THE CAYLEY TRANSFORM
AND UNIFORMLY BOUNDED REPRESENTATIONS
F. ASTENGO, M. COWLING AND B. DI BLASIO
Abstract. Let G be a simple Lie group of real rank one, with Iwasawa decomposition
KAN̄ and Bruhat big cell N M AN̄. Then G/M AN̄ may be (almost) identified with N and
with K/M , and these identifications induce the (generalised) Cayley transform C : N →
K/M . We show that C is a conformal map of Carnot–Caratheodory manifolds, and that
composition with the Cayley transform, combined with multiplication by appropriate powers
of the Jacobian, induces isomorphisms of Sobolev spaces H α (N ) and Hα (K/M ). We use
this to construct uniformly bounded and slowly growing representations of G.
1. Introduction, notation, and preliminaries
Let S n denote the unit sphere in Rn+1 , and o denote the point (0, . . . , 0, 1). There is
a conformal bijection from Rn ∪ {∞} to S n which maps ∞ to o, namely, stereographic
projection. The conformal group G of S n (or of Rn ∪ {∞}) is a simple Lie group, provided
that n ≥ 2. Given an Iwasawa decomposition KAN of G, we write M for the centraliser of
A in K, and then M AN is a parabolic subgroup, which we will abbreviate to P . We choose
the decomposition so that M A stabilises both o and −o, and so that P is the stabiliser of
o; then we may write M AN̄ for the stabiliser P̄ of −o. Then stereographic projection has a
group theoretic interpretation: S n may be identified with K/M , and Rn with N , and then
the stereographic projection is essentially the identification of the cosets nP̄ (where n ∈ N )
and k P̄ (where k ∈ K).
This paper is about an extension of this, namely, the generalised Cayley transform associated to a simple Lie group G of real rank one. In this case, Rn is replaced by a nilpotent
Lie group N of Heisenberg type, and K/M by a sphere with a privileged subbundle of the
The research for this paper was carried out at the University of New South Wales. The authors thank the
Australian Research Council, the Progetto Cofinanziato M. U. R. S. T. “Analisi Armonica”, and the School
of Mathematics of the University of the New South Wales for their support.
1
2
ASTENGO, COWLING AND DI BLASIO
tangent bundle—both are Carnot–Caratheodory manifolds. In this paper, we examine the
Cayley transform from several points of view. Our main geometric result is that the Cayley
transform is a conformal map of Carnot–Caratheodory manifolds. Our main analytic result
is that composition with the Cayley transform and multiplication by a suitable power of the
Jacobian induces isomorphisms from Sobolev spaces Hα (K/M ) to Hα (N ). This is obvious
locally, but the global result requires more work. Finally, we tie these results in to representation theory, and use them to construct uniformly bounded representations πα of G on
the spaces Hα (K/M ) and Hα (N ), and estimate the growth of the representations πα on the
space Hβ (K/M ) when α 6= β. Our estimates are those required by P. Julg [15] for his proof
of the Baum–Connes conjecture with coefficients for the groups Sp(n, 1) and F4,−20 .
In order to deal with all the real rank one simple Lie groups G simultaneously, we use the
Clifford algebra formulation of [5, 6].
Our paper is structured as follows. In the rest of this section we recall some properties
of Lorentz spaces that we will use in the sequel. In Section 2 we describe the Carnot–
Caratheodory structure of N and K/M , and we study the Cayley transform. Section 3
deals with Sobolev spaces Hα (N ) and Hα (K/M ). In Section 4 we prove that Hα (K/M ) and
Hα (N ) can be identified via the Cayley transform. Finally, in Section 5 we use our result to
construct uniformly bounded and slowly growing representations of G.
1.1. Lorentz spaces. Suppose that (Ω, µ) is a measure space and that f : Ω → C is
measurable. The nonincreasing rearrangement of f is the function f ∗ on R+
0 defined by
f ∗ (t) = inf s ∈ R+
0 : µ ({x ∈ Ω : |f (x)| > s}) ≤ t
∀t ∈ R+
0,
∗
where R+
0 = [0, ∞). The function f is nonincreasing, nonnegative, equimeasurable with |f |,
and right-continuous. Next we define
kf kLp,q (Ω)
Z ∞
q ds 1/q
q
1/p ∗
=
s f (s)
p 0
s
when 1 < p < ∞ and 1 < q < ∞, and, when 1 < p < ∞ and q = ∞
kf kLp,∞ (Ω) = sup s1/p f ∗ (s) : s ∈ R+
0 .
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
3
Definition. When 1 < p < ∞ and 1 ≤ q ≤ ∞, the Lorentz space Lp,q (Ω) consists of
those measurable functions f : Ω → C such that kf kLp,q (Ω) is finite, modulo identification of
functions which are equal almost everywhere.
It is easy to check Lp,p (Ω) coincides with the usual Lebesgue space Lp (Ω), with equality
of norms. Moreover, if q1 < q2 , then Lp,q1 (Ω) is contained in Lp,q2 (Ω) and, if 1 < q < ∞,
0
0
then the dual space of Lp,q (Ω) is Lp ,q (Ω), where p0 = p/(p − 1) and q 0 = q/(q − 1). A good
reference for Lorentz spaces is [12]. We recall a few facts from that paper.
Lemma 1 ([12, p. 271]). Suppose that 1 < p, q, r < ∞ and 1/q = 1/r + 1/p. Then there
exists a constant C(p, q, r), depending only on p, q and r, such that for every f in L p,2 (Ω)
and m in Lr,∞ (Ω),
kmf kLq,2 (Ω) ≤ C(p, q, r) kmkLr,∞ (Ω) kf kLp,2 (Ω) .
Lemma 2 ([12, p. 273]). Suppose that Ω is a unimodular locally compact topological group,
that 1 < p < q < ∞ and 1/p + 1/r = 1/q + 1. Then there exists a constant C(p, q, r),
depending only on p, q and r, such that for every f in Lp,2 (Ω) and k in Lr,∞ (Ω),
kf ∗ kkLq,2 (Ω) ≤ C(p, q, r) kkkLr,∞ (Ω) kf kLp,2 (Ω) .
We use the “variable constant convention”, according to which constants are denoted by
C, C 0 , etc., and these are not necessarily equal at different occurrences. All “constants” are
positive.
2. Heisenberg type groups and the Cayley transform
2.1. Properties of Heisenberg type groups. Let n be a real nilpotent Lie algebra, with
an inner product h·, ·i, such that n may be written as an orthogonal sum v⊕z, where [v, v] ⊂ z
and [z, z] = {0}. For Z in z, define the map JZ : v → v by the formula
hJZ X, Y i = hZ, [X, Y ]i
∀X, Y ∈ v.
Following A. Kaplan [16], we say that the Lie algebra n is H–type if
(1)
JZ2 = −|Z|2 Iv
∀Z ∈ z,
4
ASTENGO, COWLING AND DI BLASIO
where Iv is the identity operator on v. We denote the dimensions of v and z by dv and dz .
Notice that from property (1), it follows that z = [v, v], and moreover, if z is nontrivial, then
the dimension dv of v is even. A connected, simply connected Lie group N whose Lie algebra
is an H–type algebra is said to be an Heisenberg type group, or more briefly an H–type
group. The Iwasawa N –groups associated to all real rank one simple groups are H–type.
For more information about H–type groups and their connection with Iwasawa N –groups,
see [5, 6].
