Nonlinear subelliptic equations · Algebra of the rotation group SO(n) and other non-compact...

manuscripta math. 130, 251–271 (2009) © Springer-Verlag 2009

András Domokos · Juan J. Manfredi

Nonlinear subelliptic equations

Received: 22 March 2009Published online: 16 June 2009

Abstract. In this paper, we study the higher order regularity for weak solutions of a class ofquasilinear subelliptic equations. We introduce the notion of ν-closed Hörmander system ofvector fields, which includes all the previously studied nilpotent systems and extends themto some classes of non-nilpotent systems of vector fields, including those generating the LieAlgebra of the rotation group SO(n) and other non-compact semisimple and solvable Liegroups.

1. Introduction

Consider a bounded domain� ⊂ Rn , N ≤ n, and a system of smooth vector fields

X = {X1, . . . , X N } defined on �. In this paper, we propose to study in non-nil-potent structures the higher order regularity for weak solutions of the followingquasilinear subelliptic equation:

N∑

i=1

X∗i (ai (x,Xu)) = 0, in �, (1.1)

where the functions ai (x, ξ) are differentiable in each variable and, for an ε ≥ 0and p > 1, satisfy the following properties:

N∑

i, j=1

∂ai

∂ξ j(x, ξ) ηiη j ≥ l

(ε + |ξ |2

) p−22 |η|2, (1.2)

N∑

i, j=1

∣∣∣∣∂ai

∂ξ j(x, ξ)

∣∣∣∣ ≤ L(ε + |ξ |2

) p−22, (1.3)

n∑

i, j=1

∣∣∣∣∂ai

∂x j(x, ξ)

∣∣∣∣ ≤ L(ε + |ξ |2

) p−12, (1.4)

|ai (x, ξ)| ≤ L(ε + |ξ |2

) p−12. (1.5)

A. Domokos (B): Department of Mathematics and Statistics,California State University Sacramento, Sacramento, CA 95819, USAe-mail: [email protected]; URL: http://webpages.csus.edu/~domokos

J. J. Manfredi: Department of Mathematics, University of Pittsburgh, Pittsburgh,PA 15260, USA. e-mail: [email protected]; URL: http://www.pitt.edu/~manfredi

Mathematics Subject Classification (2000): 35H20, 35B65

DOI: 10.1007/s00229-009-0286-z

252 A. Domokos, J. J. Manfredi

The best known representative of Eq. (1.1) is the subelliptic p-Laplacian that

corresponds to the case of ai (x, ξ) = (ε + |ξ |2) p−22 ξi .

In this paper, we introduce the notion of ν-closed systems of Hörmander vectorfields, which includes all the previously studied nilpotent systems of vector fieldsincluding the Lie Algebra of Carnot groups and the Grušin vector fields, and allowsfor the consideration of non-nilpotent systems; for example, the Lie Algebra of therotation group SO(n) or other non-compact semisimple or solvable Lie groups.

Our starting point is Taylor’s description [17] of a subelliptic operator in the spe-cial unitary group SU(2). The following example is based on Taylor’s approach. InR

3 consider linear independent vector fields X1, X2, T satisfying the commutationrelations:

[X1, X2] = T, [X2, T ] = X1, [T, X1] = X2. (1.6)

In this case the system X = {X1, X2} generates a three dimensional Lie Algebrawhich is isomorphic to so(3), the Lie Algebra of SO(3).

The left invariant vector fields constituting the basis of so(3) can be representedby the matrices

X1 =⎡

⎣0 0 00 0 −10 1 0

⎤

⎦ , X2 =⎡

⎣0 0 10 0 0

−1 0 0

⎤

⎦ , T =⎡

⎣0 −1 01 0 00 0 0

⎤

⎦ .

The exponential mapping so(3) → SO(3) can be presented as (see [8])

eZ = I + sin ρ

ρZ + 1 − cos ρ

ρ2 Z2,

where

ρ =√

−1

2trace(Z2).

By observing the identities X1T 2 + T 2 X1 = −X1, T X1T = 0, T 2 X1T = 0,T X1T 2 = 0, T 2 X1T 2 = 0, we get

X1

(u(esT (x)

)= Ade−sT X1 u

(esT x

)= e−sT X1esT u

(esT x

)

=(

I − sin sT + (1 − cos s)T 2)

X1

×(

I + sin sT + (1 − cos s)T 2)

u(

esT x)

= X1u(esT x)−sin s X2u(esT x)+(cos s−1)X1u(esT x). (1.7)

For a weak solution u of Eq. (1.1) we have a-priori information about integra-bility of derivatives in the direction of X1 and X2, so formulas similar to (1.7) allowus to use difference quotients in the direction of T as test functions in the weakform of Eq. (1.1).

Subelliptic regularity 253

Our main goal is to prove the following result:

Theorem 1.1. Assume that X is a ν-closed system of Hörmander vector fields, thefunctions ai (x, ξ) are C∞ in both variables and a weak solution u of the Eq. (1.1)satisfies 0 < M−1 ≤ |Xu| ≤ M a.e. in �. Then u ∈ C∞(�).

