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

K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 567
where H β is the component of H given by
H β =
[Eβ,vα]¯vα
α∈Ψ + n
and Eβ is the root vector of β.
Now we fix for the rest of this section β ∈ Ψ +,(1)
c such that
β| t −
2
C = γj − γj−1
The root vector Eβ has the form Eβ = D(cj , ¯w) with w = ej,j−1 or w = ej−1,j being one of the
basis vectors {vα}. Observe that [Eβ,vα]=0 unless α is in the set Ψ1 ∪ Ψ2 ∪ Ψ3 where
Ψ1 = α ∈ Ψ + n : α| t −
Ψ2 = α ∈ Ψ + n : α| t −
2
C = γk + γj−1
2
C = γk + γj−1
Ψ3 = α ∈ Ψ + n : α| t −
γj−1
= .
C 2
Consider the Poincaré–Birkhoff–Witt decomposition
and let π be the projection
.
,k j − 1 ,
,k j ,
U(gC) = U(gC)kC + U(aC + n −
C )
π : U(gC) = U(gC)kC + U(aC + n −
C ) → U(aC + n −
C ).
The function f is now viewed as a function on G = NAK, and the group A will be identified
as (R + ) r . Under this identification, f satisfies, furthermore, the equation
R π H β f = 0,
where R is the mapping from U(aC + nC) to differential operators on NA defined by
∂
R(ξk) = tk , R(X−α) = t
∂tk
α X−α,
for ξk ∈ a, 1 k r, and X−α ∈ n identified with the corresponding left-invariant differential
operator.
We will prove that operator t − 1 2 (βj −βj−1) R(π(H β )) has analytic coefficient near t = 0 and
study the induced equation of t − 1 2 (βj −βj−1) R(π(H β ))f = 0.