Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 741095, 13 pages
doi:10.1155/2011/741095
Research Article
Littlewood-Paley g-Functions and Multipliers for
the Laguerre Hypergroup
Jizheng Huang
1, 2
1
College of Sciences, North China University of Technology, Beijing 100144, China
2
CEMA, Central University of Finance and Economics, Beijing 100081, China
Correspondence should be addressed to Jizheng Huang, [email protected]
Received 4 November 2010; Accepted 13 January 2011
Academic Editor: Shusen Ding
Copyright q 2011 Jizheng Huang. 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.
Let L −∂
2
/∂x
2
2α 1/x∂/∂xx
2
∂
2
/∂t
2
; x, t ∈ 0, ∞ × R, where α ≥ 0. Then L can
generate a hypergroup which is called Laguerre hypergroup, and we denote this hypergroup by K.
i
, 1.1
where α α
1
, ,α
d
,x
i
> 0, then define the following Littlewood-Paley function G
α
by
G
α
f
x
∞
0
t∇
α
P
α
t
f
α
t
is the Poisson semigroup associated to L
α
.In1,
the authors prove that G
α
is bounded on L
p
μ
α
for 1 <p<∞. In this paper, we consider the
following differential operator
L −
∂
2
∂x
2
2α 1
x
∂
∂x
x
2
∂
2
∂t
2
x
2α1
dxdt, α ≥ 0. 1.4
We denotes by L
p
α
K the spaces of measurable functions on K such that f
α,p
< ∞, where
f
α,p
K
f
x, t
p
dm
α
x, t
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1
2π
2π
0
f
x
2
y
2
2xy cos θ,s t xy sin θ
dθ, if α 0,
α
x,t
f
α,p
≤f
α,p
. 1.7
Let M
b
K denote the space of bounded Radon measures on K. The convolution on M
b
K
is defined by
μ ∗ ν
f
K×K
T
α
x,t
f
y, s
T
α
x,t
f
y, s
g
y, −s
dm
α
y, s
. 1.9
The following lemma follows from 1.7.
Journal of Inequalities and Applications 3
Lemma 1.1. Let f ∈ L
1
α
K and g ∈ L
p
α
K, 1 ≤ p ≤∞.Then
f ∗ g
α,p
−2α4
f
x
r
,
t
r
2
. 1.12
Then we have
f
r
α,1
f
α,1
. 1.13
We also introduce a homogeneous norm defined by x, t x
4
4t
2
1/4
cf. 7. Then we
can defined the ball centered at 0, 0 of radius r, that is, the set B
r
{x, t ∈ K : x, t <r}.
Let f ∈ L
f
ρ
cos θ
1/2
,
1
2
ρ
2
sin θ
ρ
2α3
cos θ
α
dρdθ.
1.14
If f is radial, that is, there ia a function ψ on 0, ∞ such that fx, tψx, t, then
K
f
x, t
dm
α 1
/2
2
√
π Γ
α 1
Γ
α/2 1
∞
0
ψ
ρ
ρ
2α3
dρ.
1.15
Specifically,
m
α
∂
2
∂x
2
2α 1
x
∂
∂x
x
2
∂
2
∂t
2
. 1.17
4 Journal of Inequalities and Applications
L is positive and symmetric in L
2
α
K, and is homogeneous of degree 2 with respect to the
dilations defined above. When α n − 1, L is the radial part of the sublaplacian on the
Heisenberg group H
n
. We call L the generalized sublaplacian.
Let L
α
m
be the Laguerre polynomial of degree m and order α defined in terms of the
m! Γ
α 1
Γ
m α 1
e
iλt
e
−1/2|λ|x
2
L
α
m
|
λ
|
x
2
. 1.19
The following proposition summarizes some basic properties of functions ϕ
λ,m
.
Proposition 1.2. The function ϕ
λ, m
K
f
x, t
ϕ
−λ,m
x, t
dm
α
x, t
. 1.20
It is easy to show that
f ∗ g
λ, m
R×N
g
λ, m
dγ
α
λ, m
∞
m0
Γ
m α 1
m! Γ
α 1
R
g
λ, m
|
λ
∩ L
2
α
K
. 1.23
Journal of Inequalities and Applications 5
Then the generalized Fourier transform can be extended to the tempered distributions. We
also have the inverse formula of the generalized Fourier transform.
f
x, t
R×N
f
λ, m
ϕ
λ,m
x, t
dγ
α
. 1.25
We call h
s
is the heat kernel associated to L. We have
h
s
x, t
R
λ
2sinh
2λs
α1
e
−1/2λ coth2λsx
2
e
iλt
dλ,
h
s
. 1.27
The Poisson kernel can be calculated by the subordination. In fact, we have
p
s
x, t
4s
√
π
Γ
α
5
2
∞
0
λ
sinh λ
α1
s
2
x
x, t
≤ Cs
s
2
x, t
2
−α5/2
.
