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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 Huang1, 2 College of Sciences, North China University of Technology, Beijing 100144, China CEMA, Central University of Finance and Economics, Beijing 100081, China Correspondence should be addressed to Jizheng Huang, hjzheng@163.com Received 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 /∂x2 2α 1/x ∂/∂x x2 ∂2 /∂t2 ; x, t ∈ 0, ∞ × R, where α ≥ Then L can generate a hypergroup which is called Laguerre hypergroup, and we denote this hypergroup by K In this paper, we will consider the Littlewood-Paley g-functions on K and then we use it to prove the Holmander multipliers on K ă Introduction and Preliminaries In , the authors investigated Littlewood-Paley g-functions for the Laguerre semigroup Let Lα d xi i where α ∂2 ∂xi2 αi ∂ , ∂xi − xi 1.1 α1 , , αd , xi > 0, then define the following Littlewood-Paley function Gα by Gα f x ∞ t∇α Ptα f x dt t 1/2 , 1.2 √ √ where ∇α ∂t , x1 ∂x1 , , xd ∂xd and Ptα is the Poisson semigroup associated to Lα In , the authors prove that Gα is bounded on Lp μα for < p < ∞ In this paper, we consider the following differential operator L − ∂2 ∂x2 2α ∂ x ∂x x2 ∂2 ∂t2 ; x, t ∈ 0, ∞ × R, 1.3 Journal of Inequalities and Applications where α ≥ It is well known that it can generate a hypergroup cf 2, or We will define Littlewood-Paley g-functions associated to L and prove that they are bounded on Lp K for < p < ∞ As an application, we use it to prove the Homander multiplier theorem ă on K Let K 0, ∞ × R equipped with the measure dmα x, t x2α dxdt, πΓ α α ≥ 1.4 p We denotes by Lα K the spaces of measurable functions on K such that f f f α,p K p f x, t 1/p dmα x, t , α,p < ∞, where ≤ p < ∞, 1.5 esssup x,t ∈K f x, t α,∞ α For x, t ∈ K, the generalized translation operators T x,t are defined by α T x,t f y, s ⎧ ⎪ ⎪ ⎪ ⎪ 2π ⎪ ⎨ ⎪α ⎪ ⎪ ⎪ ⎪ ⎩π 2π x2 f y2 2xy cos θ, s t xy sin θ dθ, if α 0, 2π x2 f y2 2xyr cos θ, s t xyr sin θ r − r α−1 drdθ, if α > 0 1.6 α It is known that T x,t satisfies α T x,t f α,p ≤ f α,p 1.7 Let Mb K denote the space of bounded Radon measures on K The convolution on Mb K is defined by μ∗ν f α K×K T x,t f y, s dμ x, t dν y, s It is easy to see that μ ∗ ν ν ∗ μ If f, g ∈ L1 K and μ fmα , ν α where f ∗ g is the convolution of functions f and g defined by f ∗ g x, t α K gmα , then μ ∗ ν T x,t f y, s g y, −s dmα y, s The following lemma follows from 1.7 1.8 f ∗ g mα , 1.9 Journal of Inequalities and Applications p Lemma 1.1 Let f ∈ L1 K and g ∈ Lα K , ≤ p ≤ ∞ Then α f ∗g ≤ f α,p g α,1 α,p 1.10 K, ∗, i is a hypergroup in the sense of Jewett cf 5, , where i denotes the involution defined by i x, t x, −t If α n − is a nonnegative integer, then the Laguerre hypergroup K can be identified with the hypergroup of radial functions on the Heisenberg group Hn The dilations on K are defined by rx, r t , δr x, t r > 1.11 It is clear that the dilations are consistent with the structure of hypergroup Let r − 2α fr x, t x t , r r2 f 1.12 Then we have fr α,1 f α,1 1.13 1/4 cf Then we We also introduce a homogeneous norm defined by x, t x4 4t2 can defined the ball centered at 0, of radius r, that is, the set Br { x, t ∈ K : x, t < r} Let f ∈ L1 K Set x ρ cos θ 1/2 , t 1/2ρ2 sin θ We get α K f x, t dmα x, t 2πΓ α π/2 −π/2 ∞ f ρ cos θ 1/2 , ρ2 sin θ ρ2α cos θ α dρdθ 1.14 If f is radial, that is, there