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

468 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
and μQT Q = μQT P Q (computed relative to QMQ) is concentrated on B, we get that Q
PT (B), and hence Q R.
Similarly, to prove that R Q, we must show that:
(i ′ ) RT PR = TPR,i.e.RT R = TR;
(ii ′ ) μRT PR = μRT R (computed relative to RMR) is concentrated on B.
Note that if PT (B) = 0, then R Q, so we may assume that PT (B) = 0. (i ′ ) holds, because
R(H) = P(H) ∩ PT (B)(H) is T -invariant when P(H) and PT (B)(H) are T -invariant. In order
to prove (ii ′ ), at first note that R(H) is TPT (B)-invariant. Hence
⊥
μTPT (B) = τ1(R) · μRT R + τ1 R · μR⊥TR⊥, (3.5)
where
It follows that
τ1 =
1
τ(PT (B)) · τ|PT (B)MPT (B).
c
τ1(R) · μRT R B c
μTPT (B) B = 0, (3.6)
and thus, if R = 0, then μRT R(B c ) = 0, and (ii ′ ) holds. If R = 0, then R Q is trivially fulfilled.
✷
Proposition 3.4. For every Borel set B ⊆ C,
KT (B) = KT ∗
c
B ∗⊥, (3.7)
where A ∗ := {z | z ∈ A} for A ⊆ C. Moreover, for all Borel sets A,B ⊆ C,
and
KT (A) ∩ KT (B) = KT (A ∩ B), (3.8)
KT (A ∪ B) = KT (A) + KT (B). (3.9)
Proof. Let B ∈ B(C) and let P = PT (B). Then P ⊥ is T ∗-invariant, and
∗
μP ⊥T ∗P ⊥ B = μ (P ⊥T ∗P ⊥ ) ∗(B) = μP ⊥TP⊥(B) = 0 (3.10)
(recall that μP ⊥TP⊥ is concentrated on Bc ). Thus, μP ⊥T ∗P ⊥ is concentrated on C \ B∗ , and
maximality of PT ∗(C \ B∗ ) implies that
PT (B) ⊥ = P ⊥ PT ∗
∗
C \ B . (3.11)
Since