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Then L can generate a hypergroup which is called Laguerre hypergroup, and we denote this hypergroup byK.. In this paper, we will consider the Littlewood-Paley g-functions onK and then we

Volume 2011, Article ID 741095, 13 pages

doi:10.1155/2011/741095

Research Article

the Laguerre Hypergroup

Jizheng Huang1, 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,hjzheng@163.com

Received 4 November 2010; Accepted 13 January 2011

Academic Editor: Shusen Ding

Copyrightq 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/∂x2 2α  1/x∂/∂x  x2∂2/∂t2; x, t ∈ 0, ∞ × R, where α ≥ 0 Then L can

generate a hypergroup which is called Laguerre hypergroup, and we denote this hypergroup byK.

In this paper, we will consider the Littlewood-Paley g-functions onK and then we use it to prove

the H ¨olmander multipliers onK.

1 Introduction and Preliminaries

In1, the authors investigated Littlewood-Paley g-functions for the Laguerre semigroup Let

Lαd

i1

x i ∂2

∂ x2

i

 α i  1 − x i ∂

∂ x i

where α  α1, , α d , x i > 0, then define the following Littlewood-Paley functionGαby

Gα f x 

∞

0

t∇α P α

t f x2dt

t

1/2

where∇α  ∂ t ,√

x1∂ x1, ,√

x d ∂ x d  and P α

t is the Poisson semigroup associated toLα In1, the authors prove thatGα is bounded on L p μ α  for 1 < p < ∞ In this paper, we consider the

following differential operator

L −



∂2

∂x2  2α 1

x

∂

∂x  x2∂2

∂t2



; x, t ∈ 0, ∞ × R, 1.3

where α ≥ 0 It is well known that it can generate a hypergroup cf 2,3 or 4 We will

define Littlewood-Paley g-functions associated to L and prove that they are bounded on

L p K for 1 < p < ∞ As an application, we use it to prove the H¨omander multiplier theorem

onK.

dm α x, t  1

π Γα  1 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, tp dm α x, t

1/p

, 1≤ p < ∞,

f α,∞ esssupx,t∈K f x, t. 1.5 Forx, t ∈ K, the generalized translation operators T x,t α are defined by

T x,t α f

y, s



⎧

⎪

⎪

⎪

⎪

1

2π

2π

0

f

x2 y2 2xy cos θ, s  t  xy sin θdθ, if α  0,

α

π

2π

0

1

0

f

x2 y2 2xyr cos θ, s  t  xyr sin θr

1− r2 α−1drdθ, if α > 0.

1.6

It is known that T x,t α satisfies

T x,t α 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 bK

is defined by

μ ∗ ν f 



K×KT x,t α f

y, s dμ x, tdν y, s 1.8

It is easy to see that μ ∗ ν  ν ∗ μ If f, g ∈ L1

α K and μ  fm α , ν  gm α , then μ ∗ ν  f ∗ gm α,

where f ∗ g is the convolution of functions f and g defined by

f ∗ g x, t 



KT x,t α f

y, s g

y, −s dm α

y, s 1.9 The following lemma follows from1.7

Lemma 1.1 Let f ∈ L1

α K and g ∈ L p

α K, 1 ≤ p ≤ ∞ Then

f ∗ g α,p ≤ f α,1 g α,p 1.10

K, ∗, i is a hypergroup in the sense of Jewett cf 5,6, where i denotes the involution defined by ix, t  x, −t If α  n − 1 is a nonnegative integer, then the Laguerre hypergroup

The dilations onK are defined by

δ r x, t rx, r2t

It is clear that the dilations are consistent with the structure of hypergroup Let

f r x, t  r −2α4 f



x

r ,

t

r2



Then we have

f rα,1  f α,1 1.13

We also introduce a homogeneous norm defined byx, t  x4 4t21/4cf 7 Then we can defined the ball centered at0, 0 of radius r, that is, the set B r  {x, t ∈ K : x, t < r}.

Let f ∈ L1

α K Set x  ρcos θ 1/2 , t  1/2ρ2sin θ We get



Kf x, tdm α x, t  1

2π Γα  1

π/2

−π/2

∞

0

f



ρ cos θ 1/2 ,1

2ρ

2sin 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, t  ψx, t, then



Kf x, tdm α x, t  1

2π Γα  1

π/2

−π/2 cos θ α dθ

∞

0

ψ

ρ ρ 2α3 dρ

 Γα  1/2

2√

π Γα  1Γα/2  1

∞

0

ψ

ρ ρ 2α3 dρ.

