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

570 K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580
[Eβ,vα]v ∗ α = vβ+α ¯vα ≡ (ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) −
D(vβ+α,cj ), ¯vα
= (ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) + D(cj , ¯vβ+α)vα.
To find the last term we note first that for any Jordan triple system [9],
D(vα, ¯vα), D(w, ¯cj ) = D
vα, D(cj , ¯w)vα − D D(w, ¯cj )vα, ¯vα ;
we let it act on cj and then sum over vα
vα∈Vk,j−1
D cj ,D(cj , ¯w)vα
vα = a
D(cj−1,cj−1) + D(ck,ck)
2
w
by using Proposition 5.1. It is further w, since a = 2 for type I domains. Thus
≡
(ζvβ+α + ζ ′ vβ+α )(η ¯vα + η′ ¯vα ) + (j − 2)η ¯w + (j − 2)η ′ ¯w .
vα∈Vk,j−1
k