ON THE REGULARITY OF SUBELLIPTIC p-HARMONIC FUNCTIONS ...

ON THE REGULARITY OF SUBELLIPTIC p-HARMONIC FUNCTIONS IN CARNOT GROUPS ´ DOMOKOS ANDRAS Abstract. In this paper we prove second order horizontal differentiability and C 1,α regularity results for subelliptic p-harmonic functions in Carnot groups for p close to 2.

1. Introduction This paper contains generalizations to the case of Carnot groups of the regularity results obtained by Domokos [4] and Domokos-Manfredi [5, 6] in Heisenberg groups. The starting point for this subelliptic regularity theory is the result of H¨ormander [10, 18] on the hypoellipticity of second order linear differential operators, which implies the C ∞ interior regularity of subelliptic harmonic functions in the case of H¨ormander type (or bracket-generating) vector fields. The nonlinear versions of these regularity results for weak solutions of subelliptic quasilinear equations started appearing in the early 90’s. First the H¨older continuity has been obtained by several mathematicians (see for example [3, 14]). Partial results regarding higher order regularities for subelliptic p-harmonic functions have been obtained in the case of the Heisenberg group (see [2, 4, 5, 6] and the references therein). A complete answer has not yet been found. In this paper we work in a Carnot group of step ν. First we prove the second order horizontal differentiability of the nondegenerate subelliptic p-harmonic functions for 2ν 2 ≤ p < ν−1 . Then we use the proximity of p to 2 to obtain second order horizontal differentiability and C 1,α regularity of the subelliptic p-harmonic functions. Let G be a Carnot group of step ν ≥ 2; that is, Lνa simply connected Lie group with Lie algebra g admitting a decomposition g = i=1 Vj such that [V1 , Vj ] = V1+j if 1 ≤ jP≤ ν − 1 and [V1 , Vν ] = 0. The homogeneous dimension of G is defined as ν Q = i=1 i di where di = dim Vi . © Forª simplicity we denote d = d1 and let us fix an orthonormal basis Xj , 1 ≤ j ≤ d of V1 . We call X1 , ..., Xd horizontal vector fields, because they generate the horizontal distribution for the related subriemannian geometry [16]. © ª For 2 ≤ i ≤ ν we can choose a basis Xij , 1 ≤ j ≤ di of Vi consisting of commutators of length i of the horizontal vector fields. By the fact that V1 generates g as an algebra, we deduce that the system of vector fields {X1 , ..., Xd } satisfies the H¨ormander’s condition and therefore we can associate to it a CarnotCarath´eodory metric on G [17]. We denote by B(x, r) the balls with respect to this metric. The Haar measure of B(x, r) is a constant multiple of rQ . Date: July 11, 2007. 1991 Mathematics Subject Classification. Primary 35H20, 35J70. Key words and phrases. Carnot group, p-Laplacian, weak solutions, regularity. 1

´ DOMOKOS ANDRAS

2

³ ´ We denote by Xu = X1 u, ..., Xd u the horizontal gradient and by X 2 u = (Xi Xj u)1≤i,j≤d the matrix of second order horizontal derivatives of a corresponding function u defined on G. Vν is the center of the Lie algebra, therefore the vector fields Xνj which commutes with any Xij - will have a special role. As an example of a Carnot group of step 2 we can refer to the Heisenberg group Hn , which is the nilpotent Lie group R2n+1 endowed with the group multiplication (x1 , ..., x2n , t) · (y1 , ..., y2n , τ ) = n ´ ³ 1X (xn+i yi − xi yn+i ) . = x1 + y1 , ..., x2n + y2n , t + τ − 2 i=1 The horizontal vector fields are Xi

=

Xn+i

=

∂ xn+i ∂ , − ∂xi 2 ∂t xi ∂ ∂ + . ∂xn+i 2 ∂t

Observe that

∂ , ∂t 2n and otherwise [Xi , Xj ] = 0. So, in this case g = R ⊕ R and Q = 2n + 2. [Xi , Xn+i ] = T =

