Rend. Sem. Mat. Univ. Pol. Torino
Vol. 58, 4 (2000)
M. Bramanti - L. Brandolini
L P ESTIMATES FOR UNIFORMLY HYPOELLIPTIC
OPERATORS WITH DISCONTINUOUS COEFFICIENTS ON
HOMOGENEOUS GROUPS
Abstract. Let G be a homogeneous group and let X 0 , X 1 ,. . . , X q be left invariant real vector fields on G, satisfying Hörmander’s condition. Assume that
X 1 ,. . . , X q be homogeneous of degree one and X 0 be homogeneous of degree
two. We study operators of the kind:
L=
q
X
i, j =1
ai j (x)X i X j + a0 (x)X 0
where ai j (x) and a0 (x) are real valued, bounded measurable functions belonging to the space “Vanishing Mean Oscillation”, defined with respect to the quasidistance naturally induced by the structure of homogeneous group. Moreover,
the matrix {ai j (x)} is uniformly elliptic and a0 (x) is bounded away from zero.
Under these assumptions we prove local estimates in the Sobolev space S 2, p
(1 < p < ∞) defined by the vector fields X i , for solutions to the equation Lu = f
with f ∈ L p . From this fact we also deduce the local Hölder continuity for solutions to Lu = f , when f ∈ L p with p large enough. Further (local) regularity
results, in terms of Sobolev or Hölder spaces, are proved to hold when the coefficients and data are more regular. Finally, lower order terms (in the sense of the
degree of homogeneity) can be added to the operator mantaining the same results.
1. Introduction
A classical result of Agmon-Douglis-Nirenberg [1] states that, for a given uniformly elliptic
operator in nondivergence form with continuous coefficients,
X
Lu =
ai j (x) u xi x j
i, j
one has the following L p -estimates for every p ∈ (1, ∞), on a bounded smooth domain  of
Rn :
≤ c kLuk L p () + kuk L p () .
u xi x j p
L ()
While the above estimate is false in general if the coefficients are merely L ∞ , a remarkable
extension of the above result, due to Chiarenza-Frasca-Longo [6],[7], replaces the continuity
assumption with the weaker condition ai j ∈ V M O, where V M O is the Sarason’s space of
vanishing mean oscillation functions, a sort of uniform continuity in integral sense.
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M. Bramanti - L. Brandolini
Roughly speaking, this extension relies on the classical theory of Calderón-Zygmu-nd operators, a theorem of Coifman-Rochberg-Weiss [8] (which we will recall later in detail) about
the commutator of an operator of this type with a B M O function, and the knowledge of the fundamental solution for constant coefficients elliptic operators on Rn . All these ideas admit broad
generalizations: the Calderón-Zygmund theory and the commutator theorem can be settled in the
general framework of spaces of homogeneous type, in the sense of Coifman-Weiss (see [9], [20]
and [4]); however the knowledge of the fundamental solution is a more subtle problem. Apart
from the elliptic case, an explicit fundamental solution is also known for constant coefficients
parabolic operators. This kernel is homogeneous with respect to the “parabolic dilations”, so
that the abstract Calderón-Zygmund theory can be applied to this situation to get L p -estimates
of the above kind for parabolic operators with V M O coefficients (see Bramanti-Cerutti [3]).
In recent years it has been noticed by Lanconelli-Polidoro [23] that an interesting class
of ultraparabolic operators of Kolmogorov-Fokker-Planck type, despite of its strong degeneracy, admits an explicit fundamental solution which turns out to be homogeneous with respect
to suitable nonisotropic dilations, and invariant with respect to a group of (noncommutative)
translations. These operators can be written as:
(1)
Lu =
q
X
i, j =1
ai j u xi x j +
n
X
i, j =1
xi bi j u x j − u t
n+1 , b
where (x, t)
i j is a constant real matrix with a suitable upper triangular structure,
∈R
while ai j is a q × q uniformly elliptic matrix, with q < n. The structure of space of homogeneous type underlying the operator and the knowledge of a fundamental solution well shaped
on this structure, suggest that an analog L p theory could be settled for operators of kind (1) with
ai j in V M O. This has been actually done by Bramanti-Cerutti-Manfredini [5] . (In this case,
only local estimates are proved).
The class of operators (1) contains prototypes of Fokker-Planck operators describing brownian motions of a particle in a fluid, as well as Kolmogorov operators describing systems with 2n
degrees of freedom (see [23] ), and is still extensively studied (see for instance [22], [24], [25],
[27] and references therein).
When ai j = δi j , (1) exhibits an interesting example of “Hörmander’s operator”, of the kind
Lu =
q
X
i=1
X i2 u + X 0 u
P
where X 0 = ni, j =1 xi bi j ∂x j − ∂t , and X i = ∂xi for i = 1, 2, , . . . q. This introduces us to the
point of view of hypoelliptic operators. Recall that a differential operator P with C ∞ coefficients
is said to be hypoelliptic in some open set U ⊆ R N if, whenever the equation Pu = f is satisfied
in U by two distributions u, f , then the following condition holds: if V is an open subset of U
such that f |V ∈ C ∞ (V ), then u |V ∈ C ∞ (V ). We recall the well-known
T HEOREM 1 (H ÖRMANDER , [16]). Let X 0 , X 1 ,. . . , X q be real vector fields with coefficients C ∞ (R N ). The operator
(2)
P=
q
X
i=1
X i2 + X 0
is hypoelliptic in R N if the Lie algebra generated at every point by the fields X 0 , X 1 ,. . . , X q is
R N . We will call this property “Hörmander’s condition”.
L p estimates for hypoelliptic operators
391
The operator (1) with
ai j ’s satisfies Hörmander’s condition, by the structure as constant
sumption on the matrix bi j , and is therefore hypoelliptic.
In ’75, Folland [11] proved that any Hörmander’s operator like (2) which is left invariant
with respect to a group of translations, and homogeneous of degree 2 with respect to a family
of (nonisotropic) dilations, which are group automorphisms, has a homogeneous left invariant
fundamental solution. This allows to apply the abstract theory of singular integrals in spaces of
homogeneous type, to get local L p estimates of the kind
X i X j u p 0 ≤ c kLukL p () + kukL p ()
(3)
(i, j = 1, . . . , q)
L ( )
for any p ∈ (1, ∞) , 0 ⊂⊂ .
Motivated by the results obtained by [3], [5], the aim of this paper is to extend the above
techniques and results to the homogeneous setting considered by Folland, where good properties
of the fundamental solution allow to obtain in a natural way the L p estimates, using the available
real variable machinery.
More precisely, we study operators of the kind:
L=
q
X
i, j =1
ai j (x)X i X j + a0 (x)X 0
where X 0 , X 1 ,. . . , X q form a system of C ∞ real vector fields defined in R N (N ≥ q + 1), satisfying Hörmander’s condition. We also assume that X 0 , X 1 ,. . . , X q are left invariant with respect
to a “translation” which makes R N a Lie group, and homogeneous with respect to a family of
“dilations” which are group automorphisms. More precisely, X 1 ,. . . , X q are homogeneous of
degree one and X 0 is homogeneous of degree two. The coefficients ai j (x), a0 (x) are real valued
bounded measurable functions, satisfying very weak regularity conditions (they belong to the
class V M O, “Vanishing Mean Oscillation”, defined with respect to the homogeneous distance;
in particular, they can be discontinuous); moreover, the matrix {ai j (x)} is uniformly elliptic and
not necessarily symmetric; the function a0 (x) is bounded away from zero.
Under these assumptions (see §2 for precise statements) we prove that the local L p estimates
(3) hold for p ∈ (1, ∞), every bounded domain , any 0 ⊂⊂ , and any u for which the right
hand side of (3) makes sense (see Theorem 3 for a precise statement). From this fact we also
deduce the local Hölder continuity for solutions to the equation Lu = f , when f ∈ L p () with
p large enough (see Theorem 4).
To get (3) we will first prove the following estimate:
X i X j u ≤ c kLuk p
(i, j = 1, . . . , q, 1 < p < ∞),
(4)
p
for every test function u supported in a ball with sufficiently small radius (see Theorem 2). It is
in this estimate that the V M O regularity of the coefficients plays a crucial role.
Further (local) regularity results for solutions to the equation Lu = f , in terms of Sobolev or
Hölder spaces, are proved to hold when the coefficients and data are more regular (see Theorems
5, 6). Finally, lower order terms (in a suitable sense) can be added to the operator maintaining
the same results (see Theorem 7).
Since the operator L has, in general, nonsmooth coefficients, the above definition of hypoellipticity makes no sense for L. However we will show (Theorem 8) that if the coefficients ai j (x)
are smooth, then L is actually hypoelliptic. Moreover, for every fixed x0 ∈ R N , the frozen
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M. Bramanti - L. Brandolini
operator
(5)
L0 =
q
X
i, j =1
ai j (x0 )X i X j + a0 (x0 )X 0
is always hypoelliptic and, by the results of Folland [11] (see Theorem 9 below), has a homogeneous fundamental solution, which we will prove to satisfy some uniform bounds, with respect
to x0 (Theorem 12). This perhaps justifies the (improper) name of “uniformly hypoelliptic operators” for L, which appears in the title.
We point out that the results in this paper contain as particular cases the local estimates
proved in [6], [3] and [5]. On the other side, global L p estimates on a domain are not available
for hypoelliptic operators, even in simple model cases.
A natural issue is to discuss the necessity of our homogeneity assumptions. In a famous
paper, Rothschild-Stein [28] introduced a powerful technique of “lifting and approximation”,
which allows to study a general Hörmander’s operator by means of operators of the kind studied
by Folland. As a consequence, they obtained estimates like (3) in this more general setting.
In a forthcoming paper [2], we shall use their techniques, combined with our results, to
attack the general case where the homogeneous structure underlying the Hörmander’s vector
fields is lacking.
Outline of the paper. §§2.1, 2.2, 2.5 contain basic definitions and known results. In §2.3
we state our main results (Theorems 2 to 7 ). In §2.4 we illustrate the relations between our
class of operators and the operators of Hörmander type, comparing our results with those of
Rothschild-Stein [28].
In §3 we prove Theorem 2 (that is (4)). The basic tool is the fundamental solution of the
frozen operator (5), whose existence is assured by [11] (see §3.1. The line of the proof consists
of three steps:
(i) we write a representation formula for the second order derivatives of a test function in
terms of singular integrals and commutators of singular integrals involving derivatives of the
fundamental solution (see §3.2);
(ii) we expand the singular kernel in series of spherical harmonics, to get singular integrals
of convolution-type, with respect to our group structure (see §3.3); this step is necessary due to
the presence of the variable coefficients ai j (x) in the differential operator;
(iii) we get L p -bounds for the singular integrals of convolution-type and their commutators,
applying general results for singular integrals on spaces of homogeneous type (see §3.4).
This line is the same followed in [5], which in turn was inspired by [6], [7]. While the commutator estimate needed in [6], [7] to achieve point (iii) is that proved by Coifman-RochbergWeiss in [8], the suitable extension of this theorem to spaces of homogeneous type has been
proved by Bramanti-Cerutti in [4].
The basic difficulty to overcome in the present situation, due to the class of differential operators we are considering, is that an explicit form for the fundamental solution of the frozen
operator L0 in (5) is in general unknown. Therefore we have to prove in an indirect way uniform bounds with respect to x0 for the derivatives of the fundamental solutions corresponding
to L0 (Theorem 12). This will be a key point, in order to reduce the proof of (3) to that of L p
boundedness for singular integrals of convolution type. We wish to stress that, although several
deep results have been proved about sharp bounds for the fundamental solution of a hypoelliptic
operator (see [26], [29], [19]), these bounds are proved for a fixed operator, and the dependence
of the constants on the vector fields is not apparent: therefore, these results cannot be applied in
L p estimates for hypoelliptic operators
393
order to get uniform bounds for families of operators. On the other side, a useful point of view
on this problem has been developed by Rothschild-Stein [28], and we will adapt this approach to
our situation. To make more readable the exposition, the proof of this uniform bound (Theorem
12) is postponed to §4.
To prove local estimates for solutions to the equation Lu = f , starting from our basic estimate (4), we need some properties of the Sobolev spaces generated by the vector fields X i ,
which we investigate in §5: interpolation inequalities, approximation results, embedding theorems. Some of these results appear to be new and can be of independent interest, because they
regard spaces of functions not necessarily vanishing at the boundary, whereas in [11] or [28], for
instance, only Sobolev spaces of functions defined on the whole space are considered.
In §6 we apply all the previous theory to local estimates for solutions to Lu = f . First
we prove (3) and the local Hölder continuity of solutions (see Theorems 4, 5). Then we prove
some regularity results, in the sense of Sobolev or Hölder spaces (see Theorems 5, 6), when
