Takeshi Kawazoe

Department of Mathematics,Keio University at SFC,Endo,Fujisawa,Kanagawa, 252-8520,Japan

?

H1-Estimates of the Littlewood-Paley and Lusin Functions for Jacobi Analysis II

Takeshi Kawazoe?

Department of Mathematics,Keio University at SFC,Endo,Fujisawa,Kanagawa, 252-8520,Japan

Abstract.Let(R+,?,?)be the Jacobi hypergroup.We introduce analogues of the Littlewood-Paley g function and the Lusin areafunction for the Jacobi hypergroup and consider their(H1,L1)boundedness.Although the g operator for(R+,?,?)possesses better property than the classical g operator,the Lusin area operator has an obstacle arisen from a second convolution.Hence,in order to obtain the(H1,L1)estimate for the Lusin area operator,a slight modification in its form is required.

Jacobi analysis,Jacobi hypergroup,g function,area function,real Hardy space.

AMS Subject Classifications:22E30,43A30,43A80

Analysis in Theory and Applications

Anal.Theory Appl.,Vol.32,No.1(2016),pp.38-51

1　Introduction

One of main subjects of the so-called real method in classical harmonic analysis related to the Poisson integral f?ptis to investigate the Littlewood-Paley theory.For example,in the one dimensional setting,the following singular integral operators were respectively well-known as the Littlewod-Paley g function and the Lusin area function

where χtis the characteristic function of[?t,t].These operators satisfy the maximal theorem,that is,a weak type L1estimate and a strong type Lpestimate for 1＜p≤∞. Moreover,they are bounded form H1into L1(cf.[10–12]).Our matter of concern is to extend these results to other topological spaces X.Roughly speaking,in some examples of X ofhomogeneoustype(see[2]),Poissonintegrals are generalized on X and analogousLittlewood-Paley theoryhas been developed(cf.[2,5,10]).On the otherhand,if the space X is not of homogeneous type,we encounter difficulties.As an example of X of non homogeneous type with Poisson integrals,noncompact Riemannian symmetric spaces X=G/K are well-known.Lohoue[9]and Anker[1]generalize the Littlewood-Paley g function and the Luzin area function to G/K and show that they satisfy the maximal theorem(see below).However,we know little or nothing whether they are bounded from H1into L1,because we first have to find out a suitable definition of a real Hardy space on G/K.The aim of this paper is to introduce a real Hardy space H1(?)and show that they are bounded from H1(?)into L1(?)for the Jacobi hypergroup(R+,?,?),which is a generalization of K-invariant setting on G/K of real rank one.

We briefly overview the Jacobi hypergroup(R+,?,?).We refer to[4]and[8]for a description of general context.For α≥β≥and(α,β)6=we define the weight function?on R+as

Clearly,it follows that

where2F1the hypergeometric function.Then the Jacobi transform ?f of a function f on R+is defined by

We define a generalized translation on R+by using the kernel form of the product formula of Jacobi functions:For x,y∈R+,

The kernel K(x,y,z)is non-negative and symmetric in the tree variables.Then the generalized translation Txof f is defined as

and the convolution of f,g is given by

In Jacobi analysis,the Poisson kernel pt(x),t＞0,is defined as the function such that

Then,as analogue of the classical case,we introduce a generalized Littlewood-Paley g function gσ(f)and a generalized Lusin area function Sa,h(f),which are respectively defined by

where σ,a≥0,h(x)is a positive function on R+and

This paper is organized as follows.Basic notations are given in Section 2.Especially we recall the definition of the Hardy space H1(?)and give a relation with Euclidean weighted Hardy spaces Hw1(R).In Section 3 we prove key lemmas on generalized translations.Finally,in Section 4 and Section 5 we consider(L2(?),L2(?))and(H1(?),L1(?)) boundedness of gσand Sa,hrespectively.

