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

578 K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580
Recall projection π from U(gC) onto U(aC + n −
C ). Here we will not be able to find explicit
formula for π(Yδ ) as in Proposition 6.10. Nevertheless we can compute the induced equation.
Consider the decomposition of U(aC + n −
C ) under aC,
where
is root lattice of Σ − (g, a).
U(aC + n −
C ) =
p∈Π −
U(aC + n −
C )p, (28)
Π −
= p =
β∈Σ − (g,a)
cββ, 0 cβ, cβ∈ Z
Lemma 7.4. Let α + β − γ = δ ≡ (γj + γj−1)/2 be as in (26). Decomposing π(¯vα ¯vβvγ ) ∈
U(aC + n −
C ) according to (28),
π(¯vα ¯vβvγ ) =
π(¯vα ¯vβvγ )p, (29)
p∈Π −
we have that p −(βj − βj−1)/2, for any p appearing in (29) so that π(¯vα ¯vβvγ )p = 0.
Proof. The lemma can be proved by a case by case computation of the projection by using
Lemma 6.9, and is essentially contained in [1]. We sketch another somewhat more systematic
method. We denote the Iwasawa decomposition as ¯vα = π(¯vα) + y with y ∈ kC. Thus
The Iwasawa projection of the first term is
¯vα ¯vβvγ = π(¯vα) ¯vβvγ + y ¯vβvγ .
π(¯vα)π( ¯vβvγ ).
The projection of the second term is
π
[y, ¯vβ]vγ + π ¯vβ[y,vγ ] .
Observe by Lemma 6.9 that the element y is a positive compact root vector, so that all these projections
involved are of the form π(¯vδvɛ) with vδ and vɛ being non-compact positive root vectors.
Our lemma reduces to the following claim, which can be proved easily by using Lemma 6.9. The
weights p of π(¯vδvɛ) satisfy the inequalities
p − βj − βj−1
2
+ βk − βk ′
2
p − βj − βj−1
2
if δ − ɛ = γj + γj−1
2
if δ − ɛ = γj + γj−1
2
− γk + γk ′
2
− γk,
with k>k ′ ,