A new proof of the $C^\infty$ regularity of $C^2$ conformal mappings on the Heisenberg group

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by  Alex D. Austin

A NEW PROOF OF THE C∞ REGULARITY OF C2 CONFORMAL
MAPPINGS ON THE HEISENBERG GROUP
7 ALEX D. AUSTINANDJEREMY T. TYSON
1
0
2
Dedicated to Bogdan Bojarski
n
a
J
Abstract. WegiveanewprooffortheC∞ regularityofC2 smoothconformalmappings
1
ofthesub-RiemannianHeisenberggroup. Ourproofavoidsanyuseofnonlinearpotential
1
theoryandreliesonlyonhypoellipticityofH¨ormanderoperatorsandquasiconformalflows.

This approach is inspired by prior work of Sarvas and Liu.
]
P
A
1. Introduction
.
h Inthis paperwegive anew proofof the C∞ regularity of C2 smooth conformal mappings
t
a
of the Heisenberg group.
m
Recall that Liouville’s rigidity theorem states that conformal mappings of Euclidean
[
domains in dimension at least three are the restrictions of M¨obius transformations. In
1 particular, they are C∞ smooth.
v
Liouville’s theoremhasalongandstoriedhistorywhichisclosely tiedtothedevelopment
2
of geometric mapping theory and analysis in metric spaces throughout the latter half of the
8
1 twentieth century. The first proof, for C3 diffeomorphisms, is due to Liouville in 1850.
3 Gehring’s proof [6] of the Liouville theorem for 1-quasiconformal mappings was a major
0
turning point and inaugurated a line of research aimed at identifying optimal Sobolev
.
1
regularity criteria. An extension of the Liouville theorem to 1-quasiregular mappings was
0
first obtained by Reshetnyak; see for instance his books [13] and [14]. Since that time the
7
1 topichasbeenextensively investigated bymanypeople, includingBojarski, Iwaniec, Martin
v: and others. The book by Iwaniec and Martin [8] gives an excellent overview.
i Our present work is motivated by a recent proof of Liouville’s theorem due to Liu [11].
X
In contrast with previous proofs, which relied on nonlinear PDE and the regularity theory
r
a for p-harmonic functions, Liu’s proof uses purely linear techniques, specifically, an analysis
of Ahlfors’ conformal strain operator and quasiconformal flows. An earlier paper by Sarvas
[15] used similar methods to derive Liouville’s theorem in the C2 category.
Modern developments in the theory of analysis in metric spaces motivate the study
of quasiconformal and conformal mappings beyond Riemannian environments. The sub-
Riemannian Heisenberg group Hn was historically the firstsuch space in which quasiconfor-
mal mapping theory was considered, and remains an important testing ground for ongoing
research. Mostow [12] used quasiconformal mappings of the Heisenberg group in the proof
ofhiseponymousrigiditytheoremforrankonesymmetricspaces. Kora´nyiandReimann[9],
[10]undertookacomprehensivedevelopmentofHeisenbergquasiconformalmappingtheory.
Date: January 13, 2017.
2010 Mathematics Subject Classification. Primary 30L10; Secondary 30C65, 53C17, 35J70
Key words and phrases. Heisenberg group, conformal mapping, Kor´anyi–Reimann flow.
JTT was supported by NSF Grant DMS-1600650 ‘Mappings and measures in sub-Riemannian and metric
spaces’.
1
2 ALEXD.AUSTINANDJEREMYT.TYSON
In particular, in [9] the authors prove a Liouville theorem for C4 conformal mappings of
the first Heisenberg group H = H1 via the boundary behavior of biholomorphic mappings
and the Cauchy–Riemann equations in C2.
The first proof of Liouville’s theorem for 1-quasiconformal maps of the Heisenberg group
was by Capogna [1]. Capogna’s proof, similar to those of Gehring and others in the Eu-
clidean setting, relied on nonlinear potential theory, specifically, regularity estimates for
Q-harmonic functions (here Q is the homogeneous dimension of the Heisenberg group).
More recent developments include the work of Capogna–Cowling [2] (smoothness of 1-
quasiconformal maps in all Carnot groups), Cowling–Ottazzi [4] (classification of confor-
mal maps in all Carnot groups), and Capogna–Le Donne–Ottazzi [3] (smoothness of 1-
quasiconformal maps of certain sub-Riemannian manifolds).
