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

704 D. Brydges, A. Talarczyk / Journal of Functional Analysis 236 (2006) 682–711
We now proceed to the question of regularity of ALϕ.Letϕ ∈ H+(Λ) then, in particular, ϕ is
a continuous function on [a,b] which vanishes at a and b. Assume first that b − a>4L. By
(4.14) we obtain that for y ∈ (a + 2L,b − 2L)
ALϕ(y) =
y
y−2L
From this we obtain for y ∈ (a + 2L,b − 2L)
and
2L − y + x
ϕ(x)
(2L) 2
y+2L
dx +
d
dy ALϕ(y) = 1
(2L) 2
−
y
y−2L
y
ϕ(x)dx +
ϕ(x)
y+2L
y
2L + y − x
(2L) 2
ϕ(x)dx
dx. (4.15)
(4.16)
d2 dy2 ALϕ(y) = 1
(2L) 2
ϕ(y − 2L) + ϕ(y + 2L) − 2ϕ(y) . (4.17)
If y ∈ (a, a + 2L) then, again by (4.14) we have
and
ALϕ(y) =
y
a
+
2L − y + x
ϕ(x)
(2L) 2
a+2L
dx +
y+2L
a+2L
ϕ(x)
2L + y − x
(2L) 2
d
dy ALϕ(y) = 1
(2L) 2
y
− ϕ(x)dx +
a
a+2L
y
y
y − a
ϕ(x)
2L(x − a) dx
dx, (4.18)
ϕ(x) 2L
dx +
x − a
y+2L
a+2L
ϕ(x)dx
(4.19)
d2 dy2 ALϕ(y) = 1
(2L) 2
ϕ(y + 2L) − ϕ(y) − ϕ(y) 2L
. (4.20)
y − a
Now it is easy to see, taking into account that ϕ(a) = 0, that if we let y → a + 2L all the
corresponding limits in (4.15)–(4.20) coincide. Hence ALϕ is also twice continuously differentiable
at y = a + 2L. The same reasoning shows that ALϕ is twice continuously differentiable
in (b − 2L,b) and at b − 2L, which proves the statement of the proposition in case b − a>4L.
The cases 2L b − a 4L and b − a 2L can be investigated in a similar way.