Let 1 < p < +∞, k ∈ N and Ω ⊂ G be a domain. Consider the following Sobolev space with respect to the horizontal vector fields : n o HW k,p (Ω) = u ∈ Lp (Ω) : Xi1 ...Xik u ∈ Lp (Ω) , for all {i1 , ..., ik } ⊂ {1, ..., d} . Let HW0k,p (Ω) denote the closure of C0∞ (Ω) in HW k,p (Ω). In this context the subelliptic p-Laplacian equation is defined as d X

Xi (ai (Xu)) = 0 in Ω ,

(1.1)

i=1

where ai (ξ) = |ξ|p−2 ξi , for all ξ ∈ Rd . 1,p A function u ∈ HWloc (Ω) is called subelliptic p-harmonic if it is a weak solution of equation (1.1), which means that d Z X ai (Xu(x)) Xi ϕ(x) dx = 0 (1.2) i=1

Ω

HW01,p (Ω)

for all ϕ ∈ with supp ϕ ⊂⊂ Ω. For λ > 0 let us consider the nondegenerate subelliptic p-Laplacian equation d X

¡ ¢ Xi aλi (Xu) = 0 in Ω ,

i=1

where

¢ p−2 ¡ aλi (ξ) = λ + |ξ|2 2 ξi , for all ξ ∈ Rd ,

(1.3)

REGULARITY IN CARNOT GROUPS

3

and call its weak solutions nondegenerate subelliptic p-harmonic functions. Let us summarize the results, obtained in the case of the Heisenberg group, which constitutes the basis of this paper. Theorem 1.1. [4, 5, 6] In the Heisenberg group Hn a nondegenerate subelliptic p-harmonic function uλ and a subelliptic p-harmonic function u have the following regularity properties: (a) For 1 < p < 4 we have T uλ ∈ Lploc (Ω). 2,2 (b) For 2√≤ p < 4 we have uλ ∈ HWloc (Ω). 2,p 17−1 ≤ p ≤ 2 we have u ∈ HW (c) For λ loc (Ω). 2 (d) For √ n + n 4n2 + 4n − 3 2≤p 0 sufficiently small such that x · e±sZ ∈ Ω. For 0 < α, θ < 1 define the following differences and difference quotients: 4Z,s u(x) = u(x · esZ ) − u(x) , 42Z,s u(x) = u(x · esZ ) + u(x · e−sZ ) − 2u(x) , DZ,s,θ u(x)

=

DZ,−s,θ u(x)

=

u(x · esZ ) − u(x) , sθ u(x · e−sZ ) − u(x) . −sθ

Note that

42Z,s u(x) . sα+θ We recall the following lemmas about the connection between the differentiability of a function u and the control of the Lp norm of its difference quotients. DZ,−s,α DZ,s,θ u(x) = DZ,s,θ DZ,−s,α u(x) =

Lemma 2.1. [2, 10] Let K be a compact set included in Ω, Z be a left invariant vector field and u ∈ Lploc (Ω). If there exist σ and C, two positive constants, such that sup kDZ,s,1 ukLp (K) ≤ C

0 0 we have sup kDZ,s,1 ukLp (K) ≤ 2||Zu||Lp (K) .

0 0 and r > 0 such that B(x0 , 3r) ⊂ Ω . Then there exists a positive constant c independent of u such that Z Z ¯ ¯p ¯ ¯ sup (|u|p + |Xu|p ) dx . (2.1) ¯DZ,s, 1i u(x)¯ dx ≤ c 0 0, independent of u, and a possibly different σ from that one in (2.2) such that sup ||DZ,s,β u||Lp ≤ c(||u||Lp + M ) . (2.3) β=