the coefficients are more regular, as well as the generalization of all the previous estimates to
the operator with lower order terms (Theorem 7). Observe that, since the vector fields do not
commute, estimates on higher order derivatives are not a straightforward consequence of the
basic estimate (3). Instead, we shall prove suitable representation formulas for higher order
derivatives and then apply again the machinery of §3.
2. Definitions, assumptions and main results
2.1. Homogeneous groups and Lie algebras
Following Stein (see [31], pp. 618-622) we call homogeneous group the space R N equipped
with a Lie group structure, together with a family of dilations that are group automorphisms.
Explicitly, assume that we are given a pair of mappings:
h
i
[(x, y) 7→ x ◦ y] : R N × R N → R N and x 7→ x −1 : R N → R N
that are smooth and so that R N , together with these mappings, forms a group, for which the
identity is the origin. Next, suppose that we are given an N-tuple of strictly positive exponents
ω1 ≤ ω2 ≤ . . . ≤ ω N , so that the dilations
(6)
D(λ) : (x1 , . . . ,x N ) 7→ λω1 x1 , . . . , λω N x N
are group automorphisms, for all λ > 0. We will denote by G the space R N with this structure
of homogeneous group, and we will write c(G) for a constant depending on the numbers N,
ω1 ,. . . , ω N and the group law ◦.
We can define in R N a homogeneous norm k·k as follows. For any x ∈ R N , x 6= 0, set
1 kxk = ρ ⇔ D( )x = 1,
ρ
where |·| denotes the Euclidean norm; also, let k0k = 0. Then:
(i) kD(λ)xk = λ kxk for every x ∈ R N , λ > 0;
P
(ii) the set {x ∈ R N : kxk = 1} coincides with the Euclidean unit sphere N ;
(iii) the function x 7→ kxk is smooth outside the origin;
(iv) there exists c(G) ≥ 1 such that for every x, y ∈ R N
kx ◦ yk ≤ c(kxk + kyk) and x −1 ≤ c kxk ;
(7)
394
(8)
M. Bramanti - L. Brandolini
1
|y| ≤ kyk ≤ c |y|1/ω
c
if kyk ≤ 1, with ω = max (ω1 , . . . , ω N ) .
The above definition of norm is taken from [12]. This norm is equivalent to that defined in
[31], but in addition satisfies (ii), a property we shall use in §3.3. The properties (i),(ii) and (iii)
are immediate while (7) is proved in [31], p. 620 and (8) is Lemma 1.3 of [11].
In view of the above properties, it is natural to define the “quasidistance” d:
d(x, y) = y −1 ◦ x .
For d the following hold:
(9)
d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y;
(10)
1
d(y,x) ≤ d(x, y) ≤ c d(y,x);
c
(11)
d(x, y) ≤ c d(x, z) + d(z, y)
for every x, y, z ∈ R N and some positive constant c(G) ≥ 1. We also define the balls with
respect to d as
n
o
B(x, r ) ≡ Br (x) ≡ y ∈ R N : d(x, y) < r .
Note that B(0, r ) = D(r )B(0, 1). It can be proved (see [31], p. 619) that the Lebesgue measure
in R N is the Haar measure of G. Therefore
(12)
|B(x, r )| = |B(0, 1)| r Q ,
for every x ∈ R N and r > 0, where Q = ω1 + . . . + ω N , with ωi as in (6). We will call Q the
homogeneous dimension of R N . By (12) the Lebesgue measure d x is a doubling measure with
respect to d, that is
|B(x, 2r )| ≤ c · |B(x, r )| for every x ∈ R N and r > 0
and therefore (R N ,d x, d) is a space of homogenous type in the sense of Coifman-Weiss (see
[9]). To be more precise, the definition of space of homogenous type in [9] requires d to be
symmetric, and not only to satisfy (10). However, the results about spaces of homogeneous type
that we will use still hold under these more general assumptions. (See Theorem 16).
We say that a differential operator Y on R N is homogeneous of degree β > 0 if
Y f (D(λ)x = λβ (Y f )(D(λ)x)
for every test function f , λ > 0, x ∈ R N . Also, we say that a function f is homogeneous of
degree α ∈ R if
f (D(λ)x) = λα f (x) for every λ > 0, x ∈ R N .
Clearly, if Y is a differential operator homogeneous of degree β and f is a homogeneous function
of degree α, then Y f is homogeneous of degree α − β.
Let us consider now the Lie algebra ` associated to the group G (that is, the Lie algebra of
left-invariant vector fields). We can fix a basis X 1 ,. . .,X N in ` choosing X i as the left invariant
L p estimates for hypoelliptic operators
395
vector field which agrees with ∂∂x at the origin. It turns out that X i is homogeneous of degree
i
ωi (see [11], p. 164). Then, we can extend the dilations D(λ) to ` setting
D(λ) X i = λωi X i .
D(λ) turns out to be a Lie algebra automorphism, i.e.,
D(λ) [X, Y ] = [D(λ)X, D(λ)Y ] .
In this sense, ` is said to be a homogeneous Lie algebra; as a consequence, ` is nilpotent (see
[31], p. 621-2).
Recall that a Lie algebra ` is said to be graded if it admits a vector space decomposition as
`=
r
M
i=1
Vi with Vi ,V j ⊆ Vi+ j for i + j ≤ r , Vi ,V j = {0} otherwise.
In this paper, ` will always be graded and it will be possible to choose Vi as the set of vector
fields homogeneous of degree i.
e1 , . . . , V
es
Also, a homogeneous Lie algebra is called stratified if there exist s vector spaces V
such that
`=
s
M
i=1
ei with V
e1 , V
ei = V
ei+1 for 1 ≤ i < s and V
e1 , V
es = {0}.
V
e1 is the whole `. Clearly, if ` is stratified
This implies that the Lie algebra generated by V
then ` is also graded.
Throughout this paper, we will deal with two different situations:
Case A. There exist q vector fields (q ≤ N) X 1 ,. . . , X q , homogeneous of degree 1 such that
e1 is spanned by
the Lie algebra generated by them is the whole `. Therefore ` is stratified and V
X 1 ,. . . , X q . In this case the “natural” operator to be considered is
L=
(13)
q
X
X i2 ,
i=1
which is hypoelliptic, left invariant and homogeneous of degree two.
E XAMPLE 1. The simplest (nonabelian)
example of Case A is the Kohn-Laplacian on the
3
Heisenberg group G = R , ◦, D(λ) where:
(x1 , y1 , t1 ) ◦ (x2 , y2 , t2 ) =
= (x1 + x2 , y1 + y2 , t1 + t2 + 2(x2 y1 − x1 y2 ))
and
D(λ) (x, y, t) = λx, λy, λ2 t .
X=
∂
∂
∂
∂
∂
+ 2y ; Y =
− 2x ; [X, Y ] = −4 ;
∂x
∂t
∂y
∂t
∂t
` = V1 ⊕ V2 with V1 = hX, Y i.
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M. Bramanti - L. Brandolini
The fields X, Y are homogeneous of degree 1, and the operator
L = X2 + Y 2
is hypoelliptic and homogeneous of degree two. Here the homogeneous dimension of G is
Q = 4.
Case B. There exist q + 1 vector fields (q + 1 ≤ N) X 0 , X 1 ,. . . , X q , such that the Lie algebra generated by them is the whole `, X 1 ,. . . , X q are homogeneous of degree 1 and X 0 is
homogeneous of degree 2. In this case the “natural” operator to be considered is
L=
(14)
q
X
i=1
X i2 + X 0 .
Under these assumptions ` may or may not be stratified (see examples below).
E XAMPLE2. (Kolmogorov-type
operators, studied in [23]).
Consider G = R3 , ◦, D(λ) with:
(x1 , y1 , t1 ) ◦ (x2 , y2 , t2 ) = (x1 + x2 , y1 + y2 − x1 t2 , t1 + t2 )
and
D(λ) (x, y, t) = λx, λ3 y, λ2 t .
X1 =
(15)
∂
∂
∂ ∂
; X0 =
−x
; X 0, X 1 =
;
∂x
∂t
∂y
∂y
e1 ⊕ V
e2 with V
e1 = hX 1 , X 0 i, V
e2 = h ∂ i
`=V
∂y
therefore ` is stratified; the fields X 1 , X 0 are homogeneous of degree 1 and 2, respectively, and
the operator
L = X 12 + X 0
is hypoelliptic and homogeneous of degree two. Note that in this case the stratification (15) of `
is different from the natural decomposition of ` as a graded algebra:
` = V1 ⊕ V2 ⊕ V3 with V1 = hX 1 i, V2 = hX 0 i, V3 = h
∂
i.
∂y
This is the simplest (nonabelian) example of Case B; note that Q = 6. If, keeping the same
group law ◦, we changed the definition of D(λ) setting
D(λ) (x, y, t) = λx, λ2 y, λt ,
then the fields X 0 , X 1 would be homogeneous of degree one, and we should consider the operator
L = X 12 + X 02 ,
as in Case A.
L p estimates for hypoelliptic operators
397
E XAMPLE3. This is anexample of the non-stratified case.
Consider G = R5 , ◦, D(λ) with:
(x1 , y1 , z 1 , w1 ,t1 ) ◦ (x2 , y2 , z 2 , w2 , t2 ) =
= (x1 + x2 , y1 + y2 , z 1 + z 2 , w1 + w2 + x1 y2 ,
1 2
x y1 + x1 w2 + x1 z 2 )
2 2
D(λ) (x, y, z, w, t) = λx, λy, λ2 z, λ2 w, λ3 t .
t1 + t2 − x1 x2 y1 − x1 x2 y2 −
and
The natural base for ` consists of:
X=
∂
∂
∂
∂
∂
∂
− xy ; Y =
+x
; Z=
+x ;
∂x
∂t
∂y
∂w
∂z
∂t
∂
∂
∂
+x ; T = .
∂w
∂t
∂t
W =
We can see that ` is graded setting
` = V1 ⊕ V2 ⊕ V3 with V1 = hX, Y i, V2 = hZ, W i, V3 = hT i.
The nontrivial commutation relations are:
[X, Y ] = W ; [X, Z] = T ; [X, W ] = T.
e
e
Therefore,
if we
set V1 = hX, Y, Zi, we see that
the Lie algebra generated by V1 is `; moreover
e2 = V
e1 , V
e1 = hW, T i and V
e3 = V
e1 , V
e2 = hT i, so that ` is not stratified. Noting that
V
X, Y, Z are homogeneous of degrees 1, 1, 2 respectively, we have that the operator
L = X2 + 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 0 ,
X 1 ,. . . , X q are the vector fields appearing in (13)-(14), define, for p ∈ [1, ∞]
kDuk p ≡
q
X
i=1
kX i uk p ;
q
X
2 X i X j u + kX 0 uk .
D u ≡
p
p
p
i, j =1
More in general, set
X
k X j . . . X j u
D u ≡
l
1
p
p
where the sum is taken over all monomials X j1 . . . X jl homogeneous of degree k. (Note that X 0
has weight two while the remaining fields have weight one. Obviously, in Case A the field X 0
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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, p
is finite. We shall also denote by S0
k X
h D u h=0
L p ()
() the closure of C0∞ () in S k, p ().
Since we will often consider the case k = 2, we will briefly write S p () for S 2, p () and
p
2, p
S0 () for S0 ().
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 Hölder spaces 3k,α (), for α ∈ (0, 1), k nonnegative integer, setting
|u|3α () = sup
x6 = y
x,y∈
|u(x) − u(y)|
d(x, y)α
and
kuk3k,α () = D k u 3α ()
+
k−1
X
j =0
j D 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,
Z
t
kuk2H t,2 =
|b
u (ξ )|2 1 + |ξ |2 dξ ,
RN
where b
u (ξ ) 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
Z 1
f (x) − f Bρ d x for every r > 0,
(16)
η f (r ) = sup ρ<r Bρ Bρ
where Bρ is any ball of radius ρ and f Bρ is the average of f over Bρ .
We say that f ∈ B M O if k f k∗ ≡ supr 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 replacing Bρ with Bρ ∩  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 ` its Lie algebra; let
{X i } (i = 1, 2, . . . , N) be the basis of ` 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
ai j (x)X i X j
i, j =1
or
L=
(17)
q
X
i, j =1
ai j (x)X i X j + a0 (x)X 0
where
ai j and a0 are real valued bounded measurable functions and the matrix
ai j (x) satisfies a uniform ellipticity condition:
(18)
µ |ξ |2 ≤
q
X
i, j =1
ai j (x) ξi ξ j ≤ µ−1 |ξ |2
for every ξ ∈ Rq , a.e. x,
for some positive constant µ. Analogously,
µ ≤ a0 (x) ≤ µ−1 .
(19)
Moreover, we will assume
a0 , ai 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 ∈ C0∞ R N and sprt u ⊆ Br (Br any ball of radius r )
then
2 D u ≤ c kLuk p
p
where η denotes dependence on the “V M O moduli” of the coefficients a0 , ai 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
0 ⊂⊂ . If u ∈ S p (), then
kuk S p (0 ) ≤ c kLukL p () + kukL p ()
where c = c( p, G, µ, η, , 0 ).
T HEOREM 4 (L OCAL H ÖLDER CONTINUITY FOR SOLUTIONS TO THE
Lu = f IN A DOMAIN ). Under the assumptions of Theorem 3, if u ∈ S p () for
some p ∈ (1, ∞) and Lu ∈ Ls () for some s > Q/2, then
EQUATION
kuk3α (0 ) ≤ c kLukLr () + kukL p ()
for r = max( p, s), α = α(Q, p, s) ∈ (0, 1), c = c(G, µ, p, s, , 0 ).
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 a0 , ai j ∈ S k,∞ (), u ∈ S p () and Lu ∈ S k, p () for some
positive integer k (k even, in Case B), 1 < p < ∞, then
o
n
kuk S k+2, p (0 ) ≤ c1 kLuk S k, p () + c2 kukL p ()
where c1 = c1 ( p, G, µ,η,, 0 ) and c2 depends on the S k,∞ () norms of the coefficients.
T HEOREM 6 (R EGULARITY OF THE SOLUTION IN TERMS OF H ÖLDER
Under the assumption of Theorem 3, if a0 , ai j ∈ S k,∞ (), u ∈ S p () and Lu ∈
S k,s () for some positive integer k (k even, in Case B), 1 < p < ∞, s > Q/2, then
n
o
kuk3k,α (0 ) ≤ c1 kLuk S k,r () + c2 kukL p ()
SPACES ).
where r =max( p, s), α = α(Q, p, s) ∈ (0, 1), c1 = c1 ( p, s, k, G, µ,η,, 0 ) and c2 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:
X
X
q
q
L≡
ai j (x)X i X j + a0 (x)X 0 +
ci (x)X i + c0 (x) ≡
i, j =1
i=1
≡ L2 + L1 .
∞
i) If ci ∈ L () for i = 0, 1, . . . , q, then:
if the assumptions of Theorem 3 hold for L2 , then the conclusions of Theorem 3 hold for L;
if the assumptions of Theorem 4 hold for L2 , then the conclusions of Theorem 4 hold for L.
ii) If ci ∈ S k,∞ () for some positive integer k, i = 0, 1, . . . , q, then:
if the assumptions of Theorem 5 hold for L2 , then the conclusions of Theorem 5 hold for L;
if the assumptions of Theorem 6 hold for L2 , 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 a0 ,
ai 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 a0 ,
ai j ∈ V M O.
2.4. Relations with operators of Hörmander type
Here we want to point out the relationship between our class of operators and operators of
Hörmander type (2).
T HEOREM 8. Under the assumptions of §2.3:
(i) if the coefficients ai j (x) are Lipschitz continuous (in the usual sense), then the operator
L can be rewritten in the form
q
X
L=
Yi2 + Y0
i=1
L p estimates for hypoelliptic operators
401
where the vector fields Yi (i = 1, . . . , q) have Lipschitz coefficients and Y0 has bounded measurable coefficients;
(ii) if the coefficients ai j (x) are smooth (C ∞ ), then L is hypoelliptic;
(iii) if the coefficients ai j are constant, then L is left invariant and homogeneous of degree
two; moreover, the transpose LT of L is hypoelliptic, too.
Proof. Let us split the matrix ai j (x) in its symmetric and skew-symmetric parts:
1
1
ai j (x) + a j i (x) +
ai j (x) − a j i (x) ≡ bi j (x) + e
bi j (x).
ai j (x) =
2
2
If the matrix A = ai j (x) satisfies condition (18), the same holds for B = bi 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
q
X
X
bi j ξi ξ j = η12 +
bi∗j ηi η j
i, j =1
with
i, j =2


