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

488 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
Lemma 7.5. For n ∈ N and 1 i n define fi : C n → C n+1 by
fi(z1,...,zn) = (z1,...,zn,zi).
Then for n commuting operators T1,...,Tn ∈ M, one has that
μT1,...,Tn,Ti
Proof. Given Borel sets B1,...,Bn+1 ⊆ C we must show that
Clearly,
μT1,...,Tn,Ti (B1 ×···×Bn+1) = μT1,...,Tn
= fi(μT1,...,Tn ). (7.13)
f −1
i (B1 ×···×Bn+1) . (7.14)
f −1
i (B1 ×···×Bn+1) = B1 ×···×(Bi ∩ Bn+1) ×···×Bn
so that the right-hand side of (7.14) is
τ PT1 (B1) ∧···∧PTi (Bi
∩ Bn+1) ∧···∧PTn (Bn)
= τ PT1 (B1) ∧···∧PTi (Bi) ∧ PTi (Bn+1) ∧···∧PTn (Bn) .
But this is exactly the left-hand side of (7.14) and we are done. ✷
We will not give the proof of Theorem 7.1 in full generality but rather, by way of an example,
illustrate how it goes. Consider for instance 3 commuting operators T1,T2,T3 ∈ M and the
polynomial q ∈ C[z1,z2,z3] given by
At first define φ1 : C 3 → C 5 by
q(z1,z2,z3) = 1 + 2z 2 2 + z1z2z3. (7.15)
φ1(z1,z2,z3) = (z2,z2,z1,z2,z3).
By repeated use of Lemmas 7.4 and 7.5 we find that
Next define φ2 : C 5 → C 2 by
μT2,T2,T1,T2,T3
= φ1(μT1,T2,T3 ).
φ2(z1,...,z5) = (2z1z2,z3z4z5),
and by repeated use of (7.9) and Lemma 7.4 conclude that
With φ3 : C 2 → C given by
μ 2T 2 2 ,T1T2T3 = (φ2 ◦ φ1)(μT1,T2,T3 ).
φ3(z1,z2) = z1 + z2