0 0 such that B(x0 , 3r) ⊂ Ω, and for  ν p=2  · ¸ if ν(p−2) ln(1− p ) (2.4) k= 2ν if 2 < p < ν−1  ln 2 p

there exists c > 0 depending on p, ν and r, but not on λ, such that Z Z ³¡ ´ ¢p p |T uλ (x)| dx ≤ c λ + |Xuλ (x)|2 2 + |uλ (x)|p dx . B(x0 ,

r 22k

)

(2.5)

B(x0 ,2r)

Proof. First step: Let us consider

1 , ν and let η be a cut-off function between B(x0 , 2r ) and B(x0 , r) such that the usual n conditions ||Xi1 ...Xin η||L∞ ≤ c 2rn hold for all 1 ≤ n ≤ ν + 1. For s > 0 sufficiently small we define the test function ¡ ¢ ϕ = DT,−s,γ1 η 2 DT,s,γ1 uλ ∈ HW01,p (Ω) , (2.6) γ1 =

and use it in the weak form of equation (1.3): d Z ³ X ¡ ¢´ aλi (Xuλ (x)) Xi DT,−s,γ1 η 2 DT,s,γ1 uλ (x) dx = 0 . i=1

Ω

REGULARITY IN CARNOT GROUPS

5

By the fact that Xi commutes with DT,s,γ1 and DT,−s,γ1 , we obtain d Z X i=1

Ω

¡ ¢ DT,s,γ1 aλi (Xuλ (x)) η 2 (x) DT,s,γ1 (Xi uλ (x)) dx +

d Z X Ω

i=1

¡ ¢ DT,s,γ1 aλi (Xuλ (x)) DT,s,γ1 uλ (x) 2η(x) Xi η(x) dx = 0 .

(2.7)

Using the properties of the functions aλi and Lemma 2.2 we get Z ³ ´ p−2 2 2 η 2 (x) λ + |Xuλ (x)|2 + |Xuλ (x · esT )|2 |DT,s,γ1 Xuλ (x)| dx B(x0 ,r) Z ¡ ¢p ≤c λ + |Xuλ (x)|2 2 + |uλ (x)|p dx . (2.8) B(x0 ,2r)

In a similar way to the proof of Lemma 3.1 from [4] we obtain Z ¯ ¯p ¯ ¯ ¯DT,s, 2γ1 X(η 2 uλ )(x)¯ dx p B(x0 ,r) Z ¡ ¢p ≤c λ + |Xuλ (x)|2 2 + |uλ (x)|p dx .

(2.9)

B(x0 ,2r)

Let us denote the right hand side of (2.9) by M p and use again Lemma 2.2 to get that Z ¯ ¯p ¯ ¯ (2.10) ¯DT,−s, ν1 DT,s, 2γ1 (η 2 uλ )(x)¯ dx ≤ M p . p

B(x0 ,r)

By switching between s and −s in the test function (2.6) and repeating the above estimates, we obtain that there exists σ > 0 such that ° 2 ° °4 (η 2 uλ )° T,s Lp (G) sup ≤M. (2.11) 1 2 + 0 0 we have ¯ ´ ³ ¡ ³ ´ ¢´ ¡ d ¯¯ sZ Y u x·e = Y u ◦ ResZ (x) = u ◦ ResZ x · etY ¯ dt t=0 ¯ ¯ ³ ³ ´ ´ d ¯¯ d ¯¯ tY sZ sZ −sZ tY sZ = u x · e u x · e · e = · e · e · e dt ¯t=0 dt ¯t=0 ¯ ³ ³ ´ ¡ tY ¢´ d ¯¯ sZ sZ −sZ (Y )u x · e −sZ e = Ad u x · e · Ad = e e dt ¯t=0 ³ ´ ν−1 X ¢ ¡ sk = ead(−sZ) (Y )u x · esZ = (−1)k (adZ)k (Y )u x · esZ k! k=0

(3.1) In the same way we can prove that ³ ¡ X sk ¡ ¢ ¢´ ν−1 (adZ)k (Y )u x · e−sZ . Y u x · e−sZ = k!

