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

478 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
(ii) if B is a disjoint union of sets (B (k) ) ∞ k=1
k ∈ N, i = 1,...,n, then
(iii) and for general B ∈ B(C n ),
Moreover, for every B ∈ B(C n ),
PT1,...,Tn
PT1,...,Tn
(B) =
(B) =
6. An alternative characterization of μS,T
= (B(k)
1 ×···×B(k) n ) ∞ k=1 , where B(k)
i ∈ B(C),
∞
k=1
B⊆U, U⊆C n open
(k)
PT1,...,Tn, B ; (5.10)
PT1,...,Tn (U). (5.11)
μT1,...,Tn (B) = τ PT1,...,Tn (B) . (5.12)
In this final section we are going to give a characterization of the Brown measure of two
commuting operators in M, which is different from the one we gave in Theorem 4.1. Recall
from [4] that for T ∈ M, the Brown measure of T , μT , is the unique compactly supported Borel
probability measure on C which satisfies the identity
τ log |T − λ1|
= log |z − λ| dμT (z) (6.1)
for all λ ∈ C.
We are going to prove that a similar property characterizes μS,T .
C
Theorem 6.1. Let S,T ∈ M be commuting operators. Then μS,T is the unique compactly supported
Borel probability measure on C2 which satisfies the identity
τ log |αS + βT − 1|
= log |αz + βw − 1| dμS,T (z, w) (6.2)
for all α, β ∈ C.
C 2
Remark 6.2. Let S,T ∈ M be as in Theorem 6.1. Note that if μS,T satisfies (6.2) for all α, β ∈ C,
then for all α, β, λ ∈ C,
τ log |αS + βT − λ1|
= log |αz + βw − λ| dμS,T (z, w). (6.3)
C 2
This is clear for λ = 0, and for λ = 0, (6.3) follows from the fact that two subharmonic functions
defined in C coincide iff they agree almost everywhere with respect to Lebesgue measure. It now
follows from Brown’s characterization of μαS+βT that μαS+βT is the push-forward measure να,β
of μS,T via the map (z, w) ↦→ αz + βw. On the other hand, if να,β = μαS+βT , then (6.2) holds.