ia a function ψ on 0, ∞ such that f x, t K f x, t dmα x, t 2πΓ α π/2 −π/2 cos θ Γ α /2 √ π Γ α Γ α/2 α ∞ dθ ψ x, t ψ ρ ρ2α ∞ ψ ρ ρ2α , then dρ 1.15 dρ Specifically, mα Br √ π α Γ α /2 Γ α Γ α/2 r 2α 1.16 We consider the partial differential operator L − ∂2 ∂x2 2α ∂ x ∂x x2 ∂2 ∂t2 1.17 Journal of Inequalities and Applications L is positive and symmetric in L2 K , and is homogeneous of degree with respect to the α dilations defined above When α n − 1, L is the radial part of the sublaplacian on the Heisenberg group Hn We call L the generalized sublaplacian α Let Lm be the Laguerre polynomial of degree m and order α defined in terms of the generating function by ∞ α sm Lm x 1−s m α exp − xs 1−s 1.18 For λ, m ∈ R × N, we put m! Γ α iλt − 1/2 |λ|x2 α e e Lm |λ|x2 Γm α ϕ λ,m x, t 1.19 The following proposition summarizes some basic properties of functions ϕ λ,m Proposition 1.2 The function ϕ λ,m satisfies that a ϕ λ,m ϕ λ,m 0, α,∞ α T x,t ϕ λ,m y, s , b ϕ λ,m x, t ϕ λ,m y, s c Lϕ λ,m |λ| 4m 1, ϕ λ,m 2α Let f ∈ L1 K , the generalized Fourier transform of f is defined by α f λ, m K f x, t ϕ −λ,m x, t dmα x, t 1.20 It is easy to show that f ∗g λ, m f λ, m g λ, m , 1.21 f r λ, m fr λ, m Let dγα be the positive measure defined on R × N by ∞ R×N g λ, m dγα λ, m m Γ m α m! Γ α R g λ, m |λ|α dλ 1.22 p Write Lα K instead of Lp R × N, dγα We have the following Plancherel formula: f α,2 f L2 K α , f ∈ L1 K ∩ L2 K α α 1.23 Journal of Inequalities and Applications 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 dγα λ, m 1.24 provided f ∈ L1 K α In the following, we give some basic notes about the heat and Poisson kernel whose proofs can be found in Let {H s } {e−sL } be the heat semigroup generated by L There is a unique smooth function h x, t , s hs x, t on K × 0, ∞ such that f ∗ hs x, t H s f x, t 1.25 We call hs is the heat kernel associated to L We have hs x, t α λ sinh 2λs R hs x, t ≤ Cs e− 1/2 λ coth 2λs x eiλt dλ, 1.26 −α−2 − A/s e x,t √ Let {P s } {e−s L } be the Poisson semigroup There is a unique smooth function p x, t , s ps x, t on K × 0, ∞ , which is called the Poisson kernel, such that f ∗ ps x, t P s f x, t 1.27 The Poisson kernel can be calculated by the subordination In fact, we have ps x, t 4s √ Γ α π ∞ × cos ps x, t ≤ C s s2 α λ sinh λ x, t s2 α − α 5/2 x2 λ coth λ arctan s2 2λt − 2α /4 2λt x2 λ coth λ 1.28 dλ, The heat maximal function MH is defined by MH f x, t sup H s f x, t s>0 sup f ∗ hs x, t 1.29 s>0 The Poisson maximal function MP is defined by MP f x, t sup P s f x, t s>0 sup f ∗ ps x, t s>0 1.30 Journal of Inequalities and Applications The Hardy-Littlewood maximal function is defined by MB f x, t sup r>0 mα Br α Br f T x,t sup f ∗ br x, t , y, s dmα y, s 1.31 r>0 where b x, t 1/ mα B1 χB1 x, t The following proposition is the main result of Proposition 1.3 MB MP and MB are operators on K of weak type 1, and strong type p, p for < p ≤ ∞ The paper is organized as follows In the second section, we prove that Littlewoodp Paley g-functions are bounded operators on Lα K As an application, we prove the Hormander multiplier theorem on K in the last section ă Throughout the paper, we will use C to denote the positive constant, which is not necessarily same at each occurrence Littlewood-Paley g-Function on K ∗ Let k ∈ N, then we define the following G-function and gλ -function gk f ∞ x, t ∗ gk f ∞ x, t − α K s −2 s ∂k