Theorem 1.1 generalizes similar results obtained in Heisenberg and Carnotgroups by Capogna [3,4]. On our way to prove Theorem 1.1 we will generalizeother regularity results which do not require the assumption 0 < M−1 ≤ |Xu| ≤M . As a short overview of subelliptic regularity results we mention Hörmander’sC∞-regularity result in the linear p = 2 case [10]. For p �= 2 the Hölder continu-ity of weak solutions was proved in [2,12]. These results are valid for any systemof Hörmander vector fields. However, the second order differentiability results inthe case ε > 0 proved in [5,7,14] and the Moser iterations leading to C1,α regu-larity developed in [13] are valid only in certain stratified nilpotent structures, asHeisenberg and Carnot groups and for p in certain intervals.

Let us return now to the system of vector fields X = {X1, . . . , X N }. TheX-gradient of a function u is given by Xu = (X1u, . . . , X N u). We denote the set ofall commutators of length up to a fixed ν ∈ N by B. To unify notations, elementsof X will be considered commutators of length 1, so we have X ⊂ B.

Definition 1.1. We say that X is a ν-closed Hörmander system of vector fields on� if:

(1) The set B of commutators of order at most ν spans the tangent space at everyx ∈ �.

(2) There exists a T1 ∈ B\X such that

[T1, Xi ] ∈ X ∪ {0}, for all 1 ≤ i ≤ N .

(3) If after selecting Tk = {T1, . . . , Tk} we still have B\{X ∪ Tk} �= ∅, then thereexists Tk+1 ∈ B\{X ∪ Tk} such that

[Tk+1,Y ] ∈ X ∪ Tk ∪ {0}, for all Y ∈ X ∪ Tk .

(4) By continuing the process started in (1)–(3) we cover all B\X.

Remark. The assumptions (2)–(4) of Definition 1.1 imply that for all Xi ∈ X andT ∈ B we have [Xi , T ] ∈ B ∪ {0}. This is the reason for choosing the name“ν-closed” for X.

The following examples show a variety of subelliptic structures within the scopeof Definition 1.1.

Examples. 1. The vector fields generating the Lie algebra of the Heisenberggroup: X = {X1, X2} and B = {X1, X2, T1} where

X1 = ∂

∂x− y

2

∂

∂t, X2 = ∂

∂y+ x

2

∂

∂t, T1 = ∂

∂t.


2. The Grušin vector fields of step k + 1:

X = {X1, X2} and B = {X1, X2, T1, . . . , Tk}where

X1 = ∂

∂x, X2 = ∂

∂x+ xk ∂

∂t, Ti = k(k − 1) . . . (k − i + 1)xk−i ∂

∂t.

3. The vector fields generating the Lie algebra of a Carnot group G, which has itsLie algebra g stratified as g = ⊕ν

i=1gi : X is a basis of g1 and B is the unionof bases of gi , 1 ≤ i ≤ ν. We start with a T ∈ gν and cover all vector fieldsfrom the basis of gν . Then we move to the vector fields from the basis of gν−1and continue until we reach g2.

4. All the nilpotent Lie algebras with dimension up to 6, as listed in [15], can beviewed as ν-closed Hörmander system of vector fields.

5. We can also include some nilpotent Lie algebras without stratification, as theone mentioned in [9] spanned by vector fields X1, . . . , X7 with commuta-tion relations: [X1, Xi ] = Xi+1, [X2, X3] = X6, [X2, X4] = [X5, X2] =[X3, X4] = X7. Using the notation of Definition 1.1 we have X = {X1, X2},T1 = X7, T2 = X5, T3 = X4, T4 = X6, T5 = X3.

6. The Lie algebra so(n) of the special orthogonal group SO(n) has a basis Bmade by vector fields X jk , 1 ≤ j < k ≤ n, which can be shown as elementsof an upper triangular matrix

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 X12 X13 . . . . . . . . . . . . . . . X1n

. . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 . . . X jm . . . X jk . . . . . .

0 0 0 0 . . . . . . . . . . . . . . .

0 0 0 0 0 . . . Xmk . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 0 0 0 0 0 Xn−1,n0 0 0 0 0 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and have the following nonzero commutation relations

[Xmk, X jk] = X jm, [X jk, X jm] = Xmk, [X jm, Xmk] = X jk

for all 1 ≤ j < m < k ≤ n. Consider X = {Xmn, 1 ≤ m ≤ n − 1}, which isthe set formed by the last column. Then, any other vector field X jk, 1 ≤ j <k ≤ n − 1 is a commutator of length two of elements of X and satisfy of thefollowing commutation relations:

[X jm, Xqn] =⎧⎨

⎩

0 if j �= q and q �= m−Xmn if q = j

Xmn if q = m

⎫⎬

⎭ . (1.8)

Hence, we can start the selection process of Definition 1.1 with any vector fieldnot belonging to X and continue it in an arbitrary order.

7. We can include in this list the 3-dimensional Lie algebras mentioned in [16].Among them we find a nilpotent (the Heisenberg Lie Algebra), a compactsemisimple (so(3)), a noncompact semisimple (sl(2,R)) and a solvable (theLie algebra of the motion group of the Lorentzian plane).