(3.2)

k=0

We would like to prove these formulas for weak derivatives, too. For clarity of computations consider a Carnot group of step 3 and Y, Z ∈ W1 . Then V = [Z, Y ] ∈ V2 , T = [Z, [Z, Y ]] ∈ V3 . The adjoints of these vector fields are: Y ∗ = −Y , Z ∗ = −Z, V ∗ = −V and T ∗ = −T . Also, by the fact that a Carnot group is unimodular, for any Z ∈ g the Jacobian of the transformation z = x · esZ is equal to 1. By (3.2), for any φ ∈ C0∞ (Ω) holds ³ ´ ³ ´ V φ (x · e−sZ ) = V φ(x · e−sZ ) − sT φ(x · e−sZ ) , (3.3) and hence

³ ´ s2 Y φ (x · e−sZ ) = Y φ(x · e−sZ ) − sV φ (x · e−sZ ) − T φ (x · e−sZ ) 2 ³ ´ ³ ´ s2 ³ ´ = Y φ(x · e−sZ ) − sV φ(x · e−sZ ) + T φ(x · e−sZ ) . (3.4) 2 Therefore, Z ³ ´ Y u(x · esZ ) φ(x) dx Ω Z Z =− u(x · esZ ) Y φ(x) dx = − u(x) Y φ(x · e−sZ ) dx Ω Ω Ã ! Z ³ ´ ³ ´ s2 ³ ´ =− u(x) Y φ(x · e−sZ ) − sV φ(x · e−sZ ) + T φ(x · e−sZ ) dx 2 Ω Z ³ s2 ´ = Y − sV + T u(x) φ(x · e−sZ ) dx 2 Ω ! Z Ã s2 sZ sZ sZ = Y u (x · e ) − sV u (x · e ) + T u (x · e ) φ(x) dx . 2 Ω

´ DOMOKOS ANDRAS

8

Therefore, we have the following result: Lemma 3.1. In a Carnot group of step ν consider a function u with compact support included in Ω. If for Y, Z ∈ g and for all 0 ≤ k ≤ ν − 1 the weak derivatives (adZ)k (Y )u ∈ Lp (Ω), then for any sufficiently small s > 0 we have the following identities in the weak sense: ³ ¡ X ¢´ ν−1 ¢ ¡ sk Y u x · esZ = (−1)k (adZ)k (Y )u x · esZ , k!

(3.5)

³ ¡ X sk ¢´ ν−1 ¡ ¢ −sZ Y u x·e = (adZ)k (Y )u x · e−sZ , k!

(3.6)

k=0

k=0

and ³

´ ³ ´ DZ,−s,1 DZ,s,1 u(x) = Y DZ,s,1 DZ,−s,1 u(x) à ! ³ ´ ³ 1 sZ −sZ = DZ,−s,1 DZ,s,1 Y u(x) − [Z, Y ]u x · e − [Z, Y ]u x · e s à ! ν−1 ³ ´ ³ ´ X sk−2 k k sZ k −sZ + (−1) (adZ) (Y )u x · e + (adZ) (Y )u x · e . k!

Y

k=2

(3.7) 1,p 2ν and uλ ∈ HWloc (Ω) be a weak solution of (1.3). Theorem 3.1. Let 2 ≤ p < ν−1 Consider an arbitrary x0 ∈ Ω, r > 0 such that B(x0 , 3r) ∈ Ω. Then there exist c > 0 and l ∈ N, independent of λ, such that

Z B(x0 ,

r 2l

¯2 ¡ ¢ p−2 ¯ λ + |Xuλ (x)|2 2 ¯X 2 uλ (x)¯ dx )

Z ≤c

³¡

λ + |Xuλ (x)|2

¢ p2

´ + |uλ (x)|p dx .

