L
p
estimates for hypoelliptic operators 411
By Proposition 1, k satisfies 43, with β = 1, M = MG, and
50 c
2
= cG · sup
x ∈6
N
|∇Y
km
x | .
As to 44, the left hand side equals: Z
r kykR
H
km
y d y +
Z
r y
−1
R
H
km
y d y .
The first term is a multiple of Z
6
N
H
km
y dσ y and therefore vanishes; the second term, by 42, 48 and 10 can be seen to be bounded by
cG · c
1
see for instance Remark 4.6 in [4]. Hence c
3
has the same form of c
1
. Moreover, 45 is trivially satisfied, by the vanishing property of H
km
. Finally, combining 39 with 48, 50 we get, by Theorem 16 and Remark 3, the following:
T
HEOREM
17. For every p ∈ 1,∞ there exists a constant c such that for every a ∈ B M O,
f ∈ L
p
R
N
, m = 1, 2,. . .; k = 1, . . . , g
m
kT
km
f k
L
p
R
N
≤ c k f k
L
p
R
N
C T
km
, a f
L
p
R
N
≤ c kak
∗
k f k
L
p
R
N
. Explicitly, c
= c p, G · m
N2
. We now turn to the expansion 35. Combining Theorem 17 with the uniform bound 41
on the coefficients in the expansion which crucially depends on Theorem 12, and using 38, we get Theorem 14, where the constant in 31, 32 is c p, G, µ.
4. Uniform bounds for the derivatives of fundamental solutions of families of operators
In this section we prove Theorem 12; this will complete the proof of Theorem 2. The proof of Theorem 12 is carried out repeating an argument by Rothschild-Stein contained in §6 of [28];
this, in turn, is based on several results proved by Kohn in [21]. We will not repeat the whole proof, but will state its steps, pointing out the necessary changes to adapt the argument to our
case. As in the previous section, it will be enough to write the proof for Case B.
Let A
µ
be the set of q × q constant matrices A ={a
i j
}, satisfying: µ
2
|ξ|
2
≤
q
X
i, j =1
a
i j
ξ
i
ξ
j
≤ µ
−2
|ξ|
2
for every ξ ∈ R
q
, where µ is the same as in 18, 19. Throughout this section we will consider the operator
L
A
=
q
X
i, j =1
a
i j
X
i
X
j
+ X
412 M. Bramanti - L. Brandolini
where A ={a
i j
}∈ A
µ
and the fields X
i
satisfy the assumptions stated in §2.1. Let Ŵ
A
be the fundamental solution for L
A
, homogeneous of degree 2 − Q see §3.1, and let:
T
A
: f 7→ f ∗ Ŵ
A
. By Theorem 9, T
A
L
A
= L
A
T
A
=identity. We will prove that: 51
sup
kxk=1
∂ ∂
x
β
Ŵ
A
x ≤ cβ, G, µ.
Note that, if L is the frozen operator defined in 24, L
= a x
·L
A
with A
= a
i j
x a
x
q i, j
=1
∈ A
µ
and Ŵ x ,
· = a x
· Ŵ
A
. This shows that 26 follows from 51.
The operator T
A
can be regarded as a fractional integral operator, for which the following estimates hold:
T
HEOREM
18. a If 1 p
Q 2
and
1 s
=
1 p
−
2 Q
, then kT
A
f k
s
≤ c k f k
p
with c
= c p, G · sup
6
N
Ŵ
A
. b If
Q 2
p Q and β = 2 −
Q p
hence β ∈ 0, 1, then for every f ∈ C
∞
and every x
1
, x
2
∈ R
N
|T
A
f x
1
− T
A
f x
2
| ≤ c k f k
p
dx
1
, x
2 β
with c
= c p, G · sup
6
N
Ŵ
A
+ sup
6
N
∇Ŵ
A
. c If p
Q 2
and sprt f ⊆ B
r
x for some r 0, x
∈ R
N
kT
A
f k
L
∞
B
r
≤ c k f k
L
p
B
r
with c
= c p, G,r · sup
6
N
Ŵ
A
. Note: parts b-c of Theorem 18 will be used only in §6proof of Theorem 4.
