multidimensional notations will be used, for example,
∇ f (x) · ∇g(x) γ (x) dx =
f (x) δ · ∇g(x) γ (x) dx , f, g ∈ C
2
poly
R
n
,
with δ · ∇g(x) = x · ∇g(x) − Δg(x).
For example, in this notation, the divergence of Eq. (1.1) with P = p · γ , Q =
q · γ , and p, q ∈ C
2
poly (R
n
), becomes
1
2
∇ log
p(x)
q(x)
· ∇ log
p(x)
q(x)
p(x) γ (x) dx =
1
2
log
p(x)
q(x)
δ ·
∇ log
p(x)
q(x)
p(x)
γ (x) dx .
The inner product Eq. (1.2) becomes, with P = p · γ and f, g, p ∈ C
2
poly (R
n
),
∇ f (x) · ∇g(x) p(x) γ (x) dx =
f (x)δ · ∇(g(x) p(x)) γ (x) dx .
Hermite polynomials H α = δ
α 1 provide an orthogonal basis for L
2
(γ ) such that
∂ i H α = α i H α−e i , e 1 the i-th element of the standard basis of R
n . In turn, this provides
a way to prove that there is a closure of both operator ∂ i and δ i on a domain which is
an Hilbert subspace of L
2
(γ ). Such a space is denoted by D
2 in [15]. Moreover, the
closure of ∂ i is the infinitesimal generator of the translation operator, [3, 16]. The
space D
2 is a Sobolev Space with Gaussian weight based on the L
2 norm, [1]. By
replacing that norm with a (cosh −1) Orlicz norm, one derives the applications to
IG that are presented in [13, 27].
1.2.3 Exponential Statistical Bundle
We refer to [25, 27] for the definition of maximal exponential manifold E (γ ), and
of statistical bundle SE (γ ). Below we report the results that are necessary in the
context of the present paper.