400 M. Bramanti - L. Brandolini
T
HEOREM
5 R
EGULARITY OF THE SOLUTION IN TERMS OF
S
OBOLEV SPACES
. Under the assumptions of Theorem 3, if a
, a
i j
∈ S
k, ∞
, u
∈ S
p
and Lu
∈ S
k, p
for some
positive integer k k even, in Case B, 1 p ∞, then
kuk
S
k +2, p
′
≤ c
1
n kLuk
S
k, p
+ c
2
kuk
L
p
o where c
1
= c
1
p, G, µ,η,,
′
and c
2
depends on the S
k, ∞
norms of the coefficients.
T
HEOREM
6 R
EGULARITY OF THE SOLUTION IN TERMS OF
H ¨
OLDER SPACES
. Under the assumption of Theorem 3, if a , a
i j
∈ S
k, ∞
, u
∈ S
p
and Lu
∈ S
k,s
for some positive integer k k even, in Case B, 1 p
∞, s Q2, then kuk
3
k,α
′
≤ c
1
n kLuk
S
k,r
+ c
2
kuk
L
p
o where r
=max p, s, α = αQ, p, s ∈ 0, 1, c
1
= c
1
p, s, k, G, µ,η,,
′
and c
2
depends on the S
k, ∞
norms of the coefficients.
T
HEOREM
7 O
PERATORS WITH LOWER ORDER TERMS
. Consider an operator with “lower order terms” in the sense of the degree of homogeneity, of the following kind:
L ≡
q
X
i, j =1
a
i j
xX
i
X
j
+ a xX
+
q
X
i =1
c
i
xX
i
+ c x
≡ ≡ L
2
+ L
1
. i If c
i
∈ L
∞
for i
= 0, 1, . . . , q, then: if the assumptions of Theorem 3 hold for L
2
, then the conclusions of Theorem 3 hold for L; if the assumptions of Theorem 4 hold for L
2
, then the conclusions of Theorem 4 hold for L. ii If c
i
∈ S
k, ∞
for some positive integer k, i
= 0, 1, . . . , q, then: if the assumptions of Theorem 5 hold for L
2
, then the conclusions of Theorem 5 hold for L; if the assumptions of Theorem 6 hold for L
2
, then the conclusions of Theorem 6 hold for L. R
EMARK
1. Since all our results are local, it is unnatural to assume that the coefficients a ,
a
i j
be defined on the whole R
N
. Actually, it can be proved that any function f ∈ V M O,
with bounded Lipschitz domain, can be extended to a function e f defined in
R
N
with V M O modulus controlled by that of f . For more details see [3]. Therefore, all the results of Theorems
2, 7 still hold if the coefficients belong to V M O, but it is enough to prove them for a ,
a
i j
∈ V M O.
2.4. Relations with operators of H¨ormander type
Here we want to point out the relationship between our class of operators and operators of H¨ormander type 2.
T
HEOREM
8. Under the assumptions of §2.3: i if the coefficients a
i j
x are Lipschitz continuous in the usual sense, then the operator L can be rewritten in the form
L =
q
X
i =1
Y
2 i
+ Y
L
p
estimates for hypoelliptic operators 401
where the vector fields Y
i
i = 1, . . . , q have Lipschitz coefficients and Y
has bounded mea- surable coefficients;
ii if the coefficients a
i j
x are smooth C
∞
, then L is hypoelliptic; iii if the coefficients a
i j
are constant, then L is left invariant and homogeneous of degree two; moreover, the transpose L
T
of L is hypoelliptic, too. Proof. Let us split the matrix a
i j
x in its symmetric and skew-symmetric parts: a
i j
x =
1 2
a
i j
x + a
j i
x +
1 2
a
i j
x − a
j i
x ≡ b
i j
x + e
b
i j
x. If the matrix A
= a
i j
x satisfies condition 18, the same holds for B
= b
i j
x . Therefore
we can write B = M M
T
where M = {m
i j
x } is an invertible, triangular matrix, whose entries
are C
∞
functions of the entries of B. To see this, we can use the “method of completion of squares” see e.g. [17], p. 180,
writing
q
X
i, j =1
b
i j
ξ
i
ξ
j
= η
2 1
+
q
X
i, j =2
b
∗ i j
η
i
η
j
with η
1
=
p
b
11
ξ
1
+
q
X
j =2
b
1 j
√ b
11
ξ
j
; η
i
= ξ
i
for i ≥ 2;
b
∗ ii
= b
ii
− b
2 1i
b
11
; b
∗ i j
= b
i j
for i, j = 2, . . . , q, i 6= j.
Since η
1
, . . . , η
q
are a linear invertible function of ξ
1
, . . . , ξ
q
, and the quadratic form P
q i, j
=1
b
i j
ξ
i
ξ
j
is positive on R
q
, also the quadratic form P
q i, j
=2
b
∗ i j
η
i
η
j
is positive on R
q −1
, and we can iterate the same procedure. Note that η
1
= P
q k
=1
m
1k
ξ
k
with m
1k
smooth functions of the b
i j
’s; moreover, b
∗ i j
are smooth functions of the b
i j
’s. Therefore iteration of this procedure allows us to write
q
X
i, j =1
b
i j
ξ
i
ξ
j
=
q
X
k =1
λ
2 k
with: λ
k
=
q
X
h =k
m
kh
ξ
h
and m
kh
are smooth functions of the b
i j
’s. This means that b
i j
= P
k ≥i, j
m
ki
m
k j
with m
kh
smooth functions of the b
i j
’s. Therefore we can write:
L =
q
X
i, j =1
q
X
k =1
m
ik
x m
j k
x X
i
X
j
+ X
i j
e b
i j
x X
i
,X
j
+ a xX
where the functions m
ik
x have the same regularity of the a
i j
x’s. To simplify the notation, from now on we forget the fact that m
ik
= 0 if k i. If the a
i j
x’s are Lipschitz continuous, the above equation can be rewritten as
20 L =
q
X
k =1
Y
2 k
+ Y
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