2　Notations

Let Lp(?)denote the space of functions f on R+with finite Lp-norm:

where C(λ)is Harish-Chandra’s C-function.Furthermore,the map f→?f extends to an isometry of L2(?)onto L2(R+,|C(λ)|?2dλ):

(see[4,Section 2]and[8,Theorem 3.1,Remark 3]).Let f∈L1(?).Since φλis bounded by 1 for|?λ|≤ρ(see[4,(2.17)]),?f has a holomorphic extension onWe recall that,as a function of λ,φλ(x)is the Fourier Cosine transform of a function A(x,y)supported on[0,x]:

(see[8,(2.16)]).Hence,if we define the Abel transformof f by

then we see that

Since|A(x,y)|≤ceρy(thy)2αby the explicit form(see[8],(2.18)),it follows that

and for λ∈R,

Especially,we have

where?denotes the Euclidean convolution on R.As shown in[8],Section 3,is of the form:

We now define the real Hardy space H1(?)as the subspace of L1(?)consisting of all functions with finite H1(?)-norm:

In[7],Section4wedefinearadial maximal operator M fortheJacobi hypergroup(R+,?,?) and deduce that H1(?)coincides with the space consisting of allwhose radial maximal functions Mf belong to L1(?)and

The letter c will be used to denote a positive constant which may assume different values at different places.

?We also use the fact that0＜γ＜1,corresponds to the Fourier multiplier of?i|λ|γ(sgn(λ)sin

3　Key lemmas

The following lemmas on the generalized translation Txwill play a key role in the arguments in Section 4 and Section 5.The first one is obtained in[4,(5.2)],and the second one is essentially obtained in[6,Lemma 2.2],for group cases.

Lemma 3.1(see[4]).Let f∈Lp(?),1≤p≤∞,and x∈R+.Then

Moreover,if f is positive,then the equality holds.

Lemma 3.2.Let x,y≥0.Then

where c is independent of x,y.

Proof.We may assume that x≥y.It follows from[4,(4.19)],that

and moreover,from[4,(4.20)],that

Hence we can obtain the desired estimate.

Lemma 3.3.Let x,t≥0.ThenZ

where c is independent of x,t.

Proof.Similarly as(3.1),Txχt(y)≤1.Since Txχt(y)is supported on[|x?t|,x+t],the desired result is obvious.

4　Littlewood-Paley g functions

As shown in[1,Corollary 6.2],gσis strongly bounded on Lp(?)provided σ＜2ρ/whereand g0satisfies a weak type L1estimate.We give a simple proof of the L2boundedness of gσfor σ＜ρ and consider a modified operator gρ,βwhen σ=ρ.

Theorem 4.1(see[1]).Let σ＜ρ.Then gσis(L2(?),L2(?))bounded.

Proof.Since

for λ∈R and t＞0,it follows that

where

Thus,we complete the proof.

Theorem 4.2.Let gρ,βbe the operator defined by replacing e2ρtin the definition(1.2)of gρby

Then gρ,βis(L2(?),L2(?))bounded.

Proof.We note that

is dominated by

Hence the desired result follows similarly as in Theorem 4.1.

As shownin[7],Section6,g0is boundedfrom H1(?)to L1(?).Inordertounderstand the usage of the formula(2.1)we give a sketch of the proof.

Theorem 4.3(see[7]).g0is(H1(?),L1(?))bounded.

Proof.We recall(2.1)and,for simplicity,we suppose that the integral terms vanish,that corresponds to the case of α,β∈N+.For general α,β,we refer to the arguments in[7], Section 6.Hence,we see that

where gRis the Euclidean g-function on R(see(1.1)).Since gRis bounded formto(see[12,XII,Section 3],with a slight modification by a weight function),it follows from(2.2)that

Thus,we complete the proof.

5　Lusin area functions

We shall consider strong type estimates of the modified area function Sa,h.Similarly as in the Euclidean case,the L2boundedness of Sa,his reduced to the one of gσ.

Theorem 5.1.Sa,his(L2(?),L2(?))bounded provided that a＜2 and h is the following:

(a)h=1,

(c)h=(th)γ0.

Therefore,if we can deduce that

(a)h=1:It follows from Lemma 3.1 that

Therefore,(5.1)holds for σ=(a?1)ρ.Hence,if a＜2,then σ＜ρ.