q
X
p
b1 j
√
ξj ;
η1 =  b11 ξ1 +
b11
ηi = ξi for i ≥ 2;
j =2
2
∗ = b − b1i ; b∗ = b
bii
i j for i, j = 2, . . . , q, i 6 = j.
ii
ij
b11
Since η1 , . . . , ηq are a linear invertible function of ξ1 , . . . , ξq , and the quadratic form
Pq
P
q
∗
q
i, j =1 bi j ξi ξ j is positive (on R ), also the quadratic form
i, j =2 bi j ηi η j is positive (on
P
q
Rq−1 ), and we can iterate the same procedure. Note that η1 = k=1 m 1k ξk with m 1k smooth
functions of the bi j ’s; moreover, bi∗j are smooth functions of the bi j ’s. Therefore iteration of this
procedure allows us to write
q
X
i, j =1
λk =
q
X
bi j ξi ξ j =
q
X
λ2k
with:
k=1
m kh ξh and m kh are smooth functions of the bi j ’s.
h=k
P
This means that bi j = k≥i, j m ki m kj with m kh smooth functions of the bi 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
bi j (x) X i ,X j + a0 (x)X 0
where the functions m ik (x) have the same regularity of the ai j (x)’s. (To simplify the notation,
from now on we forget the fact that m ik = 0 if k < i). If the ai j (x)’s are Lipschitz continuous,
the above equation can be rewritten as
(20)
L=
q
X
k=1
Yk2 + Y0
402
M. Bramanti - L. Brandolini
with
Yk =
Y0 =
X
i< j
q
X
m ik (x)X i
and
i=1
q X
q
X
e
bi j (x) X i ,X j + a0 (x)X 0 −
m ik (x) · X i m j k (x) X j ,
i, j =1 k=1
which proves (i). If the coefficients ai j (x) are C ∞ , the Yi ’s are C ∞ vector fields and satisfy
Hörmander’s condition, because every linear combination of the X i (i = 0, 1, . . . , q) can be
rewritten as a linear combination of the Yi and their commutators of length 2. Therefore, by
Theorem 1, L is hypoelliptic, that is (ii). Finally, if the coefficients ai j are constant, then (20)
holds with
q
X
X
e
Yk =
m ik X i and Y0 =
bi j X i ,X j + a0 X 0 ,
i=1
i< j
which means that L is left invariant and homogeneous of degree two. Moreover, since the fields
X i are translation invariant, the transpose X iT of X i equals −X i and as a consequence LT is
hypoelliptic as well. This proves (iii).
R EMARK 2. By the above Theorem, if ai 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 Hörmander type”; however,
in this case we cannot check Hörmander’s condition for the fields Yi 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
Z
Z
g(y −1 ◦ x) f (y) d y,
f (x ◦ y −1 ) g(y) d y =
( f ∗ g)(x) =
RN
RN
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 iR which
agree with ∂/∂ xi (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 iR . It can be proved that
Xi =
X iR =
N
X
∂
∂
+
qik (x)
∂ xi
∂ xk
k=i+1
N
X
∂
∂
+
e
qik (x)
∂ xi
∂ xk
k=i+1
where qik (x), e
qik (x) are polynomials, homogeneous of degree ωk −ωi (the ωi ’s are the exponents
appearing in (6)). From the above equations we find that
Xi =
N
X
cik (x) X kR
k=i
where cik (x) are polynomials, homogeneous of degree ωk −ωi . In particular, since ωk −ωi < ωk ,
cik (x) does not depend on xh for h ≥ k and therefore commutes with X kR , that is
(22)
Xi u =
N
X
k=i
X kR cik (x) u
(i = 1, . . . , N)
for every test function u. This representation of X i in terms of X iR 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 LT are both hypoelliptic.
Moreover, assume Q ≥ 3. Then there is a unique fundamental solution 0 such that:
(a) 0 ∈ C ∞ R N \ {0} ;
(b) 0 is homogeneous of degree (2 − Q);
(c) for every distribution τ ,
L (τ ∗ 0) = (Lτ ) ∗ 0 = 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
Th be the operator
Th f = f ∗ K h
and let P h be a left invariant differential operator homogeneous of degree h.
Then:
P h Th 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 vanishing property:
Z
r<kxk<R
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 αi , with −Q < αi < 0, α1 + α2 <
−Q;
(iii) for every multiindex β,
∂ β
sup
sup K i (x, y) ≤ cβ .
∂y
x∈R N kyk=1
Then, for every test function f and any x0 , y0 ∈ R N ,
( f ∗ K 1 (x0 , ·)) ∗ K 2 (y0 , ·) = f ∗ (K 1 (x0 , ·) ∗ K 2 (y0 , ·)) .
Moreover, setting K (x0 , y0 ,·) = K 1 (x0 , ·) ∗ K 2 (y0 , ·), we have the following:
(iv) for every (x0 , y0 ) ∈ R2N , K (x0 , y0 ,·) ∈ C ∞ (R N \ {0});
(v) for every (x0 , y0 ) ∈ R2N , K (x0 , y0 ,·) is homogeneous of degree α1 + α2 + Q;
(vi) for every multiindex β,
∂ β
K (x, y, z) ≤ cβ .
(23)
sup
sup ∂z
2N
kzk=1
(x,y)∈R
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 0 .
3.1. Fundamental solutions
For any x0 ∈ R N , let us “freeze” at x0 the coefficients ai j (x), a0 (x) of the operator (17), and
consider
(24)
L0 =
q
X
i, j =1
ai j (x0 )X i X j + a0 (x0 )X 0 .
By Theorem 8, the operator L0 satisfies the assumptions of Theorem 9; therefore, it has a fundamental solution with pole at the origin which is homogeneous of degree (2 − Q). Let us denote
it by 0 (x0 ; ·), to indicate its dependence on the frozen coefficients ai j (x0 ), a0 (x0 ). Also, set
for i, j = 1, . . . , q,
0i j (x0 ; y) = X i X j 0 (x0 ; ·) (y).
L p estimates for hypoelliptic operators
405
Next theorem summarizes the properties of 0 (x0 ; ·) and 0i j (x0 ;·) that we will need in the
following. All of them follow from Theorem 9 and Lemma 1.
T HEOREM 11. For every x0 ∈ R N :
(a) 0 (x0 ,·) ∈ C ∞ R N \ {0} ;
(b) 0 (x0 ,·) is homogeneous of degree (2 − Q);
(c) for every test function u and every x ∈ R N ,
Z
0 x0 ; y −1 ◦ x L0 u(y)d y;
u(x) = (L0 u ∗ 0 (x0 ;·)) (x) =
RN
moreover, for every i, j = 1, . . . , q, there exist constants αi j (x0 ) such that
(25)
X i X j u(x) = P.V.
Z
RN
0i j x0 ; y −1 ◦ x L0 u(y)d y + αi j (x0 ) · L0 u(x);
(d) 0i j (x0 ;·) ∈ C ∞ R N \ {0} ;
(e) 0i j (x0 ;·) is homogeneous of degree −Q;
( f ) for every R > r > 0,
Z
Z
0i j (x0 ; y) d y =
kyk=1
r<kyk<R
0i j (x0 ; y) dσ (y) = 0.
The above properties hold for any fixed x0 . We also need some uniform bound for 0, with
respect to x0 . Next theorem contains this kind of result.
T HEOREM 12. For every multi-index β, there exists a constant
c1 = c1 (β,G, µ) such that
∂ β
(26)
sup 0i j (x; y) ≤ c1 ,
N ∂y
x∈R
kyk=1
for any i, j = 1, . . . , q. Moreover, for the αi j ’s appearing in (25), a uniform bound holds:
(27)
sup αi j (x) ≤ c2 ,
x∈R N
for some constant c2 = c2 (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 L0 = L + (L0 − L) and then letting x be equal to x0 , we get the
following representation formula:
406
M. Bramanti - L. Brandolini
T HEOREM 13. Let u ∈ C0∞ R N . Then, for i, j = 1, . . . , q and every x ∈ R N
X
Z
q
X i X j u(x) = P.V.
0i j (x; y −1 ◦ x)
ahk (x) − ahk (y) X h X k u(y) +
h,k=1
(28)
+ a0 (x) − a0 (y) X 0 u(y) + Lu(y) d y + αi j (x) · Lu(x).
In order to rewrite the above formula in a more compact form, let us introduce the following
singular integral operators:
Z
(29)
K i j f (x) = P.V.
0i j (x;y −1 ◦ x) f (y) d y.
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 (Lu) −
(30)
q
X
h,k=1
C K i j , ahk (X h X k u) +
+C K i j , a0 (X 0 u) + αi 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:
K i j ( f ) p N ≤ c k f k p N (31)
L R
L (R )
(32)
C K i j , a ( f )
L p R N ≤ c kak∗ k f kL 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 ⊆ Br
C K i j , a ( f ) p
(33)
L (B ) ≤ c ( p, µ, G) · ε k f kL p (Br ) .
r
Finally, using the bounds (27), (31), (33) in the representation formula (30), we get Theorem
2. Note that the term X 0 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 convolution type because of the presence of the first variable x in the kernel, which comes from
the variable coefficients ai 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 Calderón-Zygmund [10], in the case of “standard” singular integrals, and has been
adapted to kernels with mixed homogeneities by Fabes-Rivière [12]. We briefly describe this
technique. (See [10] for details). Let
{Ykm }m=0,1,2,...
k=1,...,gm
be an orthonormal system of spherical harmonics in R N , complete in L2 (6 N ) (m is the degree
of the polynomial, gm 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)
0i j (x;y) =
∞
X
gm
X
m=1 k=1
cikm
j (x) Ykm (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 Theorem11. Also,
note that the integral of Ykm (y) over 6 N , for
m ≥ 1, is zero. If y ∈ R N , let y 0 = D kyk−1 y; recall that, by (ii) at page 393, y 0 ∈ 6 N . By
(34) and homogeneity of 0i j (x; ·) we have
0i j (x; y) =
∞
X
gm
X
cikm
j (x)
m=1 k=1
Ykm y 0
kyk Q
for i, j = 1, . . . ,q.
Then
(35)
K i j ( f ) (x) =
cikm
j (x) Tkm f (x)
m=1 k=1
with
(36)
gm
X
∞
X
Tkm f (x) = P.V.
Z
Hkm (y −1 ◦ x) f (y) d y
and
(37)
Hkm (x) =
Ykm x 0
kxk Q
.
We will use the following bounds about spherical harmonics:
(38)
(39)
gm ≤ c(N) · m N−2 for every m = 1, 2, . . .
∂ β
N−2
+|β|
Ykm (x) ≤ c(N) · m 2
∂x
408
M. Bramanti - L. Brandolini
for x ∈ 6 N , k = 1, . . . , gm , m = 1, 2, . . ..
P
Moreover, if f ∈ C ∞ (6 N ) and if f (x) ∼
k,m bkm Ykm (x) is the Fourier expansion of
n
o
f (x) with respect to Ykm , that is
bkm =
Z
6N
f (x) Ykm (x) dσ (x)
then, for every positive integer r there exists cr such that
∂ β
|bkm | ≤ cr · m −2r sup f (x) .
(40)
∂x
x∈6 N
|β|=2r
In view of Theorem 12, we get from (40) the following bound on the coefficients cikm
j (x) appearing in the expansion (34): for every positive integer r there exists a constant c = c(r, G, µ)
such that
−2r
(41)
sup cikm
j (x) ≤ c(r, G, µ) · m
x∈R N
for every m = 1, 2,. . .; k = 1, . . . , gm ; i, j = 1, . . . , q.
3.4. Estimates on singular integrals of convolution type and their commutators, and convergence 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 (B2r (x)) ≤ c · µ (Br (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 : Br (x) = {x}}
(here sup ∅ = 0). We say that (X, d, µ) satisfies a reverse doubling condition if there exist
c0 > 1, M > 1 such that for every x ∈ X, r > r x
µ(B Mr (x)) ≥ c0 · µ(Br (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 unbounded, assume that the reverse doubling condition holds. Let k : X × X \ {x = y} → R be a
kernel satisfying:
i) the growth condition:
(42)
|k(x, y)| ≤
c1
for every x, y ∈ X, some constant c1
µ B (x, d(x, y))
ii) the “Hörmander inequality”: there exist constants c2 > 0, β > 0, M > 1 such that for
every x0 ∈ X, r > 0, x ∈ Br (x0 ), y ∈
/ B Mr (x0 ),
|k(x0 ,y) − k(x, y)| + |k(y, x0 ) − k(y, x)| ≤
≤
(43)
c2
µ B (x0 , d(x0 , y))
·
d(x0 ,x)β
;
d(x0 ,y)β
iii) the cancellation property: there exists c3 > 0 such that for every r, R, 0 < r < R < ∞,
a.e. x
Z
Z
+
≤ c3 .
(44)
k(x,
y)
dµ(y)
k(z,
x)
dµ(z)
r<d(x,y)<R
r<d(x,z)<R
iv) the following condition: for a.e. x ∈ X there exists
Z
(45)
lim
k(x, y) dµ(y).
ε→0 ε<d(x,y)<1
For f ∈ L p , p ∈ (1, ∞), set
K ε f (x) =
Z
k(x, y) f (y) dµ(y).
ε<d(x,y)<1/ε
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) · (c1 + c2 + c3 ).
Proof. To see this, note that if k satisfies (42), (43), (44) with constants c1 , c2 , c3 , then k 0 ≡
k/(c1 + c2 + c3 ) satisfies (42), (43), (44) with constants 1, 1, 1, so that for the kernel k 0 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:
Ykm x 0
−1
.
k(x, y) = Hkm y ◦ x with Hkm (x) =
kxk Q
By homogeneity, k satisfies (42) with
c1 = c(G) · sup |Ykm (x)| .
(48)
x∈6 N
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 = c(G, f ) > 0, M = M(G) > 1 such that
| f (x ◦ y) − f (x)| + | f (y ◦ x) − f (x)| ≤ c kyk kxkλ−1
(49)
for every x, y such that kxk ≥ M kyk. Moreover
c = c(G) · sup |∇ f (z)| .
z∈6 N
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 ≤ 1/M then kx ◦ yk ≥ 1/2. Set:
F(x, y) = f (x ◦ y), L(x, y) = x ◦ y
and
n
o
K ≡ (x, y) : kxk = 1 and kyk ≤ 1/M .
By homogeneity, it is enough to prove (49) for (x, y) ∈ K . Since f (z) is smooth for kzk ≥ 1/2
and L is smooth (everywhere), by the mean value theorem
| f (x ◦ y) − f (x)| = |F(x, y) − F(x, 0)| ≤ |y| · ∇ F x, y ∗ with x, y ∗ ∈ K . But:
X
N
∂F
≤
(x,
y)
(x,y)∈K ∂ xi
sup
j =1
∂L j
· sup ∂ f (z) ≤
(x,
y)
∂x
∂z
i
j
(x,y)∈K
kzk≥ 1
sup
2
≤ c(G) · sup |∇ f (z)| ,
z∈6 N
and the same holds for sup(x,y)∈K ∂∂ yF (x, y).
i
Recalling that |y| ≤ c(G) 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 = M(G), and
c2 = c(G) · sup |∇Ykm (x)| .
(50)
x∈6 N
As to (44), the left hand side equals:
Z
Hkm (y) d y .
Hkm (y) d y + −1
r< y
<R
r<kyk<R
Z
The first term is a multiple of
Z
6N
Hkm (y) dσ (y)
and therefore vanishes; the second term, by (42), (48) and (10) can be seen to be bounded by
c(G) · c1 (see for instance Remark 4.6 in [4]). Hence c3 has the same form of c1 . Moreover, (45)
is trivially satisfied, by the vanishing property of Hkm .
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, . . . , gm
kTkm ( f )kL p R N ≤ c k f kL p (R N )
C Tkm , a ( f )
L p (R N ) ≤ c kak∗ k f kL p (R N ) .
Explicitly, c = c( p, G) · m N/2 .
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 Aµ be the set of q × q constant matrices A ={ai j }, satisfying:
µ2 |ξ |2 ≤
q
X
i, j =1
ai j ξi ξ j ≤ µ−2 |ξ |2 for every ξ ∈ Rq ,
where µ is the same as in (18), (19). Throughout this section we will consider the operator
LA =
q
X
i, j =1
ai j X i X j + X 0
412
M. Bramanti - L. Brandolini
where A ={ai j }∈ Aµ and the fields X i satisfy the assumptions stated in §2.1. Let 0 A be the
fundamental solution for L A , homogeneous of degree (2 − Q) (see §3.1), and let:
TA : f 7→ f ∗ 0 A .
By Theorem 9, T A L A = L A T A =identity. We will prove that:
∂ β
A
0 (x) ≤ c(β, G, µ).
(51)
sup kxk=1 ∂ x
Note that, if L0 is the frozen
operator defined in (24), L0 = a0 (x0 )·L A with
q
A = ai j (x0 )/a0 (x0 ) i, j =1 ∈ Aµ and 0 (x0 , ·) = a0 (x0 ) · 0 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:
1
2
1
T HEOREM 18. a) If 1 < p < Q
2 and s = p − Q , then
kT A f ks ≤ c k f k p
with
c = c( p, G) · sup 0 A .
6N
Q
∞
b) If Q
2 < p < Q and β = 2 − p (hence β ∈ (0, 1)), then for every f ∈ C 0 and every
x1 , x2 ∈ R N
|(T A f )(x1 ) − (T A f )(x2 )| ≤ c k f k p d(x1 , x2 )β
with
(
)
A
A
c = c( p, G) · sup 0 + sup ∇0 .
6N
6N
N
c) If p > Q
2 and sprt f ⊆ Br (x0 ) for some r > 0, x0 ∈ R
kT A f kL∞ (Br ) ≤ c k f kL p (Br )
with
c = c( p, G,r ) · sup 0 A .
6N
Note: parts b-c of Theorem 18 will be used only in §6(proof of Theorem 4).
Proof. Part a) follows from Proposition 1.11 in [11], or also from results about fractional integrals on general spaces of homogeneous type, see [15]. The form of the constant c depends on
the bound:
1
A (52)
.
0 (x) ≤ sup 0 A ·
kxk Q−2
6N
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:
Z
h
i
A −1
|(T A f )(x1 ) − (T A f )(x2 )| ≤
0 (y ◦ x1 ) − 0 A (y −1 ◦ x2 ) f (y) d y ≤
RN
L p estimates for hypoelliptic operators
Z
≤ 413
Z
. . . d y = I + I I.
. . . dy + y −1 ◦x 1 ≤M y −1 ◦x 1 ≥M x 2−1 ◦x 1 x 2−1 ◦x 1 By Proposition 1,
Z
I ≤ c( p, G) · sup ∇0 A x2−1 ◦ x1 | f (y)|
Q−1 d y.
y −1 ◦x 1 ≥M x 2−1 ◦x 1 y −1 ◦ x1 6N
Let p, p0 be conjugate exponents; by Hölder’s inequality and a change of variables