In this paper we return to the setting of the Heisenberg group Hn. Our aim is to give a
new proof of the following theorem.
Theorem 1.1. Every C2 smooth conformal mapping between domains of the Heisenberg
group Hn is C∞ smooth.
Our proof differs from previous proofs in the literature by making no use of nonlinear
potential theory, nonlinear PDE, or the boundary behavior of biholomorphic mappings.
Theonly tools whichweusearehypoellipticity of H¨ormanderoperators andquasiconformal
flows. Our method is inspired by, but differs in important respects from, the work of Liu
and Sarvas.
For the benefit of the reader we provide a brief sketch of the proof of Theorem 1.1 in
the setting of the lowest dimensional Heisenberg group H. Let F : U H U′ H be a
⊂ → ⊂
C2 conformal mapping between domains in the Heisenberg group, write F = (f ,f ,f ) in
1 2 3
coordinates, and let X, Y and T denote the canonical left invariant vector fields spanning
the Lie algebra of H. Let Z = 1(X iY) and Z = 1(X + iY) be the complexified left
2 − 2
invariant horizontal vector fields derived from X and Y. We differentiate the conformal
flow
s F−1(exp(sT) F)
7→ ∗
at s = 0 and use hypoellipticity of the H¨ormander operators 1(X2 + Y2) √3T to
deduce smoothness of the horizontal Jacobian J F. In fact, we sh−ow4 that u= (J±F)−1 is a
0 0
distributional solution of the equation ZZu= 0 and appeal to the identity
1(ZZZZ +ZZZZ)= 1(X2 +Y2)+√3T 1(X2 +Y2) √3T
2 −4 −4 −
(cid:18) (cid:19)(cid:18) (cid:19)
to conclude that u is smooth. We then repeat the argument for the conformal flows
s F−1(exp(sX) F)
7→ ∗
and
s F−1(exp(sY) F)
7→ ∗
to deduce that ZZ(f /J F) = 0 for j = 1,2. Hence f /J F and f /J F are smooth. It
j 0 1 0 2 0
follows that f and f are smooth, after which smoothness of f follows from the contact
1 2 3
property of Heisenberg conformal mappings.
The proof in higher dimensional Heisenberg groups follows a similar line of reasoning
but uses all possible complexified horizontal second derivatives Z Z and the symplectic
k ℓ
structure of the horizontal tangent spaces.
C2 CONFORMAL MAPS OF THE HEISENBERG GROUP ARE C∞ 3
2. Background and definitions
2.1. The Heisenberg group. We model the nth Heisenberg group Hn as R2n+1 equipped
with the nonabelian group law
′ ′ ′ ′ ′ ′ ′ ′
(x,y,t) (x,y ,t) = (x+x,y+y ,t+t 2x y +2x y),
∗ − · ·
where(x,y,t),(x′,y′,t′) Rn Rn R. Sometimesitisconvenient tointroducethecomplex
coordinate z = x+iy ∈Cn. T×hen×Hn is modeled as Cn R with group law
∈ ×
′ ′ ′ ′ ′
(z,t) (z ,t) = (z+z ,t+t +2ω(z,z )),
∗
where ω(z,z′)= Im( n z z′) is the standard symplectic form in Cn and z = (z ,...,z ).
j=1 j j 1 n
Let x = (x ,...,x ) and y = (y ,...,y ). The left invariant vector fields
1 n 1 n
P
∂ ∂ ∂ ∂
X = +2y and Y = 2x
j j j j
∂x ∂t ∂y − ∂t
j j
spana2n-dimensionalsubspaceH Hn inthefulltangentspaceT Hn atapointp = (x,y,t).
p p
The subbundle HHn is known as the horizontal bundle; it defines the accessible directions
at p. Since the distribution HHn is nonintegrable — note that
(2.1) [X ,Y ] = 4T
j j
−
for any j = 1,...,n where T = ∂ — the Chow–Raskevsky theorem implies that Hn is
∂t
horizontally connected. The Carnot–Carath´eodory metric d is defined as follows: d (p,q)
cc cc
is the infimum of the lengths of absolutely continuous horizontal curves γ joining p to q.