(3.8)

B(x0 ,2r)

Proof. By Theorem 2.1 and Remark 2.1 we can control the derivatives in the direction of Vν , the center of Lie algebra. We continue by controlling the derivatives in the direction of Vν−1 and then going downwards we gain control over the derivatives in the direction of Vν−2 , Vν−3 , ... until we reach V1 . Let Z ∈ Vν−1 . Consider s > 0 sufficiently small and a cut-off function η between r ) where k is given by (2.4). In the weak form of the B(x0 , 2r2k ) and B(x0 , 22k+1 equation (1.3) we use the test function ³ ´ ϕ = DZ,−s,γ1 η 2 DZ,s,γ1 uλ ,

(3.9)

REGULARITY IN CARNOT GROUPS

where γ1 = d Z X i=1

Ω

1 ν−1 .

9

By Lemma 3.1 we obtain that

¡ ¢ DZ,s,γ1 aλi (Xuλ (x)) η 2 (x) DZ,s,γ1 (Xi uλ (x)) dx =s −

1−γ1

i=1

d Z X i=1

Ω

+ s1−γ1

d Z X Ω

³ ´ Ti aλi (Xuλ (x)) η 2 (x) DZ,s,γ1 u(x) dx

¡ ¢ DZ,s,γ1 aλi (Xuλ (x)) DZ,s,γ1 uλ (x) 2η(x) Xi η(x) dx

d Z X Ω

i=1

¡ ¢ DZ,s,γ1 aλi (Xuλ (x)) η 2 (x) Ti uλ (x · esZ ), dx (3.10)

where Ti = [Z, Xi ]. By the properties of aλi , Theorem 2.1 and Remark 2.1 each of the three integrals on the right hand side can be controlled, so using again the notation Z ³¡ ´ ¢p p M =c λ + |Xuλ (x)|2 2 + |uλ (x)|p dx , B(x0 ,2r)

for s > 0 sufficiently small we get that Z ¡ ¢ p−2 2 λ + |Xuλ (x)|2 + |Xuλ (x · esZ )|2 2 |DZ,s,γ1 Xuλ (x)| η 2 (x) dx ≤ M p . Ω

Hence,

(3.11) Z ¯ ¯p ¯ ¯ ¯DZ,s, 2γ1 X(η 2 uλ )(x)¯ ≤ M p . Ω

p

Using Lemma 2.2, the formula ¡ ¢ ¡ ¢ Xi DZ,s,α (η 2 uλ ) (x) = DZ,s,α Xi (η 2 uλ )(x) − s1−α [Z, Xi ](η 2 uλ ) x · esZ , and the fact that we can switch between s and −s in the test function and repeat the above computations, we get that there exists σ > 0 such that ° 2 ¡ 2 ¢° °∆ ° Z,s η uλ Lp (G) sup ≤M. (3.12) 1 2 + 0 0 such that B(x0 , 2r) ⊂ Ω. Then for all δ > 0 there exists c(δ) > 0 such that

||Xu||sLs (B(x0 ,r)) ≤ δ||X 2 u||sLs (B(x0 ,r)) + c(δ)||u||sLs (B(x0 ,r)) . Theorem 4.2 makes possible to prove Theorem 4.3 in a similar way to [5, 6].

REGULARITY IN CARNOT GROUPS

11

Theorem 4.3. Let us suppose that the operator A satisfies the assumptions of Theorem 4.1 and that B(x0 , 3r) ⊂ Ω. Then ³ ´ ||X 2 u||Ls (B(x0 ,r)) ≤ c ||Au||Ls (B(x0 ,2r)) + ||u||Ls (B(x0 ,2r)) 2,s for all u ∈ HWloc (Ω).