1.15

Specifically,

m α B r  Γα  1/2

4√

π α  2Γα  1Γα/2  1 r 2α4 . 1.16

We consider the partial differential operator

L −



∂2

∂x2 2α 1

x

∂

∂x  x2∂2

∂t2



L is positive and symmetric in L2

αK, 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 groupHn 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

generating function by

∞



m0

s m L α m x  1

1 − s α1exp



− xs

1− s



Forλ, m ∈ R × N, we put

ϕ λ,m x, t  m! Γα  1

Γm  α  1 e iλt e −1/2|λ|x

2

L α m 

|λ|x2

. 1.19

The following proposition summarizes some basic properties of functions ϕ λ,m

a ϕ λ,mα,∞ ϕ λ,m 0, 0  1,

b ϕ λ,m x, t ϕ λ,m y, s  T x,t α ϕ λ,m y, s,

c Lϕ λ,m  |λ|4m  2α  2ϕ λ,m

Let f ∈ L1

α K, the generalized Fourier transform of f is defined by



f λ, m 



Kf x, tϕ −λ,m x, tdm α x, t. 1.20

It is easy to show that

f ∗ g λ, m   f λ, mgλ, m,



f r λ, m   f

r2λ, m

.

1.21

Let dγ αbe the positive measure defined onR × N by



R×Ng λ, mdγ α λ, m ∞

m0

Γm  α  1

m! Γα  1



Rg λ, m|λ| α1dλ. 1.22

Write L p α  K instead of L p R × N, dγ α We have the following Plancherel formula:

f α,2  fL2 K, f ∈ L1

α K ∩ L2

α K. 1.23

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, mϕ λ,m x, tdγ α λ, m 1.24

provided f ∈ L1

α K.

In the following, we give some basic notes about the heat and Poisson kernel whose proofs can be found in8 Let {H s }  {e −sL } be the heat semigroup generated by L There is

a unique smooth function hx, t, s  h s x, t on K × 0, ∞ such that

H s f x, t  f ∗ h s x, t. 1.25

We call h s is the heat kernel associated to L We have

h s x, t 



R



λ

2 sinh2λs

α1

e −1/2λ coth2λsx2e iλt dλ,

h s x, t ≤ Cs −α−2 e −A/sx,t2.

1.26

Let{P s }  {e −s√L } be the Poisson semigroup There is a unique smooth function px, t, s 

p s x, t on K × 0, ∞, which is called the Poisson kernel, such that

P s f x, t  f ∗ p s x, t. 1.27 The Poisson kernel can be calculated by the subordination In fact, we have

p s x, t

 √4s

π Γ



α5 2

 ∞

0



λ

sinh λ

α1

s2 x2λ coth λ2

 2λt2−2α5/4

× cos



α5 2

 arctan



2λt

s2 x2λ coth λ



dλ,

p s x, t ≤ C ss2 x, t2−α5/2

.

1.28

The heat maximal function M His defined by

M H f x, t  sup

s>0

H s f x, t  sup

s>0

 f ∗ h s x, t. 1.29

The Poisson maximal function M P is defined by

M P f x, t  sup

s>0

P s f x, t  sup

s>0

 f ∗ p s x, t. 1.30

The Hardy-Littlewood maximal function is defined by

M B f x, t  sup

r>0

1

m α B r



B r

T x,t α f y, s dm α

y, s  sup

r>0

f  ∗ b r x, t, 1.31

where bx, t  1/m α B1χ B1x, t.

The following proposition is the main result of8

1 < p ≤ ∞.

The paper is organized as follows In the second section, we prove that

Littlewood-Paley g-functions are bounded operators on L p αK As an application, we prove the

H ¨ormander multiplier theorem onK in the last section.

Throughout the paper, we will use C to denote the positive constant, which is not

necessarily same at each occurrence

2 Littlewood-Paley g-Function on K

Let k ∈ N, then we define the following G-function and g∗

λ-function

g k

f 2x, t 

∞

0



∂ k

s P s f x, t2

s 2k−1 ds,

g k∗

f 2x, t 

∞

0



Ks −α1

1 s−2 y, r 4−k

∂ s P s Tα y,r fx, t



2dm α

y, r



ds.

2.1

Then, we can prove

g k

f α,2  C k f α,2 2.2

b For 1 < p < ∞ and f ∈ L p K, there exist positive constants C1and C2, such that

C1f α,p ≤ g k

f α,p ≤ C2f 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

Proof a When k ∈ N, by the Plancherel theorem for the Fourier transform on K,

g k

f 2

α,2



K

∞

0



∂ k

s P s f x, t2

s 2k−1 ds



dm α x, t



∞

0



R×N



∂ k s P s f

λ, m2

dγ α λ, m



s 2k−1 ds



∞

0



R

∞



m0

Γm  α  1

m! Γα  1 

∂ k s P s f

λ, m2

|λ| α1dλ



s 2k−1 ds.

