L
p
estimates for hypoelliptic operators 397
E
XAMPLE
3. This is an example of the non-stratified case. Consider G
= R
5
, ◦, Dλ
with: x
1
, y
1
, z
1
, w
1
,t
1
◦ x
2
, y
2
, z
2
, w
2
, t
2
= = x
1
+ x
2
, y
1
+ y
2
, z
1
+ z
2
, w
1
+ w
2
+ x
1
y
2
, t
1
+ t
2
− x
1
x
2
y
1
− x
1
x
2
y
2
− 1
2 x
2 2
y
1
+ x
1
w
2
+ x
1
z
2
and Dλ x, y, z, w, t
= λ
x, λy, λ
2
z, λ
2
w, λ
3
t .
The natural base for ℓ consists of: X
= ∂
∂ x
− xy ∂
∂ t
; Y =
∂ ∂
y + x
∂ ∂w
; Z =
∂ ∂
z + x
∂ ∂
t ;
W =
∂ ∂w
+ x ∂
∂ t
; T = ∂
∂ t
. We can see that ℓ is graded setting
ℓ = V
1
⊕ V
2
⊕ V
3
with V
1
= hX, Y i, V
2
= hZ, W i, V
3
= hT i. The nontrivial commutation relations are:
[X, Y ] = W ; [X, Z] = T ; [X, W ] = T.
Therefore, if we set e V
1
= hX, Y, Zi, we see that the Lie algebra generated by e V
1
is ℓ; moreover e
V
2
= e
V
1
, e V
1
= hW, T i and e V
3
= e
V
1
, e V
2
= hT i, so that ℓ is not stratified. Noting that X, Y, Z are homogeneous of degrees 1, 1, 2 respectively, we have that the operator
L = X
2
+ Y
2
+ Z is hypoelliptic and homogeneous of degree two.
2.2. Function spaces
Before going on, we need to introduce some notation and function spaces. First of all, if X ,
X
1
,. . . , X
q
are the vector fields appearing in 13-14, define, for p ∈ [1, ∞]
kDuk
p
≡
q
X
i =1
kX
i
u k
p
; D
2
u
p
≡
q
X
i, j =1
X
i
X
j
u
p
+ kX u
k
p
. More in general, set
D
k
u
p
≡ X
X
j
1
. . . X
j
l
u
p
where the sum is taken over all monomials X
j
1
. . . X
j
l
homogeneous of degree k. Note that X has weight two while the remaining fields have weight one. Obviously, in Case A the field X
398 M. Bramanti - L. Brandolini
does not appear in the definition of the above norms. Let be a domain in R
N
, p ∈ [1, ∞] and
k be a nonnegative integer. The space S
k, p
consists of all L
p
functions such that
kuk
S
k, p
=
k
X
h =0
D
h
u
L
p
is finite. We shall also denote by S
k, p
the closure of C
∞
in S
k, p
.
Since we will often consider the case k = 2, we will briefly write S
p
for S
2, p
and
S
p
for S
2, p
.
Note that the fields X
i
, and therefore the definition of the above norms, are completely determined by the structure of G.
We define the H¨older spaces 3
k,α
, for α
∈ 0, 1, k nonnegative integer, setting |u|
3
α
= sup
x 6=y
x ,y ∈
|ux − uy| dx, y
α
and kuk
3
k,α
= D
k
u
3
α
+
k −1
X
j =0
D
j
u
L
∞
. In §4, we will also use the fractional but isotropic Sobolev spaces H
t,2
R
N
, defined in the usual way, setting, for t
∈ R, kuk
2 H
t,2
= Z
R
N
| b
uξ |
2
1 + |ξ|
2 t
dξ , where
b uξ denotes the Fourier transform of u.
The structure of space of homogenous type allows us to define the space of Bounded Mean Oscillation functions B M O, see [18] and the space of Vanishing Mean Oscillation functions
V M O, see [30]. If f is a locally integrable function, set 16
η
f
r = sup
ρ r
1 B
ρ
Z
B
ρ
f x − f
B
ρ
dx for every r 0, where B
ρ
is any ball of radius ρ and f
B
ρ
is the average of f over B
ρ
. We say that f
∈ B M O if k f k
∗
≡ sup
r
η
f
r ∞.
We say that f ∈ V M O if f ∈ B M O and η
f
r → 0 for r → 0.
We can also define the spaces B M O and V M O for a domain ⊂ R
N
, just replac- ing B
ρ
with B
ρ
∩ in 16.
2.3. Assumptions and main results
We now state precisely our assumptions, keeping all the notation of §§2.1, 2.2. Let G be a homogeneous group of homogeneous dimension Q
≥ 3 and ℓ its Lie algebra; let {X
i
} i = 1, 2, . . . , N be the basis of ℓ constructed as in §2.1, and assume that the conditions of
L
p
estimates for hypoelliptic operators 399
Case A or Case B hold. Accordingly, we will study the following classes of operators, modeled on the translation invariant prototypes 13, 14:
L =
q
X
i, j =1
a
i j
xX
i
X
j
or 17
L =
q
X
i, j =1
a
i j
xX
i
X
j
+ a xX
where a
i j
and a are real valued bounded measurable functions and the matrix
a
i j
x satisfies a uniform ellipticity condition:
18 µ
|ξ|
2
≤
q
X
i, j =1
a
i j
x ξ
i
ξ
j
≤ µ
−1
|ξ|
2
for every ξ ∈ R
q
, a.e. x, for some positive constant µ. Analogously,
19 µ
≤ a x
≤ µ
−1
. Moreover, we will assume
a , a
i j
∈ V M O. Then:
T
HEOREM
2. Under the above assumptions, for every p ∈ 1, ∞ there exist c = c p, µ, G
and r = r p, µ, η, G such that if u ∈ C
∞
R
N
and sprt u ⊆ B
r
B
r
any ball of radius r then
D
2
u
p
≤ c kLuk
p
where η denotes dependence on the “V M O moduli” of the coefficients a , a
i j
. T
HEOREM
3 L
OCAL ESTIMATES FOR SOLUTIONS TO THE EQUATION
Lu = f
IN A DOMAIN
. Under the above assumptions, let be a bounded domain of R
N
and
′
⊂⊂ . If u ∈ S
p
, then
kuk
S
p
′
≤ c kLuk
L
p
+ kuk
L
p
where c = c p, G, µ, η, ,
′
. T
HEOREM
4 L
OCAL
H ¨
OLDER CONTINUITY FOR SOLUTIONS TO THE EQUATION
Lu = f
IN A DOMAIN
. Under the assumptions of Theorem 3, if u ∈ S
p
for
some p ∈ 1, ∞ and Lu ∈ L
s
for some s Q2, then
kuk
3
α
′
≤ c kLuk
L
r
+ kuk
L
p
for r = max p, s, α = αQ, p, s ∈ 0, 1, c = cG, µ, p, s, ,
′
.
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