Proof. Part a follows from Proposition 1.11 in [11], or also from results about fractional inte- grals on general spaces of homogeneous type, see [15]. The form of the constant c depends on
the bound:
52 Ŵ
A
x ≤ sup
6
N
Ŵ
A
· 1
kxk
Q −2
. Part b could also be proved as a consequence of results in [15], but it is easier to prove it
directly. Let M be the same number as in Proposition 1; let us write: |T
A
f x
1
− T
A
f x
2
| ≤ Z
R
N
h Ŵ
A
y
−1
◦ x
1
− Ŵ
A
y
−1
◦ x
2
i f y
dy ≤
L
p
estimates for hypoelliptic operators 413
≤ Z
y
−1
◦x
1
≥M x
−1 2
◦x
1
. . . d y
+ Z
y
−1
◦x
1
≤M x
−1 2
◦x
1
. . . d y
= I + I I. By Proposition 1,
I ≤ c p, G · sup
6
N
∇Ŵ
A
x
−1 2
◦ x
1
Z
y
−1
◦x
1
≥M x
−1 2
◦x
1
| f y| y
−1
◦ x
1 Q
−1
d y. Let p, p
′
be conjugate exponents; by H¨older’s inequality and a change of variables I
≤ c x
−1 2
◦ x
1
k f k
p
Z
kyk≥M x
−1 2
◦x
1
1 kyk
Q −1 p
′
d y
1 p
′
≤ computing the integral, under the assumption p Q,
≤ c x
−1 2
◦ x
1 β
k f k
p
where β = 1 −
1 q
[Q − 1q − 1 − 1] = 2 −
Q p
∈ 0, 1. By 52,
I I ≤ sup
6
N
Ŵ
A
· ·
Z
y
−1
◦x
1
≤M x
−1 2
◦x
1
| f y| 1
y
−1
◦ x
1 Q
−2
+ 1
y
−1
◦ x
2 Q
−2
d y =
= I I
′
+ I I
′′
. By H¨older’s inequality and reasoning as above, we get, if p Q2,
I I
′
≤ c x
−1 2
◦ x
1 β
k f k
p
. As to I I
′′
, if y
−1
◦ x
1
≤ M x
−1 2
◦ x
1
, then y
−1
◦ x
2
≤ c x
−1 2
◦ x
1
and therefore I I
′′
can be handled as I I
′
. As to c, noting that x, y
∈ B
r
x ⇒ y
−1
◦ x ∈ B
K r
0 for some K = K G, we can
write, by H¨older’s inequality let p
′
be the conjugate exponent of p: kT
A
f k
L
∞
B
r
x
≤ k f k
L
p
B
r
x
· Ŵ
A L
p′
B
K r
≤ by 52 ≤ k f k
L
p
B
r
x
· c p, G · sup
6
N
Ŵ
A
· r
2 −Q p
, which proves the result, assuming p Q2.
Now, let S
A i j
f = X
i
X
j
T
A
f. By c of Theorem 11, setting Ŵ
A i j
= X
i
X
j
Ŵ
A
, we can write 53
S
A i j
f = P.V.
Ŵ
A i j
∗ f + α
i j
A · f.
Let us apply Theorem 16 and Remark 3 to the kernel Ŵ
A i j
. By the properties d, e, f listed in Theorem 11 and Proposition 1, we get:
414 M. Bramanti - L. Brandolini
T
HEOREM
19. For every p ∈ 1, ∞, f ∈ C
∞
R
N
, P.V.
Ŵ
A i j
∗ f
p
≤ c k f k
p
with c
= c p, G · sup
6
N
Ŵ
A
+ sup
6
N
∇Ŵ
A
. L
EMMA
2. For every p ∈ 1, ∞ and for every A
∈ A
µ
, there exists ε 0 such that if |A − A
| ε, then kE
A
f k
p
≤ 1
2 k f k
p
where |A| denotes the Euclidean norm of the matrix A and
E
A
= L
A
o
− L
A
T
A
o
. This Lemma is proved in [28] Lemma 6.5, for a different class of operators.
Proof. Let us write L
A
o
− L
A
=
q
X
i, j =1
a
o i j
− a
i j
X
i
X
j
. Then
E
A
f =
q
X
i, j =1
a
o i j
− a
i j
X
i
X
j
T
A
o
f. By 53 and Theorem 19 we get the result.
L
EMMA
3. Let p ∈ 1, Q2 and let
1 s
=
1 p
−
2 Q
. There exists c = cG, µ, p such that
for every A ∈ A
µ
54 kT
A
f k
s
≤ c k f k
p
. This Lemma is an adjustment of Lemma 6.7 proved in [28], which contains a minor mistake
it implicitly assumes Q 4. Proof. Let A
∈ A
µ
and let E
A
and ε be as in the previous lemma. For every f ∈ L
p
, if |A − A
| ε, then for every p ∈ 1, ∞, kE
A
f k
p
≤
1 2
k f k
p
, so that we can write 55
∞
X
n =0
E
n A
f = I − E
A −1
f ≡ g.