2.5

Since



∂ k s P s f

λ, m   f λ, m



−4m  2α  2|λ|

k

e −s

√

we get

g k

f 2

α,2



∞

0



R

∞



m0

Γm  α  1

m! Γα  1  f λ, m2

4m  2α  2|λ| k e −2s√

4m2α2|λ| |λ| α1dλ



s 2k−1 ds.

2.7 By

∞

0

e −2s√

we have

g k

f 2

α,2  C k



R

∞



m0

Γm  α  1

m! Γα  1  f λ, m2

|λ| α1dλ  C k f2

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 ≤ C2f α,pcf 9 For the reverse, we can prove by polarization to the identity anda cf 10

c We first prove



Kg k∗

f 2x, tψx, tdm α x, t ≤ C



Kg1

f 2x, tM B ψ x, tdm α x, t, 2.11 where 0≤ ψ ∈ L q

α K and ψ α,q ≤ 1, 1/q  2/p  1.

Since k > α  2/2, we know



K1   y, r 4−k dm α

y, r < ∞. 2.12

ByProposition 1.3,



Kg k∗

f 2x, tψx, tdm α x, t





K

∞

0



Ks −α1

1 s−2 y, r 4−k

∂ s P s Tα y,r fx, t



2dm α

y, r ds



ψ x, tdm α x, t



∞

0



Ks −α1∂ s P s f

y, r 2



KT x,t α 

1 s−2 y, r 4−k

ψ x, tdm α x, t



dm α

y, r ds

≤ C



Kg1

f 2

y, r M B ψ

y, r dm α

y, r

≤ Cg1

f 2

α,p M B ψα,q ≤ Cf2

α,p

2.13 Thereforeg∗

k f α,p ≤ Cf α,p This gives the proof ofTheorem 2.1

We can also consider the Littlewood-Paley g-function that is defined by the heat semigroup as follows: let k∈ N, we define

GH k

f 2x, t 

∞

0



∂ k

s H s f x, t2

s 2k−1 ds,

GH,∗

k

f 2x, t 

∞

0



Ks −α1

1 s−2 y, r 4−k

∂ s H s Tα y,r fx, t



2dm α

y, r



ds.

2.14 Similar to the proof ofTheorem 2.1, we can prove

GH

k f α,2  C k f α,2 2.15

b For 1 < p < ∞ and f ∈ L p K, there exist constants C1and C2, such that

C1f α,p≤ GH

k

f α,p ≤ C2f α,p 2.16

c If k > α  2/2 and f ∈ L p K, p > 2, then G H,∗

k f α,p ≤ Cf α,p

ByTheorem 2.2, we can getcf 10

Corollary 2.3 Let k ∈ N and f ∈ L2K, if G H

k f ∈ L p K, 1 < p < ∞, then f ∈ L p K and there

exists C > 0 such that

C f α,p ≤ GH

k

f α,p 2.17

3 H ¨ormander Multiplier Theorem on K

In this section, we prove the H ¨ormander multiplier theorem onK The main tool we use is

the Littlewood-Paley theory that we have proved

We first introduce some notations AssumeΨ is a function defined on R × N, then let

Δ−Ψλ, 0  Ψλ, 0 and for m ≥ 1,

Δ−Ψλ, m  Ψλ, m − Ψλ, m − 1,

ΔΨλ, m  Ψλ, m  1 − Ψλ, m. 3.1

Then we define the following differential operators:

Λ1Ψλ, m  |λ|1 mΔ−Ψλ, m  α  1ΔΨλ, m,

Λ2Ψλ, m  −12λ α  m  1ΔΨλ, m  mΔ−Ψλ, m.

3.2

We have the following lemma

α  1/2  1 times differentiable function on R2and satisfies









Λ1 2



Λ2 ∂

∂λ

j

h λ, m



 ≤C j 4m  2α  2|λ| −j 3.3

for j  0, 1, 2, , α  1/2  1 Then one has



Λ1 2



Λ2 ∂

∂λ



g λ, m

 ≤ Cmax|λ|s1 , 1 |λ|s m



where 0 <  < 1 and s > 0.

