# A Continuous Characterization of Triebel-Lizorkin Spaces Associated with Hermite Expansions

Authors: Shuli Gong; Bolin Ma; Zunwei Fu
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We study the properties of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions onRd (d≥1). In addition to the endpoint estimates, we give the continuous characterizations of these spaces via harmonic and thermic extensions from Rd into R+d+1. Based on this result, we obtain the boundedness of Riesz transform associated with the Hermite expansions.


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## Body

## 1. Introduction

For a smooth functional series{φj}-∞∞, if it satisfies the following conditions:(i)
supp⁡φj⊂{x:2j-1≤x≤2j+1}, j=1,2,3,…,(ii)
2jkDkφjx≤Ck, j∈Z, k∈N0,(iii)
∑j=-∞∞φj(x)=1, x∈Rd,then we call {φj}-∞∞ a smooth dyadic resolution of unity in Rd(d≥1). As proved in [1], it is easy to construct such a resolution of unity.LetD be a self-adjoint operator acting on L2(Rd), and let dE be its spectral resolution which is defined by(1)Df=∫0∞λdEλf.For α∈R, 0<p<∞, 0<q≤∞, and a C∞ function φ∈{φj}-∞∞, we define the Triebel-Lizorkin norm associated with D by (2)fFpα,qD=∑j=-∞∞2jαφjDfxq1/qLpRd,where(3)φjDf=φ2-jDf=∫0∞φ2-jλdEλf.Note that if D=Δ=-∑j=1d∂2/∂xj2 is the Laplacian on Rd, then the norm fFpα,q(D) in (2) is equivalent to the classical Triebel-Lizorkin norm fFpα,q(Δ); see [2]. The main goal of this paper is to study the Triebel-Lizorkin space Fpα,q(D) when D is the Hermite operator.Thed-dimensional Hermite functions on the d-dimensional Euclidean space Rd(d≥1) are defined by(4)hαx=hα1x1×hα2x2×⋯hαdxd,where α=(α1,…,αd),  αi∈0,1,…,  x=(x1,…,xd), and(5)hks=π1/22kk!-1/2Hksexp⁡-s22,k=0,1,…,are the one-dimensional Hermite functions; Hk(s) denotes the kth Hermite polynomial. We know that the Hermite functions form an orthonormal basis for L2(Rd) and that hα are eigenfunctions of the Hermite operator H=-Δ+x2; that is, (6)Hhα=2α+dhα,where α=α1+⋯+αd. Moreover, the operator H is positive and symmetric in L2 on the domain Cc∞(Rd). It is easy to check that the operator H given by (7)H∑f,hαhα=∑2α+df,hαhαis a self-adjoint extension of H on the domain (8)Dom⁡H=f∈L2Rd:∑2α+df,hα2<∞.The operator H has the discrete spectrum {2k+d:k=0,1,…} and admits the spectral decomposition(9)Hf=∑k=0∞2k+dPkf,f∈Dom⁡H,where Pk is a spectral projection that is defined to be Pkf=∑α=k‍f,hαhα. As to the other properties of the Hermite functions, the interested readers can refer to Thangavelu’s monograph [3] for a detailed description.In the past few years, many authors have studied the Triebel-Lizorkin spaces associated with the Hermite operators. In [4], Epperson studied the Triebel-Lizorkin spaces associated with the one-dimensional Hermite operators for (10)supp⁡φ⊂12,2,φλ>c>0,λ∈34,74.Using Mehler’s formula, Epperson proved that the definition