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

480 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
Repeating this argument, we find that for arbitrary θ ∈[0, 2π[,
i.e.
Since θ was arbitrary, we conclude that
and this proves (6.7). ✷
supp(μ e iθ (S+T) ) ⊆ z ∈ C Re z r ′ (S) + r ′ (T ) ,
supp(μS+T ) ⊆ z ∈ C Re e −iθ z r ′ (S) + r ′ (T ) .
supp(μS+T ) ⊆ B 0,r ′ (S) + r ′ (T ) ,
Lemma 6.4. Let S,T ∈ M be commuting operators, and let α, β ∈ C. Then μαS,βT is the pushforward
measure of μS,T via the map hα,β : C × C → C × C given by
hα,β(z, w) = (αz, βw).
Proof. Recall that μαS,βT is uniquely determined by the property that for all B1,B2 ∈ B(C),
Now, it is easily seen that for α = 0 and β = 0,
Hence,
μαS,βT (B1 × B2) = τ PαS(B1) ∧ PβT (B2) . (6.8)
PαS(B1) = PS
1
α B1
1
μαS,βT (B1 × B2) = τ PS
α B1
1
∧ PT
and PβT (B2) = PT
β B2
1
β B2
.
1
= μS,T
α B1 × 1
β B2
−1
= μS,T hα,β (B1 × B2) . (6.9)
If for instance α = 0, then PαS(B1) = 0if0/∈ B1 and PαS(B1) = 1 if 0 ∈ B1. It then follows that
(6.9) holds in this case as well. Similar arguments apply if β = 0. ✷
Proof of Theorem 6.1. As noted in Remark 6.2, it suffices to prove that for all α, β ∈ C, μαS+βT
is the push-forward measure of μS,T via the map (z, w) ↦→ αz + βw. At first we will consider
the case α = β = 1. Define a : C × C → C by
We are going to prove that for all B ∈ B(C),
a(z,w) = z + w (z,w∈ C).
−1
μS+T (B) = μS,T a (B) . (6.10)