We shall use the following properties of the map J, which are proved in [5, Section 1]:
JZ JZ 0 + JZ 0 JZ = −2hZ, Z 0 iIv
(2)
[X, JZ X] = |X|2 Z
∀Z, Z 0 ∈ z,
∀X ∈ v ∀Z ∈ z,
J[X,X 0 ] X = |X|2 PJz X X 0
∀X, X 0 ∈ v,
where PJz X X 0 is the projection of the vector X 0 on the space Jz X = {JZ X : Z ∈ z}.
Since N is a simply connected nilpotent Lie group, the exponential mapping is bijective.
For X in v and Z in z, we denote by (X, Z) the element exp(X + Z) of the group N . By
the Baker–Campbell–Hausdorff formula, the group law is given by
1
0
0
0
0
0
∀X, X 0 ∈ v,
(X, Z) (X , Z ) = X + X , Z + Z + [X, X ]
2
∀Z, Z 0 ∈ z.
The group N is unimodular and an element dn of Haar measure on N is dX dZ, where dX
and dZ are elements of the Lebesgue measures on the real vector spaces v and z respectively.
When t > 0, we define the homogeneous dilation δt on N by
δt (X, Z) = (tX, t2 Z)
∀(X, Z) ∈ N.
It is easy to check that δt is a group automorphism. We write Q for the homogeneous
dimension of N , i.e., dv + 2dz . The function N on N given by
!1/4
|X|4
N (X, Z) =
+ |Z|2
16
−1
is a homogeneous gauge. We define dN : N × N → R+
0 by dN (n1 , n2 ) = N (n1 n2 ); then dN
is a metric on N .
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
5
Definition. [5] We say that n satisfies the J 2-condition if, for any X in v and Z, Z 0 in z
such that hZ, Z 0 i = 0, there exists Z 00 in z such that
JZ JZ 0 X = JZ 00 X.
It is known [5] that an H–type group N is the Iwasawa N –group of a real rank one simple
Lie group if and only if its Lie algebra satisfies the J 2-condition. We shall henceforth assume
that this condition holds.
2.2. The Cayley transform. We write S for the unit sphere in v ⊕ z ⊕ R, i.e.,
2
2
2
S = {(X 0 , Z 0 , t0 ) ∈ v ⊕ z ⊕ R : |X 0 | + |Z 0 | + t0 = 1}.
The Cayley transform C : N → S, introduced in [6], is given by
1
|X|4
2
C(X, Z) =
+ |Z| ,
Ā(X, Z)X, 2Z, −1 +
B(X, Z)
16
where A(X, Z) and Ā(X, Z) are the linear operators on v defined by
|X|2
|X|2
0
0
0
A(X, Z)X = 1 +
+ JZ X and Ā(X, Z)X = 1 +
− JZ X 0 ,
4
4
and the real number B(X, Z) is defined by
2
|X|2
B(X, Z) = 1 +
+ |Z|2 .
4
Its inverse C −1 : S \ {o} → N is given by
C −1 (X 0 , Z 0 , t0 ) =
1
0
0
0
0
2 (2(1 − t + JZ )X , 2Z ) .
0
2
0
(1 − t ) + |Z |
Here o stands for (0, 0, 1). Note that if (X 0 , Z 0 , t0 ) = C(X, Z), then
B(X, Z) =
4
.
(1 − t0 )2 + |Z 0 |2
When the dependence on (X, Z) in N is clear, we shall just write A, Ā and B.
Because we assume the J 2 -condition, there is a simple real rank one Lie group G whose
Iwasawa N –group is N . The subgroups K, A and M of G are defined in the standard
way: KAN is an Iwasawa decomposition of G, and M is the centraliser of A in K. The
6
ASTENGO, COWLING AND DI BLASIO
sphere S may be identified with K/M , and the map C is essentially the map treated by
S. Helgason [11] (see Corollary 1.9 on p. 407 and the following sections).
There is a natural action of K on S by orthogonal transformations, which can be realized
so that M stabilizes o and −o. We will write k · ζ to indicate the result of applying k ∈ K
to ζ ∈ S.
Given a function f on S, we denote by f ] the corresponding function on K/M . Conversely,
given a function g on K/M , we denote by g [ the corresponding function on S.
We define D to be the function (B −1/4 ◦ C −1 )] on K/M . Note that D is M –invariant, and
so we may define a function dS on S × S by
dS (k1 · o, k2 · o) = D(k1−1 k2 M )
∀k1 , k2 ∈ K.
In [2], A. Banner proved that dS is a distance on S.
Lemma 3. For all n1 , n2 in N ,
dS (C(n1 ), C(n2 )) = B −1/4 (n1 ) B −1/4 (n2 ) N (n−1
1 n2 ).
Proof. We recall Banner’s explicit expression for the distance on the sphere [2], namely
(3)
2
4 dS (ζ1 , ζ2 )4 = hX10 , X20 i2 + |[X10 , X20 ]| + 2h t01 − JZ10 X10 , t02 − JZ20 X20 i + 1
0 2
0 2
+ 1 − |X1 |
1 − |X2 | − 2 (hX10 , X20 i + hZ10 , Z20 i + t01 t02 ) ,
where ζ1 = (X10 , Z10 , t01 ) and ζ2 = (X20 , Z20 , t02 ). When h = 1 or 2, let
ζh = C(nh ) = C(Xh , Zh ) =
where
Ah =
|Xh |2
1+
4
Bh−1 Ah Xh , 2Bh−1 Zh , 1 − 2Bh−1
− J Zh
and Bh =
|Xh |2
1+
4
Note that
t0h − JZh0 = 1 − 2Bh−1 Ah ,
2
|Xh |2
1+
4
+ |Zh |2 .
!!
,
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
7
so 4 dS (ζ1 , ζ2 )4 is equal to
2
hB1−1 A1 X1 , B2−1 A2 X2 i2 + [B1−1 A1 X1 , B2−1 A2 X2 ]
+ 2 h 1 − 2B1−1 A1 B1−1 A1 X1 , 1 − 2B2−1 A2 B2−1 A2 X2 i
+ 1 + 1 − B1−1 |X1 |2 1 − B2−1 |X2 |2
− 2 hB1−1 A1 X1 , B2−1 A2 X2 i − 2 h2 B1−1 Z1 , 2 B2−1 Z2 i
|X1 |2
|X2 |2
−1
−1
− 2 1 − 2B1
1+
1 − 2B2
1+
4
4
2
= B1−1 B2−1 B1−1 B2−1 hA1 X1 , A2 X2 i2 + B1−1 B2−1 [A1 X1 , A2 X2 ]
+ 2 hA1 X1 − 2 X1 , A2 X2 − 2 X2 i + 1
+ B1 − |X1 |2 B2 − |X2 |2 − 2 hA1 X1 , A2 X2 i − 8 hZ1 , Z2 i
!
|X1 |2
|X2 |2
− 2 B1 − 2 1 +
,
B2 − 2 1 +
4
4
by (3). The equality
2
hA1 X1 , A2 X2 i2 + [A1 X1 , A2 X2 ] = B1 B2 hX1 , X2 i2 + |[X1 , X2 ]|2
is equivalent to the J 2 –condition (see [5, p. 25], where A is defined slightly differently). By
writing Ah and Bh in terms of Xh and Zh and simplifying, we see that 4 dS (ζ1 , ζ2 )4 is equal
to
1
|X1 |4 + 2 |X1 |2 |X2 |2 + |X2 |4 + hX1 , X2 i2 − |X1 |2 + |X2 |2 hX1 , X2 i
4
+ 4 |Z1 |2 + |Z2 |2 + 4 hZ1 − Z2 , [X1 , X2 ]i + |[X1 , X2 ]|2 − 8 hZ1 , Z2 i
4
= 4 B1−1 B2−1 N n−1
,
1 n2
B1−1 B2−1
as required.