1/ p 0
Z
1

≤
I ≤ c x2−1 ◦ x1 k f k p 
0 dy
kyk≥M x 2−1 ◦x 1 kyk(Q−1) p
computing the integral, under the assumption p < Q,
β
≤ c x2−1 ◦ x1 k f k p
where β = 1 − q1 [(Q − 1)(q − 1) − 1] = 2 − Qp ∈ (0, 1).
By (52),
I I ≤ sup 0 A ·
6N
Z
| f (y)|
· y −1 ◦x 1 ≤M x 2−1 ◦x 1 (
1
1
+
y −1 ◦ x1 Q−2
y −1 ◦ x2 Q−2
)
dy =
= I I 0 + I I 00 .
By Hölder’s inequality and reasoning as above, we get, if p > Q/2,
β
I I 0 ≤ c x2−1 ◦ x1 k f k p .
As to I I 00 , if y −1 ◦ x1 ≤ M x2−1 ◦ x1 , then y −1 ◦ x2 ≤ c x2−1 ◦ x1 and therefore I I 00
can be handled as I I 0 .
As to c), noting that x, y ∈ Br (x0 ) ⇒ y −1 ◦ x ∈ B K r (0) for some K = K (G), we can
write, by Hölder’s inequality (let p0 be the conjugate exponent of p):
kT A f kL∞ (Br (x0 )) ≤ k f kL p (Br (x0 )) · 0 A p0
≤ (by (52))
L (B K r (0))
≤ k f kL p (Br (x0 )) · c( p, G) · sup 0 A · r 2−Q/ p ,
6N
which proves the result, assuming p > Q/2.
Now, let
SiAj f = X i X j T A f.
By (c) of Theorem 11, setting 0iAj = X i X j 0 A , we can write
(53)
SiAj f = P.V. 0iAj ∗ f + αi j (A) · f.
Let us apply Theorem 16 and Remark 3 to the kernel 0iAj . 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 ∈ C0∞ R N ,
P.V. 0iAj ∗ f ≤ c k f k p
p
with
(
)
A
A
c = c( p, G) · sup 0 + sup ∇0 .
6N
6N
L EMMA 2. For every p ∈ (1, ∞) and for every A 0 ∈ Aµ , there exists ε > 0 such that if
| A − A 0 | < ε, then
1
kE A f k p ≤ k f k p
2
where |A| denotes the Euclidean norm of the matrix A and
E A = L Ao − L A T Ao .
This Lemma is proved in [28] (Lemma 6.5), for a different class of operators.
Proof. Let us write
L Ao − L A =
Then
EA f =
q X
aioj − ai j X i X j .
i, j =1
q X
i, j =1
aioj − ai j
X i X j T Ao f.
By (53) and Theorem 19 we get the result.
2 . There exists c = c(G, µ, p) such that
L EMMA 3. Let p ∈ (1, Q/2) and let 1s = 1p − Q
for every A ∈ Aµ
kT A f ks ≤ c k f k p .
(54)
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 0 ∈ Aµ and let E A and ε be as in the previous lemma. For every f ∈ L p , if
| A − A 0 | < ε, then for every p ∈ (1, ∞), kE A f k p ≤ 12 k f k p , so that we can write
(55)
∞
X
n=0
E nA f = (I − E A )−1 f ≡ g.
Therefore
f = (I − E A )g = g − L Ao T Ao g + L A T Ao g = L A T Ao g,
that is
T A f = T Ao g.
L p estimates for hypoelliptic operators
415
Again from (55) we have
kgk p ≤
∞
X
n=0
kE A kn 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 ks = T Ao g s ≤ c(G, p, A 0 ) kgk p ≤ 2c k f k p .
Since this is true for every fixed A 0 ∈ Aµ and any matrix A such that |A − A 0 | < ε, by the
2
compactness of Aµ in Rq we can choose a constant c = c(G, p,µ) such that (54) holds for
every A ∈ Aµ .
T HEOREM 20. Let ϕ, ϕ1 ∈ C0∞ R N with ϕ1 = 1 on sprt ϕ. There exists ε = ε (G)
and, for every
t ∈ R, there exists c = c(t, µ,G, ϕ, ϕ1 ) such that for every A ∈ Aµ and every
u ∈ C0∞ R N
kϕuk H t+ε,2 ≤ c kϕ1 L A uk H t,2 + kϕ1 ukL2 .
(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 modifications 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, Br will be
a ball
centered
∞
∞
N
such that:
at the origin. Let g ∈ C0 (B2 \ B1 ) such that kgk2 ≤ 1, and let ϕ, ϕ1 ∈ C0 R
ϕ = 1 in B1/4 , sprtϕ ⊆ B1/2 , ϕ1 = 1 in B1/2, sprtϕ1 ⊆ B1 . Let f = T A g. Since L A f = g = 0
in B1 and L A is hypoelliptic, f ∈ C ∞ (B1 ).
2 < 1 < min 1 + 2 , 1 and let s be as in
Pick a positive number p such that max 21 , Q
p
Q
2
Lemma 3. Note that 1 < p < Q/2 and p < 2 < s. Then, by Lemma 3:
kϕ1 f k2 ≤ c(ϕ1 ) k f ks ≤ c(ϕ1 ,G, µ, p) kgk p ≤ c(ϕ1 ,G, µ) kgk2 ≤ c(ϕ1 ,G, µ).
Applying Theorem 20 to ϕ, ϕ1 , f , since L A f = 0 on sprtϕ1 , we get:
kϕ f k H t+ε,2 ≤ c kϕ1 f k2 ≤ c(t,ϕ,ϕ1 ,G, µ)
for every t ∈ R. Therefore, by the standard Sobolev embedding Theorems, we can bound any
(isotropic) Hölder norm C h,α of f on B1/4 with a constant c(h, G, µ); in particular, for every
differential operator P:
(56)
|P f (0)| ≤ c(P, G, µ).
416
M. Bramanti - L. Brandolini
Now, recall that f = g ∗ 0 A . If P is any left invariant differential operator,
Z
(57)
P f (0) =
P 0 A (y −1 ) g(y) d y.
Since (56) holds for every g ∈ C0∞ (B2 \ B1 ) such that kgk2 ≤ 1, from (57) we get
(58)
≤ c(P, G, µ).
P 0 A (y −1 ) 2
L (B2 \B1 )
β
in terms of left invariant vector fields, (58) gives
Now, writing any differential operator ∂∂x
k,2
A
us a bound on every H -norm of 0 on B2 \ B1 , and therefore, reasoning as above, on every
Hölder norm C h,α of 0 A on a smaller spherical shell C ≡ B7/4 \ B5/4 . In particular, we get
∂ β
A
sup 0 (x) ≤ c(β, G, µ),
x∈C ∂ x
from which (51) follows, by homogeneity of 0 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 i2 u + X 0 u + cu,
where the fields X i (i = 0, 1, . . . , q) satisfy Hörmander’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 uk2 ,
for every u ∈ C0∞ R N .
We can rephrase as follows the steps of the proof of Theorem 20 given in [21]:
(i) There exist ε = ε(G), c = c(G, µ) such that for every u ∈ C0∞ R N and every A ∈ Aµ
kuk H ε,2 ≤ c kL A uk2 + kuk2 .
(ii) For
every t ∈ R, M > 0, there exists c = c(t, M, µ, G) such that for every u ∈
C0∞ R N and every A ∈ Aµ
kuk H t+ε,2 ≤ c kL A uk 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 kϕ1 L A uk H t,2 + kϕ1 uk H −M,2 ,
where ϕ, ϕ1 ∈ C0∞ R N with ϕ1 = 1 on sprt ϕ, c = c(ϕ,ϕ1 ,t, M, µ, G).
Since k·k H −M,2 ≤ k·k2 , from point (iii) we get Theorem 20.
L p estimates for hypoelliptic operators
417
R EMARK 4 (A N ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS
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);
t /2
(c) for t ∈ R, 3t defined by \
3t u (ξ ) = 1 + |ξ |2
b
u (ξ ).
ADAPTED TO THE FIELDS
By general properties of homogeneous groups (see [31], p. 621), the vector fields X i are
linear combinations of ∂/∂ xi with polynomial coefficients (and, by Hörmander’s condition, the
∂/∂ xi ’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 0 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 C0∞ 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.
n o
Let kγ
be a family of kernels such that kγ is homogeneous of degree h − Q for some
γ ∈3
h > 0 and kγ ∈ C0∞ R N \ {0} . Let Tγ be the distribution associated to kγ and let P h be a
left invariant differential operator homogeneous of degree h. Then, Theorem 9 states that
(59)
P h Tγ = P.V. P h kγ + αγ δ.
(Observe that (25) is a particular case of (59)). With these notations, we can prove the following:
then
L EMMA 4. If kγ satisfies a uniform bound like (26), that is, for every multiindex β
∂ β
kγ (y) ≤ c (β) ,
sup sup ∂
y
γ ∈3 kyk=1
sup αγ ≤ c.
γ ∈3
Proof. Let u be a test function with u(0) 6= 0, sprtu ⊆ B1 (0). By (59)
αγ u(0) = h (P h )T u, Tγ i − hu, P.V. P h kγ i =
=
Z
Z
(P h )T u(x) kγ (x) d x − lim
P h kγ (x) u(x) d x
ε→0 kxk>ε
418
M. Bramanti - L. Brandolini
(here (·)T denotes transposition). Since kγ 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 kγ
(see Lemma 1), homogeneity and (59) we can write
Z
Z
h k (x) [u(x) − u(0)] d x ≤
h k (x) u(x) d x = P
P
γ
γ
kxk>ε
ε<kxk<1
≤ sup |∇u(y)|
kyk≤1
Z
c
Q
ε<kxk<1 kxk
· |x| d x ≤ c.
(The convergence of the last integral follows from (8)).
5. Some properties of the Sobolev spaces S k, p
We start pointing out the following interpolation inequality for Sobolev norms:
P ROPOSITION 2. Let
X be
a left invariant vector field, homogeneous of degree α > 0. Then
p
N
for every ε > 0, u ∈ S R , p ∈ [1, ∞)
(60)
2
kXuk p ≤ ε X 2 u + kuk p .
p
ε
Proof. The following argument is taken from the proof of Theorem 9.4 in [13]. Let γ (t) be the
integral curve of X with γ (0) = 0. Then, applying Taylor’s theorem to the function F(t) =
u (x ◦ γ (t)),
Z 1
u(x ◦ γ (1) ) = u(x) + Xu(x) +
(1 − t) X 2 u(x ◦ γ (t))dt.
0
Using the translation invariance of k·k p and Minkowski’s inequality, we get
kXuk p ≤ X 2 u + 2 kuk p .
p
Since X is homogeneous, by a dilation argument we get the result.
We will need a version of (60) for functions defined on a ball (not necessarily vanishing at
the boundary). For standard Sobolev norms, this result follows from the analog of (60) using an
extension theorem (see for instance [14], pp. 169-173). However, it seems not easy to construct
p