Let us consider a weak solution uλ ∈ HW 1,p (Ω) of equation (1.3). The differentiated version of equation (1.3) has the form d X

aλij Xi Xj uλ = 0 in Ω ,

(4.3)

i,j=1

where aλij (x) = δij + (p − 2)

Xi uλ (x) Xj uλ (x) . λ + |Xuλ (x)|2

Then aλij ∈ L∞ (Ω) and we can define the mapping Lλ : W02,s (Ω) → Ls (Ω) by Lλ (v)(x) =

d X

aλij (x)Xi Xj v(x) .

(4.4)

i,j=1

We remark that |Lλ v(x) − ∆X v(x)| ≤ |p − 2| |X 2 v(x)| for a.e. x ∈ Ω and for all v ∈ HW02,s (Ω). For a γ > 0 arbitrary small, but fixed, define n o c˜ = max CG,s , s ∈ [2, Q + γ] . The convexity theorem of M. Riesz and G.O. Thorin implies that c˜ < +∞. Theorem 4.4. In the case of

n 2ν 1o 2 ≤ p < min , 2+ ν−1 c˜

1,α any subelliptic p-harmonic function u has the interior regularity u ∈ Cloc (Ω).

Proof. For λ > 0 consider the nondegenerate subelliptic p-harmonic function uλ defined on a ball of radius 3r with boundary condition u − uλ ∈ HW01,p (B3r ). In this way we can suppose that locally the HW 1,p norm of uλ is controlled uniformly by a constant times the HW 1,p norm of u. 1,2 So, we can use Theorem 4.3 to conclude that that Xuλ ∈ HWloc (Br ) with uniform 1,2 q1 bounds. We use the embedding HW0 (Br ) ,→ L (Br ) where q1 =

2Q . Q−2

For η ∈ C0∞ (Br ) we have ¡ ¢ ||Lλ η 2 uλ ||Lq1 (Br ) 2n ° ³ ´° X ° ° = c°uλ Lλ (η 2 ) + aλij (x) Xj (η 2 )Xi uλ + Xi (η 2 )Xj uλ ° i,j=1

Lq1 (Br )

´ ³ ≤ c ||uλ ||Lq1 (suppη) + ||Xuλ ||Lq1 (suppη) .

(4.5)

´ DOMOKOS ANDRAS

12

2,q1 Therefore, by Theorems 4.1 and 4.3 we have that uλ ∈ HWloc (Br ) with locally 2,qk uniform bounds. Repeating this procedure k times we get that uλ ∈ HWloc (Br ) for 2Q qk = . Q − 2k We stop after m steps for the smallest m for which get Q − 2m < 4. Let us choose now β > 1 enough close to 1 such that 2, Qβ

u ∈ HWloc 2 (Br ) and Q

β ≤Q+γ. 2−β

Then we use the embedding 1, Qβ

Q

β

HWloc 2 (Br ) ,→ Lloc2−β (Br ) and inequalities similar to (4.5) to conclude that β 2,Q 2−β

uλ ∈ HWloc

(Br ) .

Moreover, the embedding β 1,Q 2−β

HWloc

2β−2

(Br ) ,→ Clocβ (Br )

1,α shows that uλ has interior regularity Cloc (Br ) where

α=

2β − 2 . β

Because of, locally, the estimates for uλ are uniform in λ, we can conclude that 1,α u ∈ Cloc (Br ). ¤ We turn now our attention to the second order horizontal differentiability of the subelliptic p-harmonic functions. Observe that the first part of the proof of 2,2 Theorem 4.4 actually gives uλ ∈ HWloc (Br ) with uniform bounds, so for p given 2,2 by Theorem 4.4 the subelliptic p-harmonic functions are in HWloc (Ω). In the following we propose another method, that can lead to similar results, with a more precise range of p. Definition 4.1. We say that A = (aij )1≤i,j≤d satisfies the Cordes condition Kε,σ if there exists ε ∈ (0, 1] and σ > 0 such that à d !2 d X X 1 1 2 a (x) ≤ 0< ≤ aii (x) , a.e. x ∈ Ω . (4.6) σ i,j=1 ij d − 1 + ε i=1 The proof of the following theorem doesn’t depend on the structure of the underlying space, so we just quote [5]. √ Theorem 4.5. Let 0 < ε ≤ 1, σ > 0 such that γ = 1 − ε CG,2 < 1 and A satisfies the Cordes condition Kε,σ . Then for all u ∈ HW02,2 (Ω) we have ||X 2 u||L2 ≤ CG,2