Horizontality of γ means that γ′(s) H Hn for a.e. s. Length of a horizontal curve is
γ(s)
∈
measured with respect to the smoothly varying family of inner products defined in the hor-
izontal subbundle making X ,Y ,...,X ,Y into an orthonormal frame. It is well known
1 1 n n
that (Hn,d ) is a geodesic and proper metric space. The metric d is topologically equiv-
cc cc
alent to the underlying Euclidean metric on R2n+1, but d is not bi-Lipschitz equivalent to
cc
any Riemannian metric on R2n+1.
The Lie algebra h of Hn can be identified with the tangent space at the origin. Abusing
n
notation, we write X , Y = X and T for the values of the corresponding vector fields
j j n+j
at the origin, and note that these elements form a basis for h . We will denote by exp the
n
exponential mapping from h to Hn. Since Hn is connected and simply connected, exp is a
n
global diffeomorphism.
The first-order differential operators X and Y are self-adjoint. The Laplacian (some-
j j
times known as the Kohn Laplacian) on Hn is the operator
n
= X2+Y2.
△0 j j
j=1
X
For any c R, the operator
∈
(2.2) := 1 +cT
Lc −4△0
∞
is of H¨ormander type and hence is hypoelliptic. That is, if u = f and f C , then
c
∞ L ∈
u C . See, e.g., [7].
∈The horizontal gradient of a function u: Hn R is
→
u= (X u,Y u,...,X u,Y u)
0 1 1 n n
∇
andthehorizontal divergenceofahorizontalvectorfieldV~ = a X +b Y + +a X +b Y
1 1 1 1 n n n n
···
is
div V~ = X (a )+Y (b )+ +X (a )+Y (b ).
0 1 1 1 1 n n n n
···
4 ALEXD.AUSTINANDJEREMYT.TYSON
Note that u= div ( u).
0 0 0
△ ∇
We make extensive use of the complexified first-order differential operators
1 1
Z = (X iY ) and Z = (X +iY ), j = 1,...,n.
j j j j j j
2 − 2
Note that
n
1 = Z Z +Z Z .
2△0 j j j j
j=1
X
Moreover,
(2.3) 4Z Z = (X X Y Y )+i(X Y +Y X ), j,k = 1,...,n.
j k j k j k j k j k
−
For notational convenience, we sometimes write x = y ,...,x = y and similarly
n+1 1 2n n
X = Y ,...X = Y . However, we continue to denote the final coordinate by t and we
n+1 1 2n n
write T = ∂ .
t
Let U be a domain in Hn. Write ′(U) for the real valued distributions on U.
D
′
Lemma 2.1. Suppose that λ (U), and that Z Z λ = 0 for all j,k = 1,...,n. Then λ
j k
∞ ∈ D
may be identified with a C (U) function.
Proof. Let be the operator defined in (2.2). An easy computation yields the identity
c
L
n
1
(Z¯ Z¯ Z Z +Z Z Z¯ Z¯ )= ,
2 j k j k j k j k L−√n(n+2)L√n(n+2)
j,k=1
X
′
see, e.g., [10, p. 76]. Since λ (U) is real valued, the hypothesis Z Z λ = 0 also implies
j k
that Z¯ Z¯ λ = 0 for all j,k. H∈enDce
j k
λ = 0.
L−√n(n+2)L√n(n+2)
Since is a product of hypoelliptic operators, it is also hypoelliptic.
L−√n(n+2)L√n(n+2)
Thus λ C∞(U) as asserted. (cid:3)
∈
Remark 2.2. The second order differential operators Z Z also arise in Kora´nyi and
j k
Reimann’s theory of Heisenberg quasiconformal flows [10, Section 5]. To wit, if
n
V := ϕT + 1 (X ϕY Y ϕX )
4 j j − j j
j=1
X
for some compactly supported ϕ C∞(Hn), then the flow maps f : Hn Hn, s 0,
s
∈ → ≥
solving the ODE ∂ f (p) = V(f (p)), f (p) = p, are quasiconformal. Specifically, f is
s s s 0 s
K(s)-quasiconformal, where K(s) satisfies
1(K(s)+K(s)−1)= 1+n(exp(s√2 M ) 1).
2 || ||HS −
∞
Here M = ( ZjZkϕ ∞)j,k is the n n matrix of L norms of the functions ZjZkϕ, and
|| || ×
A denotes the Hilbert–Schmidt norm of a matrix A.