a continuous extension operator E: S p (Br ) → S0 (B2r ). We are going to show how to bypass
this difficulty.
First we construct
a suitable family of cutoff functions. Given two balls Br1 , Br2 and a
function ϕ ∈ C0∞ R N , let us write Br1 ≺ ϕ ≺ Br2 to say that 0≤ ϕ ≤ 1, ϕ ≡ 1 on Br1 and
sprtϕ ⊆ Br2 .
L EMMA 5 (R ADIAL
CUTOFF FUNCTIONS ). For any σ ∈ (0, 1), r > 0, k positive integer,
there exists ϕ ∈ C0∞ R N with the following properties:
Bσ r ≺ ϕ ≺ Bσ 0 r
with σ 0 = (1 + σ ) /2;
L p estimates for hypoelliptic operators
j P ϕ ≤
419
c(G, j )
σ j −1 (1 − σ ) j r j
for 1 ≤ j ≤ k,
where P j is any left invariant differential monomial homogeneous of degree j .
Proof. For simplicity, we prove the assertion for k = 2. The general case is similar. Pick a
function f : [0, r ) → [0, 1] such that:
f ≡ 1 in [0, σ r ), f ≡ 0 in [σ 0r, r ), f ∈ C ∞ (0, r ) ,
c
c
, f 00 ≤
.
(1 − σ ) r
(1 − σ )2 r 2
Setting ϕ(x) = f (kxk), we can compute:
0
f ≤
X i ϕ(x) = f 0 (kxk)X i (kxk) ;
X i X j ϕ (x) = f 00 (kxk)X i (kxk) X j (kxk) + f 0 (kxk)X i X j (kxk) .
Since X i (kxk) is homogeneous of degree zero for i = 1, . . . , q, X i X j (kxk)
(for i, j = 1, . . . , q) and X 0 (kxk) are homogeneous of degree −1 and f 0 (kxk) 6= 0 for kxk >
σ r , we get the result.
Another tool we need in this context is an approximation result by suitable mollifiers. For a
fixed cutoff function ϕ, B1 (0) ≺ ϕ ≺ B2 (0), set, for every ε > 0,
1
−Q
x
ϕε (x) = c · ε
ϕ D
ε
with c =
R
R N ϕ(x) d x
−1
. Then
L EMMA 6. For u ∈ S k, p R N (k nonnegative integer, 1≤ p < ∞) and ϕε as above,
define u ε = ϕε ∗ u. Then u ε ∈ C ∞ and u ε → u in S k, p for ε → 0.
Proof. The proof follows the same line as in the Euclidean case. We just point out the following
facts:
(i) convergence in L p is established first for u ∈ 3β (G,R N ), u with bounded support.
The density of this space in L p can be proved in a general space of homogeneous type (see for
instance [4]);
(ii) since X i is right invariant, X i (u ε ) = X i u : from this remark and convergence in L p
ε
we get convergence in S k, p ;
(iii) to see that u ε ∈ C ∞ , one has to consider right invariant vector fields X iR , write
X iR (ϕε ∗ u) = X iR ϕε ∗u, and iterate. The possibility of representing any Euclidean derivative
in terms of right invariant vector fields (see §2.5) provides the conclusion.
The above Lemma is useful for us mainly in view of the following
C OROLLARY 1. If u ∈ S k, p () (1 ≤ p < ∞, k ≥ 1) and ϕ ∈ C0∞ (), then uϕ ∈
k, p
S0 ().
420
M. Bramanti - L. Brandolini
Proof. The function uϕ is compactly supported in  and, extended to zero outside , belongs to
S k, p R N . Then (uϕ)ε converges to uϕ in S k, p . Since, for ε small enough, (uϕ)ε is compactly
k, p
supported in , (uϕ)ε ∈ C0∞ () and uϕ ∈∈ S0
().
T HEOREM 21 (I NTERPOLATION INEQUALITY IN C ASE A). Assume we are in Case A. For
any u ∈ S H, p (Br ), p ∈ [1, ∞), H ≥ 2, r > 0, define the following quantities:
k k
k
8k = sup
for k = 0, 1, 2, . . . , H.
(1 − σ ) r D u p
L (Brσ )
1
2 <σ <1
Then for every integer j , 1 ≤ j ≤ H − 1, there exist positive constants c, δ0 depending on
G, j, H such that for every δ ∈ (0, δ0 ) we have
c
8 j ≤ δ 8 H + j/(H − j ) 80 .
δ
(61)
Proof. We proceed by induction on H . Let H = 2.
p
Let u ∈ S p (Br ) and ϕ a cutoff function as in Lemma 5. By Corollary 1, uϕ ∈ S0 (Br ),
hence by density we can apply Proposition 2, writing
n
o 2
kX i (uϕ)k p ≤ ε kϕ X i X i uk p + k2 X i u X i ϕk p + ku X i X i ϕk p + kϕuk p
ε
for any ε > 0. Hence
n
kDukL p (Bσ r ) ≤ ε D 2 u L p ( Bσ 0 r )
+
o
c(G)
kDukL p ( B 0 ) +
σ r
(1 − σ ) r
2
kukL p ( B 0 ) .
σ r
ε
Multiplying both sides for (1 − σ ) r and choosing ε = δσ (1 − σ ) r we find
+
(1 − σ ) r kDukL p (Bσ r ) ≤ δσ (1 − σ )2 r 2 D 2 u p
L ( Bσ 0 r )
1
kukL p ( B 0 ) ≤
+c(G)δσ (1 − σ ) r kDukL p ( B 0 ) + c(G) δ +
σ r
σ r
δσ
+
Therefore
c(G)
σ (1 − σ )2 r 2
kukL p ( B
σ 0r )
+
(noting that (1 − σ 0 ) = (1 − σ )/2)
2
≤ 4δ82 + C(G)δ81 + c(G) δ +
80 .
δ
c(G) δ + 2δ
4δ
81 ≤
8 +
80
1 − c(G)δ 2
1 − c(G)δ
which, for δ small enough, is equivalent to
(62)
81 ≤ δ82 +
c
8 .
δ 0
Assume now that (61) holds for H − 1. An argument similar to that used to obtain (62) applied
to D H −2 u yields
c
8 H −1 ≤ δ8 H + 8 H −2
δ
L p estimates for hypoelliptic operators
while, by induction,
421
c
8 H −2 ≤ η8 H −1 + H −2 80 .
η
Therefore
8 H −1 ≤ δ8 H +
δ we get
and, choosing η = 2c
c
δ
c
η8 H −1 + H −2 80
η
c
8 H −1 ≤ 2δ8 H + H −1 80 .
δ
(63)
If j = H − 1, this is exactly what we have to prove; if j < H − 1, by induction
c
8 j ≤ ε8 H −1 + j/(H −1− j ) 80 ≤ by (63)
ε
c
c
≤ ε 2δ8 H + H −1 80 + j/(H −1− j ) 80 .
δ
ε
Choosing 2δ = η1/(H − j ) and ε = η1−1/(H − j ) we get the result.
R EMARK 5. Note that the second part
of the
above proof does not hold in Case B since in
that case D k u cannot be obtained as D D k−1 u . However, the proof for H = 2 holds also in
Case B, since the field X 0 does not play any role in the definition of Du. We are going to prove
an analogous interpolation inequality, in Case B, which will hold for H even. This proof will be
achieved in several steps.
Let
L≡
q
X
i=1
X i2 + X 0 ,
and let 0 be the fundamental solution of L homogeneous of degree two; recall that the transpose
of L is just
q
X
LT ≡
X i2 − X 0 .
i=1
L EMMA 7. Let Q > 4. For every integer k ≥ 2 and any couple of left invariant differential
monomials P 2k−1 and P 2k−2 , homogeneous of degrees 2k − 1 and 2k − 2, respectively, we
can determine two kernels K (1) , K (2) (depending only on these monomials) which are smooth
outside the origin and homogeneous of degrees (1 − Q) , (2 − Q), respectively, such that for
any test function u
P 2k−1 u(x) = (L L . . . Lu) ∗ K (1) (x);
k times
(64)
P 2k−2 u(x) = (L L . . . Lu) ∗ K (2) (x).
k times
Proof. By induction on k. Let k = 2. By Lemma 1, we can write
u = Lu ∗ 0 = (L Lu ∗ 0) ∗ 0 = L Lu ∗ K ,
422
M. Bramanti - L. Brandolini
where K = 0 ∗ 0 is homogeneous of degree (4 − Q). Hence
P 3 u = L Lu ∗ P 3 K = L Lu ∗ K (1) ;
P 2 u = L Lu ∗ P 2 K = L Lu ∗ K (2) ,
with K (1) , K (2) homogeneous of degree (1 − Q), (2 − Q), respectively.
Now, assume (64) holds for k − 1, that is
!
P 2k−3 u(x) =
(L L . . . Lu) ∗ K (1) (x);
P 2k−4 u(x) =
(L L . . . Lu) ∗ K (2)
(k−1) times
(k−1) times
!
(x).
Any differential monomial P 2k−2 can be written either as X i P 2k−3 (for some i = 1, . . . , q) or
as X 0 P 2k−4 . In the first case, we can write
!
P 2k−2 u(x) = X i (L L . . . Lu) ∗ K (1) (x) =
(k−1) times
= X i (L L . . . Lu) ∗ 0 ∗ K (1) (x) =
k times
= (L L . . . Lu) ∗ X i 0 ∗ K (1) (x) =
k times
e(2) (x),
(L L . . . Lu) ∗ K
k times
e(2) homogeneous of degree ((2 − Q) + (1 − Q) + Q) − 1 = 2 − Q (we have applied
with K
Lemma 1). In the second case,
!
P 2k−2 u(x) = X 0 (L L . . . Lu) ∗ K (2) (x) =
(k−1) times
= (L L . . . Lu) ∗ X 0 0 ∗ K (2) (x) =
k times
(2)
(L L . . . Lu) ∗ e
K
(x),
k times
(2)
e homogeneous of degree (2 − Q).
with, again, e
K
Similarly, any differential monomial P 2k−1 can be written either as P 2 P 2k−3 (with P 2 =
X 0 or P 2 = X i X j for some i, j = 1, . . . , q) or as P 3 P 2k−4 (with P 3 = X i X 0 for some
i = 1, . . . , q). Reasoning as above we get the result for this case, too.
L EMMA 8. For every integer k ≥ 2 there exists a constant c(G, k) such that for every ε > 0
and every test function u,
c(G, k)
2k− j u ≤ ε D 2k u + 2k− j kuk p for j = 1, 2, k ≥ 2.
D
p
p
ε
L p estimates for hypoelliptic operators
423
Proof. Let K (1) , K (2) be as in Lemma 7. We split the kernel K (1) as
(1)
(1)
K (1) = ϕ K (1) + (1 − ϕ) K (1) ≡ K 0 + K ∞ ,
(1)
where ϕ is a cutoff function, B1 (0) ≺ ϕ ≺ B2 (0). Therefore K 0 is homogeneous of degree
(1)
(1 − Q) near the origin and has compact support, hence it is integrable, while K ∞ is homogee
neous of degree (1 − Q) near infinity and vanishes near the origin. Writing f (x) = f (x −1 ), we
can compute
Z
Z
(1)
(1)
e∞
x −1 ◦ y L L . . . Lu(y) d y =
K ∞ y −1 ◦ x L L . . . Lu(y) d y = K
=
Therefore
Z
(1) −1
e∞
LT K
(x ◦ y) (L L . . . L)u(y) d y = . . .
Z k−1 times
(1)
e∞
x −1 ◦ y u(y) d y =
LT LT . . . LT K
=
k times
∼

