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

554 K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580
is easily seen to be a maximal compact subgroup of G. The Lie algebra g of G decomposes into
g = k + p, where k, the Lie algebra of K, consists of all matrices
a 0
, a ∈ Mr,r(C), d ∈ Mr+b,r+b(C), a
0 d
∗ =−a, d ∗ =−d,
and p consists of all matrices
0 v
v∗
, v∈Mr,r+b(C). 0
The induced vector fields are given respectively by
z ↦→ az − zd, and z ↦→ ξv(z) = v − zv ∗ z.
The complex Lie algebra kC is given by the set of all matrices
a
0
0
,
d
a ∈ Mr,r(C), d ∈ Mr+b,r+b(C), tr(a) + tr(d) = 0.
Hence, kC can be written as the sum
where k (1)
C
and
and k(2)
C
kC = k (1)
C ⊕ k(2)
C ,
are the ideals consisting respectively of the matrices
a 0
0 − tr(a)
r+b Ir+b
, a ∈ Mr,r(C),
0 0
, d ∈ Mr+b,r+b(C), tr(d) = 0.
0 d
Then, identifying kC as linear transformations of V ,wehave
kC = span D(u, ¯v), u,v ∈ V
and
where the endomorphism D(u, ¯v) (1) is given by
k (1)
C = span D(u, ¯v) (1) ,u,v∈ V ,
D(u, ¯v) (1) z = uv ∗ z.