Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 498638, 13 pages
doi:10.1155/2011/498638
Research Article
Riesz Potential on the Heisenberg Group
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 24 November 2010; Accepted 17 February 2011
Academic Editor: Vijay Gupta
Copyright q 2011 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.
The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover, the Hardy-Littlewood-Sobolev inequality is established.
1. Introduction
The classical Riesz potential Iα is defined on Rn by
Iα f −Δ−α/2 f ,
0 < α < n,
1.1
where Δ is the Laplacian operator. By virtue of the equations
−Δ−α/2 f x 2π|x|−α fx,
|·|
−α
y γα2π y ,
−nα −α 1.2
where γα π n/2 2α Γα/2/Γn/2 − α/2, one can get the explicit expression of Riesz
potential
Iα f x 1
γα
Rn
f y
dy.
x − yn−α
In addition, one has the following Hardy-Littlewood-Sobolev theorem see 1.
1.3
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Journal of Inequalities and Applications
Theorem 1.1. Let 1 < α < n, 1 ≤ p < q < ∞, 1/q 1/p − α/n. One has the following.
a If p > 1, then Iα fq ≤ Ap,q fp ;
b if f ∈ L1 Rn , then for all λ > 0,
m x ∈ Rn : Iα f > λ ≤
q
Af 1
.
λ
1.4
In recent years many interesting works about the Riesz potential have been done by
many authors. Thangavelu and Xu 2 discussed the Riesz potential for the Dunkl transform.
Garofalo and Tyson 3 proved superposition principle Riesz potentials of nonnegative
continuous function on Lie groups of Heisenberg type. Huang and Liu 4 studied the HardyLittlewood-Sobolev inequality of this operator on the Laguerre hypergroup. For more results
about the Riesz potential, we refer the readers to see 5–9.
It is a remarkable fact that the Heisenberg group, denoted by H n , arises in two aspects.
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
notions of quantum mechanics. In other words, there is its genesis in the context of quantum
mechanics which emphasizes its symplectic role in the theory of theta functions and related
parts of analysis. Due to this reason, many interesting works were devoted to the theory of
harmonic analysis on H n in 10–15 and the references therein.
In present paper, we consider the Riesz potential associated with the Heisenberg
group. We will show a connection between the Riesz potential and the heat kernel, and then
get the Hardy-Littlewood-Sobolev inequality.
2. Preliminaries
The Heisenberg group H n is a Lie group with the underlying manifold Cn × R, the multiplication law is
z, t z , t z z , t t 1
Im zz ,
2
2.1
where zz nj1 zj zj . The dilation of H n is defined by δa z, t az, a2 t with a > 0. For
z, t ∈ H n , the homogeneous norm of z, t is given by
1/4
.
|z, t| z, t−1 |z|4 |t|2
2 1/4
Note that |δa z, t| |az|4 |a2 t| inequality
2.2
a|z, t|. In addition, | · | satisfies the quasi-triangle
z, t z , t ≤ |z, t| z , t .
2.3
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3
The ball of radius r centered at z, t is given by
z , t ∈ H n : z, t−1 z , t < r .
Br z, t 2.4
For 1 ≤ p ≤ ∞, let Lp H n be the space of measurable functions f on H n , such that
f p
1/p
fz, tp dz dt
< ∞,
if p ∈ 1, ∞,
Hn
f ess supfz, t < ∞,
∞
z,t∈H n
2.5
if p ∞.
Let πλ z, t z x iy, λ ∈ R∗ R/{0} be the Schrödinger representations which
acts on ϕ ∈ L2 Rn by
πλ z, tϕζ eiλt eiλx·ζ1/2x·y ϕ ζ y ,
2.6
where x · y nj1 xj yj . Suppose that f is a Schwartz function on H n , that is, f ∈ SH n . The
Fourier transform of f is defined by
fλ
Hn
fz, tπλ z, tdz dt.
2.7
fz, t πλ z, tϕ, ψ dz dt,
2.8
This means that, for each ϕ, ψ ∈ L2 Rn ,
ψ fλϕ,
Hn
where , denotes the inner product.
Let us write πλ z, t eiλt πλ z with πλ z πλ z, 0 and define
f z λ
∞
fz, teiλt dt.
2.9
f λ zπλ zdz.
2.10
gzπλ zdz,
2.11
−∞
Then 2.7 can be written as
fλ
Cn
If we set
Wλ g Cn
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Journal of Inequalities and Applications
then fλ
Wλ f λ . Let dμλ 2π−n−1 |λ|n dλ; one has the inversion of Fourier transform
fz, t ∞
−∞
dμλ,
tr πλ∗ z, tfλ
2.12
where πλ∗ z, t denotes the adjoint of πλ z, t.
