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

646 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
= (fn+1,fn+1) − γ −1
n dn+1,dn+1
+ min
t=(tk)∈Rn
γn(t − t0), (t − t0)
= (fn+1,fn+1) − γ −1
n dn+1,dn+1
. ✷
Remark 11. In fact a more general result holds. Let us denote by An+1 the real non-necessarily
symmetric matrix in R n+1 and by An its n × n block after crossing the element in the last column
and row, by vn+1 = (a1n+1,a2n+1,...,ann+1), hn+1 = (an+11,an+12,...,an+1n) vectors vn+1,
hn+1 ∈ R n . If det An = 0 then we have
an+1n+1 − A −1
n vn+1,hn+1
det An+1
= . (7)
det An
Proof. It is sufficient to use the identity (Schur–Frobenius decomposition)
The generators
An
An+1 =
v t n+1
hn+1 an+1n+1
=
Akn := A R,m
kn
An
0 Id A−1 0 1
= d
dt T R,μm B
n vt n+1
hn+1 an+1n+1
I+tEkn
t=0
. ✷
of the one-parameter groups I + tEkn have the following form (on smooth functions of compact
support):
where
k−1
Akn =
r=1
xrkDrn + Dkn, 1 k m, k < n, Akn =
m
r=1
xrkDrn, m