HS
|| ||
C2 CONFORMAL MAPS OF THE HEISENBERG GROUP ARE C∞ 5
2.2. Conformal mappings of the Heisenberg group. A reference for the material in
this section is [10, Section 2.3]. We consider mappings F : U Hn where U is a domain
in Hn. All mappings will be assumed to be diffeomorphisms wh→ich are at least C1 smooth.
We write F = (f ,...,f ) in coordinates and denote the standard contact form in Hn
1 2n+1
by
n
α= dt+2 (x dx x dx ).
k n+k n+k k
−
k=1
X
A diffeomorphism F is contact if it preserves the contact structure. In other words,
∗
(2.4) F α = λ α
F
for some nonzero real-valued function λ . We must have either λ >0 everywhere in U or
F F
λ < 0 everywhere in U; we assume that the former condition holds.
F
Contact maps preservethe horizontal distribution. Denoting by DF(p) thedifferential of
F at thepointp, wehave DF(p)(H Hn)= H Hn for all p U. Moreover, therestriction
p F(p)
∈
of DF(p) to the horizontal tangent space, denoted D F(p), is a multiple of a symplectic
0
∗
transformation. Indeed, F (dα)HHn = λFdαHHn and dα defines the standard symplectic
structure in Cn. Since α (dα)n|is a volume f|orm, the full Jacobian JF = detDF satisfies
∧
(2.5) JF = λn+1,
F
while the horizontal Jacobian J F = detD F satisfies
0 0
(2.6) J F = λn.
0 F
SinceHpHn = Kerα(p),(2.4)impliesthatα(F∗Xk) = 0fork = 1,...,2n,ormoreexplicitly,
n
(2.7) X f +2 (f X f f X f ) =0
k 2n+1 j k n+j n+j k j
−
j=1
X
for each k = 1,...,2n. Furthermore,
n
(2.8) Tf +2 (f Tf f Tf )= λ .
2n+1 j n+j n+j j F
−
j=1
X
We now come to the definition of the main objects of study in this paper.
Definition 2.3. A C1 diffeomorphism F : U U′ between domains in Hn is conformal if
→
F is contact, J F > 0, and the equation
0
(2.9) (D F)T(p)D F(p) = J F(p)1/nI
0 0 0 2n
holds for all p U.
∈
In view of (2.6), (2.9) can alternatively be written in the form
(2.10) (D F)T(p)D F(p)= λ (p)I .
0 0 F 2n
C
It is known that conformal maps satisfy a Cauchy–Riemann type equation. Defining f =
k
f +if we have
k n+k
C
(2.11) Z f = 0
ℓ k
for all 1 k,ℓ n. See [10, Theorem C].
≤ ≤
Theonly properties of conformal mappings which wewill usein the proofof Theorem 1.1
are (2.10) and (2.11), together with the facts that inverses and compositions of conformal
mappings are conformal. The latter facts are easy to see from the above definition.
6 ALEXD.AUSTINANDJEREMYT.TYSON
Now assume that F is C2. For fixed ν = 1,...,n consider equation (2.7) for the two
indices ν and n + ν. Differentiating the first using X and the second using X and
n+ν ν
subtracting yields
n
X X f +2 (f X f f X f )
n+ν ν 2n+1 j ν n+j n+j ν j
 − 
j=1
X
 
n
X X f +2 (f X f f X f ) = 0.
ν n+ν 2n+1 j n+ν n+j n+j n+ν j
−  − 
j=1
X
 
Simplifying the result using (2.8) leads to the identity
n
(2.12) λ = (X f X f X f X f ),
F n+ν n+j ν j n+j ν j n+ν
−
j=1
X
valid for each ν = 1,...,n. Equation (2.12) can also be derived from the fact that D F is
0
a multiple of a symplectic matrix.
3. Proof of Theorem 1.1
Let F = (f ,...,f ) : U U′ be a C2 conformal mapping between domains of the
1 2n+1
Heisenberg group Hn. Our goal→is to prove that f C∞(U) for each j. Let G := F−1 and
j
∈
write G= (g ,...,g ).
1 2n+1
Let W be a left invariant vector field. Fix a domain Ω ⋐ U and choose s > 0 so that
0
the conformal flow
H(p,s)= H (p,s) := G(exp(sW ) F(p))
W 0
∗
is well defined for all s with s < s and p Ω. Write H = (h ,...,h ).