(1)
e∞  (x) ≡ u ∗ K (1) (x).
= u ∗  LT LT . . . LT K
1
k times
P 2k−1 u(x) =
(1)
(L L . . . L)u ∗
k times
(1)
h
(1)
(1)
K0 + K∞
(1)
i
(x) =
= (L L . . . Lu) ∗ K 0 + u ∗ K 1 ,
where K 1 is homogeneous of degree (1 − Q − 2k) near infinity and vanishes near the origin,
(1)
(1)
hence it is integrable. Integrability of K 0 , K 1 gives
2k−1 u ≤ c(G, k) D 2k u + kuk p .
D
p
p
The same reasoning applied to K (2) gives
2k−2 u ≤ c(G, k) D 2k u + kuk p .
D
p
p
The conclusions follows from the last two inequalities and a dilation argument.
T HEOREM 22. [Interpolation inequality in Case B] For any function
u ∈ S 2k, p (Br ) (k ≥ 1, p ∈ [1, ∞), r > 0), let 8h (h = 0, 1, . . . , 2k) be the seminorms defined
in Theorem 21. Then
8 j ≤ ε82k + c(ε, k)80
for every integer j with 1 ≤ j ≤ 2k − 1 and every ε > 0.
2k, p
Proof. Let ϕ be a cutoff function as in Lemma 5. By Corollary 1 uϕ ∈ S0 (Br ), hence, by
density, we can apply Lemma 8 to uϕ. By standard arguments (see the first part of the proof of
Theorem 21) we get, for every δ > 0,
82k− j ≤ δ
2k
X
h=0
c
8h + (2k− j )/j 80 ( j = 1, 2).
δ
424
M. Bramanti - L. Brandolini
The previous inequality clearly holds if k is replaced by any integer i with 2 ≤ i ≤ k. Adding
up these inequalities for 2 ≤ i ≤ k, j = 1, 2, we get