of the corresponding space fFpα,q(H) is independent of the particular choice of the function φ. In [5], Dziubański continued this study and proved that the results in [4] also hold for the multidimensional Hermite expansions. Besides, Petrushev and Xu [6] showed that the Triebel-Lizorkin on Rd induced by Hermite expansions can be characterized in terms of the needlet coefficients and that the Hermite-Triebel-Lizorkin space is, in general, different from the respective classical spaces. In addition, Olafsson and Zheng [7] study the Triebel-Lizorkin space associated with the Peetre type maximal function for Hermite operator H, which is defined by(11)φj,s∗fx=supy∈Rd⁡φjHfy1+2j/2x-ys,j∈Z,s>0.They gave the maximal characterization of this space as follows.Theorem GS.Letφj∈{φj(x)}-∞∞,0<p<∞,0<q≤∞, andα∈R. Ifs>d/min⁡(p,q), then(12)fFpα,q~2jαφj,s∗HfLplq,whereφj,s∗(H)fis the Peetre type maximal function for Hermite operator.In [3], Thangavelu introduced Riesz transforms associated with Hermite expansions and proved that they are bounded operators on Lp(Rd),  1<p<∞. In [8], Harboure et al. gave a different proof to show that the Lp-norms of these operators are bounded by a constant not depending on the dimension d. Moreover, they also defined the Riesz transforms of higher order and obtained the free dimensional estimates of the Lp-bounds of these operators. After that, Stempak and Torrea [9] obtained the weighted Lp-inequalities for the gradient square function associated with the Poisson semigroup. They get the result proposed in [3] by using a slightly different proof and they also get the analogous result for the g-function associated with the Poisson semigroup.Although many results have been obtained about the Triebel-Lizorkin spaces associated with the Hermite operators, all of them only discuss the case of the discrete quasi-norm of the Triebel-Lizorkin spaces associated with the Hermite operators. Up to now, as we know, there are few papers related to the continuous case due to the fact that the Hermite expansions do not satisfy the general convolution property. This makes the study of the continuous case a meaningful problem. We first present a general characterization of the Triebel-Lizorkin spaces associated with the multidimensional Hermite expansions satisfied continuous quasi-norm onRd. Furthermore, analogous to the classical Triebel-Lizorkin space, we construct the Littlewood-Paley theorem on these spaces. Finally, we obtain the endpoint estimate of these spaces and establish the boundedness of the Riesz transforms associated with Hermite functions expansions.Throughout this paper,x is the Euclidean norm of x for x∈Rd;  A~B means C1A≤B≤C2A, where C1, C2 are positive constants; C denotes a positive constant that may vary at every occurrence. For simplicity, we abbreviate fFpα,q(H) to fFpα,q.


