404 M. Bramanti - L. Brandolini
the singular integral operator f
7→ P.V. f
∗ P
h
K
h
is continuous on L
p
for 1 p ∞.
To handle the convolution of several kernels, we will need also the following L
EMMA
1. Let K
1
·, ·, K
2
·, · be two kernels satisfying the following: i for every x
∈ R
N
K
i
x, · ∈ C
∞
R
N
\ {0} i = 1, 2; ii for every x
∈ R
N
K
i
x, · is homogeneous of degree α
i
, with −Q α
i
0, α
1
+ α
2
−Q; iii for every multiindex β,
sup
x ∈R
N
sup
kyk=1
∂ ∂
y
β
K
i
x, y ≤ c
β
. Then, for every test function f and any x
, y
∈ R
N
, f
∗ K
1
x ,
· ∗ K
2
y ,
· = f ∗ K
1
x ,
· ∗ K
2
y ,
· . Moreover, setting K x
, y
, · = K
1
x ,
· ∗ K
2
y ,
·, we have the following: iv for every x
, y
∈ R
2N
, K x ,
y ,
· ∈ C
∞
R
N
\ {0}; v for every x
, y
∈ R
2N
, K x ,
y ,
· is homogeneous of degree α
1
+ α
2
+ Q; vi for every multiindex β,
23 sup
x ,y ∈R
2N
sup
kzk=1
∂ ∂
z
β
K x, y, z ≤ c
β
. The above Lemma has been essentially proved by Folland see Proposition 1.13 in [11],
apart from the uniform bound on K , which follows reading carefully the proof.
3. Proof of Theorem 2
All the proofs in this section will be written for the Case B. The results in Case A which is easier simply follow dropping the term X
.
3.1. Fundamental solutions
For any x ∈ R
N
, let us “freeze” at x the coefficients a
i j
x, a x of the operator 17, and
consider 24
L =
q
X
i, j =1
a
i j
x X
i
X
j
+ a x
X .
By Theorem 8, the operator L satisfies the assumptions of Theorem 9; therefore, it has a funda-
mental solution with pole at the origin which is homogeneous of degree 2 − Q. Let us denote
it by Ŵ x ; ·, to indicate its dependence on the frozen coefficients a
i j
x , a
x . Also, set
for i, j = 1, . . . , q,
Ŵ
i j
x ; y
= X
i
X
j
Ŵ x
; ·
y.
L
p
estimates for hypoelliptic operators 405
Next theorem summarizes the properties of Ŵ x ; · and Ŵ
i j
x ;
· that we will need in the following. All of them follow from Theorem 9 and Lemma 1.
T
HEOREM
11. For every x ∈ R
N
: a Ŵ x
, · ∈ C
∞
R
N
\ {0} ;
b Ŵ x ,
· is homogeneous of degree 2 − Q; c for every test function u and every x
∈ R
N
, ux
= L u
∗ Ŵ x ;
· x = Z
R
N
Ŵ x
; y
−1
◦ x L
uyd y; moreover, for every i, j
= 1, . . . , q, there exist constants α
i j
x such that
25 X
i
X
j
ux = P.V.
Z
R
N
Ŵ
i j
x ; y
−1
◦ x L
uyd y + α
i j
x · L
ux; d Ŵ
i j
x ;
· ∈ C
∞
R
N
\ {0} ;
e Ŵ
i j
x ;
· is homogeneous of degree −Q; f for every R r 0,
Z
r kykR
Ŵ
i j
x ; y dy =
Z
kyk=1
Ŵ
i j
x ; y dσ y = 0.
The above properties hold for any fixed x . We also need some uniform bound for Ŵ, with
respect to x . Next theorem contains this kind of result.
