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

φpq(t; tqq) =
Since
=
S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 665
R p+1
exp i
p p
k=1
r=k
xkrtrk
ykn + tqqyqn
dμ(x,y)
exp i a(x) + T,y
dμ(x,y) = exp − 1
C a(x) + T ,a(x)+ T
2
dμI (x).
C a(x) + T ,a(x)+ T = Ca(x), a(x) + 2 a(x),CT + (CT , T ),
we have
φpq(t; tqq) = exp − 1
(CT , T )
2
exp − 1
Ca(x), a(x) + 2 a(x),CT
2
To calculate the latter integral we use (56). Let us introduce the notation
We show that
for some
We have
where
Further
a(x),CT =
d(t) = drk(t)
X = (x12; x13,x23; ...; x1p,...; xp−1p) ∈ R (p−1)(p−2)
2 .
Ca(x), a(x) + 2 a(x),CT = C(t)X,X + 2 d(t),X
d(t) ∈ R (p−1)(p−2)
(p − 1)(p − 2)
2 and C(t) ∈ Mat
, R .
2
p
ak(x)(CT )k =
k=1
=
1k