1.28
The heat maximal function M
H
is defined by
M
H
f
x, t
sup
s>0
H
sup
s>0
P
s
f
x, t
sup
s>0
f ∗ p
s
x, t
. 1.30
6 Journal of Inequalities and Applications
The Hardy-Littlewood maximal function is defined by
M
B
dm
α
y, s
sup
r>0
f
∗ b
r
x, t
, 1.31
where bx, t1/m
α
B
1
χ
B
1
x, t.
The following proposition is the main result of 8.
Proposition 1.3. M
∞
0
∂
k
s
P
s
f
x, t
2
s
2k−1
ds,
g
∗
k
f
P
s
T
α
y,r
f
x, t
2
dm
α
y, r
ds.
2.1
Then, we can prove
Theorem 2.1. a For k ∈ N and f ∈ L
2
K, there exists C
k
> 0 such that
≤ C
2
f
α,p
. 2.3
c If k>α 2/2 and f ∈ L
p
K,p>2, then there exists a constant C>0 such that
g
∗
k
f
α,p
≤ Cf
α,p
. 2.4
Journal of Inequalities and Applications 7
Proof. a When k ∈ N, by the Plancherel theorem for the Fourier transform on K,
g
k
f
2
α,2
K
∞
0
R×N
∂
k
s
P
s
f
λ, m
2
dγ
α
λ, m
f
λ, m
2
|
λ
|
α1
dλ
s
2k−1
ds.
2.5
Since
∂
k
s
P
s
f
2
α,2
∞
0
R
∞
m0
Γ
m α 1
m!Γ
α 1
f
λ, m
e
−2s
√
4m2α2|λ|
s
2k−1
ds C
k
4m 2α 2
|
λ
|
−k
, 2.8
we have
g
k
f
2
α,2
C
k
R
2
α,2
. 2.9
Therefore
g
k
f
α,2
C
k
f
α,2
. 2.10
b As {P
s
} is a contraction semigroup cf. Proposition 5.1 in 3, we can get
g
k
f
α,p
≤ C
2
f
α,p
cf. 9. For the reverse, we can prove by polarization to the identity
and acf. 10.
c We first prove
x, t
M
B
ψ
x, t
dm
α
x, t
, 2.11
where 0 ≤ ψ ∈ L
q
α
K and ψ
α,q
≤ 1, 1/q 2/p 1.
8 Journal of Inequalities and Applications
Since k>α 2/2, we know
K
1
y, r
K
∞
0
K
s
−α1
1 s
−2
y, r
4
−k
∂
s
P
∞
0
K
s
−α1
∂
s
P
s
f
y, r
2
K
T
α
x,t
g
1
f
2
y, r
M
B
ψ
y, r
dm
α
y, r
≤ Cg
1
f
2
α,p
M
B
0
∂
k
s
H
s
f
x, t
2
s
2k−1
ds,
G
H,∗
k
f
2
x, t
y,r
f
x, t
2
dm
α
y, r
ds.
2.14
Similar to the proof of Theorem 2.1, we can prove
Theorem 2.2. a For k ∈ N and f ∈ L
2
K, there exists C
k
> 0 such that
G
H
k
f
. 2.16
c If k>α 2/2 and f ∈ L
p
K,p>2,thenG
H,∗
k
f
α,p
≤ Cf
α,p
.
By Theorem 2.2, we can get cf. 10
Journal of Inequalities and Applications 9
Corollary 2.3. Let k ∈ N and f ∈ L
2
K,ifG
H
k
f ∈ L
p
K, 1 <p<∞,thenf ∈ L
p
K and there
exists C>0 such that
Cf
α,p
≤G
H
k
,
Δ
Ψ
λ, m
Ψ
λ, m 1
− Ψ
λ, m
.
3.1
Then we define the following differential operators:
Λ
1
Ψ
λ, m
1
|
λ
|
Ψ
λ, m
mΔ
−
Ψ
λ, m
.
3.2
We have the following lemma.
