New real variable characterizations of Hardy spaces associated with twisted convolution

R E S E A R C H

Open Access

New real-variable characterizations of

Hardy spaces associated with twisted

convolution

Jizheng Huang

_{and Zhou Xing}

*_{Correspondence:}

[email protected]

College of Sciences, North China University of Technology, Beijing, 100144, P.R. China

Abstract

In this paper, we give some new real-variables characterizations of the Hardy space associated with twisted convolution, including Poisson maximal function, area integral, and Littlewood-Paleyg-function.

MSC: 42B30; 42B25; 42B35

Keywords: twisted convolution; Hardy space; Poisson maximal function; area integral; Littlewood-Paleyg-function

1 Introduction

In this paper, we consider the nlinear differential operators

Zj=

∂ ∂zj

+

z¯j, Z¯j= ∂ ∂¯zj

–

zj onC

n_{,j= , , . . . ,n. ()}

Together with the identity they generate a Lie algebrahn_{which is isomorphic to the n+ }

dimensional Heisenberg algebra. The only nontrivial commutation relations are

[Zj,Z¯j] = –



I, j= , , . . . ,n. ()

The operatorLdefined by

L= – 

n

j=

(ZjZ¯j+Z¯jZj)

is nonnegative, self-adjoint, and elliptic. Therefore it generates a diffusion semigroup

{TL

t}t>={e–tL}t>. The operators in () generate a family of ‘twisted translations’τwon

Cn_{defined on measurable functions by}

(τwf)(z) =exp

 

n

j=

(wjzj+w¯jz¯j)

f(z)

=f(z+w)exp

i

Im(z· ¯w)

.

The ‘twisted convolution’ of two functionsf andgonCn_{can now be defined as}

(f ×g)(z) =

Cn

f(w)τ–wg(z)dw

=

Cnf(z–w)g(w)ω(z,¯ w)dw,

whereω(z,w) =exp(_iIm(z· ¯w)). More about twisted convolution can be found in [–]. In [], the authors defined the Hardy spaceH

L(Cn) associated with a twisted

convo-lution. They gave several characterizations ofH

L(Cn) via maximal functions, the atomic

decomposition, and the behavior of the local Riesz transform. As applications, the bound-edness of Hömander multipliers on Hardy spaces is considered in []. The ‘twisted can-celation’ and Weyl multipliers were introduced for the first time in []. Recently, Huang and Wang [] defined the Hardy spaceH_Lp(Cn_{) associated with a twisted convolution for}

n

n+<p≤. Huang gave the characterizations of the Hardy space associated with twisted

convolution by the Lusin area integral function and the Littlewood-Paley function defined by the heat kernel in [] and established the boundedness of the Weyl multiplier by these characterizations in []. Recently, Huang and Liu gave the molecular characterization of Hardy space associated with twisted convolution in []. The purpose of this paper is to give some new real-variable characterizations forH_Lp(Cn), including the Poisson maxi-mal function, the Lusin area integral, and the Littlewood-Paleyg-function defined by the Poisson kernel.

We first give some basic notations concerningH_Lp(Cn_{). LetBdenote the class ofC}∞

-functionsϕonCn_{, supported on the ballB(, ) such thatϕ}

∞≤ and∇ϕ∞≤. For

t> , letϕt(z) =t–nϕ(z/t). Givenσ> ,  <σ≤+∞, and a tempered distributionf, define

the grand maximal function

Mσf(z) =sup

ϕ∈B

sup

<t<σ

ϕt×f(z).

Then the Hardy spaceH_Lp(Cn_{) can be defined by}

H_Lp Cn=f∈S Cn:M∞f ∈Lp Cn.

For anyf ∈H_Lp(Cn_{), definef}

Hp_L(Cn₎=M∞fLp.

Definition  Let _n+n <p≤≤q≤ ∞andp=q. A functiona(z) is aH_Lp,q-atom for the

Hardy spaceH_Lp(Cn_{) associated to a ballB(z} ,r) if

() suppa⊂B(z,r);

() aq≤B(z,r)/q–/p;

()

Cna(w)ω(z¯ ,w)dw= .