The following result was proved by Banner [2].
Corollary 4. The Cayley transform C is 1–quasiconformal.
8
ASTENGO, COWLING AND DI BLASIO
Proof. It suffices to show that
lim
r→0
sup{ dS (C(n1 ), C(n2 )) : N (n−1
1 n2 ) = r }
=1
−1
inf{ dS (C(n1 ), C(n2 )) : N (n1 n2 ) = r }
∀n1 ∈ N.
−1
But if N (n−1
1 n2 ) = N (n1 n3 ) = r, then
B 1/4 (n3 )
dS (C(n1 ), C(n2 ))
= 1/4
,
dS (C(n1 ), C(n3 ))
B (n2 )
and this can be made arbitrarily close to 1 by taking r small.
d
z
v
2.3. Vector fields and the Jacobian. We fix orthonormal bases {Ej }dj=1
of v and {Uk }k=1
of z. Given V in n, we also write V for the associated left-invariant vector field, i.e.,
d
V f (n) = f (n exp tV )
∀n ∈ N ∀f ∈ C ∞ (N ).
dt
t=0
It is easy to check that, for all smooth functions f on N and (X, Z) in N ,
dz
1X
Ej f (X, Z) = ∂xj f (X, Z) +
hJUk X, Ej i ∂zk f (X, Z)
2
(4)
k=1
Uk f (X, Z) = ∂zk f (X, Z),
(1)
where j = 1, . . . , dv and k = 1, . . . , dz . At any point n of N , we write Tn (N ) for the span of
(2)
the tangent vectors at n associated to the Ej , and Tn (N ) for the corresponding subspace
associated to the Uk .
We endow the sphere S with the SO(dv + dz + 1)–invariant Euclidean metric. The tangent
space To (S) at the point o decomposes into the orthogonal sum v ⊕ z with respect to the
Euclidean metric. Since K acts by orthogonal transformations [6, Theorem 6.1], and Ad(M )
preserves v and z, the tangent space Tζ (S) at any point ζ decomposes into the orthogonal
(1)
(2)
(1)
(2)
sum Tζ (S) ⊕ Tζ (S), where Tζ (S) = k∗ v and Tζ (S) = k∗ z when ζ = k · o.
The Carnot–Caratheodory structure of N and S is reflected in the following lemma, linking
the metrics and the special subspaces of the tangent spaces.
Lemma 5. Suppose that n ∈ N and V ∈ Tn (N ), and that γ : I → N is a smooth curve such
(1)
that γ(0) = n and γ̇(0) = V . Then dN (n, γ(s)) = O(s) as s → 0 if and only if V ∈ Tn (N ).
In this case, dN (n, γ(s)) = |s||V |/2 + o(s) as s → 0.
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
9
Suppose that ζ ∈ S and V ∈ Tζ (S), and that γ : I → S is a smooth curve such that
(1)
γ(0) = ζ and γ̇(0) = V . Then dS (ζ, γ(s)) = O(s) as s → 0 if and only if V ∈ Tζ (S). In
this case, dS (ζ, γ(s)) = |s||V |/2 + o(s) as s → 0.
Proof. We prove only the second part of the lemma, as the proof of the first part is similar
(1)
but easier. The metric dS and the subspace Tζ (S) are both K–invariant, so it suffices to
consider the case where ζ = o.
Suppose that X 0 ∈ v and Z 0 ∈ z. Then from (3),
4 dS ((0, 0, 1), (X 0, Z 0 , (1 − |X 0 |2 − |Z 0 |2 )1/2 ))4
= 2 − |X 0 |2 − 2(1 − |X 0 |2 − |Z 0 |2 )1/2
1
1
0 2
0 2
0 2
0 2 2
0 2
0 2 3
0 2
= 2 − |X | − 2 1 − (|X | + |Z | ) − (|X | + |Z | ) + O((|X | + |Z | )
2
8
1
= |Z 0 |2 + (|X 0 |2 + |Z 0 |2 )2 + O (|X 0 |2 + |Z 0 |2 )3 .
4
(1)
It follows that dS (o, γ(s)) = O(s) as s → 0 if and only if V ∈ To (S), and when this is
satisfied, lims→0 |s|−1 dS (o, γ(s)) = |V |/2 as s → 0.
(1)
Lemma 6. Suppose that n ∈ N and that ζ = C(n). For every vector V in Tn (N ) the vector
(1)
C∗ V lies in Tζ (S), and vice versa. Moreover
|C∗ V | = B −1/2 (n) |V |
∀V ∈ Tn(1) (N ).
Proof. Take V in Tn (N ), and a curve γ : I → N such γ(0) = n and γ̇(0) = V . From
Lemma 3, dS (C(n), C(γ(s))) = O(s) as s → 0 if and only if dN (n, γ(s)) = O(s) as s → 0.
(1)
(1)
From Lemma 5, this implies that V ∈ Tn (N ) if and only if C∗ V ∈ Tζ (S). Further, in the
case when both these conditions hold,
dN (n, γ(s))
dS (C(n), C(γ(s)))
= 2B(n)−1/2 lim
= B(n)−1/2 |V |.
s→0
s→0
|s|
|s|
|C∗ V | = 2 lim
10
ASTENGO, COWLING AND DI BLASIO
The Jacobian determinant JC −1 of the map C −1 was computed in [6, p. 234], and is given
by
d(C −1 ζ)
=
JC −1 (ζ) =
dσ(ζ)
4
(1 − t0 )2 + |Z 0 |2
Q/2
∀ζ = (X 0 , Z 0 , t0 ) ∈ S \ {o},
−Q/2
where σ denotes the standard measure on the sphere. Clearly JC = JC−1
. This
−1 ◦ C = B
result may also be deduced from the previous lemma and the fact that the Jacobian of a
1-quasiconformal map is the Qth power of the dilation factor.
2.4. Sublaplacians. Define the sublaplacian ∆ on N by
dv Z
X
(∆f | f )N =
|Ej f (n)|2 dn
∀f ∈ Cc∞ (N ),
j=1
N
where (· | ·)N denotes the usual L2 inner product on N . Since the fields Ej are divergence
free,
∆=−
dv
X
Ej2 .
j=1
Similarly, we define the sublaplacian L on the sphere S by
dv Z
X
(5)
(Lf | f )S =
|Wj f |2 dσ
∀f ∈ C ∞ (S),
j=1
S
v
is any collection of vector fields defined almost everywhere on S which form
where {Wj }dj=1
an orthonormal basis for Tζ1 (S) at almost every point ζ, and (· | ·)S denotes the usual L2 inner
product on S. In what follows, we will define Wj by
1/Q
Wj = |C∗ Ej |−1 C∗ Ej = JC −1 C∗ Ej
j = 1, . . . dv .
Note that L is K–invariant, because the action of K preserves T (1) (S), while the vector fields
Wj are not K–invariant. The fields Wj are not divergence free either.
Remark. In the real case, i.e., when v = 0, the operator L is the Laplace–Beltrami
operator on the real Euclidean sphere. In the complex case, i.e., when N is the Heisenberg
group, the operator L is the laplacian considered by D. Geller in [10] on the complex sphere.
