402 M. Bramanti - L. Brandolini
with Y
k
=
q
X
i =1
m
ik
xX
i
and Y
= X
i j
e b
i j
x X
i
,X
j
+ a xX
−
q
X
i, j =1
q
X
k =1
m
ik
x ·
X
i
m
j k
x X
j
, which proves i. If the coefficients a
i j
x are C
∞
, the Y
i
’s are C
∞
vector fields and satisfy H¨ormander’s condition, because every linear combination of the X
i
i = 0, 1, . . . , q can be
rewritten as a linear combination of the Y
i
and their commutators of length 2. Therefore, by Theorem 1, L is hypoelliptic, that is ii. Finally, if the coefficients a
i j
are constant, then 20 holds with
Y
k
=
q
X
i =1
m
ik
X
i
and Y =
X
i j
e b
i j
X
i
,X
j
+ a X
, which means that L is left invariant and homogeneous of degree two. Moreover, since the fields
X
i
are translation invariant, the transpose X
T i
of X
i
equals −X
i
and as a consequence L
T
is hypoelliptic as well. This proves iii.
R
EMARK
2. By the above Theorem, if a
i j
∈ C
∞
, our class of operators is contained in that studied by Rothschild-Stein [28], so in this case our results follow from [28], without assuming
the existence of a structure of homogeneous group. If the coefficients are less regular, but at least Lipschitz continuous, our operators can be written as “operators of H¨ormander type”; however,
in this case we cannot check H¨ormander’s condition for the fields Y
i
and therefore our estimates do not follow from known results about hypoelliptic operators. Finally, if the coefficients are
merely V M O, we cannot even write L in the form 20.
2.5. More properties of homogeneous groups
We recall some known results which will be useful later. First of all, we define the convolution of two functions in G as
f ∗ gx =
Z
R
N
f x ◦ y
−1
gy d y =
Z
R
N
gy
−1
◦ x f y dy, for every couple of functions for which the above integrals make sense. From this definition we
read that if P is any left invariant differential operator, P f
∗ g = f ∗ Pg provided the integrals converge. Note that, if G is not abelian, we cannot write f
∗ Pg = P f ∗g. Instead, if X and X
R
are, respectively, a left invariant and right invariant vector field which agree at the origin, the following hold see [31], p. 607
21 X f
∗ g = f ∗ X
R
g ; X
R
f ∗ g =
X
R
f ∗ g.
In view of the above identities, we will sometimes use the right invariant vector fields X
R i
which agree with ∂∂ x
i
and therefore with X
i
at the origin i = 1, . . . , N, and we need some prop-
L
p
estimates for hypoelliptic operators 403
erties linking X
i
to X
R i
. It can be proved that X
i
= ∂
∂ x
i
+
N
X
k =i+1
q
k i
x ∂
∂ x
k
X
R i
= ∂
∂ x
i
+
N
X
k =i+1
e q
k i
x ∂
∂ x
k
where q
k i
x, e
q
k i
x are polynomials, homogeneous of degree ω
k
−ω
i
the ω
i
’s are the exponents appearing in 6. From the above equations we find that
X
i
=
N
X
k =i
c
k i
x X
R k
where c
k i
x are polynomials, homogeneous of degree ω
k
−ω
i
. In particular, since ω
k
−ω
i
ω
k
, c
k i
x does not depend on x
h
for h ≥ k and therefore commutes with X
R k
, that is 22
X
i
u =
N
X
k =i
X
R k
c
k i
x u i
= 1, . . . , N for every test function u. This representation of X
i
in terms of X
R i
will be useful in §6. T
HEOREM
9. See Theorem 2.1 and Corollary 2.8 in [11]. Let L be a left invariant
differential operator homogeneous of degree two on G, such that L and L
T
are both hypoelliptic. Moreover, assume Q
≥ 3. Then there is a unique fundamental solution Ŵ such that: a Ŵ
∈ C
∞
R
N
\ {0} ;
b Ŵ is homogeneous of degree 2 − Q;
c for every distribution τ , L τ ∗ Ŵ = Lτ ∗ Ŵ = τ.
T
HEOREM
10. See Proposition 8.5 in [13], Proposition 1.8 in [11]. Let K
h
be a kernel which is
C
∞
R
N
\ {0} and homogeneous of degree h
− Q, for some integer h with 0 h Q; let T
h
be the operator T
h
f = f ∗ K
h
and let P
h
be a left invariant differential operator homogeneous of degree h. Then:
P
h
T
h
f = P.V.
f ∗ P
h
K
h
+ α f for some constant α depending on P
h
and K
h
; the function P
h
K
h
is C
∞
R
N
\ {0} , homogeneous of degree
−Q and satisfies the van- ishing property:
Z
r kxkR
P
h
K
h
x d x = 0 for 0 r R ∞;
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