2. Vector fields and their flows

Consider a smooth vector field defined on �:

T =n∑

i=1

βi (x)∂

∂xi. (2.1)

The formal adjoint of T is defined as

T ∗ = −T −n∑

i=1

∂βi

∂xi,

and satisfies the equation∫

�

T u(x) φ(x) dx =∫

�

u(x) T ∗φ(x) dx,

for all u ∈ L1loc(�) with T u ∈ L1

loc(�) and φ ∈ C∞0 (�).

We denote by etT x the flow associated to the vector field T which means thatthe function fT (t,x) = etT x satisfies the initial value problem

⎧⎨

⎩

∂ fT

∂t(t,x) = T fT (t,x)

fT (0,x) = x.(2.2)

Let us remind that for each x0 ∈ � we can find an open neighborhood U ⊂ �

of x0 and an open interval I containing 0 such that fT is smooth on I × U and forall t ∈ I the map f (t, ·) is a diffeomorphism of U onto its image [11].

We denote by gT (t,x) = e−tT x the inverse of the flow etT which satisfieslocally the equations

fT (t, gT (t,x)) = gT (t, fT (t,x)) = x (2.3)

and⎧⎨

⎩

∂gT

∂t(t,x) = −T gT (t,x)

gT (0,x) = x.(2.4)

We follow now a point of view that can be derived from the methods presented in[1,10]. We think about a vector field X as a linear mapping X : C∞

0 (�) → C∞0 (�)

defined by

Xφ(x) = d

dt

∣∣∣∣t=0

φ(et Xx).

For φ ∈ C∞0 (�) and t > 0 sufficiently small let us define

et Xφ (x) = φ(et Xx).


The Taylor series expansion at t = 0 gives the following formal power seriesrepresentation:

et Xφ =∞∑

k=0

tk

k! Xkφ. (2.5)

The following equations need be understood in the sense that when we evaluateboth the left and right sides for a φ ∈ C∞

0 (�)we consider s and t sufficiently smallsuch that es Z ◦ et X ◦ e−s Zφ ∈ C∞

0 (�). Consider the operators

Ades Z (et X ) = es Z ◦ et X ◦ e−s Z ,

Ades Z (X) = d

dt

∣∣∣∣t=0

Ades Z (et X ) = es Z ◦ X ◦ e−s Z ,

and

adZ (X) = [Z , X ].In this way we get that

d

dsAdes Z (X) = d

ds(es Z ◦ X ◦ e−s Z )

= es Z ◦ Z ◦ X ◦ e−s Z − es Z ◦ X ◦ Z ◦ e−s Z = Ades Z ◦ adZ (X),

which shows the identity

Ades Z = es adZ (2.6)

Lemma 2.1. Consider an arbitrary φ ∈ C∞0 (�) and x ∈ �. Then for sufficiently

small s > 0 we have

X(

es Zφ(x))

=∞∑

k=0

(−1)ksk

k!((adZ)k(X) φ

) (es Zx

), (2.7)

X(

e−s Zφ(x))

=∞∑

k=0

sk

k!((adZ)k(X) φ

) (e−s Z x

). (2.8)

Proof. We prove (2.7) and then (2.8) follows in a similar way.

X(

es Zφ (x))

= d

dt

∣∣∣∣t=0

es Zφ(

et Xx)

= d

dt

∣∣∣∣t=0

φ(

es Z et Xx)

= d

dt

∣∣∣∣t=0

φ(

es Z et X e−s Z es Z x)

= d

dt

∣∣∣∣t=0

e−s Z ◦ et X ◦ es Zφ(

es Zx)


= d

dt

∣∣∣∣t=0

Ade−s Z (et X ) φ(

es Zx)

= Ade−s Z (X) φ(

es Z x)

= ead(−s Z)(X) φ(

es Zx)

=∞∑

k=0

(−1)ksk

k!(

adZ)k(X) φ) (

es Zx). (2.9)

�Lemma 2.2. Consider a ν-closed Hörmander system X of vector fields and anarbitrary φ ∈ C∞

0 (�) and x ∈ �.

(i) If T ∈ B\X such that [T, Xi ] ⊂ X ∪ {0} for all 1 ≤ i ≤ N, then there exists�i (s) = 〈ψi,1(s), . . . , ψi,N (s)〉, where ψi,k are analytic function in s, suchthat

Xi

(e±sTφ(x)

)= Xiφ

(e±sT x

)± s�i (±s) · Xφ

(e±sT x

). (2.10)

(ii) Suppose that we already selected a set of vector fields T = {T1, . . . , Tm} andcontinue the process by selecting Z ∈ B such that [Z ,Y ] ∈ X ∪ T ∪ {0} forall Y ∈ X ∪ T. Then, for every 1 ≤ i ≤ N there exist two vectors �i and i

of analytic functions in s, such that

Xi

(e±s Zφ(x)

)= Xiφ

(e±s Zx

)± s�i (±s) · Xφ

(e±s Zx

)

± s i (±s) · Tφ(

e±s Zx). (2.11)

Proof. We sketch the proof of (i) and leave the proof of (ii) to the reader. The vectorfields (ad T )k(Xi ) belong to X for all 1 ≤ k < ∞, so we can rearrange (2.7) and(2.8) in sums of finitely many convergent series. The analytic functions ψi, j (s)appear as coefficients of X j and they are 0 or have a series expansion

