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

700 D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711
with
hL(x) = Ld−2
|BL|
=
∂BL
1
|B|L 2 |x| d
because |x − z| 2 =|x| 2 − 2x · z + L 2 . ✷
L 2 −|x − z| 2
∂BL
|x| d
1|x−z|LσL(dz)
2x · z −|x| 2 1|x−z|LσL(dz)
The function hL of this proposition can be written in a more explicit form, namely:
Proposition 4.2. In the notation of Proposition 4.1, ifd = 1 then
hL(x) = 1
1 −
2L
|x|
1|x|2L. (4.5)
2L
If d = 2 then
If d = 3 then
hL(x) = 1
π 2 L 2
= 1
π 2 L 2
hL(x) =
1
|x|/2L
2L
√ 1 − r 2
|x|
r 2
dr1|x|2L
2 − 1 − arccos |x|
1|x|2L. (4.6)
2L
3
8πL|x| 2
1 − |x|
2 1|x|2L. (4.7)
2L
These expressions show that the kernel hL of AL is singular at x = 0, but the singularity is
integrable. In Section 5 we will see that AL increases the smoothness (away from the boundary)
by at least one derivative.
Proof. Assume first that d = 1. Then the right-hand side of (4.2) equals
1
|B|L 2 |x| d
∂BL
2
2xz − x 1|x−z|LσL(dz) = 1
2L2 1 2
2xz − x
|x| 2
1|x−z|L
z=±L
= 1
4L2 2
2L|x|−|x|
|x|
1|x−z|L =
z=±L
1
1 −
2L
|x|
2L
1|x|2L.
Case d 2. Referring to (4.2) let α be the angle between x and z so that x · z = L|x| cos α
and the constraint |x − z| L is equivalent to 0 α arccos(|x|/2L). Then