Paley and Hardy ’ s inequalities for the Fourier-Dunkl expansions

Purpose – Paley ’ sandHardy ’ sinequalityareprovedonaHardy-typespacefortheFourier – Dunklexpansions based on a complete orthonormal system of Dunkl kernels generalizing the classical exponential system defining the classical Fourier series. Design/methodology/approach – Although the difficulties related to the Dunkl settings, the techniques usedbyK.Satowerestillefficientinthiscasetoestablishtheinequalitieswhichhaveexpectedsimilaritieswiththeclassicalcase,andHardyandPaleytheoremsfortheFourier – Bessel expansions due to the fact that the Bessel transform is the even part of the Dunkl transform. Findings – Paley ’ sinequalityandHardy ’ sinequalityareprovedonaHardy-typespacefortheFourier – Dunkl expansions. Research limitations/implications – This work is a participation in extending the harmonic analysis associatedwiththeDunkloperatorsanditshowstheutilityofBMOspacestoestablishsomeanalyticalresults. Originality/value – DunkltheoryisageneralizationofFourieranalysisandspecialfunctiontheoryrelatedto root systems. Establishing Paley and Hardy ’ s inequalities in these settings is a participation in extending the Dunkl harmonic analysis as it has many applications in mathematical physics and in the framework of vector valued extensions of multipliers.


Introduction
Dunkl operators are differential-difference operators on R N related to finite reflection groups.They can be regarded as a generalization of partial derivatives and they lead to a generalization of the classical tools of harmonic analysis.For further details on the corresponding basic theory, one can see Refs [1][2][3].
In rank-one case, we consider the Dunkl operator D α associated with the reflection group Z 2 on R, given by The Fourier-Dunkl expansions admits a unique solution, denoted by E α ðiλ$Þ expressed in terms of the normalized spherical Bessel functions j α and j αþ1 , namely where j β ðzÞ ¼ 8 > < > : J β being the Bessel function of the first kind and order β (see Ref. [4]).For α ¼ −1=2, it is clear that D −1=2 ¼ d=dx and E −1=2 ðizÞ ¼ e iz .
For α ≥ − 1=2, λ ∈ R and z ∈ C the estimate holds.In particular, we have As a generalization of the classical Fourier transform, the Dunkl transform F α of order α ≥ − 1=2 is defined by for f ∈ L 1 ðR; dμ α Þ the space of integrable functions with respect to the Haar measure dμ α ðxÞ ¼ ð2 αþ1 Γðα þ 1ÞÞ −1 jxj 2αþ1 dx.
The aim of the present work is to obtain the analog of Paley and Hardy's inequalities for the Fourier-Dunkl expansions.We recall that if RH 1 is the real Hardy space consisting of the boundary functions f ðθÞ ¼ lim r→1 RFðre iθ Þ where F ∈ H 1 ðDÞ the Hardy space on the unit disc D which consists of the analytic functions FðzÞ on D satisfying and kf k RH 1 ¼ kFk H 1 with real Fð0Þ, then the Paley's inequality is given by (see Ref. [5]): where fn k g ∞ k¼1 is an Hadamard sequence, that is, a sequence of positive integers such that n kþ1 =n k ≥ ρ with a constant ρ > 1.And Hardy's inequality is where Analogs of these inequalities were established in Refs [6,7] for the Fourier-Jacobi expansions, and with respect to the Fourier-Bessel expansions in Ref. [8].Although the difficulties related to the Dunkl settings, the obtained results have strong similarities with (4) and (5), since for α ¼ −1=2, we cover the classical case results.As we also cover the inequalities established in Ref. [8] due to the fact that the Bessel transform is the even part of the Dunkl transform.Now, let us introduce the Fourier-Dunkl expansions and recall the definition of the nonperiodic real Hardy space.It is wellknown that the Bessel function J αþ1 ðxÞ has an increasing sequence of positive zeros fs n g n≥1 .Then, the real function ImðE α ðixÞÞ ¼ x 2ðαþ1Þ j αþ1 ðxÞ is odd and it has the infinite sequence of zeros fs n g n∈Z (with 0 < s 1 < s 2 < :::, s −n ¼ −s n and s 0 ¼ 0).In Ref. [9], for α > − 1, the authors normalized the Dunkl kernel E α to obtain a sequence of functions defining a complete orthonormal system in L 2 ðΔ; jxj 2αþ1 dxÞ, where Δ ¼ ð−1; 1Þ.In this work, we define a new sequence of functions fe α;n ðixÞg n∈Z presenting a complete orthonormal system of L 2 ðΔÞ, given by e α;n ðixÞ ¼ d α;n js n xj αþ1=2 E α ðis n xÞ; n ∈ Znf0g; x ∈ Δ; where js n j αþ1=2 jj α ðs n Þj and e α;0 ðixÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi This orthonormal system is a generalization of the classical exponential system defining Fourier series, and we define the Fourier-Dunkl expansion of a function f ðxÞ on Δ, by We should mention that the theory of Hardy spaces on R d was initiated by Stein and Weiss [10].Then, real variable methods were introduced in Ref. [11] and led to a characterization of Hardy spaces via the so-called "atomic decomposition", obtained by Coifman [12] when n ¼ 1, and in higher dimensions by Latter [13].A real-valued function a on Δ, is a Δ-atom if there exists a subinterval I ⊂ Δ, satisfying the following conditions: (1) supp ðaÞ ⊂ I, , where jI j is the length of the interval I.
The function aðxÞ ¼ 1  2 x; x ∈ Δ, is a Δ-atom.The nonperiodic real Hardy space is defined to be the set of functions representable in the form: where The Fourier-Dunkl expansions and every a n is a Δ-atom.The series in (7) converges in L 1 ðΔÞ (the set of integrable functions on Δ with respect to the Lebesgue measure) and also a.e.
The Hardy space HðΔÞ is endowed with the norm k:k HðΔÞ , given by where the infimum is taken over all those sequences fλ n g ∞ n¼0 ⊂ C such that f is given by ( 7) for certain Δ-atoms fa n g.Then HðΔÞ is a Banach space and kf k L 1 ðΔÞ ≤ kf k HðΔÞ .Now, we state our theorem: where fn k g ∞ k¼1 is a Hadamard sequence, and where the constant C is independent of f.This paper is organized as follows.In Section 2 we state some technical lemmas needed for the proof of Theorem 1.1.In section 3 we recall the duality property between BMO and Hardy spaces, which plays an important role to prove a technical proposition for the proof of (8).In the last section, we give the proof of Theorem 1.1 and we finalize with some remarks.

