Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

A. Olofsson / Journal of Functional Analysis 236 (2006) 517–545 533
where y = DT ∗ n Inx − Tna0 ∈ DT ∗. Applying the operator θTn to this last equality we find that
n
Inx
a0
= θTn
0 DT
=
y
∗ n y
−T ∗ n y
.
This makes evident that every solution x ∈ H and a0 ∈ DTn of Eq. (3.4) is of the form
x = I −1
n DT ∗ n y,
a0 =−T ∗ n y
for some element y ∈ DT ∗ n . Also, if x ∈ H and a0 ∈ DTn are given by (3.5) for some element
y ∈ DT ∗, then, by property (3.2) of defect operators, Eq. (3.4) holds. We have thus shown that
n
the solutions of (3.4) are parametrized by (3.5). By (3.5) we now have that
f = a0 + S ′ nVnx =−T ∗ n y + S′ −1
nVnIn DT ∗ n y,
where y ∈ DT ∗ n .
Let us now prove the norm equality that f 2 An =y2 n .Letx ∈ H and a0 ∈ Dn,T be given
by (3.5). By Theorem 2.2 we have that
f 2 An =a0 2 n−1
+
k=0
(−1) k
n T
k + 1
k x 2 =a0 2 +x 2 n = T ∗ n y 2 +DT ∗ n y2 n =y2 n ,
where the last equality follows by (3.1). This completes the proof of the theorem. ✷
Let T ∈ L(H) be an n-hypercontraction. Notice that by Lemma 3.2 the operator TnT ∗ n in
L(Hn) acts as
TnT ∗ n x = TT∗
n−1
(−1) k
k=0
n
T
k + 1
∗k T k
x, x ∈ Hn.
Since the operator Tn ∈ L(H, Hn) is a contraction by Lemma 3.3, the operator
TT ∗
n−1
(−1) k
k=0
n
T
k + 1
∗k T k
in L(H)
is self-adjoint in L(Hn) and has its spectrum contained in the closed unit interval [0, 1]. We
denote by Qn,T the operator
Qn,T =
I − TT ∗
n−1
(−1) k
k=0
n
T
k + 1
∗k T k
1/2 in L(H),
(3.5)