The convolution of f and g is defined by
f ∗ gz, t fz, t−w, −sgw, sdw ds.
Hn
2.13
g λ. In addition, we have the generalized Yong inequality
It is clear that f
∗ gλ fλ
f ∗ g ≤ f g ,
r
p
q
2.14
where 1/r 1/p 1/q − 1. More details about the harmonic analysis on Heisenberg
group can be found in 14–16.
Let T be a mapping from Lp H n to Lq H n , 1 ≤ p, q ≤ ∞. Then T is of type p, q if
T f Lq H n ≤ AfLp H n ,
f ∈ Lp H n ,
2.15
where A does not depend on f. Similarly, T is of weak type p, q if
m z, t ∈ H n : T f z, t > λ ≤
q
Af p
λ
,
when q < ∞,
2.16
where A does not depend on f or λ λ > 0.
Let Sn be the unit sphere in H n and Σn−1 the unit Euclidean sphere in Rn . Suppose that
f is a measurable function on H n , and we have see 10
fz, tdz dt ∞ Hn
−∞
∞
Σ2n−1
f ρζ, t ρ2n−1 dρ dζ dt.
2.17
0
We set ρ2 r 2 cos θ, t r 2 sin θ, then
fz, tdz dt Hn
Σ2n−1
π/2 ∞ f rcos θ1/2 ζ, r 2 sin θ cos θn−1 r 2n1 dr dθ dζ.
−π/2
2.18
0
Hence
f dσ Sn
Σ2n−1
π/2 f cos θ1/2 ζ, sin θ cos θn−1 dθ dζ.
−π/2
2.19
Journal of Inequalities and Applications
5
A direct calculation shows that the area of Sn is
σSn 4π n1/2 Γ1/2
.
ΓnΓn 1/2
2.20
In addition, we have the volume of unit ball in H n
mBn 2π n1/2 Γ1/2
,
n 1ΓnΓn 1/2
2.21
and thus the volume of Br z, t
mBr z, t r 2n2 mBn .
2.22
For a radial function f, we have
fz, tdz dt σS n
Hn
∞
frr 2n1 dr.
2.23
0
The Hardy-Littlewood maximal operator M is defined on H n by
1
Mfz, t sup
mB
r
r>0
fz, t−w, −sdw ds,
Br
2.24
which is of type p, p for 1 < p ≤ ∞ and is of weak type 1,1 see 17, 18.
3. The Sublaplacian and the Heat Kernel on the Heisenberg Group
As it is known, the following vector fields
Xj ∂
1 ∂
− yj ,
∂xj 2 ∂t
j 1, 2, . . . , n,
Yj ∂
1 ∂
xj ,
∂yj 2 ∂t
j 1, 2, . . . , n,
3.1
∂
∂t
T
form a basis for the Lie algebra of left-invariant vector fields on H n . The sublaplacian is
defined by
L−
n Xj2 Yj2 ,
j1
3.2
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Journal of Inequalities and Applications
which also has another explicit form
1
L −Δz − |z|2 ∂2t N∂t ,
4
3.3
where Δz is the standard Laplacian on Cn and
N
n
j1
∂
∂
xj
− yj
∂yj
∂xj
.
3.4
For the Schrödinger representations πλ one easily calculates that
πλ∗ Xj ϕζ iλζj ϕζ,
∂
πλ∗ Yj ϕζ ϕζ.
∂ζj
3.5
So that πλ∗ L −Δ λ2 |ζ|2 Hλ.
Let hk t k ∈ Z {0, 1, 2, . . .} be the normalized Hermite functions given by
−1/2
√
2
Hk te−1/2t ,
hk t 2k πk!
3.6
where Hk t −1k dk /dtk {e−t }et . For α α1 , α2 , . . . , αn ∈ Zn and ζ ∈ Rn , we define
2
2
Φα ζ n
hαj ζj .
3.7
j1
Then {Φα } forms an orthonormal basis for L2 Rn .
We set Φλα ζ |λ|n/4 Φα |λ|1/2 ζ and denote
HλΦλα 2|α| n|λ|Φλα .
3.8
β
β
L πλ z, tΦλα , Φα 2|α| n|λ| πλ z, tΦλα , Φα .
3.9
Lfλ
fλHλ.
3.10
Therefore,
Moreover, one has
Now let Leiλt fz eiλt Lλ fz, then Lλ has the form
1
Lλ −Δz λ2 |z|2 iλN.
4
3.11
Journal of Inequalities and Applications
7
From 3.9 we know that the functions
Φλα,β z 2π−n/2 |λ|n/2 πλ zΦλα , Φλβ
3.12
are eigenfunctions of the operator Lλ :
Lλ Φλα,β z 2|α| n|λ|Φλα,β z.
