Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 132186, 12 pages
doi:10.1155/2010/132186
Research Article
The Calderón Reproducing Formula Associated
with the Heisenberg Group Hd
Jinsen Xiao and Jianxun He
Department of Mathematics, School of Mathematics and Information Sciences, Guangzhou University,
Guangzhou 510006, China
Correspondence should be addressed to Jianxun He, h jianxun@hotmail.com
Received 9 October 2009; Accepted 20 May 2010
Academic Editor: Victoria Vampa
Copyright q 2010 J. Xiao and J. He. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We establish the Calderón reproducing formula for functions in L2 on the Heisenberg group Hd .
Also, we develop this formula in Lp Hd with 1 < p < ∞.
1. Introduction
The classical Calderón reproducing formula reads
f
∞
f ∗ φt ∗ ψt
0
dt
,
t
1.1
where φt x t−1 φx/t, ψt x t−1 ψx/t, and ∗ denotes the convolution on R. The
Calderón reproducing formula is a useful tool in pure and applied mathematics see 1–
4, particularly in wavelet theory see 5, 6. We always call 1.1 an inverse formula of
wavelet transform. In 7, the authors generalized 1.1 to Rn when φ and ψ are sufficiently
nice normalized radial wavelet functions. The generalization of 1.1 involving nonradial
wavelets φ and ψ can be written in the following form:
∞
f
dγ
SOn
0
f ∗ φγ,t ∗ ψγ,t
dt
,
t
1.2
where φγ,t and ψγ,t are rotated versions of φ and ψ on Rn . The authors in 8, 9 established
1.1 for f ∈ Lp R. Holschneider 10 studied the formula 1.2 in case f ∈ Lp R2 and gave
an inversion formula of the Radon transform in Lp -space by using wavelets. Furthermore,
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Mathematical Problems in Engineering
Rubin 4 developed the Calderón reproducing formula, windowed X-ray transforms, the
Radon transforms, and k-plane transforms in Lp -spaces on Rn .
It is a remarkable fact that the Heisenberg group, denoted by Hd , arises in two
fundamental but different setting in analysis. On the one hand, it can be realized as the
boundary of the unit ball in several complex variables. On the other hand, an important
aspect of the study of the Heisenberg group is the background of physics, namely, the
mathematical ideas connected with the fundamental nations of quantum mechanics. In
other words, there is its genesis in the context of quantum mechanics which emphasizes its
symplectic role in connection with the Fourier transform, pseudodifferential operators, and
related matters see 11. Due to this reason, many interesting works were devoted to the
theory of harmonic analysis on Hd in 11–13 and the references therein. Also, the researches
of wavelet analysis on Hd are concerned increasingly; for this we refer readers to 14–16.
And the inversion formula of the Radon transform by using inverse wavelet transform on Hd
was established in 17. Our goal of the present article is to study the Calderón reproducing
formula on the Heisenberg group in Lp -space with 1 < p < ∞. In the sequel we will develop
the theory of inverse Radon transform on Hd .
The Heisenberg group Hd is a Lie group with the underlying manifold Cd × R, and the
multiplication law is given by
z, t z , t z z , t t 2 Im zz ,
1.3
where zz dj1 zj zj . The dilation of Hd is defined by ρz, t ρz, ρ2 t with ρ > 0. For
z, t ∈ Hd , the homogeneous norm of z, t is given by see 11
|z, t| z, t−1 max |z|, |t|1/2 .
1.4
Notice that |ρz, t| max{|ρz|, |ρ2 t|1/2 } ρ|z, t|. In addition, | · | satisfies the quasitriangle
inequality:
z, t z , t ≤ |z, t| z , t .
1.5
The homogeneous dimension of Hd is 2d 2, and the volume of a ball Bz, t, r {z , t ∈
Hd : |z, t−1 z , t | ≤ r} is c r 2d2 , where c is a constant.
Let P {z, t, ρ : z, t ∈ Hd , ρ > 0}; then P is a locally compact nonunimodular group
with the group law
z, t, ρ z , t , ρ z ρz , t ρ2 t 2ρ Im zz , ρρ .