Some technical lemmas
We begin this section by collecting three asymptotic formulas which will be needed later: (1) Let fs n g ∞ n¼1 be the sequence of the successive positive zeros of J αþ1 ðxÞ, the Bessel function of the first kind of order α þ 1.Then we have, (see Ref. [4]) (2) An estimation of the constant d α;n as stated in (6), is (3) Using the asymptotic formula for the Bessel function J α ðxÞ, the Bessel function of the first kind of order α ∈ R, when x → þ ∞, given by We begin with two auxiliary results interesting in themselves.We will denote by C a positive constant which is not necessary the same in each occurrence.
Lemma 2.1.Let α ≥ − 1=2, then there exists a constant C such that where , and the inequality ( 13) is obvious in this case.
(1) If ju 2 − u 1 j ≤ 1, ju 1 j ≤ 1 and ju 2 j ≤ 1, the power series representation of the Bessel function leads to the power series of the Dunkl kernel where The Fourier-Dunkl expansions So E α ðiuÞ is an entire function and we have where C is independent of u 1 and u 2 (1) For the case ju 2 − u 1 j < 1, ju 1 j < 1 and ju 2 j > 1, we divide the matter in two parts at the points 1 or −1 and we use the results established in the previous cases.
(2) The case where −1 ≤ a < 0 < b ≤ 1, is a consequence from the first and the second cases, since we can write The integrals on the right hand side of the last inequality cover respectively the second and the first cases' conditions.So there exist two positive constants C 1 and C 2 , such that

Duality between BMO and Hardy spaces
The duality between bounded mean oscillation ðBMOÞ and Hardy spaces was studied extensively in Refs [10,[14][15][16] and others.The nonperiodic BMOðΔÞ space is defined to be the space of functions f ∈ L 1 ðΔÞ, verifying with where the supremum is taken over all subintervals I of Δ and The space BMOðΔÞ endowed with the norm kf k BMO is a Banach space and its duality with the Hardy space ðHðΔÞÞ * ¼ BMOðΔÞ, plays an essential role in the proof of Theorem1.for a positive integer N. Then with a constant C independent of N and the sequence fr k g ∞ k¼1 .
Proof.Knowing that ; to prove (20), it is enough to show that where the constant C is independent of I ; N and the sequence fr k g ∞ k¼1 .According to Remark 3.1, it is sufficient to verify that for every subinterval I ⊂ Δ, there exists a constant c I such that 1 jI j : x 2 be a subinterval of Δ, then if jI j > 1=n 1 , we have The Fourier-

Dunkl expansions
If there exists a positive integer M, such that 1=n M þ1 < jI j < 1=n M , we show inequality (21) with c I ¼ g M ðx 1 Þ.We write g N ðxÞ ¼ g M ðxÞ þ E M ;N ðxÞ, with Using Schwarz's inequality and Lemma 2.1, we get Since fn k g is a Hadamard sequence, it is possible to choose M such that jI j n M ≤ 1, so that Now we estimate the second integral on the right-hand side of (22), we have For the last inequality we used the fact that jI jn l > 1 for l ≥ M þ 1.Also, we have 1=ðjI jn k Þ ≤ ð1=ρÞ kÀl .So we deduce that there exist two constants C and σ, with 0 < σ < 1, such that m ðixÞe α;n ðixÞdx :

I
ðe α;n l ðixÞ À e α;−n l ðixÞÞðe α;n k ðixÞ À e α;−n k ðixÞÞdx≤ À2cm X N l;k¼Mþ1 jr l kr k j jI j Â U n l ;n k þ U n l ;−n k þ U −n l ;n k þ U −n l ;−n k Ã ;whereU p;q ¼ Z I e α;p ðixÞe α;q ðixÞdx : Under the assumption n l ≤ n k and by Lemma 2.2, we obtain 1 jI j Z I ðe α;n l ðixÞ À e α;−n l ðixÞÞðe α;n k ðixÞ À e α;−n kðixÞÞdx ≤ C & n l n k δ þ log þ ðn k jI jÞ jI jn k þ 1 jI jn k ' :Since fn k g is a Hadamard sequence, we have n lIf we fix a positive number μ, with 0 < μ < 1, then there exists a constant C μ verifying: log þ ðn k jI jÞ jI jn k

Iσ
ðe α;n l ðixÞ À e α;−n l ðixÞÞðe α;n k ðixÞ À e α;−n k ðixÞÞdx ≤ Cσ jkÀl j ;for l; k ≥ M þ 1.As a consequence, there exists a constant C for which jkÀl j jr l kr k j !1=2 1.In particular, if g ∈ L ∞ ðΔÞ ⊂ BMOðΔÞ and f ∈ HðΔÞ, we have the following inequality The next proposition is the key tool to prove the Paley's inequality.