0 1 2n+1
| | ∈
Denote by π : Hn Cn the projection map π(z,t) = z and write F˜ = π F =
→ ◦
(f ,...,f ). By an application of the chain rule we find that
1 2n
X h (p,s) = g (exp(sW ) F(p)) X F˜(p)
ℓ k 0 k 0 ℓ
∇ ∗ ·
for each k,ℓ = 1,...,2n and all p Ω. Consequently,
∈
(3.1) X h (p,0) = g (F(p)) X F˜(p) = X (g F)(p) = δ ,
ℓ k 0 k ℓ ℓ k kℓ
∇ · ◦
where δ denotes the Kronecker delta.
kℓ
We now define the matrix-valued flow
M(p,s) = M (p,s) := λ (p,s)−1(D H)T(p,s)(D H)(p,s).
W H 0 0
Since H(,s) is conformal for each s, M(p,s) = I for all p Ω and s < s ; see (2.10).
2n 0
· ∈ | |
Hence, if m (p,s) denotes the (k,ℓ) entry of the matrix M(p,s), we have
k,ℓ
′
m (p,0) = 0.
k,ℓ
Here and henceforth we use primes to denote differentiation with respect to the time pa-
rameter s. On the other hand,
′ ′ ′
λ ((X h )(X h ) +(X h )(X h )) λ (X h X h )
m′ (p,s) = H m k m ℓ m k m ℓ m − H m k m ℓ m .
k,ℓ λ2
P H P (cid:12)(p,s)
(cid:12)
Since λ (p,0) = 1, we may use (3.1) to conclude that (cid:12)
H (cid:12)
′ ′ ′ ′
m (p,0) = (X h )(p,0)+(X h )(p,0) δ λ (p,0).
k,ℓ ℓ k k ℓ − kℓ H
C2 CONFORMAL MAPS OF THE HEISENBERG GROUP ARE C∞ 7
′
We compute λ (p,0) using (2.12). We only need this value in the case 1 k = ℓ n. We
H ≤ ≤
choose ν = k = ℓ in (2.12) and obtain
′ ′ ′
λ (p,0) = (X h )(p,0)+(X h )(p,0).
H n+k n+k k k
Hence
′ ′
(X h )(p,0)+(X h )(p,0), k = ℓ,
′ ℓ k k ℓ
m (p,0) = 6
k,ℓ ′ ′
((Xkhk)(p,0) (Xn+khn+k)(p,0), 1 k = ℓ n.
− ≤ ≤
′
In the following lemma we identify the value of (X h )(p,0) for each k and ℓ. For a left
k ℓ
invariant vector field W, we denote by W˜ the right invariant mirror of W, i.e., the unique
right invariant vector field whose value at the origin agrees with that of W. For instance,
if W = X = ∂ +2y ∂ then W˜ = ∂ 2y ∂ . Observe that
j xj j t xj − j t
X˜ = X 4y T, Y˜ = Y +4x T, and T˜ = T,
j j j j j j
−
and that any of the vector fields X˜ , Y˜ and T˜ commute with all of the left invariant
j j
horizontal vector fields X ,Y . To see why the latter claim is true, it suffices to verify that
k k
X˜ commutes with Y and that Y˜ commutes with X . In fact,
j j j j
[X˜ ,Y ] = X Y 4y TY Y X +Y (4y T)= 0
j j j j j j j j j j
− −
and a similar computation shows that [Y˜,X ] = 0.
j j
Lemma 3.1. For any k,ℓ = 1,...,2n and with H(p,s) = H (p,s)= (h ,...,h )(p,s),
W 1 2n+1
we have
(X h )′(p,0) = X (W˜ g F)(p).
k ℓ k ℓ
◦
Proof of Lemma 3.1. First, we show that for a real-valued function u on U and a point
q Hn, we have
∈
d
(u(exp(sW ) q)) = (W˜ u)(exp(sW ) q).