1.6
The left and right Haar measures on P are given by
dz dt dρ
dμl z, t, ρ ,
ρ2d3
where dz denotes the Lebesgue measure on Cd .
dz dt dρ
,
dμr z, t, ρ ρ
1.7
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3
Let Lp Hd be the space of measurable functions f on Hd , such that
f p d L H Hd
1/p
fz, tp dz dt
< ∞,
f p d ess sup fz, t < ∞,
L H z,t∈Hd
if p ∈ 1, ∞,
if p ∞.
1.8
Let Z {0, 1, 2, . . .} and α α1 , α2 , . . . , αd ∈ Z d . The Fock space H is the space of
holomorphic functions F on Cd such that
F2H d 2
2
|Fζ|2 e−2|ζ| dζ < ∞.
π
Cd
1.9
√
√
From 18 we know that {Eα ζ 2ζα / α! : α ∈ Z d } is an orthonormal basis of the
Hilbert space H. For λ ∈ R \ {0}, let πλ be the Bargmann-Fock representation of Hd which
acts on H by
√
⎧
√ 2
⎨e−iλt−λ|z| 2 λζz F ζ − λz ,
√
πλ z, tFζ ⎩e−iλtλ|z|2 −2 |λ|ζz F ζ √λz ,
if λ > 0,
if λ < 0.
1.10
The group Fourier transform of a function f ∈ L1 Hd is defined by
fλ
Hd
1.11
fz, tπλ z, tdz dt.
Let Sp H 1 ≤ p < ∞ be the classes of Schatten-von Neumann operators on Hilbert
space H, and let S∞ H denote the algebra of all bounded operators, that is, S∞ H BH.
For T ∈ Sp H, let T p trT ∗ T 1/p p/2 denote the Sp -norm of T. If p 2, T 2 is just the
Hilbert-Schmit norm of T, that denotes T HS . Let T ∞ denote the usual operator norm of
T in S∞ H. For 1 ≤ p ≤ ∞, let Lp be the Banach space consisting of all weak measurable
operator value functions F, which also satisfy Fλ ∈ Sp H|λ| , a.e. λ ∈ R \ {0}, and
FLp 1
π d1
1/p
R\{0}
p
Fλp |λ|d dλ
< ∞,
if 1 ≤ p < ∞,
1.12
FLp ess sup Fλ∞ < ∞,
if p ∞.
λ∈R\{0}
For f, g ∈ L2 Hd , the Parseval formula is
2d−1
f, g L2 Hd d1
π
∞
−∞
tr g λ∗ fλ
|λ|d dλ,
1.13
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Mathematical Problems in Engineering
where g λ∗ denotes the adjoint of g λ. The Plancherel formula is
2
f 2
L Hd 2d−1
d1
π
∞ 2
fλ |λ|d dλ.
−∞
HS
1.14
As a consequence of 1.13, one has the inversion of the Fourier transform:
fz, t 2d−1
π d1
∞
−∞
tr πλ z, t∗ fλ
|λ|d dλ.
1.15
Suppose ρ > 0, and let
z t
fρ z, t ρ−2d2 f ρ−1 z, t ρ−2d2 f
, 2 .
ρ ρ
1.16
By a direct computation, we have
fρ λ f ρλ .
1.17
Let f ∗ g be the convolution of f and g, that is,
f ∗ gz, t Hd
−1
f z , t g z , t z, t dz dt .
1.18
Then
g λ.
f
∗ gλ fλ
1.19
t fz, t−1 f−z, −t, then
We should notice the following facts: if fz,
∗.
fλ
fλ
1.20
g λ f−λ.
1.21
And if gz, t f−z, −t, then
The further detail of harmonic analysis on Hd can be found in 11, 12.
Mathematical Problems in Engineering
5
2. Calderón Reproducing Formula
The authors in 14, 15, 18 studied the theory of continuous wavelet associated with the
concept of square integrable group representations. The unitary representation of P on L2 Hd is defined by
Uz,t,ρ f z , t ρ
−d1
f
z − z t − t − 2 Im zz
,
ρ
ρ2
.