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
11
Lemma 7. For every β in C,
∆JCβ (X, Z)
=
β+2/Q
JC
(X, Z)
2
dv
2 Q
βQ + |X| β
2
4
1
1
−β + −
2 Q
∀(X, Z) ∈ N.
Proof. This is a routine calculation, based on the properties (2) of the map J and the relations
−2/Q
Ej (JC
)(X, Z) = Ej (B)(X, Z) = hAX, Ej i,
(6)
Ej (AX) = AEj +
where (X, Z) is in N .
1
hX, Ej i + J[X,Ej ] X,
2
Theorem 8. Let L be the sublaplacian on the sphere and let ∆ be the sublaplacian on N .
Then
1/2−1/Q
1/2−1/Q
∆ JC
ψ ◦ C | JC
ϕ◦C
= ((L + b)ψ | ϕ)S
∀ψ, ϕ ∈ C ∞ (S),
N
where b = (Q − 2)dv /4.
Proof. From (5), (Lψ | ϕ)S is equal to
dv X
1/Q
JC −1
C ∗ Ej ψ |
1/Q
JC −1
C ∗ Ej ϕ
j=1
=
dv X
1/2−1/Q
JC
Ej (ψ
◦
S
1/2−1/Q
C) | JC
Ej (ϕ
◦ C)
j=1
=
dv X
1/2−1/Q
JC
Ej (ψ
j=1
=−
dv h
X
j=1
◦ C) | Ej
1/2−1/Q
JC
ϕ
N
◦C −
1/2−1/Q
E j JC
1/2−1/Q
1/2−1/Q
E j JC
Ej (ψ ◦ C) | JC
ϕ◦C
ϕ◦C
N
N
i
1/2−1/Q
1/2−1/Q
ϕ◦C
+ JC
Ej (ψ ◦ C) | Ej JC
N
=−
dv X
j=1
1/2−1/Q 2
1/2−1/Q
1/2−1/Q
Ej (ψ ◦ C) + JC
Ej (ψ ◦ C) | JC
ϕ◦C
2 E j JC
N
1/2−1/Q
1/2−1/Q
1/2−1/Q
1/2−1/Q
ψ ◦ C | JC
ϕ◦C
= ∆ JC
ψ ◦ C | JC
ϕ◦C
− ∆ JC
N
N
12
ASTENGO, COWLING AND DI BLASIO
The computation now follows by Lemma 7, where β = 1/2 − 1/Q, and by a change of
variables.
We now determine a fundamental solution of the “conformal sublaplacian” (L + b), i.e., a
distribution u on the sphere such that (L + b)u is the Dirac delta at the point o.
1/2−1/Q
Theorem 9. The fundamental solution of L + b is a positive multiple of J C −1
.
Proof. We first note that, for every β in C and ζ = (X 0 , Z 0 , t0 ) in S \ {o},
dv Q
1
Q2
1
1
1
β+2/Q
β
β
(7) (L + b)JC −1 (ζ) = −
β− +
JC −1 (ζ) − β
β− +
|X 0 |2 JC −1 (ζ).
2
2 Q
4
2 Q
This formula is a straightforward consequence of the intertwining property of Theorem 8
applied to the function JCβ−1 , and Lemma 7.
Note that the function JCβ−1 is regular away from the pole o. Moreover, when (X 0 , Z 0 , t0 )
is in a small neighbourhood of o on the sphere, t0 = (1 − |X 0 |2 − |Z 0 |2 )1/2 , and arguing as in
Lemma 5,
2
2
JC −1 (X 0 , Z 0 , t0 ) = d−2Q
(o, (X 0 , Z 0 , (1 − |X 0 | − |Z 0 | )1/2 ))
S
−Q/2
1 04
0 4
0 2
0 2
= |Z | + |X | + O(|X | + |Z | )
4
N −2Q (X 0 , Z 0 ).
Therefore JCβ−1 is integrable on the sphere if Re(β) < 1/2. Similarly, one can prove that
β+2/Q
(X 0 , Z 0 , t0 ) 7→ |X 0 |2 JC −1
(X 0 , Z 0 , t0 ) is integrable if Re(β) < 1/2 − 1/Q.
As in [4, Lemma 1.1], we deduce that the distribution-valued functions β 7→ JCβ−1 and
β+2/Q
β 7→ |X 0 |2 JC −1
are holomorphic when Re(β) < 1/2 and Re(β) < 1/2 − 1/Q, and that they
extend meromorphically to C with simple poles in {1/2+k/Q : k ∈ N} and {1/2+(k −1)/Q :
k ∈ N} respectively.
When Re(β) < 1/2 − 1/Q, we can evaluate the sublaplacian of the distribution JCβ−1
by ordinary differentiation. The conclusion follows by evaluating the limit as β tends to
1/2 − 1/Q of both sides in formula (7) and checking that the residue of the function β 7→
β+2/Q
|X 0 |2 JC −1
at 1/2 − 1/Q is a positive multiple of the Dirac delta at the point o. This can
be done as in [4, Remark 1.10].
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
13
3. Sobolev spaces on N and on K/M
3.1. Sobolev spaces on N. In this subsection, we recall some properties of Sobolev spaces
on N . For further details the reader can refer to [4, 9]. In [4], the proofs are given for
only some Iwasawa N groups, but these extend to all Heisenberg type groups with minor
modifications (see [1]).
Let γ be in C. A function f on N is said to be homogeneous of degree γ if
f ◦ δ t = tγ f
∀t ∈ R+ .
We say that a distribution K on N is a kernel of type γ if it coincides with a homogeneous
function fK of degree γ − Q on N \ {0}. In the sequel, we shall use the same notation K for
the kernel K and the associated function fK .
The sublaplacian ∆ is a densely defined, essentially self-adjoint, positive operator on
L2 (N ). Hence it has a spectral resolution given by
Z ∞
∆=
λ dAλ .
0
G. B. Folland [9, Proposition 3.9] proved that 0 is not an eigenvalue of ∆, and so we may
define the operators ∆α , for every α in C, by the formula
Z ∞
α
∆ =
λα dAλ .
0
Proposition 10. [9, Theorem 3.15, Proposition 3.17, 3.18] The operators ∆α have the following properties:
(i) ∆α is closed on L2 (N ) for every α in C;
(ii) if f is in Dom(∆α ) ∩ Dom(∆α+β ), then ∆α f is in Dom(∆β ) and ∆β ∆α f = ∆α+β f .
In particular, ∆−α = (∆α )−1 ;
(iii) if 0 < Re(α) < Q, then there exists a kernel Rα of type α, smooth in N \ {(0, 0)},
−α/2
such that if f is in Dom(∆
−α/2
), then ∆
f = f ∗ Rα .
Definition. For α in (−Q/2, Q/2), we define the homogeneous Sobolev space H α (N ) to be
the completion of the space of smooth functions with compact support on N with respect
14
ASTENGO, COWLING AND DI BLASIO
to the norm
1/2
α/2
kf kHα (N ) = (∆α f | f )N = k∆
f k2
∀f ∈ Cc∞ (N ).
The spaces Hα (N ) and H−α (N ) are dual with respect to the usual L2 inner product.
We need the following characterization of Sobolev spaces, proved by Folland for the nonhomogeneous Sobolev spaces Hα (N ) ∩ L2 (N ) for positive α; his proof can be adapted without
substantial changes to the case of homogeneous Sobolev spaces.