k
2k−1
k
2i
X
X
X
X

82i−1 + 82i−2 =
8i ≤ 2δ
8h  + c(δ, k) · 80 ≤
i=2
i=2
≤ 2δk
i=2
2k
X
h=0
h=0
8h + c(δ, k) · 80 .
Adding also (62) (which holds in Case B, too, as noted in Remark 5):
81 ≤ δ82 +
c
8
δ 0
we can write
2k−1
X
h=0
8h ≤ 2δk
and, finally, for every ε > 0,
2k−1
X
h=0
2k
X
h=0
8h + c(δ, k) · 80
8h ≤ ε · 82k + c(ε, k) · 80
which proves the result.
Next, we need the following Sobolev-type embedding theorem:
p
T HEOREM 23. Let u ∈ S0 (Br ) for some r > 0. Then:
1
2
1
a) if 1 < p < Q
2 and p ∗ = p − Q , then
kuk p ∗ ≤ c ( p, G) D 2 u ;
p
Q
b) if Q
2 < p < Q and β = 2 − p , then
kuk3β (Br ) ≤ c( p, G, r ) D 2 u L p (Br )
.
Proof. Let L , 0 be as in the proof of Lemma 7. Then for any u ∈ C0∞ (Br ) we can write
u = Lu ∗ 0.
Theorem 18 then gives the assertion.
The results contained in Theorem 23 have been proved by Folland in [11], where a more
complete theory of Sobolev and Hölder spaces defined by the vector fields X i is developed (see
§§4, 5 in [11], in particular Theorems 4.17 and 5.15). However, Folland’s theory relies on a deep
analysis of the sub-Laplacian on stratified groups, and therefore does not cover completely the
cases we are considering here: remember that under our assumptions, the group G is graded but
not necessarily stratified.
L p estimates for hypoelliptic operators
425
6. Local estimates for solutions to the equation Lu = f in a domain
In this section we will prove Theorems 3 to 7, as a consequence of the basic estimate contained
in Theorem 2 and the properties of Sobolev spaces expounded in §5. For convenience of the
reader, we recall the statement of each Theorem before its proof.
T HEOREM 3 (L OCAL L p - ESTIMATES FOR SOLUTIONS TO THE EQUATION Lu = f IN
A DOMAIN ). Under the assumptions stated in §2.3, let  be a bounded domain of R N and
0 ⊂⊂ . If u ∈ S p (), then
kuk S p (0 ) ≤ c kLukL p () + kukL p ()
where c = c( p, G, µ, η, , 0 ).
Proof of Theorem 3. Let u ∈ S p (), Br ⊂ , and fix r0 = r0 ( p, G, µ, η) in such a way
that for r < r0 Theorem 2 holds. Let ϕ be a cutoff function as in Lemma 5. By Corollary 1,
p
uϕ ∈ S0 (Br ); then, by density, we can apply Theorem 2 to uϕ:
X i X j (uϕ) ≤ c kL (uϕ)k p .
p
From the above inequality we get:
n
c(G)
X i X j u p
kLukL p ( B 0 ) +
kDukL p ( B 0 ) +
≤
c
L (Bσ r )
σ r
σ r
(1 − σ ) r
+
c(G)
σ (1 − σ )2 r 2
kukL p ( B
σ 0r )
o
.
Multiplying both sides for (1 − σ )2 r 2 , adding to both sides (1 − σ ) r kDukL p (Bσ r ) , reasoning
like in the first part of the proof of Theorem 21 and applying (62) we get
o
n
82 + 81 ≤ c r 2 kLukL p (Br ) + kukL p (Br ) .
Hence
that is
r 2 D 2 u Lp
Br/2
n
o
2
+ r kDukL p B ≤ c r kLukL p (Br ) + kukL p (Br ) ,
r/2
kuk S p (Br/2 ) ≤ c kLukL p (Br ) + kukL p (Br ) ,
with c = c( p, G, µ, η, r ), r < r0 ( p, G, µ, η). The last estimate and a covering argument give
the result.
T HEOREM 4 (L OCAL H ÖLDER 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 ∈ Ls () for some s > Q/2, then
n
o
kuk3α (0 ) ≤ c kLukLr () + kukL p ()
for r = max( p, s), α = α(Q, p, s) ∈ (0, 1), c = c(G, µ, p, s, , 0 ).
426
M. Bramanti - L. Brandolini
Proof of Theorem 4. Let u ∈ S p (B2R ) (for some p ∈ (1, ∞)) be a solution to Lu = f in a ball
B2R , with f ∈ Ls (B2R ) for some s > Q/2. We can always assume p, s < Q.
p
Let ϕ be a cutoff function, B R/2 ≺ ϕ ≺ B R . By Corollary 1, uϕ ∈ S0 (B R ). If p > Q/2,
then by (b) of Theorem 23, u ∈ 3α (B R/2 ). If p ≤ Q/2 (we can assume p < Q/2), then by (a)
∗
∗
of Theorem 23, uϕ ∈ L p (B R ), and u ∈ L p (B R/2 ).
Assume p∗ < s. Then by Theorem 3, kuk S p∗ B is bounded. Hence, we can fix ϕ1
R/4
∗∗
with B R/8 ≺ ϕ1 ≺ B R/4 and repeat the argument finding that u ∈ S p (B R/16 ), and so on.
After a number k of iterations depending only on p, Q, we find that u ∈ S p̄ B R/4k for some
p̄ > Q/2: more precisely, k must be the integer belonging to the interval 2Qp − 1, 2Qp . This
k exists provided Q/2 p is not an integer, what we can always assume, replacing p, if necessary,
with a slightly smaller p1 . Therefore, by (b) of Theorem 23, u ∈ 3α (B R/22k+1 ) for some
α ∈ (0, 1).
∗
If u ∈ L p (B R/2 ) with p∗ ≥ s, Theorem 3 implies that u ∈ S s B R/4 and since s > Q/2,
u ∈ 3α B R/8 for some α ∈ (0, 1).
In any case, the following estimate holds:
kuk3α B R/K ≤ c k f kLs (B R ) + kuk S p (B R ) ≤
by Theorem 3
≤ c k f kLs (B R ) + k f kL p (B2R ) + kukL p (B2R ) .
By a covering argument, we get the result for a bounded domain .
T HEOREM 5 (R EGULARITY OF THE SOLUTION IN TERMS OF S OBOLEV SPACES ). Under
the assumptions of Theorem 3, if a0 , ai j ∈ S k,∞ (), u ∈ S p () and Lu ∈ S k, p () for some
positive integer k (k even in Case B), 1 < p < ∞, then
n
o
kuk S k+2, p (0 ) ≤ c1 kLuk S k, p () + c2 kukL p ()
where c1 = c1 ( p, G, µ, η, , 0 ) and c2 depends on the S k,∞ () norms of the coefficients.
R EMARK 6. Similarly to the proof of the interpolation inequality contained in Theorems
21 and 22, also the proof of Theorem 5 is more difficult in Case B than in Case A. The reason of
this appears clear if one tries to adapt our proof of Case A also to Case B. In doing so, the main
difficulty is the presence of the field X 0 , which has weight two but cannot be seen as composition
of two fields of weight one. This is also why the definition of the space S k, p in Case B is not
optimal when k is odd, as already noted in [28]. Therefore our proof of Theorem 5 in Case B will
follow a different line and will require the restriction k even. This also (but not only) depends
on the restriction k even that appears in Theorem 22. We also note that in the commutative case
(that is L uniformly elliptic, in Case A, or uniformly parabolic, in Case B) this Theorem follows
immediately from Theorem 3 (just differentiating the equation Lu = f ).
Proof of Theorem 5 in Case A. The proof is divided into two parts: first we prove an estimate
for the derivatives of order k + 2 of a test function u supported in a small ball, in terms of Lu (as
in Theorem 2); then we derive from this result a local estimate for any function in S k+2, p ()
(as in the proof of Theorem 3).
L p estimates for hypoelliptic operators
427
Part I. The idea of this proof, in the case of the Kohn-laplacian on the Heisenberg group, is
contained in [31]. Recall that X iR (i = 1, . . . , N) are the right invariant vector fields which
agree with X i (and therefore with ∂/∂ xi ) at the origin. Here we will use the properties of X iR
stated in §2.5. Moreover, since X 1R ,. . .,X qR generate the Lie algebra of right invariant vector
fields, for every k > q we can write
X kR =
(65)
X
1≤i j ≤q
ϑi1 ...iω X iR1 X iR2 . . . X iRω
k
k
for suitable constants ϑi1 ...iω depending only on G.
k
Let us consider a test function u. We are going to establish a representation formula for X i u
in terms of D (Lu). For a fixed point x0 ∈ R N , let us write
Z
u(x) =
0 x0 ; y −1 ◦ x L0 u(y)d y.
RN
Then, for every i = 1, . . . , q:
Z
X i u(x) =
=
Z
N
X
R N k=i
=
+
Z
N
X
Z
=
+
R N k=i
R N k=q+1 1≤i ≤q
j
Z
N
X
Z
by (22)
X kR cik (·) 0 (x0 ;·) y −1 ◦ x L0 u(y) d y =
q
X
X
by (21)
RN
X i 0 x0 ; y −1 ◦ x L0 u(y)d y =
by (65)
X kR cik (·) 0 (x0 ;·) y −1 ◦ x L0 u(y) d y +
ϑik1 ...iω X iR1 X iR2 . . . X iRω cik (·) 0 (x0 ;·) y −1 ◦ x L0 u(y) d y =
k
k
q X
R N k=i
X
R N k=q+1 1≤i ≤q
j
=
cik (·) 0 (x0 ;·)
y −1 ◦ x X k L0 u(y) d y +
ϑik1 ...iω X iR2 . . . X iRω cik (·) 0 (x0 ;·) y −1 ◦ x X i1 L0 u(y) d y =
k
k
Z
q
X
R N k=1
e
0 k,i x0 ; y −1 ◦ x X k L0 u(y) d y,
where the kernels e
0 k,i (x0 ; ·) satisfy properties analogous to those of 0 (x0 ; ·):
k,i
e
• 0 (x0 ; ·) is homogeneous of degree (2 − Q), since cik (·) is homogeneous of degree
ωk − ωi and ωi = 1 for i ≤ q;
•e
0 k,i (x0 ; ·) is smooth outside the origin;
• the derivatives of any order of e
0 k,i (x0 ; ·) satisfy the uniform bound expressed by Theok
rem 12, because the functions ci (·) are smooth and do not depend on x0 .
428
M. Bramanti - L. Brandolini
The whole previous reasoning can be iterated, getting, for every positive integer k and any
left invariant differential monomial homogeneous of degree k,
P k u(x) =
(66)
Z
q
X
R N i ,i ,...,i =1
k
1 2
e
0 i1 ,i2 ,...,ik x0 ; y −1 ◦ x X i1 X i2 . . . X ik L0 u(y) d y,
where the kernels e
0 i1 ,i2 ,...,ik (x0 ; ·) satisfy the same properties of e
0 k,i (x0 ; ·).
Differentiating twice the representation formula (66), applying Theorem 9, writing L0 =
(L0 − L) + L and letting finally x = x0 we get
X j X h P k u(x) =


q
X
i1 ,i2 ,...,ik =1
P.V.

X X . . . X ik Lu(y) + X i1 X i2 . . . X ik 
 i1 i2
Z
X j Xh e
0 i1 ,i2 ,...,ik x ; y −1 ◦ x ·
q
X
r,s=1


[ars (x) − ars (y)] X r X s u (y) d y

+α i1 ,i2 ,...,ik (x) X i1 X i2 . . . X ik Lu(x)
for j, h ∈ {1, . . . , q}, where the functions α i1 ,i2 ,...,ik (x) are uniformly bounded, by Lemma 4.
Reasoning like in the proof of Theorem 2 and setting
h D a = max D h ars r,s
∞
∞
we conclude that


k 

X
k+2− j j k
k+2 u D a ≤
u ≤ c D (Lu) +
D
D

p
∞
p
p
j =1
≤ c D k (Lu) + kak S k,∞ kuk S k+1, p
(67)
p
k+2, p
with c = c( p, G, µ, η, k). By density, (67) holds for every u ∈ S0
(Br ).
Part II. Now, let u ∈ S k+2, p (), Br ⊂
 with
r small enough so that (67) holds
for
every
k+2, p
1
∞
N
function in S0
(Br ). For any σ ∈ 2 , 1 , pick a cutoff function ϕ ∈ C0 R
like in
Lemma 5, Bσ r ≺ ϕ ≺ Bσ 0r (σ 0 =(1 + σ )/2). Applying (67) to uϕ and using Lemma 5 we get,
with some computation:
(1 − σ )k+2 r k+2 D k+2 u L p (Bσ r )
≤


k+1


X
.
≤ c (1 − σ )2 r 2 kLuk S k, p ( B 0 ) + kak S k,∞ ( B 0 ) ·
(1 − σ )h r h D h u p
σ r
σ r