We define the atomic Hardy spaceH_Lp,q(Cn_{) to be the set of all tempered distributions of}

the form_jλjaj(the sum converges in the topology ofS(Cn)), whereajareHLp,q-atoms

The atomic quasi-norm inH_Lp,q(Cn_{) is defined by}

fL-atom=inf

j

|λj|p

/p

,

where the infimum is taken over all decompositionsf=_jλjajandajareHLp,q-atoms.

The following result has been proved in [] and [].

Proposition  Let_n+n <p≤.Then for a tempered distribution f onCn_{,the following are}

equivalent:

(i) M∞f ∈Lp(Cn).

(ii) For someσ, <σ< +∞,Mσf ∈Lp_(Cn_).

(iii) For some radial functionϕ∈S,such that_Cnϕ(z)dz= ,we have

sup

<t<

ϕt×f(z)∈Lp Cn

.

(iv) f can be decomposed asf =_jλjaj,whereajareHLp,q-atoms and

j|λj|p< +∞.

Corollary  Let _n+n <p≤and <q≤ ∞. Then H_Lp,q(Cn) =H_Lp(Cn)with equivalent

norms.

Let{P_tL}t>be the Poisson semigroup generated by the operatorL. Then, forf ∈L(Cn),

the functione–t

√

L_{f has the special Hermite expansion (cf.[])}

e–t

√

L_{f(z) = (π)}–n

∞

k=

e–

√

k+nt_f×ϕ k(z),

whereϕkare Laguerre functions. Thereforee–t

√

L_{f is given by the twisted convolution with}

the kernel

Pt(z) = (π)–n

∞

k=

e–

√

k+nt_ϕ

k(z). ()

The Poisson maximal function is defined by

MP(f)(z) =sup t>

_Pt×f(z).

We can characterize the Hardy spaceH_L(Cn_{) as follows.}

Theorem  f∈H_L(Cn_{)if and only if f ∈L}_(Cn_{)and M}

P(f)∈L(Cn).Moreover,we have

f_H

L∼MP(f)L.

We define the area integral associated to{PL t}t>by

S_Lkf(z) =

+∞



|z–w|<t

Dk_tf(w)dw dt tn+

/

the Littlewood-Paleyg-function by

Gk L(f)(z) =

∞



Dk_tf(z)dt t

/

,

and we consider thegλ∗-function associated withLdefined by

g_λ∗_,kf(z) =

∞



Cn

t t+|z–w|

λn

Dk tf(w)

dw dt

tn+

/

,

whereDk

tf(z) =tk(∂tkPtLf)(z).

Now we can prove the main result of this paper.

Theorem 

(a) A functionf ∈H_L(Cn_{)if and only if its Lusin area integralS}k

Lf ∈L(Cn)and

f∈L_(Cn_{).Moreover,we have}

f_H

L∼S

k LfL.

(b) A functionf ∈H_L(Cn_{)if and only if its Littlewood-Paleyg-functionG}k

Lf∈L(Cn)and

f∈L_(Cn_{).Moreover,we have}

f_H

L∼G

k LfL.

(c) A functionf ∈H

L(Cn)if and only if itsgλ∗-functiongλ∗,kf ∈L(Cn)andf ∈L(Cn), whereλ> .Moreover,we have

f_H

L∼g

∗

λ,kfL.

Remark  In this paper, we just give the proofs of our results forp= . In fact, we can prove

the case_n+n <p<  under more conditions (such as thatfvanishes weakly at infinity). The proofs of the case _n+n <p<  are quite similar to the casep= , so we omit them.

Throughout the article, we will useCto denote a positive constant, which is independent of the main parameters and may be different at each occurrence. ByB∼B, we mean that

there exists a constantC>  such that_C ≤B B ≤C.

2 Preliminaries

In this section, we give some preliminaries that we will use in the sequel. LetKt(z) be the heat kernel of{TtL}t>. Then we can get (cf.[])

Kt(z) = (π)–n(sinht)–ne– 

|z|(cotht)_{. ()}

It is easy to prove that the heat kernelKt(z) has the following estimates (cf.[]).