If f is in Hα (N ), we shall write ∂xj f and ∂zh f for the distributional derivatives of f .
Theorem 11. [9, Theorem 4.10] Let α be in [1, Q/2). Then f is in Hα (N ) if and only if Ej f
Pv
kEj f kHα−1 (N )
is in Hα−1 (N ) for every j = 1, . . . , dv ; moreover, the norms kf kHα (N ) and dj=1
are equivalent.
The following Sobolev immersion properties hold.
Proposition 12. If f is a continuous and compactly supported function on N , then f is in
the Sobolev space Hα (N ) for every α in (−Q/2, 0], and moreover
kf kHα (N ) ≤ Cα kf kLp,2 (N ) ,
where 1/p = 1/2 − α/Q.
Proof. If α = 0, the proposition is clearly true. If α is in (−Q/2, 0), then by item (iii) of
Proposition 10
α/2
∆
f = f ∗ R−α ,
where R−α = Ω N −(α+Q) and Ω is homogeneous of degree 0 and smooth away from the
identity. Since |R−α | ≤ CN −(α+Q) , and (N −(α+Q) )∗ (t) = C t−(1+α/Q) , where t > 0, we
deduce that R−α is in Lr,∞ (N ) when 1/r = 1 + α/Q.
If 1/p = 1/2 − α/Q, then 1 < p < 2 and 1/p + 1/r = 1/2 + 1. Therefore, by Lemma 2 and
the unimodularity of N ,
α/2 ∆ f L2 (N )
as required.
= kf ∗ R−α kL2 (N ) ≤ Cα kR−α kLr,∞ (N ) kf kLp,2 (N ) ,
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
15
By duality, we obtain the following result.
Corollary 13. If 0 < α < Q/2 and 1/p = 1/2 − α/Q, then H α (N ) is contained in Lp,2 (N ),
and moreover
kf kLp,2 (N ) ≤ Cα kf kHα (N )
∀f ∈ Hα (N ).
The rest of this subsection deals with multiplier theorems on Sobolev spaces on N . Theorem 14 below was first proved in [4] for the Iwasawa N groups corresponding to the real, complex and quaternionic hyperbolic spaces. The methods arise in works of R. S. Strichartz [19]
and N. Lohoué [17].
For every positive integer k, denote by D k a differential operator of the form
D k = E j1 E j2 · · · E jk ,
where j1 , j2 , . . . , jk ∈ {1, 2, ..., dv },
and let D 0 be the identity operator.
For every real number γ and every nonnegative integer d, denote by Mγ,d (N ) the space
of functions m in C d (N \ {0}) such that
k D m ≤ Ck N −γ−k
k = 0, 1, 2, . . . , d.
For a real number a, define dae to be the integer such that a − dae is in (−1, 0], and, for α
and β real, define the nonnegative integer d(α,β) by


max(0, dαe)
if β ≥ 0
d(α,β) =

max(0, d−βe) if β < 0.
Theorem 14. Suppose that −Q/2 < α ≤ β < Q/2, γ = β − α and d = d(α,β) , and that m
is in Mγ,d (N ). Then pointwise multiplication by the function m defines a bounded operator
from Hβ (N ) to Hα (N ).
Proof. For m in Mγ,d (N ), define the operator Λ(m) by Λ(m)f = mf , for every measurable
function f on N . The proof may be divided into five steps.
16
ASTENGO, COWLING AND DI BLASIO
(i) If α ≤ 0, β ≥ 0 and γ = β − α, then Λ(m) is bounded from Hβ (N ) to Hα (N ) for all
m in Mγ,d (N ). This is an immediate consequence of Lemma 1, Proposition 12 and
Corollary 13.
(ii) If α = β = 1, then Λ(m) is bounded on H1 (N ) for all m in M0,1 (N ). This follows
from Theorem 11, step (i), and the equality
Ej (mf ) = (Ej m) f + m (Ej f )
j = 1, . . . , dv
f ∈ Cc∞ (N ).
(iii) If −1 ≤ α ≤ 1, then Λ(m) is bounded on Hα (N ) for all m in M0,1 (N ), by complex
interpolation and duality.
(iv) If 0 < α < Q/2 and d = dαe, then Λ(m) is bounded on Hα (N ) for all m in M0,d (N ).
This follows from Leibniz’ rule,
d X
d
d
D (mf ) =
(D s m) (D d−s f )
s
s=0
f ∈ Cc∞ (N ),
Theorem 11, Proposition 12, Corollary 13, and the preceding steps.
(v) The case where −Q/2 < α ≤ β < Q/2 follows by duality and complex interpolation.
Further details can be found in [1].
Corollary 15. Suppose that −Q/2 < α ≤ β < Q/2, γ is in C and Re(γ) = β − α. If m is in
C ∞ (N \ {0}) and is homogeneous of degree −γ, then pointwise multiplication by m defines
a bounded operator from Hβ (N ) to Hα (N ).
Proof. Under the hypotheses of the corollary, m is in MRe(γ),d (N ) for every nonnegative
integer d. The result follows by Theorem 14.
We shall be interested in the case where the multiplier function is a power of the Jacobian
of the Cayley transform multiplied by a polynomial.
We define the degree of a polynomial in the variables x1 , x2 , . . . xdv , z1 , z2 , . . . zdz , by
saying that the monomial xrj zis has degree r + 2s, when r and s are nonnegative integers.
Lemma 16. Let γ be a complex number and k be a nonnegative integer. Then
X (γ+k+r)/2Q
γ/2Q
=
JC
Pr , where Pr is a polynomial of degree r;
(i) D k JC
0≤r≤3k
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
17
k γ/2Q (Re(γ)+k)/2Q
(ii) D JC
;
≤ C k JC
γ/2Q (iii) D k JC
≤ Ck N − Re(γ)−k on N \ {0}, when Re(γ) ≥ −k.
Proof. This is a routine calculation, based on formulae (6).
Corollary 17. Suppose that z is in C, t in R, and −Q/2 < α < Q/2. If mz,t =
z/Q
−z/Q
JC ◦ δ e t JC
, then there exists a constant C, independent from z and t, such that
kmz,t f kHα (N ) ≤ C 1 + e−2t Re(z) kf kHα (N )
∀f ∈ Hα (N ).
Proof. Recall that JC = B −Q/2 . First we prove that mz,t is in M0,d (N ) for every nonnegative
integer d. For every (X, Z) in N ,
z/Q
−z/Q
|mz,t (X, Z)| = JC ◦ δet JC
(X, Z)
 Re(z)/2
2
|X|2
2
1+ 4
+ |Z| 

= 

2t
2 2
4t |Z|2
+
e
1 + e |X|
4
Re(z)/2
 2
|X|2
2
+
|Z|
1
+
4


= e−2 Re(z)t  .
2

2
|X|
2
−2t
+ |Z|
e + 4
Considering separately the four cases for the signs of t and Re(z), we see that |mz,t | ≤
e−2 Re(z)t if Re(z) t < 0, and |mz,t | ≤ 1 if Re(z) t ≥ 0.
