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

D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711 705
Now assume that f ∈ H−(Λ) ∩ Cb(Λ) and b − a>4L. We will show that A ′ Lf is twice
continuously differentiable in (a + 2L,b − 2L) but does not have to be differentiable at a + 2L.
(Similar argument can be applied to show that the same happens at b − 2L.) Note that
A ′ Lf(x)=
and by (4.10) we have that for x ∈ (a + 2L,b − 2L)
A ′ L f(x)=
x
x−2L
Similarly as in (4.16) we obtain
R
2L + y − x
(2L) 2
f(y)dy+
d
dx A ′ 1
Lf(x)= (2L) 2
−
x
x−2L
aL(x, y)f (y) dy
x+2L
x
f(y)dy+
2L − y + x
(2L) 2
f(y)dy.
x+2L
x
f(y)dy . (4.21)
Clearly, the second derivative also exists in (a +2L,b−2L). On the other hand, if x ∈ (a, a +2L)
then
A ′ L f(x)=
Hence for x ∈ (a, a + 2L) we have
a
x
y − a
2L(x − a) f(y)dy+
a
x+2L
x
2L − y + x
(2L) 2
f(y)dy.
d
dx A ′ 1
Lf(x)= (2L) 2
x
x+2L
2L(y − a)
−
f(y)dy+ f(y)dy . (4.22)
(x − a) 2
Letting x → a + 2L in (4.21) and (4.22) we see that left and right derivatives of A ′ Lf at a + 2L
are equal if and only if
which is not true in general. ✷
a+2L
a
f(y)dy=
a+2L
a
y − a
2L f(y)dy,
Proposition 4.4 can be extended to the case when the operator −B ′ B generates a onedimensional
non-degenerate diffusion process. Operator AL can be written in terms of the scale
function of the diffusion process.
x