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

420 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
This shows that R ∗ is an other solution of (19) and thus P1 := Λ −1 (P ∗ Λ − 2α) is an other
solution of (17). Since P ∼−τΛ −1/2 ,wehaveP1 ∼−τΛ −1/2 , and therefore the spectrum of
P1 has a negative real part for τ large enough. From the uniqueness part, we deduce that P1 = P ,
which means R ∗ = R. Hence R is Hermitian.
Knowing that R is Hermitian, we may recast (19) into (18). ✷
Fixing η ∈ R (say η = 0 since the case η = 0 is rather trivial), we consider the maximal
subinterval Jη = (α, +∞) of (−h(η) 2 , +∞) on which there is an analytical function z ↦→ P(z),
such that P(z)is a stable solution of (17) with τ 2 = z. Lemma 3.1 ensures that Jη is non-void. By
analyticity, the properties stated in Lemma 3.1 remain valid over Jη. In particular P(z) remains
uniformly bounded over this interval, because of (18).
Differentiating (17) along Jη, wehave
ΛP dP
dz
+ ΛdP
dz P + (A∗ − A) dP
dz
= In.
Because of Lemma 3.1, part (1), this equation may be written as a Lyapunov equation for the
unknown ΛP ′ :
P ∗ Λ dP
dz
+ ΛdP
dz P = In. (20)
Since P is a stable matrix, this equation has a unique solution, given by
This formula shows the following monotonicity property.
Λ dP
dz =−
+∞
e xP∗
e xP dx. (21)
0
Lemma 3.2. The map z ↦→ ΛP (z) + A∗ η is monotonous decreasing for the natural order of
Hermitian matrices.
Equation (21) may be viewed as an ODE for z ↦→ P(z), with domain the set of stable matrices.
Notice that every solution of (21) satisfies Eq. (17) for z = τ 2 and some constant matrices A
and Σ, the latter being Hermitian. To see this, differentiate P(z) ∗ ΛP (z) and deduce that Σ :=
P(z) ∗ ΛP (z) − zIn is constant. Then
ΛP − (zIn + Σ)P(z) −1 = ΛP (z) − P(z) ∗ Λ
is skew-Hermitian and constant (differentiate again). Thus there exists an A such that the lefthand
side equals A − A ∗ . Notice that ΛP (z) + A ∗ is Hermitian.
From the above remark and Lemma 3.1, it becomes clear that (Jη; z ↦→ P) is the maximal
solution of the ODE (21) that is defined up to +∞. Because of the bound given by (18), this
solution remains uniformly bounded and therefore Jη equals (−h(η) 2 , +∞).
We summarize our results in the following statement.