Lemma  There exists a positive constant C> such that

(i) |Kt(z)| ≤Ct–ne–C |z|

t ;

(ii) |∇Kt(z)| ≤Ct–n–  _e–C|z|



LetQk

t(z) be the twisted convolution kernel ofQkt =tk∂skTsL|s=t. Then

Qk_t(z) =tk∂_skKs(z)|s=t.

We have the following estimates [].

Lemma  There exist constants C,Ck> such that

(i) |Qk

t(z)| ≤Ckt–ne–Ct –_|z|

; (ii) |∇Qk

t(z)| ≤Ckt–n–e–Ct –_|z|

.

By the subordination formula, we can give the following estimates as regards the Poisson kernel.

Lemma  There exist constants Ck> ,A> such that

(a)

 <Pt(z)≤Ck

t

(t_+A|z|₎(n+)/; ()

(b)

_∇Pt(z)≤Ck

√

t

(t_+A|z|₎(n+)/. ()

Lemma  Let Dk

t(z)be the integral kernel of the operator Dkt.Then there exist constants

Ck> ,A> ,such that

(a)

Dk_t(z)≤Ck

t

(t_+A|z|₎(n+)/;

(b)

_∇Dk

t(z)≤Ck

√

t (t_+A|_z|₎(n+)/.

We also need some basic properties about the tent space (cf.[]).

Let  <p<∞, and ≤q≤ ∞. Then the tent spaceTqpis defined as the space of functions

f onCn×R+, so that

(z)

f(w,t)qdw dt tn+

/q

∈Lp Cn, when ≤q<∞

and

sup

(w,t)∈ (z)

f(w,t)∈Lp Cn, whenq=∞,

where (z) is the standard cone whose vertex isz∈Cn_,i.e.,

AssumeB(z,r) is a ball inCn, its tentBˆis defined byBˆ={(w,t) :|w–z| ≤r–t}. A function

a(z,t) supported in a tentB,ˆ Ba ball inCn_{, is said to be an atom in the tent spaceT}p q if and

only if it satisfies

ˆ

B

a(z,t)dz dt t

/

≤ |B|/–/p.

The atomic decomposition ofTqpis stated as follows.

Proposition  When <p≤,then for any f ∈T_p can be written as f =λkak,where

akare atoms and

|λk|p≤Cfp_Tp



.

3 The proofs of the main results

Let

MHf(z) =sup t>

Kt×f(z), f ∈L Cn

be the heat maximal function. Then we can characterizeH

L(Cn) by the maximal function

MHf as follows (cf.[] or []).

Lemma  f ∈H_L(Cn)if and only if MHf ∈L(Cn)and f ∈L(Cn).

Now, we give the proof of Theorem .

Proof of Theorem Iff∈H_L(Cn_{), then, by Lemma , we getM}

Hf∈L(Cn). Since

Pt(z) =



√

π

∞



K_t_/μ(z)e–μμ–/dμ,

we haveMP(f)L≤CM_H(f)_L,i.e.,M_Pf ∈L(Cn).

For the reverse, there exists a functionηdefined on (,∞) that is rapidly decreasing at

∞and satisfies the moment conditions (cf.[])

∞



η(t)dt= ,

∞



tkη(t)dt= , k= , , . . . .

Let

(z) =

∞



η(t)Pt(z)dt. ()

Since

 +s–(n+)/=

k<R

aksk+OsR

, ≤s<∞

for appropriate binomial coefficientsak, we have

t

(t_+A|_z|₎(n+)/=

k<R

akt|z|––n

t

|z| k

By () and Lemma , we know thatand any derivative ofare rapidly decreasing. Thus ∈Sand

Cn(z)dz=

∞



η(t)dt= .

Therefore,

M(f)(z)≤MP(f)(z)

˙

∞



η(t)dt≤CMP(f)(z).

This proves thatMP(f)∈L(Cn) impliesf ∈HL(Cn) and the proof of Theorem  is

com-plete.

In order to get our results, we need the following lemma (cf.Lemma  in []).

Lemma 

(i) The operatorsSk

LandGLkare isometries onL(Cn)up to constant factors.Exactly,

_Gk

LfL∼ fL, Sk_Lf

L∼ fL.