P s f x, t s y, r −k s2k−1 ds, s ∂s P T α y,r 2.1 f x, t dmα y, r ds Then, we can prove Theorem 2.1 a For k ∈ N and f ∈ L2 K , there exists Ck > such that gk f α,2 Ck f α,2 2.2 b For < p < ∞ and f ∈ Lp K , there exist positive constants C1 and C2 , such that C1 f c If k > α α,p ≤ gk f α,p ≤ C2 f α,p 2.3 /2 and f ∈ Lp K , p > 2, then there exists a constant C > such that ∗ gk f α,p ≤C f α,p 2.4 Journal of Inequalities and Applications Proof a When k ∈ N, by the Plancherel theorem for the Fourier transform on K, ∞ α,2 gk f K ∞ ∂k P s f s R×N ∞ ∞ Rm ∂k P s f x, t s s2k−1 ds dmα x, t λ, m Γm α m!Γ α s2k−1 ds dγα λ, m ∂k P s f s λ, m 2.5 |λ|α dλ s2k−1 ds Since ∂k P s f s λ, m − 4m 2α 4m f λ, m 2α |λ| k e−2s k |λ| e−s √ 4m 2α |λ| , 2.6 we get gk f α,2 ∞ ∞ Rm Γm α f λ, m m!Γ α √ 4m 2α |λ| |λ|α dλ s2k−1 ds 2.7 By ∞ e−2s √ 4m 2α |λ| 2k−1 s ds Ck 4m 2α |λ| −k , 2.8 we have gk f α,2 ∞ Ck Rm Γ m α f λ, m m!Γ α |λ|α dλ Ck f α,2 2.9 Therefore gk f α,2 Ck f α,2 2.10 b As {P s } is a contraction semigroup cf Proposition 5.1 in , we can get gk f α,p ≤ C2 f α,p cf For the reverse, we can prove by polarization to the identity and a cf 10 c We first prove K ∗ gk f x, t ψ x, t dmα x, t ≤ C q where ≤ ψ ∈ Lα K and ψ α,q ≤ 1, 1/q K 2/p g1 f x, t MB ψ x, t dmα x, t , 2.11 Journal of Inequalities and Applications Since k > α /2, we know K −k y, r dmα y, r < ∞ 2.12 By Proposition 1.3, K ∗ gk f x, t ψ x, t dmα x, t ∞ K K ≤C K ∞ K s− α s− α s−2 −k y, r ∂s P s f y, r g1 f ≤ C g1 f 1 ∂s P s T s−2 α K α y,r T x,t f x, t dmα y, r ds ψ x, t dmα x, t −k y, r ψ x, t dmα x, t dmα y, r ds y, r MB ψ y, r dmα y, r α,p MB ψ α,q ≤C f α,p 2.13 ∗ Therefore gk f α,p ≤C f α,p This gives the proof of Theorem 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 H Gk f H,∗ Gk f ∞ x, t K s− α ∞ x, t 1 s−2 ∂k H s f x, t s −k y, r s2k−1 ds, ∂s H s T α y,r f x, t dmα 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 ∈ L2 K , there exists Ck > such that H Gk f α,2 Ck f α,2 2.15 b For < p < ∞ and f ∈ Lp K , there exist constants C1 and C2 , such that C1 f c If k > α α,p H ≤ Gk f α,p ≤ C2 f H,∗ /2 and f ∈ Lp K , p > 2, then Gk f By Theorem 2.2, we can get cf 10 α,p α,p 2.16 ≤C f α,p Journal of Inequalities and Applications H Corollary 2.3 Let k ∈ N and f ∈ L2 K , if Gk f ∈ Lp K , < p < ∞, then f ∈ Lp K and there exists C > such that C f α,p H ≤ Gk f α,p 2.17 ă Hormander Multiplier Theorem on K In this section, we prove the Hormander multiplier theorem on K The main tool we use is ă the Littlewood-Paley theory that we have proved We first introduce some notations Assume Ψ is a function defined on R × N, then let Ψ λ, and for m ≥ 1, Δ− Ψ λ, Δ− Ψ λ, m Ψ λ, m − Ψ λ, m − , Δ Ψ λ, m Ψ λ, m 3.1 − Ψ λ, m Then we define the following differential operators: Λ1 Ψ λ, m mΔ− Ψ λ, m |λ| Λ2 Ψ λ, m −1 α 2λ Δ Ψ λ, m , α 3.2 Δ Ψ λ, m m mΔ− Ψ λ, m We have the following lemma Lemma 3.1 Let g λ, m 4m 2α |λ| e− 4m 2α |λ|s h λ, m , where k ∈ N, h λ, m is a α /2 times differentiable function on R2 and satisfies Λ1 for j 0, 1, 2, , α Λ1 where < Λ2 /2 Λ2 j ∂ ∂λ h λ, m ≤ Cj 4m 2α |λ| −j 3.3 Then one has ∂ ∂λ g λ, m ≤ C max ,1 |λ|s m e− |λ|s 4m 2α |λ|s , 3.4 < and s > Proof Without loss of the generality, we can assume that λ > when m Λ1 Λ2 ∂ ∂λ ∂ ∂λ 0, we have 3.5 10 Journal of Inequalities and Applications It is easy to calculate ∂ g λ, ∂λ ≤C − e λs 4m 2α λs 3.6 When m ≥ 1, we have ∂ ∂λ Λ2 Λ1 m ∂ − Δ−1 ∂λ λ 3.7 Since m ∂ − Δ−1 g λ, m ∂λ λ 4m |λ| e− 4m 2α 2α |λ|s ∂ 4m 2α |λ| e− 4m ∂λ m − Δ−1 f m g m − , λ m ∂ − Δ−1 h λ, m ∂λ λ 2α |λ|s h λ, m 3.8 we get ∂ m − Δ−1 g λ, m ∂λ λ ≤C m − e λs 4m 2α λs 3.9 Then Lemma 3.1 is proved Then we can prove Hormander multiplier theorem on the Laguerre hypergroup K ă Theorem 3.2 Let h , m be a Λ1 Λ2 α /2 ∂ ∂λ times differentiable function on R2 and satisfies j h λ, m ≤ Cj 4m 2α |λ| −j for j 0, 1, 2, , α /2 and T is an operator which is defined by T f λ, m p then T is bounded on Lα K , where < p < ∞ 3.10 h λ, m f λ, m , Proof We just prove the theorem for < p < ∞, for < p < 2; we can get the result by the dual theorem By Theorem 2.2, Corollary 2.3 and the note that T f ∈ L2 K , it is sufficient to prove the following: H,∗ H G2 T f x, t ≤ CG1 f x, t , x, t ∈ K 3.11 Journal of Inequalities and Applications Let us H s f and Us 11 H s T f , then we can get Us Gt ∗ us x, t , t 3.12 where Gt λ, m e−2 2m α |λ|t h λ, m Differentiating 3.12 with respect to t and s, then assuming that t s, we can get Fs ∗ ∂s H s f, ∂2 H 2s T f s 3.13 where Fs λ, m − 4m ∂2 H 2s T f x, t s ≤ 2α |λ| e− 4m 2α |λ|s h λ, m 3.14 Therefore α K Fs y, r T x,t ∂s H s f y, r dmα y, r 3.15 By the Cauchy-Schwartz inequality, ∂2 H 2s T f x, t s ≤A s K s−2 s−2 x, t −1 y, r α T x,t ∂s H s f y, r dmα y, r , 3.16 where As K |Fs x, t |2 dmα 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 > s A1 s s−2 x, t 1 A2 s s−2 x, t |Fs x, t |2 dmα x, t |Fs x, t |2 dmα x, t 3.19 12 Journal of Inequalities and Applications For A1 s , we can easily get A1 s ≤ C K C |Fs x, t |2 dmα x, t 4m R×N ∞ ≤C Rm Cs−α−4 ≤ Cs−α−4 Rm ∞ 2α −2 Fs λ, m dγα λ, m 4α |λ|s h λ, m dγα λ, m 2α |λ| e− 8m 4m 4m Γm α m!Γ α 4m R×N |λ| e− 8m 2α Γm α m!Γ α ∞ C 2α 4α |λ|s |λ| e− 8m |λ|α dλ 4α |λ| 3.20 |λ|α dλ ≤ Cs−α−4 m For A2 s , we have A2 s ≤ Cs−2 Cs−2 Cs−2 K x4 |Fs x, t |2 dmα x, t 4t2 2it − |x|2 Fs x, t K Λ1 R×N Λ2 dmα x, t 3.21 ∂ ∂λ Fs λ, m dγα λ, m By Lemma 3.1, Λ1 where < So Λ2 ∂ ∂λ ≤ C max Fs λ, m ,1 |λ|s m e− |λ|s 4m 2α |λ|s , 3.22 < A2 s ≤ Cs−2 R×N Cs−α−4 e− R×N 8m 4α |λ|s e− dγα λ, m 8m 4α |λ| 3.23 dγα λ, m ≤ Cs−α−4 Therefore 3.18 holds Then ∂2 H 2s T f x, t s ≤ Cs−α−4 K s−2 y, r −1 α T x,t ∂s H s f y, r dmα y, r 3.24 Journal of Inequalities and Applications 13 Integrating the both sides of the above inequality with s3 ds, we have H,∗ H G2 x, t ≤ CG1 f x, t 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 C E Guti´ rrez, A Incognito, and J L Torrea, “Riesz transforms, g-functions, and multipliers for the e Laguerre semigroup,” Houston Journal of Mathematics, vol 27, no 3, pp 579–592, 2001 M M Nessibi and K Trim` che, “Inversion of the Radon transform on the Laguerre hypergroup by e using generalized wavelets,” Journal of Mathematical Analysis and Applications, vol 208, no 2, pp 337– 363, 1997 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 K Trim` che, Generalized Wavelets and Hypergroups, Gordon and Breach 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there exists Ck > such that H Gk f α,2 Ck f α,2 2.15 b For < p < ∞ and f ∈ Lp K , there exist constants C1 and. .. Incognito, and J L Torrea, “Riesz transforms, g-functions, and multipliers for the e Laguerre semigroup,” Houston Journal of Mathematics, vol 27, no 3, pp 579–592, 2001 M M Nessibi and K Trim`... Inequalities and Applications 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