Lemma 3.1. Let gλ, m4m 2α 2|λ|e
−4m2α2|λ|s
hλ, m,wherek ∈ N, hλ, m is a
α 1/21 times differentiable function on R
2
and satisfies
Λ
1
2
Λ
1
2
Λ
2
∂
∂λ
g
λ, m
≤ C max
1
|
λ
|
s
, 1
∂
∂λ
g
λ, 0
≤ C
1
λs
e
−4m2α2λs
. 3.6
When m ≥ 1, we have
Λ
1
2
Λ
2
∂
∂λ
2
e
−4m2α2|λ|s
∂
∂λ
−
m
λ
Δ
−1
h
λ, m
∂
∂λ
4m 2α 2
|
λ
|
e
−4m2α2|λ|s
Δ
−1
g
λ, m
≤ C
1
m
λs
e
−4m2α2λs
. 3.9
Then Lemma 3.1 is proved.
Then we can prove H
¨
ormander multiplier theorem on the Laguerre hypergroup K.
Theorem 3.2. Let hλ, m be a α 1/21 times differentiable function on R
2
and satisfies
|
−j
3.10
for j 0, 1, 2, ,α1/21 and T is an operator which is defined by
Tfλ, mhλ, m
fλ, m,
then T is bounded on L
p
α
K,where1 <p<∞.
Proof. We just prove the theorem for 2 <p<∞,for1<p<2; we can get the result by the
dual theorem. By Theorem 2.2, Corollary 2.3 and the note that Tf ∈ L
2
K,itissufficient to
prove the following:
G
H
2
Tf
x, t
≤ CG
H,∗
1
, 3.12
where
G
t
λ, me
−22mα1|λ|t
hλ, m.
Differentiating 3.12 with respect to t and s, then assuming that t s, we can get
∂
2
s
H
2s
Tf
F
s
∗ ∂
s
H
s
f, 3.13
where
F
s
λ, m
≤
K
F
s
y, r
T
α
x,t
∂
s
H
s
f
y, r
K
1 s
−2
y, r
4
−1
T
α
x,t
∂
s
H
s
f
y, r
x, t
|
2
dm
α
x, t
. 3.17
In the following, we prove
A
s
≤ Cs
−α−3
. 3.18
We write
A
s
x,t≤
√
s
x, t
4
|
F
s
x, t
|
2
dm
α
x, t
A
1
s
A
2
s
F
s
λ, m
2
dγ
α
λ, m
C
R×N
4m 2α 2
|
λ
|
2
e
|
2
e
−8m4α4|λ|s
|
λ
|
α1
dλ
Cs
−α−4
R
∞
m0
Γ
m α 1
m!Γ
α 1
4m 2α 2
|
λ
≤ Cs
−2
K
4t
2
x
4
|
F
s
x, t
|
2
dm
α
x, t
Cs
−2
K
Λ
1
2
Λ
2
∂
∂λ
F
s
λ, m
2
dγ
α
λ, m
.
3.21
By Lemma 3.1,
|
λ
|
s
e
−4m2α2|λ|s
, 3.22
where 0 <<1.
So
A
2
s
≤ Cs
−2
R×N
e
−8m4α4|λ|s
dγ
α
λ, m
Cs
−α−4
R×N
≤ Cs
−α−4
K
1 s
−2
y, r
4
−1
T
α
x,t
∂
s
H
s
f
y, r
. 3.25
Then Theorem 3.2 is proved.
Acknowledgments
This Papers supported by National Natural Science Foundation of China under Grant
no. 11001002 and the Beijing Foundation Program under Grants no. 201010009009, no.
2010D005002000002.
References
1 C. E. Guti
´
errez, A. Incognito, and J. L. Torrea, “Riesz transforms, g-functions, and multipliers for the
Laguerre semigroup,” Houston Journal of Mathematics, vol. 27, no. 3, pp. 579–592, 2001.
2 M. M. Nessibi and K. Trim
`
eche, “Inversion of the Radon transform on the Laguerre hypergroup by
using generalized wavelets,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 337–
363, 1997.
3 K. Stempak, “An algebra associated with the generalized sub-Laplacian,” Polska Akademia Nauk.
Instytut Matematyczny. Studia Mathematica, vol. 88, no. 3, pp. 245–256, 1988.
4 K. Trim
`
eche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, Amster-
dam, The Netherlands, 1997.
5 R I. Jewett, “Spaces with an abstract convolution of measures,” Advances in Mathematics, vol. 18, no.
1, pp. 1–101, 1975.
6 W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, vol. 20 of de
Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, Germany, 1995.
7 K. Stempak, “Mean summability methods for Laguerre series,” Transactions of the American
Mathematical Society, vol. 322, no. 2, pp. 671–690, 1990.
8 J. Huang and H. Liu, “The weak type 1, 1 estimates of maximal functions on the Laguerre
Copyright: Tài liệu đại học ©