ψi, j (s) =∞∑

m=1

(−1)kmskm−1

km ! ,

where (km) is an increasing sequence of natural numbers. �Example. If we return to the vector fields X1, X2, T with commutation relationspresented in (1.6) we find that the relations involving higher order commutators ofT are

(ad T )2k+1(X1) = (−1)k X2, (ad T )2k(X1) = (−1)k X1,

and hence we have

X1

(esTφ(x)

)= X1φ

(esT x

)+ s

− sin s

sX2φ

(esT x

)

+ scos s − 1

sX1φ

(esT x

), (2.12)

and


X1

(e−sTφ(x)

)= X1φ

(e−sT x

)+ s

sin s

sX2φ

(e−sT x

)

+ scos s − 1

sX1φ

(e−sT x

). (2.13)

Therefore, we can consider �1(s) = [ cos s−1s , − sin s

s ].For s > 0 and 0 < γ ≤ 1 define the following difference quotients:

DZ ,s,γ u(x) = u(es Zx)− u(x)

sγ,

DZ ,−s,γ u(x) = u(x)− u(e−s Z x)

sγ.

Lemma 2.3. (i) Consider the vector field T from (i) of Lemma 2.2. If u ∈ L p(�)

has compact support and Xi u ∈ L p(�) for all 1 ≤ i ≤ N, then we have thefollowing identity in the weak sense:

Xi(DT,±s,γ u(x)

)= DT,±s,γ (Xi u(x))± s1−γ�i (±s) · Xu(e±sT x

). (2.14)

(ii) Consider the vector field Z from (ii) of Lemma 2.2. If in addition to (i) wesuppose that Tj u ∈ L p(�) for all 1 ≤ j ≤ m then we have the followingidentity in the weak sense:

Xi(DZ ,±s,γ u(x)

) = DZ ,±s,γ (Xi u(x))± s1−γ�i (±s) · Xu(

e±sT x)

± s1−γ i (±s) · Tu(

e±s Zx). (2.15)

Proof. The key step is to prove that (2.10) and (2.11) hold for weak derivativesof functions u with compact support in �. We approximate u in the appropriateSobolev space XW k,p(�)where k is the highest order of the horizontal derivativesinvolved in the commutators, by functions φn ∈ C∞

0 (�). Then (2.10) and (2.11)hold for φn and, to finish the proof, we just have to let n tend to ∞. �

3. Differentiability in the non-horizontal directions

In what follows X denotes always a ν-closed Hörmander system of vector fields.Also, in Sects. 3 and 4 we suppose ε > 0.

For p > 1 and k ∈ N define the following subelliptic Sobolev space:

XW k,p(�) = {u ∈ L p(�) : Xi1 . . . Xik u ∈ L p(�), for all 1 ≤ i j ≤ N

}.

Let XW k,p0 (�) be the closure of C∞

0 (�) in XW k,p(�) with respect to its usualnorm.

A function u ∈ XW 1,ploc (�) is a weak solution of the Eq. (1.1) if

N∑

i=1

∫

�

ai (x,Xu(x)) Xiϕ(x)dx = 0, for all ϕ ∈ C∞0 (�). (3.1)


By density arguments we can use test functions ϕ ∈ XW 1,p0 (�) with

suppϕ ⊂⊂ �.

Theorem 3.1. Consider a weak solution u ∈ X W 1,ploc (�) of (1.1), and let T ∈ B\X

such that [T, Xi ] ⊂ X ∪ {0} for all 1 ≤ i ≤ N. Suppose that T is a commutator oflength m of the horizontal vector fields. If 2 ≤ p < 2m

m−1 , x0 ∈ � and r > 0 suchthat B(x0, 3r) ⊂ �, then there exist numbers k ∈ N and c > 0 depending only onp, m, and dist(x0, ∂�), such that

∫

B(x0,r/k)

|T u|pdx ≤ c∫

B(x0,2r)

(ε + |Xu|2

) p2 + |u|pdx, (3.2)

and∫

B(x0,r/k)

(ε + |Xu(x)|2

) p−22 |T Xu(x)|2 dx

≤ c∫

B(x0,2r)

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx, (3.3)

which imply that T u ∈ L ploc(�) and (ε + |Xu|2) p−2

4 |T Xu| ∈ L2loc(�).

Proof. Denote γ = 1m and let η be a cut-off function between the C-C metric balls

B(x0,r2 ) and B(x0, r). For simplicity let’s just use the notation Br . Consider the

test function

ϕ = DT,−s,γ

(η2 DT,s,γ u

)

to get

N∑

i=1

∫

�

ai (x,Xu(x)) Xi

(DT,−s,γ

(η2 DT,s,γ u

))(x)dx = 0.

By Lemma 2.3 we obtain that

N∑

i=1

∫

�

ai (x,Xu(x)) DT,−s,γ Xi

(η2 DT,s,γ u

)(x) dx

=N∑

i=1

∫

�

ai (x,Xu(x)) s1−γ �i (−s) · X(η2 DT,s,γ u

) (e−sT x

)dx. (3.4)

On the left side we move the difference quotient to ai (x,Xu(x)). We denotethe Jacobian determinant of the transformation x �→ esT x by J+

T (s,x) and use thefacts that for small s > 0 it can be bounded in the following way:

0 < a ≤ ∣∣J+T (s,x)

∣∣ ≤ b.