Proof Without loss of the generality, we can assume that λ > 0 when m 0, we have

Λ1 2



Λ2 ∂

∂λ



 2∂

It is easy to calculate



∂λ ∂ g λ, 0

 ≤ C λs1 e −4m2α2λs 3.6

When m≥ 1, we have

Λ1 2



Λ2 ∂

∂λ



 2



∂

∂λ−m

λΔ−1



Since



∂

∂λ−m

λΔ−1



g λ, m  4m  2α  2|λ|e −4m2α2|λ|s

∂

∂λ−m

λΔ−1



h λ, m

 ∂

∂λ



4m  2α  2|λ|e −4m2α2|λ|s

h λ, m

−m

λΔ−1f mgm − 1,

3.8

we get



∂λ ∂ −m

λΔ−1



g λ, m

 ≤ C1 m

λs



ThenLemma 3.1is proved

Then we can prove H ¨ormander multiplier theorem on the Laguerre hypergroupK.









Λ1 2



Λ2 ∂

∂λ

j

h λ, m



 ≤C j 4m  2α  2|λ| −j 3.10

for j  0, 1, 2, , α1/21 and T is an operator which is defined by  Tf λ, m  hλ, m  f λ, m,

then T is bounded on L p α K, where 1 < p < ∞.

Proof We just prove the theorem for 2 < p < ∞, for 1 < p < 2; we can get the result by the

dual theorem ByTheorem 2.2,Corollary 2.3and the note that Tf ∈ L2K, it is sufficient to

prove the following:

GH

2

Tf x, t ≤ CG H,∗

1

f x, t, x, t ∈ K. 3.11

Let u s  H s f and U s  H s Tf, then we can get

U s t  G t ∗ u s x, t, 3.12

where G t λ, m  e −22mα1|λ|t h λ, m.

Differentiating 3.12 with respect to t and s, then assuming that t  s, we can get

∂2H 2s

Tf  F s ∗ ∂ s H s f, 3.13 where

F s λ, m  −4m  2α  2|λ|e −4m2α2|λ|s h λ, m. 3.14 Therefore



∂2

s H 2s

Tf x, t ≤

KF s

y, r T x,t α ∂ s H s f

y, r dm α

y, r 3.15

By the Cauchy-Schwartz inequality,



∂2

s H 2s

Tf x, t2

≤ As



K



1 s−2 y, r 4−1T α

x,t ∂ s H s f

y, r 2dm α

y, r , 3.16

where

A s 



K



1 s−2x, t4

|F s x, t|2dm α x, t. 3.17

In the following, we prove

A s ≤ Cs −α−3 3.18

We write

A s 



x,t≤√s



1 s−2x, t4

|F s x, t|2dm α x, t





x,t>√s



1 s−2x, t4

|F s x, t|2dm α x, t

 A1s  A2s.

3.19

For A1s, we can easily get

A1s ≤ C



K|F s x, t|2dm α x, t  C



R×N



 F s λ, m2

dγ α λ, m

 C



R×N4m  2α  2|λ|2e −8m4α4|λ|s h2λ, mdγ α λ, m

≤ C



R

∞



m0

Γm  α  1

m! Γα  1 4m  2α  2|λ|2e −8m4α4|λ|s |λ| α1dλ

 Cs −α−4

R

∞



m0

Γm  α  1

m! Γα  1 4m  2α  2|λ|2e −8m4α4|λ| |λ| α1dλ

≤ Cs −α−4∞

m0

4m  2α  2−2≤ Cs −α−4

3.20

For A2s, we have

A2s ≤ Cs−2

K



4t2 x4

|F s x, t|2dm α x, t

 Cs−2

K



2it − |x|2

F s x, t2

dm α x, t

 Cs−2

R×N



Λ1 2



Λ2 ∂

∂λ



F s λ, m

2dγ α λ, m.

3.21

ByLemma 3.1,



Λ1 2



Λ2 ∂

∂λ



F s λ, m

 ≤ Cmax|λ|s1 , 1 m

|λ|s



where 0 <  < 1.

So

A2s ≤ Cs−2

R×Ne −8m4α4|λ|s dγ α λ, m

 Cs −α−4

R×Ne −8m4α4|λ| dγ α λ, m

≤ Cs −α−4

3.23

Therefore3.18 holds Then



∂2

s H 2s

Tf x, t2

≤ Cs −α−4

K



1 s−2 y, r 4−1T α

x,t ∂ s H s f

y, r 2dm α

y, r 3.24

Integrating the both sides of the above inequality with s3ds, we have

GH

2 x, t ≤ CG H,∗

1

f x, t. 3.25 ThenTheorem 3.2is 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

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