T
HEOREM
12. For every multi-index β, there exists a constant c
1
= c
1
β ,G, µ such that
26 sup
x ∈R
N
kyk=1
∂ ∂
y
β
Ŵ
i j
x ; y
≤ c
1
, for any i, j
= 1, . . . , q. Moreover, for the α
i j
’s appearing in 25, a uniform bound holds: 27
sup
x ∈R
N
α
i j
x ≤ c
2
, for some constant c
2
= c
2
G, µ. We postpone the proof of the above Theorem to §4. The proof of Theorem 2 from Theorems
11, 12 proceeds in three steps, which are explained in §§3.2, 3.3, 3.4.
3.2. Representation formula and singular integrals
Let us consider 25. Writing L = L + L
− L and then letting x be equal to x , we get the
following representation formula:
406 M. Bramanti - L. Brandolini
T
HEOREM
13. Let u ∈ C
∞
R
N
. Then, for i, j = 1, . . . , q and every x ∈ R
N
X
i
X
j
ux = P.V.
Z Ŵ
i j
x ; y
−1
◦ x
q
X
h,k =1
a
hk
x − a
hk
y X
h
X
k
uy +
28 +
a x
− a y
X uy
+ Luy d y
+ α
i j
x · Lux.
In order to rewrite the above formula in a more compact form, let us introduce the following singular integral operators:
29 K
i j
f x = P.V . Z
Ŵ
i j
x;y
−1
◦ x f y dy. Moreover, for an operator K and a function a
∈ L
∞
R
N
, define the commutator C[K , a] f
= K a f − a · K f . Then 28 becomes
X
i
X
j
u = K
i j
L u
−
q
X
h,k =1
C K
i j
, a
hk
X
h
X
k
u +
30 +C
K
i j
, a X
u + α
i j
· Lu for i, j
= 1, . . .,q. Now the desired L
p
-estimate on X
i
X
j
u depends on suitable singular integral estimates. Namely, we will prove the following:
T
HEOREM
14. For every p ∈ 1, ∞ there exists a positive constant c = c p, µ, G such
that for every a ∈ B M O, f ∈ L
p
R
N
, i, j = 1, . . . , q:
31 K
i j
f
L
p
R
N
≤ c k f k
L
p
R
N
32 C
K
i j
, a
f
L
p
R
N
≤ c kak
∗
k f k
L
p
R
N
. The estimate 32 can be localized in the following way see [6] for the technique of the
proof: T
HEOREM
15. If the function a belongs to V M O, then for every ε 0 there exists r 0, depending on ε and the V M O modulus of a, such that for every f
∈ L
p
, with sprt f ⊆ B
r
33 C
K
i j
, a f
L
p
B
r
≤ c p, µ, G · ε k f k
L
p
B
r
. Finally, using the bounds 27, 31, 33 in the representation formula 30, we get Theorem
2. Note that the term X u can be estimated either by the same method used for X
i
X
j
u for i, j
= 1, . . . , q, or by difference. So the proof of Theorem 2 relies on Theorem 14 which will follow from §§3.3, 3.4, and
Theorem 12 which will follow from §4.
L
p
estimates for hypoelliptic operators 407
3.3. Expansion in series of spherical harmonics and reduction of singular integrals “with variable kernel” to singular integrals of convolution type
To prove Theorem 14, we have to handle singular integrals of kind 29, which are not of con- volution type because of the presence of the first variable x in the kernel, which comes from
the variable coefficients a
i j
x of the differential operator L. To bypass this difficulty, we can apply the standard technique of expanding the kernel in series of spherical harmonics. This idea
dates back to Calder´on-Zygmund [10], in the case of “standard” singular integrals, and has been adapted to kernels with mixed homogeneities by Fabes-Rivi`ere [12]. We briefly describe this
technique. See [10] for details. Let
{Y
km
}
m =0,1,2,...