Also, the formal adjoint of DT,s,γ is given by the formula

(DT,−s,γ

)∗u(x) = −DT,s,γ u(x)+ 1 − |J+

T (s,x)|sγ

u(esT x). (3.5)

By the fact that for a vector field T given in (2.1) we have

lims→0

J+T (s,x)− 1

s=

n∑

i=1

∂βi

∂xi(x),

it follows that for 0 < γ ≤ 1 the function1−|J+

T (s,x)|sγ is bounded for small s on

compact subsets of �.Similarly, we denote the Jacobian determinant of the transformation x �→

e−sT x by J−T (s,x) and have the formula for the formal adjoint

(DT,s,γ

)∗u(x) = −DT,−s,γ u(x)+ |J−

T (s,x)| − 1

sγu(e−sT x). (3.6)

Let’s return to (3.4) and continue as

N∑

i=1

∫

�

DT,s,γ ai (x,Xu(x)) Xi

(η2 DT,s,γ u

)(x) dx

=N∑

i=1

∫

�

1 − |J+T (s,x)|sγ

ai (esT x,Xu(esT x)) Xi

(η2 DT,s,γ u

)(x) dx

−N∑

i=1

∫

�

ai (x,Xu(x)) s1−γ�i (−s) · X(η2 DT,s,γ u

) (e−sT x

)dx.

The product rule on the left side gives

N∑

i=1

∫

�

DT,s,γ ai (x,Xu(x)) η2(x) Xi DT,s,γ u(x) dx

= −N∑

i=1

∫

�

DT,s,γ ai (x,Xu(x)) 2η(x) Xiη(x) DT,s,γ u(x) dx

+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ


(η2 DT,s,γ u

)(x) dx

−N∑

i=1

∫

�

ai (x,Xu(x)) s1−γ�i (−s) · X(η2 DT,s,γ u

) (e−sT x

)dx.


We continue the computations by switching Xi and DT,s,γ on the left side andby changing variables in the last line of the right side.

N∑

i=1

∫

�

DT,s,γ ai (x,Xu(x)) η2(x) DT,s,γ Xi u(x) dx

= −N∑

i=1

∫

�

DT,s,γ ai (x,Xu(x)) η2(x) s1−γ�(s) · Xu(esT x) dx

−N∑

i=1

∫

�


+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ


(η2 DT,s,γ u

)(x) dx

−N∑

i=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x)) s1−γ�i (−s) · X(η2 DT,s,γ u

)(x)dx.

The product rule in the last two lines of the right hand side gives

N∑

i=1

∫

�


= −N∑

i=1

∫

�


−N∑

i=1

∫

�


+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ

ai (esT x,Xu(esT x)) Xi (η

2)(x)DT,s,γ u(x) dx

+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ

ai (esT x,Xu(esT x)) η2(x) Xi DT,s,γ u(x) dx

−N∑

i=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x)) s1−γ�i (−s)

· X(η2)(x)DT,s,γ u(x) dx

−N∑

i=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x)) s1−γ η2�i (−s) · XDT,s,γ u(x) dx.


Finally, another switch between Xi and DT,s,γ in the fourth and sixth lines ofthe right side gives:

N∑

i=1

∫

�


= −N∑

i=1

∫

�


−N∑

i=1

∫

�


+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ

ai (esT x,Xu(esT x)) Xi (η

2)(x)DT,s,γ u(x) dx

+N∑

i=1

∫

�

1 − |J+T (s,x)|sγ

ai (esT x,Xu(esT x)) η2(x)DT,s,γ Xi u(x) dx

+N∑

i=1

∫

�

1−|J+T (s,x)|sγ

ai (esT x,Xu(esT x)) η2(x)s1−γ�i (s) · Xu(esT x) dx

−N∑

i=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x))s1−γ�i (−s) ·X(η2)(x)DT,s,γ u(x)dx

−N∑

i=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x)) s1−γ η2�i (−s) · DT,s,γXu(x) dx

−N∑

i, j,k=1

∫

�

|J−T (s,x)| ai (e

sT x,Xu(esT x)) s2−2γ η2ψi j (−s)ψ jk(s)Xku(x)dx.

For the following estimates we use the properties (1.2)–(1.5), the relation

DT,s,γ ai (x,Xu(x)) = ai (esT x,Xu(esT x)− ai (esT x,Xu(x))

sγ

+ ai (esT x,Xu(x)− ai (x,Xu(x))

sγ,

and the fact that | esT x−xsγ | is bounded on compact subsets of � for small s when

γ ≤ 1.


Left Side ≥ l∫

Br

η2(x)(ε + |Xu(x)|2 + |Xu(esT x)|2

) p−22 |DT,s,γXu|2 dx

− c∫

Br

η2(x)(ε + |Xu(x)|2 + |Xu(esT x)|2

) p2

dx.

Right Side line 1

≤ δ

∫

Br

η2(x)(ε + |Xu(x)|2 + |Xu(esT x)|2


+ c∫

Br

η2(x)(ε + |Xu(x)|2 + |Xu(esT x)|2

) p2

dx.