L ( Bσ 0 r ) 
h=0
Therefore:
k+2
X
h=0


k+1


X
8h
8h ≤ c r 2 kLuk S k, p (Br ) + kak S k,∞ (Br ) ·


h=0
L p estimates for hypoelliptic operators
429
with c = c( p, G, µ, η, k), r < r0 ( p, G, µ, η). By Theorem 21, we get
k+2
X
h=0
o
n
8h ≤ c1 r 2 kLuk S k, p (Br ) + c2 · 80
with c1 = c1 ( p, G, µ,η,k), c2 = c2 kak S k,∞ (Br ) , and finally
o
n
kuk S k+2, p (Br/2 ) ≤ c 1 r 2 kLuk S k, p (Br ) + c2 kukL p (Br )
which implies the desired result, by a covering argument.
Proof of Theorem 5 in Case B. Again, the proof is divided in two parts. First we prove inequality (67) for any even integer k, in Case B. Then the Theorem follows by the same argument of
the proof in Case A, Part II, applying the suitable interpolation inequality (Theorem 22).
In proving (67), we will consider in detail the case k = 2, and then we will briefly show
how to iterate the argument.
Let u be a test function, fix two points x0 , y0 ∈ R N and write Lx0 , L y0 to indicate the
operators “frozen” at x0 , y0 , respectively (see §§3.1, 3.2). Let us write:
u(x) = Lx0 u ∗ 0(x0 , ·) (x) = L y0 (Lx0 u) ∗ 0(y0 , ·) ∗ 0(x0 , ·) (x) =
= L y0 Lx0 u ∗ K (x0 , y0 ,·) (x),
where K (x0 , y0 ,z) = 0(y0 , ·) ∗ 0(x0 , ·) (z) and we have applied Lemma 1 with α1 = α2 =
2 − Q. Note that condition α1 + α2 < −Q in Lemma 1 gives, in our case, Q > 4. This condition
certainly holds since the homogeneous group corresponding to the simplest noncommutative Lie
algebra satisfying the assumptions of Case B is Q = 6 (see Example 2).
Now, let P 4 be any left invariant differential operator homogeneous of degree 4. By Lemma
1 and Theorem 9, P 4 K (x0 , y0 ,·) is a singular kernel satisfying conditions (d), (e), ( f ) of Theorem 11 and a uniform bound (23); reasoning like in §§3.1, 3.2 we can write
Z
P 4 u(x) = P.V.
P 4 K x0 , y0 , z −1 ◦ x L y0 (Lx0 u)(z)dz+
RN
+α(x0 , y0 ) · L y0 (Lx0 u)(x) =
= P.V.
Z
(writing Lx0 = (Lx0 − L) + L and setting finally x0 = x)
X
q
P 4 K x, y0 , z −1 ◦ x L y0
ahk (x) − ahk (z) X h X k u(z) +
RN
h,k=1
+ a0 (x) − a0 (z) X 0 u(z) + Lu(z) dz + α(x, y0 ) · L y0 (Lu)(x),
where α is a bounded function by Lemma 4.
radius r small enough,
4 P u ≤ c L y0 (Lu) p + D 2 a p
Now, if the support of u is contained in a ball with
1
2 D u + kDak∞ D 3 u + D 4 u .
p
p
p
∞
2
430
Therefore
M. Bramanti - L. Brandolini
4 D u ≤ c D 2 (Lu) + kak S 2,∞ kuk S 3, p
p
p
which is exactly (67), for k = 2. From now on Part II of the proof of Theorem 5 in Case A can
be repeated, to get
n
o
kuk S 4, p B ≤ c 1 r 2 kLuk S 2, p (Br ) + c2 kukL p (Br ) .
r/2
We now briefly discuss how to iterate the above proof for any even integer k. Assume, for
instance that we want to bound the L p -norm of P 6 u. If we tried to write
u(x) = L y0 L y0 Lx0 u ∗ 0(y0 , ·) ∗ 0(y0 , ·) ∗ 0(x0 , ·) (x)
the problem would be to assure the existence of the convolution of three fundamental solutions
(this would lead to the restriction Q > 6, by Lemma 1). Instead, we must perform part of the
differentiation P 6 before doing the third convolution. First of all, let us split P 6 as a composition
of the kind P 4 P 2 or P 3 P 3 (note that the possible presence of X 0 , which has weight two, implies
that one of these cases occurs). If P 6 = P 4 P 2 we proceed as follows:
u(x) = L y0 (Lx0 u) ∗ K (x0 , y0 ,·) (x),
with K as above;
P 2 u(x) = L y0 (Lx0 u) ∗ P 2 K (x0 , y0 ,·) (x),
where P 2 K is homogeneous of degree (2 − Q);
h
i
P 2 u(x) = L y0 L y0 Lx0 u ∗ 0(y0 , ·) ∗ P 2 K (x0 , y0 ,·) (x) =
e(x0 , y0 ,·) (x),
= L y0 L y0 Lx0 u ∗ K
e is homogeneous of degree (4 − Q); now applying P 4 to both sides of the last equality
where K
we can repeat the proof of the case k = 2. A similar reasoning applies to the case P 6 = P 3 P 3 .
This completes the proof of Theorem 5.
T HEOREM 6 (R EGULARITY OF THE SOLUTION IN TERMS OF H ÖLDER SPACES ). Under
the assumption of Theorem 3, if a0 , ai j ∈ S k,∞ (), u ∈ S p () and Lu ∈ S k,s () for some
positive integer k (k even in Case B), 1 < p < ∞, s > Q/2, then
n
o
kuk3k,α (0 ) ≤ c1 kLuk S k,r () + c2 kukL p ()
where r = max( p, s), α = α(Q, p, s) ∈ (0, 1), c1 = c1 ( p, s, k, G, µ, η, , 0 ) and c2 depends
on the S k,∞ () norms of the coefficients.
Proof of Theorem 6. Let us note that (b) of Theorem 23 implies the following embedding estimate:
k+2, p
Q
let u ∈ S0
(Br ) for some r > 0. Then if Q
2 < p < Q and β = 2 − p ,
kuk3k,β (G,Br ) ≤ c( p, k, G, µ, r ) kuk S k+2, p (Br ) .
(It’s enough to apply Theorem 23 to D k u). Then, Theorem 6 follows from Theorem 5 as Theorem 4 follows from Theorem 3, with (b) of Theorem 23 replaced by the above inequality. We
omit the details.
L p estimates for hypoelliptic operators
431
Finally, let us come to the proof of:
T HEOREM 7 (O PERATORS WITH LOWER ORDER TERMS ). Consider an operator with
lower order terms, of the following kind:

 

q
q
X
X
ci (x)X i + c0 (x) ≡ L2 + L1 .
ai j (x)X i X j + a0 (x)X 0  + 
L≡
i, j =1
i=1
i) If ci ∈ L∞ () for i = 0, 1, . . . , q, then:
if the assumptions of Theorem 3 hold for L2 , then the conclusions of Theorem 3 hold for
L;
if the assumptions of Theorem 4 hold for L2 , then the conclusions of Theorem 4 hold for
L.
ii) If ci ∈ S k,∞ () for some positive integer k, i = 0, 1, . . . , q, then:
if the assumptions of Theorem 5 hold for L2 , then the conclusions of Theorem 5 hold for
L;
if the assumptions of Theorem 6 hold for L2 , then the conclusions of Theorem 6 hold for
L.
p
Proof. Under the assumptions of Theorem 2, we can write, for every u ∈ S0 (Br ), with r small
enough,
n
o
X i X j u ≤ c kLuk p + kL1 uk p .
p
If ci ∈ L∞ () for i = 0, 1, . . . , q, then
kL1 uk p ≤ c kuk S 1, p ≤
Therefore
by Proposition 2
c
≤ ε D 2 u + kuk p .
p
ε
X i X j u ≤ c kLuk p + kuk p .
p
Using the last inequality instead of Theorem 2, we can repeat the proof of Theorem 3 for the
complete operator L. The same is true for Theorem 5, assuming ci ∈ S k,∞ (). Finally, Theorems 4, 6 follow from Theorems 3, 5, respectively, without taking into account the form of the
operator L. We omit the details.
References
[1] AGMON S., D OUGLIS A. AND N IRENBERG L., Estimates near the boundary for solutions
of elliptic partial differential eqautions satisfying general boundary conditions, Comm.
Pure Appl. Math., 12 (1959), 623–727 and 17 (1964), 35–92.
[2] B RAMANTI M. AND B RANDOLINI L., L p -estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. of the Amer. Math. Soc., 352 (2000), 781–822.
[3] B RAMANTI M. AND C ERUTTI C., W p1,2 -solvability for the Cauchy-Dirichlet problem for
parabolic equations with VMO coefficients, Comm. in Part. Diff. Eq., 18 9-10 (1993),
1735–1763.
432
M. Bramanti - L. Brandolini
[4] B RAMANTI M. AND C ERUTTI C., Commutators of singular integrals in homogeneous
spaces, Boll. Unione Mat. Italiana, 7 10B (1996), 843–883.
[5] B RAMANTI M., C ERUTTI C. AND M ANFREDINI M., L p -estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. and Appl., 200 (1996),
332–354.
[6] C HIARENZA F., F RASCA M. AND L ONGO P., Interior W 2, p -estimates for nondivergence
elliptic equations with discontinuous coefficients, Ricerche di Mat. XL (1991), 149–168.
[7] C HIARENZA F., F RASCA M. AND L ONGO P., W 2, p -solvability of the Dirichlet problem
for non divergence elliptic equations with VMO coefficients, Trans. Am. Math. Soc. 336 1
(1993), 841–853.
[8] C OIFMAN R., ROCHBERG R. AND W EISS G., Factorization theorems for Hardy spaces
in several variables, Annals Math. 103 (1976), 611–635.
[9] C OIFMAN R. AND W EISS G., Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin-HeidelbergNew York 1971.
[10] C ALDER ÓN A.P. AND Z YGMUND A., Singular integral operators and differential equations, Am. Journ. of Math. 79 (1957), 901–921.
[11] F OLLAND F.B., Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv
for Math. 13 (1975), 161–207.
[12] FABES E.B. AND R IVI ÈRE N., Singular integrals with mixed homogeneity, Studia Mathematica 27 (1966), 19–38.
[13] F OLLAND F.B. AND S TEIN E.M., Estimates for the ∂ b complex and analysis on the
Heisenberg group, Comm. Pure and Appl. Math. XXVII (1974), 429–522.
[14] G ILBARG D. AND T RUDINGER N.S., Elliptic Partial Differential Equations of Second
Order, Springer-Verlag, Berlin Heidelberg New York Tokio 1983 (2nd ed.).
[15] G ATTO A.E. AND V ÀGI S., Fractional Integrals on Spaces of Homogeneous Type. Analysis and Partial Differential Equations, ed. by Cora Sadosky, Lecture Notes in Pure and
Applied Math. 122 (1990), 171–216.
[16] H ÖRMANDER L., Hypoelliptic second order differential equations, Acta Mathematica, 119
(1967), 147–171.
[17] I LYIN V.A. AND P OZNYAK E.G., Linear Algebra, MIR Publisher, Moscow 1986.
[18] J OHN F. AND N IRENBERG N., On functions of bounded mean oscillation, Comm. Pure
Appl. Math., 14 (1961), 175–188.
[19] J ERISON D. AND S ANCHEZ -C ALLE A., Estimates for the heat kernel for a sum of squares
of vector fields, Indiana Univ. Math. Jour. 35 4 (1986), 835–854.
[20] K NAPP A.W. AND S TEIN E.M., Intertwining operators for semi-simple groups, Ann. of
Math. 93 (1971), 489–578.
[21] KOHN J.J., Pseudo-differential operators and hypoellipticity, Proc. Symp. Pure Math.
Amer. Math. Soc. 23 (1973), 61–69.
[22] L ANCONELLI E. AND L ASCIALFARI F., A boundary value problem for a class of quasilinear operators of Fokker-Planck type, Ann. Univ. Ferrara Sez. VII 40 (1995), 65–84.
[23] L ANCONELLI E. AND P OLIDORO S., On a class of hypoelliptic evolution operators, Rend.
Sem. Mat. Polit. Torino 51 4 (1993), 137–171.
L p estimates for hypoelliptic operators
433
[24] M ANFREDINI M., The Dirichlet problem for a class of ultraparabolic equations, Advances
in Differential Equations 2 5 (1997), 831–866.
[25] M ANFREDINI M. AND P OLIDORO S., Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients, Boll. Unione Mat.
Ital. Sez. B Artic. Ric. Mat. 8 1 (1998), no. 3, 651–675.
[26] A. NAGEL A., S TEIN E.M. AND WAINGER S., Balls and metrics defined by vector fields
I: Basic properties, Acta Mathematica 155 (1985), 130–147.
[27] M ANFREDINI M. AND P OLIDORO S., A global lower bound for the fundamental solution
of Kolmogorov-Fokker-Planck equations, Arch. Rational Mech. Anal. 137 4 (1997), 321–
340.
[28] ROTHSCHILD L.P. AND S TEIN E.M., Hypoelliptic differential operators and nilpotent
groups, Acta Math. 137 (1976), 247–320.
[29] S ANCHEZ -C ALLE A., Fundamental solutions and geometry of sum of squares of vector
fields Inv. Math. 78 (1984), 143–160.
[30] S ARASON D., Functions of vanishing mean oscillations Trans. of Am. Math. Soc. 207
(1974), 391–405.
[31] S TEIN E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory
Integrals, Princeton Univ. Press. Princeton, New Jersey 1993.
AMS Subject Classification: 35H05, 35R05, 43A85
Marco BRAMANTI
Dipartimento di Matematica
Politecnico di Milano
P.zza Leonardo da Vinci 32
20133 Milano, ITALIA
e-mail: bramanti@unical.it
Luca BRANDOLINI
Dipartimento di Ingegneria
Università degli Studi di Bergamo
viale Marconi 5
24044 Dalmine (BG), ITALIA
e-mail: brandolini@unibg.it
Lavoro pervenuto in redazione il 20.03.1999 e in forma definitiva il 01.02.2000.
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M. Bramanti - L. Brandolini