k =1,...,g
m
be an orthonormal system of spherical harmonics in R
N
, complete in L
2
6
N
m is the degree of the polynomial, g
m
is the dimension of the space of spherical harmonics of degree m in R
N
. For any fixed x
∈ R
N
, y ∈ 6
N
, we can expand: 34
Ŵ
i j
x;y =
∞
X
m =1
g
m
X
k =1
c
km i j
x Y
km
y for i, j
= 1, . . . ,q. We explicitly note that for m
= 0 the coefficients in the above expansion are zero, because of the vanishing property f of Theorem 11. Also, note that the integral of Y
km
y over 6
N
, for m
≥ 1, is zero. If y ∈ R
N
, let y
′
= D kyk
−1
y; recall that, by ii at page 393, y
′
∈ 6
N
. By 34 and homogeneity of Ŵ
i j
x ; · we have
Ŵ
i j
x ; y =
∞
X
m =1
g
m
X
k =1
c
km i j
x Y
km
y
′
kyk
Q
for i, j = 1, . . . ,q.
Then 35
K
i j
f x =
∞
X
m =1
g
m
X
k =1
c
km i j
x T
km
f x with
36 T
km
f x = P.V.
Z H
km
y
−1
◦ x f y dy and
37 H
km
x =
Y
km
x
′
kxk
Q
. We will use the following bounds about spherical harmonics:
38 g
m
≤ cN · m
N −2
for every m = 1, 2, . . .
39 ∂
∂ x
β
Y
km
x ≤ cN · m
N −2
2
+|β|
408 M. Bramanti - L. Brandolini
for x ∈ 6
N
, k = 1, . . . , g
m
, m = 1, 2, . . ..
Moreover, if f ∈ C
∞
6
N
and if f x ∼
P
k,m
b
km
Y
km
x is the Fourier expansion of f x with respect to
n Y
km
o , that is
b
km
= Z
6
N
f x Y
km
x dσ x then, for every positive integer r there exists c
r
such that 40
|b
km
| ≤ c
r
· m
−2r
sup
x ∈6
N
|β|=2r
∂ ∂
x
β
f x .
In view of Theorem 12, we get from 40 the following bound on the coefficients c
km i j
x ap- pearing in the expansion 34: for every positive integer r there exists a constant c
= cr, G, µ such that
41 sup
x ∈R
N
c
km i j
x ≤ cr, G, µ · m
−2r
for every m = 1, 2,. . .; k = 1, . . . , g
m
; i, j = 1, . . . , q.
3.4. Estimates on singular integrals of convolution type and their commutators, and con- vergence of the series
We now focus our attention on the singular integrals of convolution type defined by 36, 37 and their commutators. Our goal is to prove, for these operators, bounds of the kind 31, 32;
moreover, we need to know explicitly the dependence of the constants on the indexes k, m, appearing in the series 35. To this aim, we apply some abstract results about singular integrals
in spaces of homogeneous type, proved by Bramanti-Cerutti in [4]. To state precisely these results, we recall the following:
D
EFINITION
1. Let X be a set and d : X × X → [0, ∞. We say that d is a quasidistance
if it satisfies properties 9, 10, 11. The balls defined by d induce a topology in X ; we assume that the balls are open sets, in this topology. Moreover, we assume there exists a regular Borel
measure µ on X , such that the “doubling condition” is satisfied:
µ B
2r
x ≤ c · µ B
r
x for every r 0, x
∈ X, some constant c. Then we say that X, d, µ is a space of homogenous type.
Let X, d, µ be an unbounded space of homogenous type. For every x ∈ X, define
r
x
= sup {r 0 : B
r
x = {x}}
here sup ∅ = 0. We say that X, d, µ satisfies a reverse doubling condition if there exist
c
′
1, M 1 such that for every x ∈ X, r r
x
µ B
Mr
x ≥ c
′
· µB
r
x.