0 q 0
ds ∗ ∗
We use the identity exp(W ) q = q exp(W˜ ) to compute
0 q
∗ ∗
d u(exp(δW ) exp(sW ) q) u(exp(sW ) q)
0 0 0
(u(exp(sW ) q)) = lim ∗ ∗ − ∗
0
ds ∗ δ→0 δ
u(exp(sW ) q exp(δW˜ )) u(exp(sW ) q)
0 q 0
= lim ∗ ∗ − ∗
δ→0 δ
= (W˜ u)(exp(sW ) q).
q 0
∗
We apply the preceding identity with u = g and q = F(p) to conclude that
ℓ
h′(p,0) = (W˜ g F)(p).
ℓ ℓ ◦
Finally, since W˜ commutes with X we have
k
(X h )′(p,0) = X (W˜ g F)(p).
k ℓ k ℓ
◦
The proof is complete. (cid:3)
We now return to the proof of the theorem. The previous discussion has implied that
X (W˜ g F)(p)+X (W˜ g F)(p), 1 k = ℓ 2n,
′ ℓ k k ℓ
m (p,0) = ◦ ◦ ≤ 6 ≤
k,ℓ (Xk(W˜ gk F)(p) Xn+k(W˜ gn+k F)(p), 1 k = ℓ n.
◦ − ◦ ≤ ≤
We now suppose that there exists a real-valued function ψ on U such that
(3.2) W˜ g F = X ψ and W˜ g F = X ψ
ℓ n+ℓ n+ℓ ℓ
◦ ◦ −
8 ALEXD.AUSTINANDJEREMYT.TYSON
for all ℓ =1,...,n. Then for k,ℓ = 1,...,n we have
′
m (p,0) = (X X +X X )ψ(p)
k,ℓ ℓ n+k n+ℓ k
and
′
m (p,0) = (X X X X )ψ(p).
k,n+ℓ n+k n+ℓ− k ℓ
Recalling (2.3) we conclude that
Z Z ψ(p) = 0 for all p Ω and all 1 k,ℓ n.
k ℓ
∈ ≤ ≤
Exhausting U with a sequence of compactly contained subdomains Ω , we conclude that
ν
Z Z ψ = 0 in U for all 1 k,ℓ n.
k ℓ
≤ ≤
∞
By Lemma 2.1, ψ is a C function.
In order to take advantage of the preceding discussion, we must find an appropriate
potential function ψ corresponding to each of the right invariant vector fields X˜ , Y˜ and
j j
T˜ = T.
First, we consider W˜ = T˜ = T. We claim that
1
ψ = λ−1
−4 F
verifies (3.2) for this choice of W˜ . Since λ−1 = λ F it suffices to prove that
F G ◦
1
(3.3) Tg F = X (λ F)
n+ℓ ℓ G
◦ 4 ◦
and
1
(3.4) Tg F = X (λ F).
ℓ n+ℓ G
◦ −4 ◦
We verify (3.3). First
n
X (λ F)= X Tg +2 (g Tg g Tg ) F
ℓ G ℓ 2n+1 j n+j n+j j
◦  − ◦ 
(cid:18) j=1 (cid:19)
X
 n 
= X (Tg F)+2 X (g F)(Tg F) X (g F)(Tg F)
ℓ 2n+1 ℓ j n+j ℓ n+j j
◦ ◦ ◦ − ◦ ◦
j=1(cid:18)
X
+(g F)X (Tg F) (g F)X (Tg F)
j ℓ n+j n+j ℓ j
◦ ◦ − ◦ ◦
(cid:19)
= ( Tg F) X F˜
0 2n+1 ℓ
∇ ◦ ·
n
+2 Tg g Tg g +g Tg g Tg F X F˜.
n+j 0 j j 0 n+j j 0 n+j n+j 0 j ℓ
∇ − ∇ ∇ − ∇ ◦ ·
j=1(cid:18) (cid:19)
X
Since T commutes with we may rewrite this in the form
0
∇
X (λ F)= (T g F) X F˜
ℓ G 0 2n+1 ℓ
◦ ∇ ◦ ·
n
+2 Tg g Tg g +g Tg g Tg F X F˜.
n+j 0 j j 0 n+j j 0 n+j n+j 0 j ℓ
∇ − ∇ ∇ − ∇ ◦ ·
j=1(cid:18) (cid:19)
X
Since G is a contact map,
n
g +2 (g g g g )= 0
0 2n+1 j 0 n+j n+j 0 j
∇ ∇ − ∇
j=1
X
C2 CONFORMAL MAPS OF THE HEISENBERG GROUP ARE C∞ 9
and so
n
X (λ F)= 2 T (g g g g ) F X F˜
ℓ G j 0 n+j n+j 0 j ℓ
◦ − ∇ − ∇ ◦ ·
j=1 (cid:18) (cid:19)
X
n
+2 Tg g Tg g +g Tg g Tg F X F˜.