18
ASTENGO, COWLING AND DI BLASIO
By Lemma 16, for every nonnegative integer d, on N \ {(0, 0)}
d X
z
− z d
D s JCQ ◦ δet D d−s JC Q |D d (mz,t )| = s
s=0
d X
− z d ts s Qz e
D JC ◦ δet D d−s JC Q =
s
s=0
≤ C |mz,t |
d
X
e
ts
s=0
≤ C 1 + e−2t Re(z)
s
2Q
JC ◦ δ e t
d
X
d−s
JC2Q
d−s
N −s JC2Q
s=0
≤ C 1+e
−2t Re(z)
N −d .
Following the the argument of step (iii) of the proof of Theorem 14, it is easy to see that,
when 0 < α < Q/2 and d = dαe,
kmz,t f kHα (N ) ≤ C
d
X
kD s (mz,t )kL(Q/s),∞ (N ) + kmz,t kL∞ (N )
s=1
≤ C
d
−s X
−2t Re(z)
N 1+e
L(Q/s),∞ (N )
!
kf kHα (N )
+ kmz,t kL∞ (N )
s=1
!
≤ C 1 + e−2t Re(z) kf kHα (N ) .
The case where −Q/2 < α < 0 can be treated by duality.
kf kHα (N )
3.2. The Fourier transform on K/M . We refer to [8, 14, 13] for further details on the
subject of this subsection.
If f and g are integrable functions on K/M , and g is M –invariant, then we may define
their convolution f ? g by
f ? g(k1 M ) =
Z
f (kM ) g(k −1 k1 M ) dk,
K
where dk is an element of the Haar measure on K, normalized so that
R
K
dk = 1. We
recall that the pair (K, M ) is a Gelfand pair, i.e., the convolution algebra of M –bi-invariant
integrable functions on K is commutative.
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
19
We will be interested in the Fourier transform of the powers of the function D, which are
M –invariant functions on K/M .
If τ is an irreducible representation of the group K, we denote by V τ the vector space of
the representation and by VMτ the subspace of M –invariant vectors, i.e.,
VMτ = {v ∈ V τ : τ (m)v = v
∀m ∈ M }.
Then either VMτ = {0} or VMτ is one-dimensional; in the latter case, we say that τ is class
b M , and we define the spherical function φτ by
one, we write τ ∈ K
φτ (k) = hv0 | τ (k)v0 i
∀k ∈ K.
where v0 is a unit vector in VMτ . The Fourier transform of an M –invariant function f on
K/M is given by
fb(τ ) =
Z
f (kM ) φτ (k) dk
K
bM .
∀τ ∈ K
Let τ be a class one representation and v be in the representation space V τ ; we define the
function fv on K by
fv (k) = hv | τ (k)v0 i
∀k ∈ K.
The function fv is continuous and M –right-invariant, therefore it can be identified with
a continuous function on K/M . This gives a correspondence between vectors in V τ and
continuous functions on K/M , and henceforth we will think of V τ as a space of functions on
K/M . Then one can prove that
(8)
L2 (K/M ) =
M
V τ.
bM
τ ∈K
P
Consequently, any f in L2 (K/M ) may be written as f = τ ∈Kb M aτ Yτ , where the aτ are
P
complex numbers and the Yτ are unit vectors in V τ , and τ ∈Kb M |aτ |2 = kf k2L2 (S) .
In the degenerate case where z = {0}, provided that dv ≥ 2, the real case, it is well-known
that the subspaces of spherical harmonics of degree d are K–invariant and irreducible. For
every d in N, we denote by τd the corresponding class one representation.
In the other cases, the space of spherical harmonics of a given degree are not always irreducible; one has to restrict attention to “bigraded” spherical harmonics. Roughly speaking,
20
ASTENGO, COWLING AND DI BLASIO
to any class one representation there corresponds a pair of integers (d, h). When dim z = 1
(the complex case) we consider (d, h) in N2 . When dim z = 3 (the quaternionic case) or
dim z = 7 (the octonionic case), we consider only pairs of integers (d, h) where d ≥ h ≥ 0.
One of the main results of K. D. Johnson and N. R. Wallach [14, 13] is the following.
For every α in C \ {0, −1, −2, . . .}, define
Γ dv +d2 z +1 Γ(α)
d +2+2α .
Aα =
Γ Q+2α
Γ v 4
4
In the real case, the Fourier transforms of the powers of D can be thought of as functions of
d in N, given by
(9)
(D 2α−Q ) b (τd ) = Aα
d
Y
Q
2
Q
2
j=1
−α+j−1
+α+j−1
!
= Aα
In the other cases,
(D
2α−Q
(10)
) b (τd,h ) = Aα
= Aα
d
Y
Q
2
Q
2
j=1
Γ
Γ
Q
4
Q
4
− α + 2j − 2
!
Γ
−α+d
Γ Q2 + α
.
Γ Q2 − α
Γ Q2 + α + d
Q
2
h Y
dv − 2α + 4j − 2
+ α + 2j − 2 j=1 dv + 2α + 4j − 2
dv +2+2α
Γ
− α2 + d Γ Q4 + α2 Γ dv +2−2α
+
h
4
4
dv +2−2α
+
h
Γ
+ α2 + d Γ Q4 − α2 Γ dv +2+2α
4
4
for (d, h) in N2 or (d, h) such that d ≥ h ≥ 0.
3.3. Sobolev spaces on K/M . Define the sublaplacian L] on K/M by
L] f ] = (Lf )]
By Theorem 9,
(L + b) b = C
]
f ∈ C ∞ (S).
D
2−Q
−1
b
,
where the constant C depends on dv and dz and can be computed by evaluating the Fourier
transform of (L] + b)f when f (kM ) = 1 for all k ∈ K . Using formulae (9) and (10), we see
that, in the real case,
(11)
(L + b) b (τd ) =
]
Q
+d
2
Q
+d−1 ;
2
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
21
while in the other cases,
(L + b) b (τd,h ) =
]
(12)
Q
− 1 + 2d
2
dv
+ 2h .
2
Since L] is M –invariant, it acts on V τ by scalar multiples, say
L ] Yτ = λ τ Yτ
then
bM
τ ∈K
Yτ ∈ V τ ;
λτ = (L] + b) b (τ ) − b.
Let α be real. We can define real powers of the conformal sublaplacian by requiring that
(L] + b)α Yτ = (λτ + b)α Yτ
Let f be in L2 (K/M ). According to (8), we write f as
in V τ and aτ in C. Suppose that
X
∀ Yτ ∈ V τ .
P
bM
τ ∈K
aτ Yτ , with the Yτ unit vectors
|aτ |2 (λτ + b)2α < ∞.
bM
τ ∈K
Then
(L] + b)α f =
X
(λτ + b)α aτ Yτ .
bM
τ ∈K
Definition. For real α, we define the Sobolev space Hα (K/M ) to be the completion of the
space of smooth functions on K/M , with respect to the norm
kf kHα (K/M ) = (L] + b)α/2 f L2 (K/M )
∀f ∈ C ∞ (K/M ).
We note that if α < β, then Hβ (K/M ) is continuously embedded in Hα (K/M ), for all
the eigenvalues λτ are nonnegative. The spaces Hα (K/M ) and H−α (K/M ) are dual with
respect to the L2 inner product.
Proposition 18. Suppose that α ≥ 2. Then f is in Hα (K/M ) if and only if f is in
L2 (K/M ) and L] f is in Hα−2 (K/M ); moreover the norms kf kHα (K/M ) and kf kL2 (K/M ) +
] L f α−2
are equivalent.
H
(K/M )
Proof. This is routine.
