1 IG of Poincaré inequalities …
3
The inverse function ψ = φ
−1 has the same properties (1) to (3) as φ, so that its
primitive
Ψ (y) =
y
0
ψ(v) dv , y ≥ 0 ,
is again a Young function. The couple (Φ, Ψ ), is a couple of conjugate Young
functions. The relation is symmetric and we write both Ψ = Φ ∗ and Φ = Ψ ∗ . The
Young inequality holds true,
Φ(x) + Ψ (y) ≥ x y , x, y ≥ 0 ,
and the Legendre equality holds true ,
Φ(x) + Ψ (φ(x)) = xφ(x) , x ≥ 0 .
Here are the specific cases we are going to use:
Φ(x) =
x
p
p
, Ψ(y) =
y
q
q
, p, q > 1 ,
1
p
+
1
q
= 1 ;
(1.3)
exp 2 (x) = e
x
− 1 − x , (exp 2 ) ∗ (y) = (1 + y) log(1 + y) − y ;
(1.4)
(cosh −1)(x) = cosh x − 1 , (cosh −1) ∗ (y) =
y
0
sinh
−1
(v) dv ;
(1.5)
gauss 2 (x) = exp
1
2
x
2
− 1 .
(1.6)
Given a Young function Φ, and a probability measure μ, the Orlicz space L Φ (μ)
is the Banach space whose closed unit ball is
f ∈ L
0
(μ)
Φ(| f |) dμ ≤ 1
. This
defines the Luxemburg norm,
f L Φ (μ) ≤ α if, and only if,
Φ(α
−1
| f |) dμ ≤ 1 .
From the Young inequality, it holds
|uv| dμ ≤
Φ(|u|) dμ +
Φ ∗ (|v|) dμ .
This provides a separating duality u, v μ =
uv dμ of L Φ (μ) and L Φ ∗ (μ) such
that
u, v μ ≤ 2 u L Φ (μ) v L Φ∗ (μ) .
From the conjugation between Φ and Ψ , an equivalent norm can be defined, namely,
the Orlicz norm
f L Φ (μ)
∗ = sup
f, g μ
f L Ψ (μ) ≤ 1
.