L
p
estimates for hypoelliptic operators 409
T
HEOREM
16. See [4]. Let X, d, µ be a space of homogenous type and, if X is un- bounded, assume that the reverse doubling condition holds. Let k : X
× X \ {x = y} → R be a kernel satisfying:
i the growth condition: 42
|kx, y| ≤ c
1
µ B x, dx, y
for every x, y ∈ X, some constant c
1
ii the “H¨ormander inequality”: there exist constants c
2
0, β 0, M 1 such that for every x
∈ X, r 0, x ∈ B
r
x , y
∈ B
Mr
x ,
|kx ,y
− kx, y| + |ky, x − ky, x| ≤
43 ≤
c
2
µ B x
, dx
, y
· dx
,x
β
dx ,y
β
; iii the cancellation property: there exists c
3
0 such that for every r, R, 0 r R ∞,
a.e. x 44
Z
rdx ,yR
kx, y dµy +
Z
rdx ,zR
kz, x dµz ≤ c
3
. iv the following condition: for a.e. x
∈ X there exists 45
lim
ε →0
Z
ε dx ,y1
kx, y dµy. For f
∈ L
p
, p ∈ 1, ∞, set
K
ε
f x =
Z
ε dx ,y1ε
kx, y f y dµy. Then K
ε
f converges strongly in L
p
for ε → 0 to an operator K f satisfying
46 kK f k
p
≤ c k f k
p
for every f ∈ L
p
, where the constant c depends on X, p and all the constants involved in the assumptions.
Finally, for the operator K the commutator estimate holds: 47
kC [K, a] f k
p
≤ c kak
∗
k f k
p
for every f ∈ L
p
, a ∈ B M O, and c the same constant as in 46.
R
EMARK
3. The constant c in 46, 47 has the following form: c p, X, β, M
· c
1
+ c
2
+ c
3
. Proof. To see this, note that if k satisfies 42, 43, 44 with constants c
1
, c
2
, c
3
, then k
′
≡ kc
1
+ c
2
+ c
3
satisfies 42, 43, 44 with constants 1, 1, 1, so that for the kernel k
′
c =
c p, X, β, M.
410 M. Bramanti - L. Brandolini
Let us apply Theorem 16 to our case. By 12, our space satisfies also the reverse doubling condition. Consider the kernels:
kx, y = H
km
y
−1
◦ x with H
km
x =
Y
km
x
′
kxk
Q
. By homogeneity, k satisfies 42 with
48 c
1
= cG · sup
x ∈6
N
|Y
km
x | .
To check condition 43 we need the following: P
ROPOSITION
1. Let f ∈ C
1
R
N
\ {0} be homogeneous of degree λ 1. There exist
c = cG, f 0, M = MG 1 such that
49 | f x ◦ y − f x| + | f y ◦ x − f x| ≤ c kyk kxk
λ −1
for every x, y such that kxk ≥ M kyk. Moreover
c = cG · sup
z ∈6
N
|∇ f z| . Proof. This proposition is essentially proved in [11], apart from the explicit form of the constant
c. Choose M 1 such that if
kxk = 1 and kyk ≤ 1M then kx ◦ yk ≥ 12. Set: Fx, y
= f x ◦ y, Lx, y = x ◦ y and
K ≡
n x, y :
kxk = 1 and kyk ≤ 1M o
. By homogeneity, it is enough to prove 49 for x, y
∈ K . Since f z is smooth for kzk ≥ 12 and L is smooth everywhere, by the mean value theorem
| f x ◦ y − f x| = |Fx, y − Fx, 0| ≤ |y| · ∇F x, y
∗
with x, y
∗
∈ K . But: sup
x ,y ∈K
∂ F
∂ x
i
x, y ≤
N
X
j =1
sup
x ,y ∈K
∂ L
j
∂ x
i
x, y · sup
kzk≥
1 2
∂ f
∂ z
j
z ≤
≤ cG · sup
z ∈6
N
|∇ f z| , and the same holds for sup
x ,y ∈K
∂ F
∂ y
i
x, y .
Recalling that |y| ≤ cG kyk when kyk ≤ 1 see 8, and repeating the argument for
| f y ◦ x − f x|, we get the result.
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