Therefore f
= I − E
A
g = g − L
A
o
T
A
o
g + L
A
T
A
o
g = L
A
T
A
o
g, that is
T
A
f = T
A
o
g.
L
p
estimates for hypoelliptic operators 415
Again from 55 we have kgk
p
≤
∞
X
n =0
kE
A
k
n
k f k
p
= 2 k f k
p
, hence by Theorem 18.a, if p, s are as in the statement of the theorem,
kT
A
f k
s
= T
A
o
g
s
≤ cG, p, A kgk
p
≤ 2c k f k
p
. Since this is true for every fixed A
∈ A
µ
and any matrix A such that |A − A
| ε, by the compactness of A
µ
in R
q
2
we can choose a constant c = cG, p,µ such that 54 holds for
every A ∈ A
µ
. T
HEOREM
20. Let ϕ, ϕ
1
∈ C
∞
R
N
with ϕ
1
= 1 on sprt ϕ. There exists ε = ε G and, for every t
∈ R, there exists c = ct, µ,G, ϕ, ϕ
1
such that for every A ∈ A
µ
and every u
∈ C
∞
R
N
kϕuk
H
t +ε,2
≤ c kϕ
1
L
A
u k
H
t,2
+ kϕ
1
u k
L
2
. Recall that the norm of H
t,2
R
N
has been defined is §2.3. This Theorem is proved in [21] for a different class of operators and without taking into
account the exact dependence of the constant on the parameters. To point out the slight modi- fications which are necessary to adapt the proof to our case, we will state the main steps of the
proof of Theorem 20. Before doing this, however, we show how from Lemma 3 and Theorem 20, the uniform bound 26 follows. This, again, is an argument contained in [28], which we
include, for convenience of the reader, to make more readable the exposition. Moreover, a minor correction is needed here to the proof of [28].
Proof of 26 from Lemma 3 and Theorem 20. Throughout the proof, B
r
will be a ball centered at the origin. Let g
∈ C
∞
B
2
\ B
1
such that kgk
2
≤ 1, and let ϕ, ϕ
1
∈ C
∞
R
N
such that: ϕ
= 1 in B
14
, sprtϕ ⊆ B
12
, ϕ
1
= 1 in B
12
, sprtϕ
1
⊆ B
1
. Let f = T
A
g. Since L
A
f = g = 0
in B
1
and L
A
is hypoelliptic, f ∈ C
∞
B
1
. Pick a positive number p such that max
1 2
,
2 Q
1 p
min
1 2
+
2 Q
, 1
and let s be as in Lemma 3. Note that 1 p Q2 and p 2 s. Then, by Lemma 3:
kϕ
1
f k
2
≤ cϕ
1
k f k
s
≤ cϕ
1
,G, µ, p kgk
p
≤ cϕ
1
,G, µ kgk
2
≤ cϕ
1
,G, µ. Applying Theorem 20 to ϕ, ϕ
1
, f , since L
A
f = 0 on sprtϕ
1
, we get: kϕ f k
H
t +ε,2
≤ c kϕ
1
f k
2
≤ ct,ϕ,ϕ
1
,G, µ for every t
∈ R. Therefore, by the standard Sobolev embedding Theorems, we can bound any isotropic H¨older norm C
h,α
of f on B
14
with a constant ch, G, µ; in particular, for every differential operator P:
56 |P f 0| ≤ cP, G, µ.
416 M. Bramanti - L. Brandolini
Now, recall that f = g ∗ Ŵ
A
. If P is any left invariant differential operator, 57
P f 0 =
Z P Ŵ
A
y
−1
gy d y. Since 56 holds for every g
∈ C
∞
B
2
\ B
1
such that kgk
2
≤ 1, from 57 we get 58
P Ŵ
A
y
−1 L
2
B
2
\B
1
≤ cP, G, µ. Now, writing any differential operator
∂ ∂
x β
in terms of left invariant vector fields, 58 gives us a bound on every H
k,2
-norm of Ŵ
A
on B
2
\ B
1
, and therefore, reasoning as above, on every H¨older norm C
h,α
of Ŵ
A
on a smaller spherical shell C ≡ B
74
\ B
54
. In particular, we get sup
x ∈C
∂ ∂
x
β
Ŵ
A
x ≤ cβ, G, µ,
from which 51 follows, by homogeneity of Ŵ
A
. Now we come to Theorem 20, which is proved by Kohn in [21] for an operator of the kind
Pu ≡
q
X
i =1
X
2 i
u + X
u + cu,
where the fields X
i
i = 0, 1, . . . , q satisfy H¨ormander’s condition. Reading carefully the
paper [21], one can check that the whole proof can be repeated replacing the operator P with L
A
; moreover, the constants depend on the matrix A only through the number µ. Actually, the matrix A is involved in the proof only through the boundedness of its coefficients and the
following elementary inequality: |L
A
u, u | ≥ µ
q
X
i =1
kX
i
u k
2
, for every u
∈ C
∞
R
N
. We can rephrase as follows the steps of the proof of Theorem 20 given in [21]:
i There exist ε = εG, c = cG, µ such that for every u ∈ C
∞
R
N
and every A ∈ A
µ
kuk
H
ε, 2
≤ c kL
A
u k
2
+ kuk
2
. ii For every t
∈ R, M 0, there exists c = ct, M, µ, G such that for every u ∈ C
∞
R
N
and every A ∈ A
µ
kuk
H
t +ε,2
≤ c kL
A
u k
H
t,2
+ kuk
H
−M,2
, where ε is the same of i.