(ii) Whenλ> ,there exists a constantC> ,such that

C–fL≤g_λ∗_,kf

L≤CfL.

We define the new Lusin type area integral operator by

S_L,kαf

(z) =

+∞



|z–w|<αt

Dk_tf(w)dw dt tn+

/

,

whereα> .

Lemma  It is easy to see that the above definition of the area integral operator is

indepen-dent ofαin the sense of(Sα

Lf)Lp∼ (Sβ

Lf)Lp,for <α<β<∞and <p<∞(cf.[]). In the following,we use Sk

Lto denote SkL,.

Proof of Theorem (a) By Lemma , we can prove that there exists a constantC>  such that for any atoma(z) ofH

L(Cn), we have

S_Lka_L≤C. ()

In the following, we will show thatf ∈H

L(Cn) whenSLkf ∈L(Cn) andf ∈L(Cn).

We first assume thatf ∈L(Cn)∩L(Cn). WhenS_Lkf ∈L(Cn), we knowDk_tf ∈T_. By Proposition , we get

Dk_tf(z) =

j

whereaj(z,t) are atoms ofTand

j|λj|<∞. By the spectrum theorem (cf.[]), we can

prove

f(z) = 

∞



Dk_t Dk_tf(z)dt

t . ()

By () and (), we get

f(z) = 

+∞



Dk_t

j

λjaj(z,t)

dt

t =C

j

λj

+∞



Dk_taj(z,t)

dt t .

Therefore, it is sufficient to proveαj=

+∞

 Dktaj(z,t)dt_t ,i= , , . . . , are bounded inHL(Cn)

uniformly,i.e., there exists a constantC>  such that for any atoma(z,t) inT ,

α_H

L=

+∞



Dk_ta(z,t)dt t

H_L

≤C.

We assume thata(z,t) is supported inB(zˆ ,r), whereB(zˆ ,r) denotes the tent of the ball

B(z,r), then

sup t> e–t √ L_α(z) L≤

sup

t>

e–t

√

L_α(z)χ B∗

L+

sup

t>

e–t

√

L_α(z)χ (B∗)c

L=I+I,

whereB∗=B(z, r).

By the Hölder inequality, we get

I≤B∗/

Cn sup t> e–t √

L_α(z)_dz

/

≤B∗/α_L.

By the self-adjointness ofDk

t and Lemma , we can get

α_L = sup

β_L≤

Cnα(z)

¯

β(z)dz

= sup

β_L≤

Cn

+∞



Dk_ta(z,t)dt t

¯

β(z)dz

= sup

β_L≤

+∞



CnD

k

ta(z,t)β¯(z)dz

dt t

= sup

β_L≤

+∞



Cna(z,t)D

k tβ¯(z)dz

dt t

≤ sup

β_L≤

Cn

+∞



a(z,t)dz dt t / × Cn +∞ 

Dk_tβ(z)¯ dz dt t

/

≤ |B|–/β_L≤ |B|–/.

By Lemma , we can prove sup s> e–s √ L +∞ 

Dk_ta(z,t)dt t =sup s> e–s√L

+∞



(–t√L)k_e–t√L_a(z,t)dt

t =sup s> +∞ 

(–t√L)ke–(s+t)

√

L_a(z,t)dt

t =sup s> +∞  t s+t

k

–(s+t)√Lke–(s+t)

√

L_a(z,t)dt

t =sup s> +∞  t s+t

k

Cn

Dk_s+t(z–w)a(w,t)dw dt t ≤sup s> +∞  t s+t

Cn

s+t

((s+t)_+A|z–w|₎(n+)/a(w,t)

dw dt t ≤sup s> r  B

(s+t)–n

 +A|z–w|



(s+t)

–(n+)

t s+t



dw dt t

/

× r



B

a(w,t)dw dt t

/

≤ |B|–/|z–z|–(n+)

r



B

t dw dt /

≤Cr|z–z|–(n+).

Then we get

I≤Cr

(B∗)c|

z–z|–(n+)dz≤C.

