12
G. Pistone
f −
f (y) γ (y) dy
L (cosh −1)( γ )
≤ C 3 |∇ f || L gauss 2 (γ ) .
(1.20)
Other equivalent norms could be used in the inequalities above. For example,
L (exp 2 ) ∗ (γ ) ↔ L (cosh −1) ∗ (γ ) and L gauss 2 (γ ) ↔ L
2
cosh −1 (γ ). We do not care in the
present paper to define explicitly the relevant Gauss-Sobolev spaces as in [27]. But
notice the special relevance of the space based on the norm f → |∇ f || L
2
cosh −1 (γ ) .
1.3.2 Generator of the Ornstein-Uhlenbeck Semi-group
We consider now a further set of inequalities which are based on the use of infinitesimal generator −δ · ∇ of the Ornstein-Uhlenbeck semigroup, see Eqs. (1.11) and
(1.12). Compare, for example, [21, Sect. 1.3.7].
We have, for all f ∈ C
2
poly (R
n
), that
f (x) − f = −
∞
0
d
dt
P t f (x) dt =
∞
0
δ · ∇ P t f (x) dt .
(1.21)
Note that
∇ P t f (x) = ∇
f (e
−t x +
1 − e −2t y) γ (y) dy =
e
−t
∇ f (e
−t x +
1 − e −2t y) γ (y) dy = e
−t P t ∇ f (x) ,
so that
P t δ · ∇ f (x) = δ · ∇ P t f (x) = e
−t
δ · P t ∇ f (x) .
Now, Eq. (1.21) becomes
f (x) − f =
∞
0
e
−t
δ · P t ∇ f (x) dt .
(1.22)
As
δ · ∇ f (x) γ (x) dx = 0 ,
the covariance of f, g ∈ C
0
poly (R
n
) is
Cov γ ( f, g) =
( f (x) − f )g(x) γ (x) dx =
( f (x) − f )(g(x) − g) γ (x) dx .