22
ASTENGO, COWLING AND DI BLASIO
Proposition 19. Suppose that α is in (−Q/2, Q/2) \ Z. Then the operator
f 7→ f ? D 2α−Q
is bounded with a bounded inverse from H−α (K/M ) to Hα (K/M ). Its norm is bounded by a
constant multiple of |c(α)|−1 , and its inverse is a constant multiple of
f 7→ c(α) c(−α) f ? D −2α−Q ,
where c(α) =
Γ((Q−2α)/4) Γ((dv +2−2α)/4)
.
Γ(α)
Proof. The Fourier transforms of D 2α−Q and (L] + b)α can be evaluated using formulae (9),
(10), (11) and (12). It is easy to check that
2α−Q b
]
α b
(τ ) (L + b)
(τ ) ≤ C |c(α)|−1
D
bM .
∀τ ∈ K
where C depends on dv and dz . Suppose that f is in C ∞ (K/M ) and f =
f ? D 2α−Q =
X
bM
τ ∈K
Moreover,
D 2α−Q b (τ ) aτ Yτ .
P
bM
τ ∈K
aτ Yτ . Then
2 X 2α−Q b 2
]
α/2 b
f ? D 2α−Q 2 α
=
(τ ) D
(τ ) |aτ |2
(L + b)
H (K/M )
bM
τ ∈K
=
X
bM
τ ∈K
2
|λτ + b|2α D 2α−Q b (τ ) |λτ + b|−α |aτ |2
≤ C |c(α)|−2
X
|λτ + b|−α |aτ |2
bM
τ ∈K
= C |c(α)|−2 (L] + b)−α/2 f L2 (K/M )
= C |c(α)|−2 kf k2H−α (K/M ) .
The inverse may be evaluated by observing that the product of the Fourier transforms of
D 2α−Q and D −2α−Q is
Γ((dv +dz +1)/2) 2
,
c(α)c(−α)
D
2α−Q
?D
i.e.,
−2a−Q
Γ ((dv + dz + 1)/2)2
δ,
=
c(α)c(−α)
where δ denotes the Dirac delta at the identity.
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
23
Analogous results can be proved for the group N . The Fourier transforms of powers of the
homogeneous norm and of the sublaplacian ∆ on N are well known [4, 7]. Similar arguments
to those we used for K/M yield the following proposition.
Proposition 20. Suppose that α is in (−Q/2, Q/2) \ Z. The map
F 7→ F ∗ N 2α−Q
is an invertible bounded operator from H−α (N ) to Hα (N ). Its norm is bounded by a constant
multiple of |c(α)|−1 , and its inverse is a constant multiple of
F 7→ c(α) c(−α) F ∗ N −2α−Q ,
where c(α) =
Γ((Q−2α)/4) Γ((dv +2−2α)/4)
.
Γ(α)
Theorem 21. i) If 0 ≤ α < Q/2, then Hα (K/M ) is contained in Lp,2 (K/M ), where 1/p =
1/2 − α/Q, and moreover
kf kLp,2 (K/M ) ≤ Cα kf kHα (K/M )
∀f ∈ Hα (K/M ).
ii) If f is a continuous function on K/M , then f is in the Sobolev space H α (K/M ) for every
α in (−Q/2, 0] and moreover
kf kHα (K/M ) ≤ Cα kf kLp,2 (K/M ) ,
where 1/p = 1/2 − α/Q.
Proof. The proofs of these results are variants of the proofs of Theorem 12. By using the
calculations of Lemma 19, one can prove that there exist constants c1 and c2 such that
c1 f ? D −2α−Q L2 (K/M) ≤ kf kHα (K/M ) ≤ c2 f ? D −2α−Q L2 (K/M )
∀f ∈ Hα (K/M ),
when α is in (−Q/2, 0]. We only need to know that D −2α−Q is in Lr,∞ (S) when 1/r = 1+α/Q.
This can be proved as in Theorem 9.
24
ASTENGO, COWLING AND DI BLASIO
4. The main result
The purpose of this section is to show that the operator Tα , defined by
]
1/2−α/Q
Tα F = JC −1
F ◦ C −1
∀F ∈ Cc∞ (N ),
is bounded with a bounded inverse from Hα (N ) to Hα (K/M ), when −Q/2 < α < Q/2.
Formally,
1/2−α/Q
Tα−1 f = JC
f[ ◦ C
∀f ∈ C ∞ (K/M ).
It is straightforward that T0 is a multiple of an isometry. Moreover, by Theorem 8, the
operator T1 is a multiple of an isometry.
The preceding formulae make sense also for complex values of α, and occasionally we shall
consider the operators Tα for complex α.
Lemma 22. Suppose that α is in C and Re(α) < 0. Then
T−α F ∗ N −2α−Q = σ(S) (Tα F ) ? D −2α−Q
∀F ∈ Cc∞ (N ).
Proof. Suppose that α is in C and Re(α) < 0. By Lemma 3, for every F in Cc∞ (N ),
Z
1/2+α/Q
−1
−2α−Q [
(ζ) = JC −1
(ζ)
F (n1 ) N −2α−Q (n−1
(ζ)) dn1
T−α F ∗ N
1 C
N
Z
1/2+α/Q
=
F (n1 ) JC
(n1 ) dS−2α−Q (C(n1 ), ζ) dn1
N
Z
1/2−α/Q
= JC −1
(ζ1 ) (F ◦ C −1 )(ζ1 ) d−2α−Q
(ζ1 , ζ)dσ(ζ1).
S
S
Therefore
T−α F ∗ N
−2α−Q
(kM ) = σ(S)
Z
K
Tα F (k1 M ) D −2α−Q (k1−1 kM ) dk1
= σ(S) (Tα F ) ? D −2α−Q ,
as required.
Our main result is the following relation between homogeneous Sobolev spaces on N and
Sobolev spaces on K/M . The real case of this theorem was proved by T. Branson [3].
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
25
Theorem 23. Suppose that −Q/2 < α < Q/2. Then the map
Tα : F 7→
J
α
1
−Q
2
−1
C
F ◦C
−1
]
is a bounded invertible operator from Hα (N ) to Hα (K/M ).
Proof. We suppose that Q > 4. When Q ≤ 4, matters can be simplified.
Let F be a smooth function with compact support on N . By Theorem 8 and Leibniz’
rule,
[
]
(L + b)(Tα F ) ◦ C =
(13)
−1/2−1/Q
JC
∆
1/2−1/Q
JC
[
(Tα F ) ◦ C
−1/2−1/Q
(α−1)/Q
= JC
∆ JC
F
"
= Tα−2
∆F + m(2)
α F +
dv
X
(1)
mα,j Ej F
j=1
!#[
◦ C,
where
2/Q
m(2)
α (X, Z) = (α − 1)JC
(1)
mα,j (X, Z)
(2)
= (1 −
(X, Z)
dv
2
+
Q
8
−
2/Q
α)JC (X, Z)hAX, Ej i,
α
4
|X|2
∀(X, Z) ∈ N.
(1)
Note that mα and mα,j are in M2,d and M1,d respectively, for every nonnegative integer d.
We extend the definition of Tα to obtain an analytic family of operators Tz for Re(z) in
[0, Q/2), by
]
1/2−z/Q
Tz F = JC −1
F ◦ C −1
∀F ∈ Cc∞ (N ).
It is easy to prove that, when Re(z) = 0, the operators Tz are multiples of isometries from
L2 (N ) to L2 (K/M ).