Right Side line 2

≤ δ

∫

Br

η2(ε + |Xu(x)|2 + |Xu(esT x)|2


+ c∫

Br

((ε + |Xu(x)|2 + |Xu(esT x)|2

) p2 + |DT,s,γ u|p

)dx.

Right Side lines 3 and 6

≤ c∫

Br

((ε + |Xu(x)|2 + |Xu(esT x)|2

) p2 + |DT,s,γ u|p

)dx.


≤ δ

∫

Br

η2(ε + |Xu(x)|2 + |Xu(esT x)|2


+ c∫

Br

(ε + |Xu(x)|2 + |Xu(esT x)|2

) p2

dx.


≤ c∫

Br

(ε + |Xu(x)|2 + |Xu(esT x)|2

) p2

dx.

Using the fact that by Hörmander’s result [10, Theorem 4.3] for γ = 1/m wehave

∫

Br

|DT,s,γ u(x)|p dx ≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx,

we find that


∫

Br

η2(x)(ε + |Xu(x)|2 + |Xu(esT x)|2

) p−22 ∣∣DT,s,γXu(x)

∣∣2 dx

≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx. (3.7)

Therefore, we obtain∫

Br

∣∣∣∣DT,s, 2γpX(η2u)(x)

∣∣∣∣p

dx ≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx.

The switch between DT,s, 2γp

and X brings terms which can be embedded into

the right side, so we get that∫

Br

∣∣∣∣XDT,s, 2γp(η2u)(x)

∣∣∣∣p

dx ≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx. (3.8)

Using again Hörmander’s result [10, Theorem 4.3] we find that there existsσ > 0 such that

sup0<|s|≤σ

∫

Br

∣∣∣∣DT,−s, 1m

DT,s, 2γp(η2u)(x)

∣∣∣∣p

dx

≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx.

If

1

m+ 2γ

p< 1,

then by [6] it follows that

sup0 �=|s|≤σ

∫

B(x0,r)

∣∣∣∣DT,s, 1m + 2γ

p(η2u)

∣∣∣∣p

dx

≤ c∫

B2r

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx. (3.9)

Hence, if we begin the proof with γ0 = 1m we obtain γ1 = 1

m + 2mp . We can

restart the proof with γ1 and by an iteration process we get the sequence (γk) definedby

γk = 1

m+ 2

mp+ · · · + 2k−1

mpk−1 = 1

m

1 −(

2p

)k

1 − 2p

.

We can push γk over 1 if 2 ≤ p < 2mm−1 and this finishes the proof. �


For the next theorem let us suppose that, based on Definition 1.1, the vectorfields T = {T1, . . . , Tm} have been already chosen and the maximum length of thecommutators used is mT.

Theorem 3.2. Consider a weak solution u ∈ X W 1,ploc (�) of (1.1) and suppose that

the conclusions of Theorem 3.1 hold for the vector fields T1, . . . , Tm. By followingDefinition 1.1, let Z ∈ B\{X ∪ T} be the next element in the selection processand suppose that Z is a commutator of length m Z of the horizontal vector fields.If 2 ≤ p < min{ 2m Z

m Z −1 ,2mT

mT−1 }, x0 ∈ � and r > 0 such that B(x0, 3r) ⊂ �, thenthere exist numbers l ∈ N and c > 0 depending only on p, dist(x0, ∂�), mT andm Z such that

∫

B(x0,r/ l)

|Zu|p dx ≤ c∫

B(x0,2r)

((ε + |Xu|2

) p2 + |u|p

)dx, (3.10)

and∫

B(x0,r/ l)

(ε + |Xu(x)|2

) p−22 |ZXu(x)|2 dx

≤ c∫

B(x0,2r)

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx, (3.11)

which imply that Zu ∈ L ploc(�) and

(ε + |Xu|2) p−24 |ZXu| ∈ L2

loc(�).

Proof. Consider an s > 0 sufficiently small and a well-chosen cut-off function η.We use the test function

ϕ = DZ ,−s,γ

(η2 DZ ,s,γ u

), (3.12)

where γ = min{ 1m Z, 1

mT}, and get

N∑

i=1

∫

�

ai (x,Xu(x)) Xi

(DZ ,−s,γ

(η2 DZ ,s,γ u

))(x)dx = 0.

By Lemma 2.3 it follows that

N∑

i=1

∫

�

ai (x,Xu(x)) DZ ,−s,γ Xi

(η2 DZ ,s,γ u

)(x) dx

=N∑

i=1

∫

�

ai (x,Xu(x)) s1−γ�i (−s) · X(η2 DZ ,s,γ u

) (e−s Z x

)dx

+N∑

i=1

∫

�

ai (x,Xu(x)) s1−γ i (−s) · T(η2 DZ ,s,γ u

) (e−s Zx

)dx.


Integration by parts on the left side and change of variables on the right sidelead to

N∑

i=1

∫

�

DZ ,s,γ ai (x,Xu(x)) Xi

(η2 DZ ,s,γ u

)(x) dx

=N∑

i=1

∫

�

1 − |J+Z (s,x)|sγ

ai (es Zx,Xu(es Z x)) Xi

(η2 DZ ,s,γ u

)(x) dx

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Z x,Xu(es Zx)) s1−γ �i (−s) · X(η2 DZ ,s,γ u

)(x)dx

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Z x,Xu(es Zx)) s1−γ i (−s) · T(η2 DZ ,s,γ u

)(x)dx.