2.1
Let R denote the set of all positive real numbers, R− −R . Let Pα α ∈ Z d be the
projection from L2 Rd to 1-dimensional subspace spanned by Eα , and let σ or −,
Hασ f ∈ L2 Hd : fλ
fλP
∈ Rσ .
α , and fλ 0 if λ /
2.2
From 15, Theorem 1, we have
L2 Hd Hα ⊕ Hα− .
2.3
α∈Z d
Let α ∈ Z d and σ or −, φ ∈ Hασ ; if φ / 0 and satisfies
Cφ ∗ φλ
dλ Eα , Eα
φλ
|λ|
Rσ
H
< ∞,
2.4
then we call φ an admissible wavelet and write φ ∈ AWσα . Let φ ∈ AWσα , f ∈ Hασ ; the
continuous wavelet transform of f with respect to φ is defined by
Wφ f z, t, ρ f, Uz,t,ρ φ L2 Hd .
2.5
And the following Calderón reproducing formula holds in the weak sense:
f z , t Cφ−1
∞ 0
dz dt dρ
Wφ f z, t, ρ Uz,t,ρ φ z , t
.
ρ2d3
Hd
2.6
2.1. Calderón Reproducing Formula in L2 Hd By 1.16 and 1.18, we can rewrite 2.5 and 2.6 as follows:
ρ z, t,
Wφ f z, t, ρ ρd1 f ∗ φ
∞
−1
ρ z, t dρ .
fz, t Cφ
f ∗ φρ ∗ φ
ρ
0
2.7
2.8
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Mathematical Problems in Engineering
For 0 < ε < η < ∞, let
η
ρ z, t dρ ,
f ∗ φρ ∗ φ
ρ
ε
η
ρ z, t dρ ,
Φε,η z, t Cφ−1
φρ ∗ φ
ρ
ε
fε,η z, t Cφ−1
2.9
2.10
then fε,η z, t f ∗ Φε,η z, t. We are now in a position to show that fε,η converges to f in
L2 -space when ε → 0 and η → ∞. The result in this paper is an extension of that of Mourou
and Trimèche 19.
∈ S∞ H. Let Φε,η be defined by
Lemma 2.1. Suppose that φ ∈ AWσα and φ ∈ Hασ satisfies φλ
2.10. Then one has Φε,η ∈ L2 Hd .
Proof. By Hölder’s inequality, we have
Φε,η 2 ≤ C−2
φ
η 2 dρ η dρ
.
φρ ∗ φρ z, t
ρ ε ρ
ε
2.11
Thus
Hd
Φε,η 2 dz dt ≤ C−2
φ
η
2
dρ
ρ z, t dz dt dρ
.
φρ ∗ φ
ρ
ε ρ
Hd
η ε
2.12
By 1.14, 1.19, and 1.20, we have
∞ 2
2
d−1
∗
ρ z, t dz dt 2
φρ ∗ φ
φρλφρλ
|λ|d dλ
d1
HS
d
π
−∞
H
2d−1 ∞ ∗ ∗ d
tr φ ρλ φ ρλ φ ρλ φ ρλ |λ| dλ.
d1
π
−∞
2.13
Noticing that
∗ ∗
∗
∗
tr φ ρλ φ ρλ φ ρλ φ ρλ φ ρλ φ ρλ Eα , φ ρλ φ ρλ Eα ,
H
α
2.14
we obtain
2 ∗
∗
∗
tr φ ρλ φ ρλ φ ρλ φ ρλ ≤ φ ρλ tr φ ρλ φ ρλ .
∞
2.15
Mathematical Problems in Engineering
7
Thus we get
2
2d−1 ∞ ∗
2
φ
∗
φ
dz
dt
≤
tr
t
φ ρλ φ ρλ |λ|d dλ
φρ
ρ z, ∞
d1
L
d
π
−∞
H
∞ d−1 2
∗ 2 tr φ ρλ φ ρλ |λ|d dλ
d1 φ ∞
L
π
−∞
2 2
φ ∞ φρ L2 Hd .