Suppose now that Re(z) = 2, i.e., z = 2 + iy where y is in R. Since T0 is bounded on L2
spaces, by Theorem 11, Theorem 14 and formula (13), we obtain
iy/Q iy/Q
= L ] + b T 2 JC F kT2+iy F kH2 (K/M ) = T2 JC F 2
H0 (K/M )
H (K/M)
!
dv
X
(1)
iy/Q
iy/Q
(2)
iy/Q
m2,j Ej JC F
= T 0 ∆ JC F + m 2 JC F +
j=0
L2 (K/M )
26
ASTENGO, COWLING AND DI BLASIO
≤C
iy/Q
∆(JC F )
iy/Q ≤ C JC F L2 (N )
H2 (N )
(2) iy/Q + m 2 JC F L2 (N )
≤ C kF kH2 (N ) .
dv X
(1)
iy/Q
+
m2,j Ej (JC F )
j=0
!
L2 (N )
By Stein’s complex interpolation theorem [18], the operators Tα are bounded from Hα (N )
to Hα (K/M ) for every α in [0, 2].
Q/2−1
We write [0, Q/2) = ∪h=0 [h, h + 1) and we proceed by induction on h. We have just
proved that Tα is bounded when α is in [0, 2]. Now suppose that Tα is bounded when α is
in [0, h + 1), and let α be in [h + 1, h + 2). By (13), the inductive hypothesis, Theorem 11
and Theorem 14,
kTα F kHα (K/M ) ≤ C
≤C
k∆F kHα−2 (N )
kF kHα (N ) +
+ m(2)
α F
dv
X
dv X
(1)
m
E
F
+
α,j j Hα−2 (N )
kEj F kHα−1 (N )
j=1
!
j=1
Hα−2 (N )
!
≤ C kF kHα (N ) .
This proves that Tα is bounded when α is in [0, Q/2).
Now let α be in (−Q/2, 0) \ Z and let F be in Hα (N ). By Proposition 19 and Lemma 22,
(Tα F ) = C c(α) c(−α) T−α F ∗ N −2α−Q ? D 2α−Q .
We have just proved that T−α is bounded from H−α (N ) to H−α (K/M ), therefore by Propositions 19 and 20, Tα is bounded from Hα (N ) to Hα (K/M ) when α is in (−Q/2, 0) \ Z. More
precisely,
kTα F kHα (K/M ) = C |c(α) c(−α)| T−α F ∗ N −2α−Q ? D 2α−Q Hα (K/M )
≤ C |c(−α)| F ∗ N −2α−Q H−α (N )
≤ C kF kHα (N ) .
In order to prove that Tα is bounded when α is an integer and −Q/2 < α < 0, we use
complex interpolation as before.
−1
Finally, denote by Tα∗ the adjoint of Tα . Notice that Tα∗ = T−α
. By duality, it follows that
Tα−1 is bounded from Hα (K/M ) to Hα (N ) when α is in (−Q/2, Q/2).
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
27
5. Representations
We now apply our work on the Cayley transformation to representation theory. Recall
that the simple group G acts on the quotient space G/P̄ , where P̄ is a parabolic subgroup
of G, and that G/P̄ may be identified with the one-point compactification N ∪ {∞} of N
or with K/M . This gives rise to (non-measure-preserving) actions of G on N ∪ {∞} and on
K/M (we will just write g · n and g · kM to indicate the result of applying g to n or to kM ).
In turn these actions give rise to representations of G on spaces of functions on N ∪ {∞}
and K/M .
For λ in C, where −Q/2 < Re(λ) < Q/2, we define representations πλ and π̃λ on functions
on N and on K/M by the formulae
πλ (g)f (n) =
π̃λ (g)f (kM ) =
d(g −1 · n)
dn
1/2−λ/Q
d(g −1 · kM )
dkM
f (g −1 · n)
1/2−λ/Q
f (g −1 · kM ),
where the “derivatives” are to be interpreted in the sense of Radon–Nikodym.
Now the action of G on K/M induces an action on S (denoted similarly), and clearly
d(g −1 · kM )
dσ(g −1 · ζ)
=
.
dσ(ζ)
d(kM )
Further, the Cayley transform C maps N ∪ {∞} bijectively to S, and if ζ = Cn, then
dσ(g −1 · ζ)
dσ(g −1 · Cn)
=
dσ(ζ)
dσ(Cn)
dσ(g −1 · Cn)
=
d(g −1 · n)
dσ(Cg −1 · n)
=
d(g −1 · n)
= JC (g
−1
dσ(Cn)
dn
dσ(Cn)
dn
· n)JC (n)
−1 d(g
−1
−1
−1
d(g −1 · n)
dn
d(g −1 · n)
dn
· n)
dn
.
Consequently, the representations πλ and π̃λ are linked by the equality
π̃λ (g)Tλ = Tλ πλ (g)
∀g ∈ G.
28
ASTENGO, COWLING AND DI BLASIO
Consider the Cartan decomposition G = KAK of the simple group G. Denote by at ,
where t is real, the element in A which acts on N by the dilation
at · (X, Z) = δet (X, Z) = (et X, e2t Z)
∀(X, Z) ∈ N.
Theorem 24. Suppose that λ ∈ C and α ∈ (−Q/2, Q/2). Then there exists a constant C
such that
kπ̃λ (kat k 0 ) f kHα (K/M) ≤ C cosh ((t(Re(λ) − α)) kf kHα (K/M )
∀t ∈ R
∀k, k 0 ∈ K,
for every f in C ∞ (K/M ).
Proof. It is clear that kπ̃λ (k) f kHα (K/M ) = kf kHα (K/M ) for all k in K, so it suffices to estimate
kπ̃λ (at ) f kHα (K/M ) . It easy to check that, for any at in A,
πλ (at ) F = e−Qt(1/2−λ/Q) F ◦ δe−t
∀t ∈ R,
F ∈ Cc∞ (N ).
Since ∆(F ◦ δe−t ) = e−2t (∆F ) ◦ δe−t ,
α/2 −Qt(1/2−λ/Q)
e
F ◦ δe−t kπλ (at ) F kHα (N ) = ∆
L2 (N )
α/2
−Qt(1/2−Re(λ)/Q)−αt −t
=e
∆ F ◦ δe L2 (N )
= et(Re(λ)−α) kF kHα (N ) .
Note that, for any f in C ∞ (K/M ),
π̃λ (at ) f = Tα πλ (at ) mλ−α,t Tα−1 f
,
(λ−α)/Q
−(λ−α)/Q
where mλ−α,t = JC
◦ δ e t JC
. Therefore, by Theorem 23 and Corollary 17,
kπ̃λ (at ) f kHα (K/M ) ≤ C et(Re(λ)−α) mλ−α,t Tα−1 f Hα (N )
≤ C et(Re(λ)−α) + e−t(Re(λ)−α) Tα−1 f Hα (N )
≤ C cosh(t(Re(λ) − α)) kf kHα (K/M ) ,
as required.
CAYLEY TRANSFORM AND UNIFORMLY BOUNDED REPRESENTATIONS
29
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30
ASTENGO, COWLING AND DI BLASIO
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Dipartimento di Matematica, Università di Genova, 16146 Genova, Italia
E-mail address: [email protected]
School of Mathematics, UNSW, Sydney NSW 2052, Australia
E-mail address: [email protected]
Dipartimento di Matematica, Università di Roma “Tor Vergata”, 00133 Roma, Italia
E-mail address: [email protected]