1 ||α||L∞ ||Au||L2 , 1−γ

(4.7)

REGULARITY IN CARNOT GROUPS

where

13

Pd α(x) =

i=1 aii (x) . ||A(x)||2

The operators Lλ satisfies the assumptions of Theorem 4.5 with ε independent on λ if r ³ r ³   ´ ´ 2 2 −1 +1 1 + d (d − 1) C − 1 + 1 d (d − 1) C 1 − G,2 G,2    . (4.8) , p−2 ∈  2 2   (d − 1)CG,2 − 1 (d − 1)CG,2 − 1 Also, in the case of Lλ we have d+p−2 , d so, in a similar way to the proof of Theorem 4.4, we can use Theorem 4.5 to prove that uλ has locally uniform HW 2,2 bound and in this way get the following result: α(x) ≤

Theorem 4.6. Let

r ³   ´  2 − 1 + 1   d (d − 1) C 1 +   2ν G,2 2 ≤ p < min , 2+ . 2   ν−1 (d − 1)CG,2 − 1    

2,2 Then any subelliptic p-harmonic function is in HWloc (Ω).

Remark 4.2. In the case√1 < p < 2 and a Carnot group of step 2 identical methods to [4, 5, 6] shows that for 17−1 ≤ p < 2 the nondegenerate p-harmonic functions have 2 second order horizontal derivatives and for p close to 2 the subelliptic p-harmonic 2,2 1,α functions we have HWloc (Ω) and Cloc (Ω) regularities. References [1] S. Campanato, On the condition of nearness between operators, Ann. Mat. Pura Appl., CLXVII(1994), 243 - 256. [2] L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50(1997), 867- 889. [3] L. Capogna, D. Danielli, N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794. [4] A. Domokos, Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group, J. Differential Equations 204 (2004), 439-470. [5] A. Domokos and J. J. Manfredi, Subelliptic Cordes estimates, Proc. Amer. Math. Soc. 133 (2005), 1047-1056. [6] A. Domokos and J.J. Manfredi, C 1,α -regularity for p-harmonic functions in the Heisenberg group for p near 2, Contemp. Math. 370(2005), 17-23. [7] A. Domokos, Weighted function spaces of fractional derivatives for vector fields, Electron. J. Diff. Eqns., 17(2007), pp. 1-8. [8] J. J. Duistermaat and J. A. Kolk, Lie groups, Springer-Verlag Berlin Heidelberg, 2000. [9] G. B. Folland, Applications of analysis on nilpotent groups to partial differential equations, Bull. Amer. Math. Soc. 83(1977), 912-930. ormander, Hypoelliptic second order differential equations, Acta Math. 119(1967), 147[10] L. H¨ 171. [11] E. Giusti, Direct methods in the Calculus of Variations, World Scientific, 2003. [12] A. W. Knapp and E. M. Stein, Interwining operators for semi-simple groups, Ann. of Math., 93(1971), 489-578.

14

´ DOMOKOS ANDRAS

[13] A. Kor´ anyi and S. V´ agi, Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Sc. Norm. Sup. Pisa 25(1971), 575-648. [14] G. Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat. 40(1996), 310-329. [15] G. Lu, Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups, Acta Math. Sin. (Engl. Ser.), 16(2000), 405-444. [16] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, AMS Mathematical Surveys and Monographs, Volume 91, 2002. [17] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math 155(1985), 103-147. [18] L. P. Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. Department of Mathematics and Statistics, California State University Sacramento, Sacramento, 95819, USA E-mail address: [email protected]