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

476 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
As mentioned in Remark 5.2, every open set U ⊆ C2 may be written as a union of countably
many mutually disjoint sets from K. Thus, we have now proved existence of PS,T (U) for every
such U, and for general B ∈ B(C2 ) we will define
PS,T (B) :=
PS,T (U). (5.7)
Then again, P := PS,T (B) satisfies that
(a) P is S- and T -invariant.
Moreover, we prove that with K = P(H),
B⊆U,U⊆C 2 open
(b) μS|K,T |K is concentrated on B, and
(c) P is maximal with respect to the properties (a) and (b).
These properties will entail that when B happens to be a union of countably many mutually
disjoint sets from K, then (5.7) agrees with the previous definition of PS,T (B) (cf. (5.5)).
Now, to see that (b) holds, note that μS|K,T |K is regular (cf. [6, Theorem 7.8]), and hence
μS|K,T |K (B) = inf μS|K,T |K (U) | B ⊆ U, U ⊆ C2 open . (5.8)
Let U be any open subset of C 2 containing B. Write U as a union of countably many mutually
disjoint sets from K:
Then, according to (5.6),
μS|K,T |K
(U) =
U =
∞
k=1
∞ (k)
B
k=1
1
τP MP PS
and using Proposition 2.8 and Lemma 3.3 we find that
μS|K,T |K (U) = trP MP
= τP MP
∞
k=1
∞
k=1
= τP MP
= τP MP (P )
= 1,
eS|K
PS|K
× B(k)
2 .
(k)
B 1 ∧ PT
PS,T (U) ∧ P
(k)
B 2 ∧ P ,
(k) (k)
B 1
eT |K B 2
(k) (k)
B 1 ∧ PT |K B 2
where PS,T (U) is given by (5.5). Hence by (5.8), μS|K,T |K is concentrated on B.