Let us expand these lines by using the product rule and by switching Xi andthe difference quotients.

N∑

i=1

∫

�

DZ ,s,γ ai (x,Xu(x)) η2(x)DZ ,s,γ Xi u(x) dx

= −N∑

i=1

∫

�

DZ ,s,γ ai (x,Xu(x)) η2 s1−γ [�i (s) · Xu(es Zx)+ i · Tu(es Zx)]

dx

−N∑

i=1

∫

�

DZ ,s,γ ai (x,Xu(x)) Xi (η2(x))DZ ,s,γ u(x) dx

+N∑

i=1

∫

�

1 − |J+Z (s,x)|sγ

ai (es Zx,Xu(es Z x)) Xi (η

2(x))DZ ,s,γ u(x) dx

+N∑

i=1

∫

�

1 − |J+Z (s,x)|sγ

ai (es Zx,Xu(es Z x)) η2(x)DZ ,s,γ Xi u(x) dx

+N∑

i=1

∫

�

1 − |J+Z (s,x)|sγ

ai (es Z ,Xu(es Z x))η2(x)s1−γ

×[�i (s) · Xu(es Zx)+ i · Tu(es Zx)

]dx

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Zx,Xu(es Z x)) s1−γ�i (−s) ·X(η2(x))DZ ,s,γ u(x)dx

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Zx,Xu(es Z x)) s1−γ η2�i (−s) · DZ ,s,γXu(x) dx


−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Zx,Xu(es Z x)) s2−2γ η2(x)

×⎛

⎝∑

j,k

ψi, j (−s)ψ j,k(s)Xk(es Zx)+

∑

j,l

ψi, j (−s)φ j,l(s)Tl(es Z x)

⎞

⎠ dx

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Z x,Xu(es Z x)) s1−γ i (−s)·T(η2 DZ ,s,γ u

)(x) dx.

Except the last line of the right side, each line can be estimated in the same wayas in the proof of Theorem 3.1. So let us focus our attention on the last line. For thislet us suppose that T = [T1, . . . , TM ] and use the formal adjoint T ∗

j = −Tj + b j .Therefore,

−N∑

i=1

∫

�

|J+Z (s,x)| ai (e

s Z x,Xu(es Z x)) s1−γ i (−s) · T(η2 DZ ,s,γ u

)(x) dx

= −N ,M∑

i, j=1

∫

�

|J+Z (s,x)| ai (e

s Z x,Xu(es Z x)) s1−γ φi, j (−s)Tj

(η2 DZ ,s,γ u

)(x) dx

= s1−γ φi, j (−s)N ,M∑

i, j=1

∫

�

Tj

((|J+

Z (s,x)|)

ai (es Z x,Xu(es Z x)) η2(x)DZ ,s,γ u(x) dx

+ s1−γ φi, j (−s)N ,M∑

i, j=1

∫

�

|J+Z (s,x)| Tj

(ai (e

s Z x,Xu(es Z x)))η2(x)DZ ,s,γ u(x) dx

− s1−γ φi, j (−s)N ,M∑

i, j=1

∫

�

b j (x)|J+Z (s,x)| ai (e

s Z x,Xu(es Z x)) η2(x)DZ ,s,γ u(x) dx.

Our assumptions on the vector fields T1, . . . , Tm combined with the facts thatTj (|J+

Z (s,x)|) is bounded for small s and

Tj

(ai (e

s Zx,Xu(es Z x)))

=n∑

k=1

∂ai

∂xk(es Z x,Xu(es Z x))Tj (e

s Zx)k

+N∑

k=1

∂ai

∂ξk(es Zx,Xu(es Z x))Tj (Xku(es Zx)),

imply that the following estimate holds:∫

B(x0,r)

η2(x)(ε + |Xu(x)|2 + |Xu(es Zx)|2

) p−22 ∣∣DZ ,s,γXu(x)

∣∣2 dx

≤ c∫

B(x0,2r)

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx. (3.13)


We can finish the proof now by iteration on the order of the fractional differencequotients in the same manner as for Theorem 3.1. �

4. Second order horizontal differentiability

The intersection of all the intervals for p in Theorems 3.1 and 3.2 is [2, 2νν−1 ). Before

we attempt to use test functions involving different quotients of the horizontal vec-tor fields Xi , we have to control the derivatives in all the non-horizontal directions.Therefore, we must ask for

2 ≤ p <2ν

ν − 1.

With the observation that the proof of Theorem 3.2 can be now repeated forγ = 1 and Z = Xi , 1 ≤ i ≤ N , we conclude that:

Theorem 4.1. Consider a weak solution u ∈ X W 1,ploc (�) of (1.1). If 2 ≤ p < 2ν

ν−1 ,x0 ∈ � and r > 0 such that B(x0, 3r) ⊂ �, then there exist numbers l ∈ N andc > 0 depending only on dist(x0, ∂�), ν and p such that

∫

B(x0,r/ l)

(ε + |Xu(x)|2

) p−22∣∣∣X2u(x)

∣∣∣2

dx

≤ c∫

B(x0,2r)

((ε + |Xu(x)|2

) p2 + |u(x)|p

)dx. (4.1)

5. The case 0 < M−1 ≤ |Xu| ≤ M

In this section we suppose that for a positive constant M and for a.e. x ∈ � wehave

0 < M−1 ≤ |Xu(x)| ≤ M. (5.1)

This condition allows us to include also the degenerate case ε = 0 in our results.By the following theorem we generalize the results from [4], and moreover, give analternate proof for second the order horizontal differentiability of weak solutionsto equation (1.1).

