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

672 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
Ξ 33
= max
33
Mξ (t) 2
max
∂φ3(t)
t∈R 2
This proves (47) for (p, q) = (3, 3).
By analogy we have for general n:
= − 1
t33∈R
∂t33
2
e1(t)=e2(t)=0
Ψ 33 .
+ ∂(C1(t) −1 1
d(t), d(t)) exp(− 2 [(CT , T ) − (C1(t)
∂tnn
−1d(t), d(t))])
√ ,
det C1(t)
∂φn(t) ∂(CT,T)
∂tnn 2 ∂tnn
∂φn(t)
= −en(t) +
∂tnn
∂(C1(t) −1 1
d(t), d(t)) exp(− 2 [(CT , T ) − (C1(t)
∂tnn
−1d(t), d(t))])
√ .
det C1(t)
When trk = trr, n r k 2, we have by (65)
tr C(t) =
1k