Let x≥y.Since

it follows that

Let x＜y and x≥1.Since

it follows from Lemma 3.2 that

Let x＜y and αt＜x＜1 for sufficiently small α＞0.Since y＜x+at＜x and x＜1,

and thus,Z

Let x＜y,x＜1 and x＜αt.Since y＜x+at＜(α+a)t,it follows from Lemma 3.3 that

We note that,if t≤1,then J(t)≤c(α+a)γ0and if t＞1,thenTherefore, for a＜2,we can take a sufficiently small α＞0 for which α+a?1＜1.

Therefore,in each case,if a＜2,then there exists 0＜σ＜ρ for which(5.1)holds.

(c)h=(th)γ0:Similarly as in(b),we divide the integral(5.1)over[0,∞).

Let x≥y.Since

it follows that

Let x＜y and x≥1.Clearly(5.2)and thus,(5.3)hold.

Let x＜y and αt＜x＜1 for α＞0.Since y＜x+at＜x,(5.2)and thus,(5.3)hold.

Let x＜y,x＜1 and x＜αt.Since y＜x+at＜(α+a)t and≤c for x＜1, J(t)in the case of(b)is replaced by

Hence,if t≤1,then J(t)≤c(α+a)γ0and if t＞1,then J(t)≤ce?2ρtat≤c.Therefore,in each case,if a＜2,then there exists 0＜σ＜2 for which(5.1)holds.

Theorem 5.2.Sa,his(H1(?),L1(?))bounded provided that a and h are the following:

(b)h=(th)γ0and a≤.

Proof.Similarly as in the proof of Theorem 4.3,for simplicity,we may suppose that the integral terms in(2.1)vanish(see[7,Section 6],for general case).Then we see that Sa,h(f)(x)is dominated as

We note that,for f∈H1(?),eachbelongs toand Ptbehaves similarly as the Euclidean Poisson kernel.Therefore,if we can deduce that

and

where c is independent of x,y,t,then we can apply the arguments used in the Euclidean case(see[12,Proposition 1.2]).Then(5.5)is dominated byand thus,

The proof of(5.6):We divide the integral(5.6)over[0,∞).Let x＞y or x≤y and x≥1 or x≤y and＜x＜1.In these cases,similarly as in the proof of(b)in Theorem 5.1,it follows that

and thus,(5.6)is dominated byLet x≤y,x＜1 and x＜.Since y≤x+at＜at, it follows that

Hence we see from Lemma 3.3 that

Therefore,in each case,if a≤1,then(5.6)holds.

The proof of(5.7):We divide the integral(5.7)over[0,∞).

Let x＞y,t＞1 and y＞1.Since

it follows from Lemma 3.2 that

Let x＞y,t＞1 and y≤1.Since x≤y+at≤1+at,it follows that

Hence,replacing?(x)by ca?(y),we can deduce that

Let x＜y and 1＜x.Since

it follows that

Let x＜y and 2at＜x＜1.Since y≤x+at≤x,we see that

and thus,we can obtain the above estimate.

Let x＜y,x＜1 and x＜2at.Since y≤x+at≤3at,it follows that

Therefore,we see from Lemma 3.3 that

Therefore,in each case,(5.7)holds if a≤1.

(b):h=(th)γ0and a≤The integrand of(5.6)and(5.7)is the following.

Since x?y≤at,this is dominated by

Hence it follows from the previous arguments in(a)that the integrals in(5.6)and(5.7) are dominated by

Remark 5.1.In the definition of Sa,hin(1.2)we can insert the term e2σtas in the one of gσ.

Then it is easy to see that the condition a＜2 in Theorem 5.1 is replaced by

and the conditions a≤1 and a≤in Theorem 5.2(a),(b)are respectively replaced by

Acknowledgements

The author is partly supported by Grant-in-Aid for Scientific Research(C)No.24540191, Japan Society for the Promotion of Science.

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10.4208/ata.2016.v32.n1.4

17 November 2014;Accepted(in revised version)28 October 2015

?Corresponding author.Email address:kawazoe@sfc.keio.ac.jp(T.Kawazoe)