Whenf ∈L_(Cn_{), we can proceed similarly to Proposition  in []. In fact, we letf} s=

TL

–sf,s≥. Then, byf ∈L(Cn) and Lemma , we knowfs∈L(Cn) andSkLfs≤ SkLf.

By the above proof, we get

fsH_L(Cn₎S_Lkfs_L≤SkLfL.

By the monotone convergence theorem, we have

fs–fnH_L≤SLk(fs–fn)_L→, whens,n→+∞.

Therefore,{fs}is a Cauchy sequence inHL(Cn) and there existsg∈HL(Cn) such that

lim

s→+∞fs=g inH  L Cn

.

As

lim

s→+∞fs=f in (BMOL) ∗_,

we knowf =g∈H_L(Cn_{) andf}

This gives the proof of Theorem (a).

(b) Firstly, by Lemma , we can prove that there exists a positive constantCsuch that for any atoma(z) ofH

L(Cn), we have

_Gk

LaL≤C.

For the reverse, by (a), it is sufficient to prove

S_Lk+f_L≤CGLkfL. ()

Our proof is motivated by []. Let

F(z)(t) = ∂_tke–t

√

L_{f(z), V(z,s) =e}–s√L_F(z).

Then

V(z,s)(t) =e–s

√

L _∂k te–t

√

L_{f(z) = ∂}k te–(s+t)

√

L_f(z).

Therefore

+∞



V(z,s)(t)tk–dt=

+∞



∂_tke–(s+t)

√

L_f(z)

tk–dt

=

+∞

s

∂k te–t

√

L_f(z)_(t–s)k–_dt.

Hence

sup

s>

+∞



V(z,s)(t)tk–dt≤

+∞



tk∂_tke–t

√

L_f(z)dt

t = G

k Lf(z)



.

Let X =L_((,∞),tk–_{dt). Then}

sup

s>

e–s

√

L_F(z)

X=G

k

Lf(z)∈L Cn

.

ThereforeF∈H

X(Cn), hereHX(Cn) can be seen as a vector-valued Hardy space. This

shows thatS

LF(z)∈L(Cn), where

S_LF(z) =

+∞



|z–w|<t

D_tF(w)_Xdw dt tn+

/

.

By

S_LF(z)=

+∞



|z–w|<t

D_t(z)_Xdw dt tn+

=

+∞



|z–w|<t

+∞



(–t√L)e–t

√

L_F(w)(s)_sk–_dsdw dt

tn+

=

+∞



+∞



|z–w|<t

(–√L)k+e–(s+t)

√

L_f(w)

= +∞  +∞ s

|z–w|<(t–s)

(–√L)k+e–t

√

L_f(w)_(t–s)–n_sk–_{dw dt ds}

= +∞  t 

|z–w|<(t–s)

(–√L)k+e–t

√

L_f(w)

(t–s)–nsk–dw ds dt

≥ +∞  t/ 

|z–w|<(t–s)

(–√L)k+e–t

√

L_f(w)_(t–s)–n_sk–_{dw ds dt}

≥ +∞



t/



|z–w|<t

(–√L)k+e–t

√

L_f(w)

t–nsk–dw ds dt

=  kk

+∞



|z–w|<t

(–t√L)k+e–t

√

L_f(w)

t––ndw dt

=  kk

+∞



|z–w|<t

Dk+_t f(w)dw dt tn+ =

 kk S

k+ L f(z)



,

we getSk+

L f ∈L(Cn). Thenf∈HL(Cn) follows from (a).

This completes the proof of Theorem (b). (c) BySk_Lf(z)≤(_)λn_g∗

λ,kf(z), we knowf∈HL(Cn) whengλ∗,kf∈L(Cn) andf ∈L(Cn). In

the following, we show there exists a constantC>  such that for any atoma(z) ofH L(Cn),

we have

gλ∗,kaL≤C.

Without loss of generality, we may assumea(z) is supported inB(,r), then

gλ∗,ka(z) =

∞  Cn t t+|z–w|

λn

Dk_ta(w)dw dt tn+

=

∞



|z–w|<t

t t+|z–w|

λn

Dk_ta(w)dw dt tn+ + ∞ i= ∞ 

i–_{t≤|z–w|<}i_t

t t+|z–w|

λn

Dk_ta(w)dw dt tn+

≤CS_La(z)+

∞

i=

–iλnSk_L,ia(z).