2.16
L
Therefore,
Hd
η 2
dρ η dρ
2 φ ∞ φρ L2 Hd φ
L
ρ ε ρ
ε
η
2 dρ η dρ
2
Cφ−2 φ ∞ φρ L2 Hd ρ−2d1
L
ρ ε ρ
ε
Φε,η 2 dz dt ≤ C−2
2.17
< ∞.
Then we complete the proof of this lemma.
Theorem 2.2. Let φ ∈ AWσα and φ ∈ Hασ satisfy φλ
∈ S∞ H. Then for f ∈ Hασ , one has
fε,η ∈ L2 Hd and limε → 0,η → ∞ fε,η f.
Proof. Notice that
fε,η z, t f ∗ Φε,η z, t,
2.18
and by 1.19 and Lemma 2.1 we deduce fε,η ∈ L2 Hd . Then by 1.17 and 1.19, we have
lim
ε→0
η→∞
Hd
fε,η z, tπλ z, tdz dt Eα , Eα
H
lim
ε→0
η→∞
Hd
f ∗ Φε,η z, tπλ z, tdz dt Eα , Eα
H
η
∗ dρ
−1
Eα , Eα
lim Cφ
fλφ ρλ φ ρλ
ε→0
ρ
ε
H
2.19
η→∞
lim Cφ−1
ε→0
η→∞
η
ε
fλE
α , Eα
∗ dλ
Eα , Eα
φασ λ φασ λ
|λ |
H
H
fασ λEα , Eα
,
where σ if λ > 0, otherwise σ −. By 1.11 we get the desired result.
H
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Mathematical Problems in Engineering
2.2. Calderón Reproducing Formula in Lp Hd with 1 < p < ∞
For f ∈ Lp Hd with 1 < p < ∞, the continuous wavelet transform of f with respect to a
wavelet φ can be defined by formula 2.7 under certain conditions on φ. In this part we will
show that fε,η converges to f.
Let f be a measurable function on Hd ; for z, t ∈ Hd , define
f ∗ z, t ess sup f z , t : z , t ∈ Hd , z , t > |z, t| .
2.20
It is easy to see that f ∗ is nonnegative and radially decreasing, that is,
0 ≤ f ∗ z1 , t1 ≤ f ∗ z2 , t2 only if |z1 , t1 | > |z2 , t2 |,
2.21
and f ∗ ≥ |f| a.e. on Hd .
Let g ∈ L1 R . Then for any z, t ∈ Hd , we define
∗
G z, t sup ρ
∞
gsds : ρ > |z, t| ,
ρ
−2d2 2.22
and thus we have the following lemma.
Lemma 2.3. Let g ∈ L1 R and G∗ z, t be defined by 2.22. Then one has
Hd \B
G∗ z, tdz dt ≤ c
∞
gs ln s ds,
2.23
1
where c is a positive constant, B B0, 0, 1.
Proof. First we let
Gz, t |z, t|
−2d2
∞
|z,t|
gsds.
2.24
It is obvious that G∗ z, t ≤ Gz, t for any z, t ∈ Hd \ {0, 0}. Since Gz, t is nonnegative
and radially decreasing, from 11, page 542, we know that there exists a positive constant c
such that
Hd
Gz, tdz dt c
∞
0
Grr 2d1 dr.
2.25
Mathematical Problems in Engineering
9
Thus,
∗
Hd \B
G z, tdz dt ≤
c
Hd \B
∞
Gz, tdz dt
Grr 2d1 dr
1
c
∞
r
−2d2
1
c
gsds r 2d1 dr
2.26
r
∞ s
gsds dr
r
1
1
c
∞
∞
gs ln s ds.
1
This completes the proof.
Let k be a radial function in Lp Hd 1 < p < ∞ and let
Kz, t |z, t|−2d2
|z,t|
ksωs2d1 ds
2.27
0
for any z, t ∈ Hd \ {0, 0}, where ω is a unit vector of Hd . Then we have the following
lemma.
Lemma 2.4. Let k be a radial function in Lp Hd 1 < p < ∞ and let Kz, t be defined by k in
2.27. If k satisfies Hd kz, tdz dt 0, and kz, t ln |z, t| ∈ L1 Hd , then K ∈ L1 Hd .
Proof. Analogous to 2.20 we define
K ∗ z, t ess sup K z , t : z , t ∈ Hd , z , t > |z, t| ,
and then K ∗ ≥ |K| a.e. on Hd .