3.13
z be the Laguerre functions defined on Cn by
Let ϕn−1
k
ϕn−1
k z
Ln−1
k
1 2 −|z|2 /4
|z| e
2
3.14
√
z ϕn−1
λz for λ ∈ R∗ . Then from 19, 2.3.26 we have
and set ϕn−1
k,λ
k
|λ|n/2
|α|k
Φλα,α z 2π−n/2 ϕn−1
k,λ z.
3.15
In view of this equation we have the following.
Proposition 3.1. One has
Wλ ϕn−1
2πn |λ|−n Pk λ,
k,λ
3.16
where Pk λ stands for the projection of L2 Rn onto the kth eigenspace of Hλ, that is,
Pk λϕ |α|k
ϕ, Φλα Φλα .
3.17
Now we consider the heat equation associated to the sublaplacian
∂s Fz, t; s −LFz, t; s
3.18
with the initial condition Fz, t; 0 fz, t. In fact, the function qs given by
qs z, t cn
∞
−∞
e
−iλt
λ
sinh λs
n
2
e−1/4λ coth λs|z| dλ
3.19
is just the solution of the heat equation and satisfies
Fz, t; s e−sL f z, t qs ∗ fz, t.
3.20
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Journal of Inequalities and Applications
Moreover, we have the Fourier transform of qs z, t see 18, page 86
qs λ 2π−n |λ|n
∞
e−2kn|λ|s Wλ ϕn−1
k,λ .
3.21
k0
4. Riesz Potential on the Heisenberg Group
In Section 1 we have recalled some properties about the Riesz potential on Rn ; now we are
going to discuss the Riesz potential on the Heisenberg group.
Definition 4.1. For 0 < γ < 2n 2, the Riesz potential Iγ is defined on SH n by
Iγ fz, t L−γ/2 fz, t.
4.1
From above definition and 3.10 it is easy to see that
−γ/2
.
I
γ fλ fλHλ
4.2
If γ, τ > 0, γ τ < 2n 2, then we have
Iγ Iτ f λ Iτ f λHλ−γ/2
−γτ/2
fλHλ
Iγτ f λ,
4.3
which suggests that Iγ Iτ f Iγτ f. Especially, for 2 ≤ γ < 2n 2, one has
L Iγ f Iγ Lf Iγ−2 f.
4.4
At present we do not prepare to gain the expression of Iγ analogues to 1.3 because
it is hard to calculate the Fourier transform of |z, t|. But the following theorem will give us
another expression of Iγ , which provides a bridge to discuss the boundedness of the Riesz
potential.
Theorem 4.2. Let qs z, t be the heat kernel on H n . For 0 < γ < 2n 2, one has for f ∈ SH n ∞ −1
γ
Iγ fz, t Γ
sγ/2−1 qs ·ds ∗ fz, t.
2
0
4.5
Journal of Inequalities and Applications
9
Proof. By 3.17, 3.21, and Proposition 3.1 we have
∞ −1
γ
γ/2−1
λ
λ
fλ
Γ
s
qs λds Φα , Φβ
2
0
∞ −1
∞
γ
n
−n
γ/2−1
−2kn|λ|s
n−1
λ
λ
fλ
Γ
s
e
Wλ ϕk,λ ds Φα , Φβ
2π |λ|
2
0
k0
∞ ∞ −1
γ
γ/2−1 −2kn|λ|s
λ
λ
Γ
s
e
ds P k λΦα , Φβ
fλ
2
k0 0
∞
−γ/2
λ
λ
Pk λΦ , Φ
fλ
2k n|λ|s
α
4.6
β
k0
−γ/2 λ
Φα , Φλβ
fλHλ
λ
λ
I
γ fλΦα , Φβ .
Then we get the desired result.
Lemma 4.3. The heat kernel qs z, t satisfies the estimate
1/2
qs z, t ≤ Cs−n−1 eA/s|z,t|
4.7
with some positive constants C and A.
1/2
Proof. Since |z|4 |t|2 ≤ |z|2 |t|, then by 19, Proposition 2.8.2 we obtain this lemma.
The following theorem is an immediate consequence of Theorem 4.2 and Lemma 4.3.
Theorem 4.4. The Riesz potential Iγ satisfies the estimate
Iγ fz, t ≤ Cf ∗ |w, s|γ−2n2 z, t,
4.8
where C is a positive constant.
Using Theorems 4.2 and 4.4, we get the Hardy-Littlewood-Sobolev theorem on the
Heisenberg group.
Theorem 4.5. Let 0 < γ < 2n 2, 1 ≤ p < q < ∞, and 1/p 1/q γ/2n 2. For f ∈ Lp H n ,
one has the following.
a If p > 1, then Iγ is of type p, q.
b If p 1, then Iγ is of weak type 1, q.