Therefore,

gλ∗,kaL≤CSLaL+

∞

i=

–iλnSk_L,ia_L.

By part (a), we haveSk

LaL≤C. In the following, we will prove that

S_L,k ia_L≤Cin. ()

First, by Lemma , we can obtain

S_L,k ia_L_(B(,i+_r))≤B , i+r /

Letz∈/B(, i+_{r). We have}

Sk_L,ia(z)≤

∞



|z–w|<i_t

B(,r)

Dk_t(w–v) –Dk_t(w)a(v)dv _{dw dt}

tn+

≤

|z|

i+



|z–w|<i_t(· · ·)

dw dt

tn+ +

_∞

|z|

i+

|z–w|<i_t(· · ·)

dw dt

tn+

=I+I.

Forz∈/B(, i+_{r), when|z–w|< }i_t≤|z|

, we have|w| ∼ |z|. By Lemma , we get

I≤C

|z|

i+



|z–w|<i_t

B(,r)

√

t

(t_+A|_w|₎(n+)/|v|a(v)dv



dw dt tn+

≤Cin |z|

i+



t–n

|z| t

–(n+)

r t



dt t ≤C

in–i r

|z|n+.

By Lemma  again, we get

I≤C

∞

|z|

i+

|z–w|<i_t

B(,r)

t–n

r t

a(v)dv 

dw dt tn+

≤Cin

∞

|z|

i+

t–n

r t



dt t ≤C

i(n+) r

|z|(n+).

Thus,

|z|≥i+_r

S_L,k ia(z)dz≤Cin+i

|z|≥i+_r

r

|z|n+dz≤C

in_{. ()}

Therefore, whenλ> , we provegλ∗,kaL≤C. Then Theorem (c) is proved.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

This paper is supported by National Natural Science Foundation of China (11471018), the Beijing Natural Science Foundation (1142005).

Received: 2 February 2015 Accepted: 8 May 2015

References

1. Anderson, RF: The multiplicative Weyl functional calculus. J. Funct. Anal.9, 423-440 (1972)

2. Grassman, A, Loupias, G, Stein, EM: An algebra of pseudo-differential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble)18, 343-368 (1969)

3. Peetre, J: The Weyl transform and Laguerre polynomials. Matematiche27, 301-323 (1972)

4. Mauceri, G, Picardello, M, Ricci, F: A Hardy space associated with twisted convolution. Adv. Math.39, 270-288 (1981) 5. Mauceri, G, Picardello, M, Ricci, F: Twisted convolution, Hardy spaces and Hörmander multipliers. In: Proceedings of

the Seminar on Harmonic Analysis (Pisa, 1980). Rend. Circ. Mat. Palermo (2), pp. 191-202 (1981) 6. Mauceri, G: The Weyl transform and bounded operators onLp_(Rn_{). J. Funct. Anal.39, 408-429 (1980)} 7. Huang, J, Wang, J:Hp_{-Boundedness of Weyl multiplier. J. Inequal. Appl.2014, 422 (2014)}

9. Huang, J: The boundedness of Weyl multiplier on Hardy space associated with twisted convolution. Bull. Sci. Math.

133, 588-596 (2009)

10. Huang, J, Liu, Y: The molecular characterization of Hardy space associated with twisted convolution. J. Funct. Spaces

2014, 1-6 (2014)

11. Thangavelu, S: Lectures on Hermite and Laguerre Expansions. Math. Notes, vol. 42. Princeton University Press, Princeton (1993)

12. Coifman, RR, Meyer, Y, Stein, EM: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal.62, 304-335 (1985)

13. Stein, EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) 14. Stein, EM: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies.

Princeton University Press, Princeton (1970)

15. Auscher, P, Russ, E: Hardy spaces and divergence operators on strongly Lipschitz domain ofRn_{. J. Funct. Anal.201,} 148-184 (2003)