2.28
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Mathematical Problems in Engineering
Let z, t, z , t ∈ Hd and 0 < |z, t| < |z , t |; by 2.25 we have
K z , t ≤
≤
|z ,t |
2d1
ksωs
ds
|z , t |2d2 0
1
|z ,t |
1
|z , t |2d2
1
c|z , t |
|ksω|s2d1 ds
0
2d2
B0,0,|z ,t |
k z , t dz dt
22d2
2d2
c|z, t| |z , t |
Bz,t,|z,t||z ,t |
2.29
k z , t dz dt
≤ 22d2 Mkz, t,
where Mk is the Hardy-Littlewood maximal function of k. From the definition of K ∗ , we
have K ∗ ≤ 22d2 Mk. Because k ∈ Lp Hd , we get K ∗ ∈ Lp Hd .
By the hypothesis k is radial and Hd kz, tdz dt 0; together with the definition of K,
we have
∞
2d1
ksωs
ds
|Kz, t| .
|z, t|2d2 |z,t|
1
2.30
Since kz, tln |z, t| ∈ L1 Hd , it follows from Lemma 2.3 that Hd \B K ∗ z, tdz dt < ∞,
that is, K ∗ ∈ L1loc Hd . On the other hand, K ∗ ∈ Lp Hd ⊂ L1loc Hd , and thus K ∗ ∈ L1 Hd ,
which implies that K ∈ L1 Hd .
Without loss of generality, we assume that Cφ 1; then 2.9 and 2.10 can be written
as
Φε,η z, t fε,η z, t η
η
ε
ρ z, t dρ ,
φρ ∗ φ
ρ
2.31
ρ z, t dρ .
f ∗ φρ ∗ φ
ρ
2.32
ε
In fact, Φε,η is always stated under conditions on k : φ∗φ rather than under conditions
on φ for convenience see 4, 10. By Lemma 2.4 we have the following theorem.
Theorem 2.5. Let k be in the conditions of Lemma 2.4 and let K be defined by 2.27. Suppose φ ∗ φ k and Hd Kz, tdz dt 1. Then for f ∈ Lp Hd , one has limε → 0,η → ∞ fε,η f.
Proof. From 2.31 we have
Φε,η z, t Φε,∞ z, t − Φη,∞ z, t.
2.33
Mathematical Problems in Engineering
11
t, and φz,
t φ−z, −t, we deduce
Since kz, t φ ∗ φz,
kρ z, t ρ−2d2
ρ−4d4
ρ−4d4
Hd
Hd
−1 φ z , t φ ρ−1 z, ρ−2 t
z , t dz dt
d ρ z d ρ2 t
φ z , t φ ρ−1 z, t−1 ρz , ρ2 t
Hd
φ ρ
z , t φ ρ−1 z, t−1 z , t
dz dt
2.34
−1
ρ z, t.
φρ ∗ φ
Then we have
∞
dρ
ρ
ε
∞ z, t dρ
k
ρ
ρ2d3
ε
∞
dρ
1
z, t
k
2d2
2d3
ρ|z,
t|
ρ
|z, t|
ε/|z,t|
|z,t|/ε ρz, t 2d1
1
k
dρ
ρ
|z, t|
|z, t|2d2 0
|z,t|/ε
ε−2d1
k ρω ρ2d1 dρ
2d2
|z, t/ε|
0
Φε,∞ z, t kρ z, t
2.35
Kε z, t.
By Lemma 2.4 together with the approximation of the identity, we have
lim fε,η z, t lim Φε,η ∗ fz, t lim Kε ∗ fz, t − lim Kη ∗ fz, t fz, t.
ε→0
η→∞
ε→0
η→∞
ε→0
η→∞
2.36
Then we complete the proof of this theorem.
Acknowledgments
The work for this paper is supported by the National Natural Science Foundation of China
no.s 10671041 and 10971039 and the Doctoral Program Foundation of the Ministry of
Education of China no. 200810780002.
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Mathematical Problems in Engineering
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