n+j 0 j j 0 n+j j 0 n+j n+j 0 j ℓ
∇ − ∇ ∇ − ∇ ◦ ·
j=1(cid:18) (cid:19)
X
Using again the fact that T commutes with we conclude that
0
∇
n
X (λ F)= 4 Tg g Tg g F X F˜
ℓ G n+j 0 j j 0 n+j ℓ
◦ ∇ − ∇ ◦ ·
j=1(cid:18) (cid:19)
X
n
= 4 (Tg ) F X (g F) (Tg F)X (g F)
n+j ℓ j j ℓ n+j
◦ ◦ − ◦ ◦
j=1(cid:18) (cid:19)
X
= 4Tg F.
n+ℓ
◦
by (3.1). Theproofof (3.4)is similar. As previously discussed, this shows that thefunction
ψ = 1λ−1, and hence λ itself, is a C∞ function.
−4 F F
We now consider theright invariant vector field X˜ , for which we claim that thepotential
ℓ
function ψ = f λ−1 verifies (3.2). We use (2.10) to deduce that
n+ℓ F
(3.5) X g F = λ−1X f
ℓ k ◦ F k ℓ
and we use (2.11) to deduce that
(3.6) X f = X f
k ℓ n+k n+ℓ
for all 1 k,ℓ n. Thus
≤ ≤
X˜ g F = X g F 4f Tg F = λ−1X f +f X (λ−1),
ℓ k ◦ ℓ k ◦ − n+ℓ k ◦ F k ℓ n+ℓ n+k F
where the first line follows from the definition of X˜ and the second line uses (3.5) and the
ℓ
previous formula for Tg F. Using (3.6) we conclude that
k
◦
X˜ g F = λ−1X f +f X (λ−1) = X (f λ−1).
ℓ k ◦ F n+k n+ℓ n+ℓ n+k F n+k n+ℓ F
A similar computation shows that
X˜ g F = X (f λ−1)
ℓ n+k ◦ − k n+ℓ F
as claimed. As in the previous argument we conclude that f λ−1, and hence also f ,
n+ℓ F n+ℓ
∞
are C for each 1 ℓ n. Repeating the argument for the right invariant vector fields
Y˜ = X˜ shows th≤at th≤e components f are C∞ for each 1 ℓ n.
ℓ n+ℓ ℓ
≤ ≤
Finally, the contact equation
n
f +2 (f f f f )= 0
0 2n+1 j 0 n+j n+j 0 j
∇ ∇ − ∇
j=1
X
∞ ∞
implies that f is a C vector field, and hence f = div ( f ) C .
0 2n+1 0 2n+1 0 0 2n+1
∇ △ ∞ ∇ ∈
Hypoellipticity of the Kohn Laplacian now implies that f is C . We have shown that
2n+1
∞
all of the components of F are C smooth. This completes the proof of Theorem 1.1.
10 ALEXD.AUSTINANDJEREMYT.TYSON
Remark 3.2. It would be interesting to know if the methods introduced here could be
extended to relax the C2 regularity assumption to C1 regularity or even to the Sobolev reg-
ularity natural for quasiconformal mappings. Such extension is not without its challenges:
for one thing, we differentiate in the vertical and right-invariant directions, and a horizontal
Sobolev assumption gives no a priori regularity along these paths. The matter is somewhat
subtle, in that one should not be tempted to use the nonlinear theory it was our purpose
to avoid. It may be possible to recast the argument, first smoothing some or all of the
objects, then justifying the correct limits. Mollification in the sub-Riemannian context has
thedifficulty thata smoothedcontact mappingis likely nolonger contact. Depite this, such
arguments have been made to work before, and the interested reader might like to consult
[5] as a useful starting point.
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ADA: Departmentof Mathematics, University of California, Los Angeles, Box 951555, Los
Angeles, CA 90095-1555
E-mail address: [email protected]
JTT: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West
Green St., Urbana, IL 61801
E-mail address: [email protected]