iii Localization of the above estimate. kϕuk
H
t +ε,2
≤ c kϕ
1
L
A
u k
H
t,2
+ kϕ
1
u k
H
−M,2
, where ϕ, ϕ
1
∈ C
∞
R
N
with ϕ
1
= 1 on sprt ϕ, c = cϕ,ϕ
1
,t, M, µ, G. Since
k·k
H
−M,2
≤ k·k
2
, from point iii we get Theorem 20.
L
p
estimates for hypoelliptic operators 417
R
EMARK
4 A
N ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS ADAPTED TO THE FIELDS
X
i
. For the reader who is interested in reviewing the proof of [21], we point out that, under our
assumptions, many of the arguments of [21] can be simplified and made more self-contained by the following remark. We can precisely define an algebra of pseudodifferential operators, acting
on the Schwarz’ space S of smooth functions with fast decay at infinity. Consider the following kinds of operators:
a multiplication by a polynomial; b X
i
i = 0, 1, . . . , q;
c for t ∈ R, 3
t
defined by \ 3
t
u ξ
= 1
+ |ξ|
2 t 2
b uξ .
By general properties of homogeneous groups see [31], p. 621, the vector fields X
i
are linear combinations of ∂∂ x
i
with polynomial coefficients and, by H¨ormander’s condition, the ∂∂
x
i
’s are linear combinations of the X
i
’s and their commutators, with polynomial coefficients. Therefore X
i
maps S into itself, while the same is true for the operators a and c. The transpose of an operator of kind a, c is the operator itself, while, since the fields X
i
are translation invariant, the transpose of X
i
is −X
i
. This fact also simplifies many of the arguments in [21]; in particular, note that X
u, u = 0. Let P be the algebra generated by operators a, b, c
under sums, composition and transpose. This algebra is the suitable context where the whole proof can be carried out. On the contrary, in [21] some technical problems arise, since X
i
are defined only on C
∞
R
N
. To complete the proof of Theorem 12 we have now to prove estimate 27. We actually
prove a more general result which will be useful in §6. Let
n k
γ
o
γ ∈3
be a family of kernels such that k
γ
is homogeneous of degree h − Q for some
h 0 and k
γ
∈ C
∞
R
N
\ {0} . Let T
γ
be the distribution associated to k
γ
and let P
h
be a left invariant differential operator homogeneous of degree h. Then, Theorem 9 states that
59 P
h
T
γ
= P.V. P
h
k
γ
+ α
γ
δ. Observe that 25 is a particular case of 59. With these notations, we can prove the following:
L
EMMA
4. If k
γ
satisfies a uniform bound like 26, that is, for every multiindex β sup
γ ∈3
sup
kyk=1
∂ ∂
y
β
k
γ
y ≤ c β ,
then sup
γ ∈3
α
γ
≤ c. Proof. Let u be a test function with u0
6= 0, sprtu ⊆ B
1
0. By 59 α
γ
u0 = h P
h T
u, T
γ
i − hu, P.V. P
h
k
γ
i = =
Z P
h T
ux k
γ
x d x − lim
ε →0
Z
kxkε
P
h
k
γ
x ux d x
418 M. Bramanti - L. Brandolini
here ·
T
denotes transposition. Since k
γ
is locally integrable, the first integral is bounded, uniformly in γ by 59. As to the second term, by the vanishing property of the kernel P
h
k
γ
see Lemma 1, homogeneity and 59 we can write Z
kxkε
P
h
k
γ
x ux d x =
Z
ε kxk1
P
h
k
γ
x [ux − u0] dx
≤ ≤ sup
kyk≤1
|∇uy| Z
ε kxk1
c kxk
Q
· |x| dx ≤ c. The convergence of the last integral follows from 8.
5. Some properties of the Sobolev spaces S
