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

S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681 647
is the image of the standard Fourier transform F m in the space L 2 (R m ,dx),i.e.Fmn =
U(C (n) ) −1 F m U(B (n) ), where
U B (n)
L
1/2
dμB (n)(x)
=
dx
2 (Rm ,μB (n))
L 2 (R m ,dx)
Fmn
F m
L2 (Rm ,μC (n))
U C (n)
dμC (n)(x)
=
dx
L 2 (R m ,dx).
Since the standard Fourier transform F m is defined as follows:
m 1
F f (y) = √ exp i(y,x)f(x)dx,
(2π) m
and, for D = B (n) respectively D = C (n)
dμD(x)
U(D)=
dx
we have finally for Fmn:
R m
1/2
=
R m
1
((2π) m
exp
det D) 1/4
− 1
4
D −1 x,x
,
(Fmnf )(y) = U C (n)−1 m (n)
F U B f (y)
1
=
((2π) m det C (n)
1
(n)
exp C
) 1/4 4
−1
y,y
1
√
(2π) m
× exp i(y,x)f(x) (2π) m det B (n)
1/4
exp − 1
(n)
B
4
−1
x,x
dx
= exp( 1
4 ((C(n) ) −1y,y))
(2π) m det C (n)
Rm
exp
i(y,x) − 1
4
Using Fourier transform Fm we obtain for Akn = FmAkn(Fm) −1 :
Akn = i
where
k−1
r=1
xrkyrn + ykn
B (n) −1 x,x
f(x)dx.
1/2
m
, 1 k m