Theorem 5.1. Let 2 ≤ p < ∞ and u ∈ X W 1,ploc (�) be a weak solution of (1.1)

satisfying (5.1). If x0 ∈ � and r > 0 such that B(x0, 3r) ⊂ �, then there existl ∈ N and c > 0 depending on dist(x0, ∂�), M and ν, but independent of ε, suchthat

∫

B(x0,r/ l)

∣∣∣X2u(x)∣∣∣2

dx ≤ c∫

B(x0,2r)

(ε + |Xu(x)|2 + |u(x)|2

)dx. (5.2)


Proof. We combine the proofs of the Theorems 3.1, 3.2 and 4.1. Let’s start withthe proof of Theorem 3.1 and follow it until (3.7), which implies now the estimate

∫

B(x0,r)

∣∣∣DT,s,γX(η2u)(x)∣∣∣2

dx ≤ c∫

B(x0,2r)

(ε+|Xu(x)|2+|u(x)|2

)dx. (5.3)

Continuing the proof of Theorem 3.1 with p replaced by 2 we get control of thefractional derivatives with order γk = k

m , which can be pushed over 1 after k + 1iterations. These estimates are independent of p and ε. In the same way, we canfollow the proof of Theorem 3.2 until (3.13) and then the estimates become againindependent of p and ε. Hence, we get that for all commutators Z of order at mostν we have

∫

B(x0,r/ l)

|Zu|2 dx ≤ c∫

B(x0,2r)

(ε + |Xu|2 + |u|2

)dx, (5.4)

and∫

B(x0,r/ l)

|ZXu(x)|2 dx ≤ c∫

B(x0,2r)

(ε + |Xu(x)|2 + |u(x)|2

)dx, (5.5)

which implies that Zu ∈ L2loc(�) and |ZXu| ∈ L2

loc(�).We can use now differencequotients in the directions of the horizontal vector fields and conclude the proof.

�For the last part of this paper, we use the following function spaces:

– The space of Hölder continuous functions with respect to the Carnot–Caratheodory distance

�α(�) ={

u : � → R | supx�=y

|u(x)− u(y)|d(x, y)α

< ∞}.

– The Morrey space M p,λ(�) which contains the functions u ∈ L ploc(�) such

that

−∫

B(x,r)∩�|u(y)|p dy ≤ cr p(λ−1)

for all x ∈ � and 0 < r < min{R, diam(�)}.The important fact given by these function spaces is that Xu ∈ M p,λ(�) impliesthat u ∈ �λloc(�).

We are ready to state the following theorem:

Theorem 5.2. Let u be a weak solution of (1.1) satisfying (5.1). Then u has itsEuclidean gradient

∇u ∈ XW 1,2loc (�)

⋂�αloc(�). (5.6)


Proof. We follow the ideas from section 4 of [4]. First, by selecting a test functionϕ(x) = η2(x)(u(x)− u(x0)) we get that Xu ∈ M2,λ(�).

As a second step, an essential new detail appears which has to be taken intoconsideration. Namely, that our Lie algebra might not have a center, so we have tobe careful with the first vector field which we use for differentiating equation (1.1).It is logical that we have to select the vector fields in the order given by Defini-tion 1.1. By differentiating equation (1.1) with respect to T1 we get that w = T1uis a weak solution of the equation

N∑

i, j=1

X∗i

(∂ai

∂ξ j(x,Xu)X jw + βi

)= β, (5.7)

where

βi =n∑

k=1

∂ai

∂xk(x,Xu)Tixk +

N∑

j=1

∂ai

∂ξ j(x,Xu)[T1, X j ]u (5.8)

and

β = [T1, Xi ]ai (x,Xu)+ (T1gi )ai (x,Xu) (5.9)

We used the notation X∗i =−Xi−gi . By Definition 1.1 we have that [T1, Xi ]=0

or [T1, Xi ] ⊂ X, in which case the commutator term from β can be embedded intothe left side. During this embedding the product of a C∞ function with ai appears,but that is controllable. Therefore, the conditions for Theorem 6.32 in [4] are sat-isfied and we can conclude that T1u ∈ �λloc(�) and XT1u ∈ M2,λ(�).

We can now differentiate (1.1) with respect to T2 and using the fact that thecommutators of T2 with the horizontal vector fields are 0 or T1 or an element of X,we can find that T2u ∈ �λloc(�) and XT2u ∈ M2,λ(�).

We continue in this way until we cover all the vector fields from B and by thisfinish the proof. �

Once we have the C1,α interior regularity of weak solutions, we can start usingthe method elaborated in [18] to iteratively obtain higher and higher differentiabilityproperties of u and ultimately finish the proof of Theorem 1.1.

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