Proof. Let M be the maximal operator defined by 2.24. We claim that
f ∗ |·|γ−2n2 z, t ≤ c Mfz, t p/q f 1−p/q .
p
4.9
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Journal of Inequalities and Applications
Let BR be the ball of radius R centered at 0,0, and let χBR be its characteristic function. We
set
|w, s|γ−2n2 g1 g2
4.10
with g1 w, s |w, s|γ−2n2 χBR w, s. Obviously, g1 > 0 is radial and decreasing, then we
can write
g1 N
aj χBj ,
4.11
j1
where aj > 0 and Bj is the ball centered at origin. By 2.23 and 2.24 we have
f ∗ g1 z, t Hn
N
fz, t−w, −s aj χB w, sdw ds
j
j1
N
m Bj
fz, t−w, −sdw ds
aj m Bj Bj
j1
≤
N
aj m Bj Mfz, t
4.12
j1
σS n
R
r γ−2n2 r 2n1 drMfz, t
0
c1 Rγ Mfz, t.
Let p be the conjugate exponent of p. Since 1/p − 1/q γ/2n 2, we have
g2 p
∞
1/p
σSn r γ−2n2p r 2n1 dr
R
−2n2/q
c2 R
4.13
.
Then by Hölder’s inequality we get
f ∗ g2 z, t ≤ c2 R−2n2/q f .
p
4.14
f ∗ |w, s|γ−2n2 z, t ≤ c1 Rγ Mfz, t c2 R−2n2/q f .
p
4.15
Therefore,
Journal of Inequalities and Applications
11
We choose R such that
c1 Rγ Mfz, t c2 R−2n2/q f p .
4.16
That is, R c3 Mfz, t/fp −p/2n2 . Substituting this in the above then gives 4.9.
Now if p > 1, we obtain a by the virtue of the type p, p of the maximal operator M.
If p 1 and q 2n 2/2n 2 − γ, we have
{z,t∈H n :|Iγ fz,t|>λ}
dz dt ≤
1−1/q
≤
≤
{z,t:cMfz,t1/q f1
dz dt
>λ}
1−q
{z,t:Mfz,t>λ/cq f1 }
dz dt
4.17
c q
f .
1
λ
This proves our main theorem.
Theorem 4.6. Let 0 < γ < 2n 2 and 1 ≤ p < q < ∞. For f ∈ Lp H n , one has the following.
a If p 1, then the condition 1/q γ/2n 2 1 is necessary and sufficient for the weak
type 1, q of Iγ .
b If 1 < p < q < ∞, then the condition 1/p 1/q γ/2n 2 is necessary and sufficient
for the type p, q of Iγ .
Proof. The sufficiency follows from Theorem 4.5. We now begin to prove the necessity.
Suppose that f ∈ SH n , and let fa z, t fδa z, t. Note that
qa2 s z, t a−2n2 qs δa−1 z, t.
4.18
Then by 4.9 we get
∞ −1
γ
Γ
sγ/2−1 qs z , t dsf −z , −t az, a2 t dz dt
Iγ fa z, t 2
Hn 0
∞ −1 γ/2−1
γ
a−2 s
Γ
qa−2 s z , t da2 sf δa −a−1 z , −a−2 t z, t dz dt
2
Hn 0
∞ −1
γ
Γ
a−γ sγ/2−1 qs δa−1 z , t dsf δa δa−1 z , t z, t da−1 z da−2 t
2
Hn 0
a−γ Iγ f a z, t.
4.19
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Journal of Inequalities and Applications
Thus for 1 ≤ q < ∞, we have
Iγ fa a−γ−2n2/q Iγ f q
q
1/q
1/q
m z, t ∈ H n : Iγ fa z, t > λ
a−γ−2n2/q m z, t ∈ H n : Iγ fz, t > λ
.
4.20
Case 1 p 1. It follows from the hypothesis that
m z, t ∈ H n : Iγ fz, t > λ aqγ2n2 m z, t ∈ H n : Iγ fa z, t > λ
q
Afa 1
qγ2n2
≤a
λ
q
Af 1
qγ2n2−q2n2
a
.
λ
4.21
If 1/qγ/2n2 > 1, then m{z, t ∈ H n : |Iγ fz, t| > λ} 0 as a → 0. If 1/qγ/2n2 < 1,
then m{z, t ∈ H n : |Iγ fz, t| > λ} 0 as a → ∞. Thus we have 1/q γ/2n 2 1.
Case 2 1 < p < q < ∞. Similarly, we have
Iγ f aγ2n2/q Iγ fa q
q
≤ caγ2n2/q−2n2/p f p ,
4.22
which implies that 1/p 1/q γ/2n 2.
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. 10971039 and the Doctoral Program Foundation of the Ministry of China no.
200810780002. The authors are also grateful for the referee for the valuable suggestions.
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