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

650 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
+
exp
r
l=1
s (n)
l Aln
Aqn − xpqDpn +
m
k=1,k=q
α (n)
k Akn
where we have used the equality ξ − Mξ 2 =ξ 2 −|Mξ| 2 :
r
Dpn exp
l=1
s (n)
l Aln
− Mξ rp
(n)
n s
1
2
1
,
=Dpn1 2 − rp
(n)
Mξn s 2 = c (n)
pp − rp
(n)
Mξn s 2 .
Definition. We shall say that two series
n an and
n bn with positive members are equivalent
and shall denote this by
n an ∼
n bn if they are convergent or divergent simultaneously. We
note that if an > 0, bn > 0, n ∈ N, then we have
an
∼ an
. (13)
min
t∈R N
an + bn
n∈N
bn
n∈N
Using (5) we get, setting b = (Mξ rp
n (s (n) )) m+1+N
n=m+1 ∈ RN , N ∈ N,
∼
m+1+N
n=m+1
m+1+N
n=m+1
r
tn exp s
l=1
(n)
l Aln
m
α
k=1
(n)
k Akn
2
− xpq 1
(t, b) =−1
|Mξ rp
n (s (n) )| 2
c (n)
pp −|Mξ rp
n (s (n) )| 2 +(Aqn − xpqDpn + m k=1,k=p α (n)
2
k
Akn)1 2
−1
. ✷
DuetoRemark9weshallwriteC (respectively Ĉ) instead of C (n) (respectively Ĉ (n) ), where
⎛
Ĉ (n) ⎜
= ⎜
⎝
c (n)
11
c (n)
12
c (n)
1m
⎛
C (n) ⎜
= ⎜
⎝
c (n)
11
c (n)
11
c (n)
12
c (n)
1m c(n)
2m
c (n)
12 ... c (n)
1m
c (n)
22 ... c (n)
⎞
⎟
2m ⎟
.
⎟
.. ⎠ ,
... c(n)
mm
c (n)
12 ... c (n)
+ c(n)
22 ...
1m
c (n)
. ..
2m
c (n)
2m ... c (n)
11 + c(n)
22 +···+c(n)
⎞
⎟
⎠
mm
.
Remark 14. To simplify the further computations we assume that the measures μ B (n) for 2
n m are standard: B (n) = I . Since μ m B = ∞ n=2 μ B (n) this assumption, which only concerns
finitely many of the μ B (n)’s, does not change the equivalence class of the initial measure μ m B and
the equivalence class of the corresponding representation T R,μm B .