## 2. The Characterizations ofFpα,q

First, we give the definition of the Triebel-Lizorkin spaces associated with Hermite operators. Letφj∈{φj(x)}-∞∞,  φj(x)=φ(2-jx); the Triebel-Lizorkin spaces associated with Hermite operator H, denoted by Fpα,q,  α∈R,  0<p, and  q<∞, are defined by the quasi-norm (13)fFpα,q=∑j=-∞∞2jαφjHfxq1/qLpRd,where(14)φjHfx=∑k=0∞φ2-j2k+dPkfx.The results of Dziubański [5] show that the definition of Fpα,q does not depend on the particular choice of φ. Next, we will give the general characterization of Triebel-Lizorkin spaces associated with the Hermite expansions.Theorem 1.
Letα∈R,  1<p,  q<∞, and φ∈S(Rd); let φj(x)=φ(2-jx). If x∈Rd∖{0} and j∈Z, then (15)∫0∞t-αφtHf·qdtt1/qLpRd(with usual modification when q=∞) is an equivalent quasi-norm in Fpα,q.This theorem shows that any “discrete” quasi-norm of the Triebel-Lizorkin spaces associated with the Hermite expansions of type (13) has a “continuous” counterpart (15).In order to prove our theorem, we need some necessary lemmas.Lemma 2.
Letεk,  vk be two sequences of complex numbers. Then, for any l, we have (16)∑kεkvk=∑kΔlεkAkl-1Skl-1,where(17)Skl-1=Akl-1-1∑i=1kAk-il-1viis the (l-1)th linear mean of first k terms of vk and Δm is the mth difference operator.Since the proof of Lemma2 is easy, we omit the proof for the sake of compactness of the paper.In [3], a basic estimate for the kernel of Riesz means for the Hermite expansions has been obtained. We describe the result in the following lemma.Lemma 3.
Forα>(3d-2)/6, the following estimate is valid: (18)SRαx,y≤CRd/21+R1/2x-y-α+3d-2/6-1+1+R1/2x+y-α+3d-2/6-1,where SRα(x,y) is the kernel of the Riesz means SRαf for the Hermite expansions, which is defined by (19)SRαf=∑k=0∞1-2k+dR+αPkf.Lemma 4.
Assume thatf∈Lp(Rd),  1≤p<∞, and α>(9d-8)/6. We have the pointwise inequality (20)S∗αfx≤CMpfx,where (21)S∗αfx=supR>0⁡SRαfxis the maximal operator and(22)Mpfx=supr>0⁡1Brx∫Brxfxpdy1/pis the Hardy-Littlewood maximal function of fp raised to the power 1/p and Br(x) is the ball of radius r centered at x.Proof.
The proof is based on the estimate of Lemma3:(23)SRαfx=∫RnSRαx,yfydy≤∫RdSRαx,yq1/qfyp1/p≤C∫RdRd/21+R1/2x-y-α+3d-2/6-1qdy1/q·fLp+C∫RdRd/21+R1/2x+y-α+3d-2/6-1qdy1/q·fLp=I+II,where 1/p+1/q=1. Consider a partition of Rd into dyadic annuli Bj=y:2j≤x-y≤2j+1. For a given function f, we set fj(y)=f(y) for 2j≤x-y≤2j+1 and fj(y)=0 otherwise. Then(24)I≤C∑j=-∞∞∫BjRn/21+R1/2x-y-α+3d-2/6-1qdy1/q·fjLp≤C∑j=-∞∞Rd/21+R1/22j-α+3d-2/6-1Bj1/q+1/pMpfx≤C∑j=-∞∞R1/22jd1+R1/22j-α+3d-2/6-1Mpfx=GR1/2,where (25)Gt=∑j=-∞∞t2jd1+t2j-α+3d-2/6-1.It is obvious that G(t) is locally bounded. Since G(2it)=G(t), it is bounded on (0,∞) (see [3]). The boundedness of II can also be obtained by replacing y in the proof of the boundedness of I by -y and following from the same estimate tactics as above. Hence, we have S∗αf(x)≤CMpf(x).We give the proof of Theorem1 in what follows.Proof.
Consider the following.
Step 1. Let f∈Fpα,q. We will prove that fFpα,q in (13) can be estimated from above by the quasi-norm in (15). Let ρj∈φjx-∞∞ and l-1>(9d-8)/6. It follows from Lemma 4 that(26)ρjHfx=ρjHφλHφλHfx=∑kΔlρj2k+dφλ2k+dAkl-1Skl-1φjλHf·x≤∑kΔlρj2k+dφλ2k+dAkl-1·MφjλHfx.For the first term of the right-hand side of (26), taking the supremum with respect to λ on 1<λ<2 yields(27)sup1≤λ≤2⁡∑kΔlρj2k+dφjλ2k+dAkl-1≤C∑k2-j+1l∑i=0lCliρl-i2-j2k+d·1φ2-j2k+dikl-1≤C∑k22-j+1l·1+2-j2k+db1+2-j2k+dakl-1≤C∑k22-j+1l2a-bj·11+2k+da-bkl-1.We choose a-b=2l>l-1>0; then we have (28)sup1≤λ≤2⁡∑kΔlρj2k+dφλ2k+dAkl-1≤C.
Afterwards, integrating the modified inequality with respect toλ that appears now only in the maximal function yields (29)ρjHfx≤C∫12MφjλHfxdλ,where M(f) is the Hardy-Littlewood maximal function.
For1<p<∞ and 1<q<∞, we first multiply (29) by 2jα; and then we apply the lq-norm with respect to j; afterwards, we apply the Lp-norm with respect to x. That gives(30)∑j=-∞∞2jαρjHfxq1/qLp≤C∑j=-∞∞∫12M2αjφjλHfxdλq1/qLp≤C∑j=-∞∞2αqj∫12∑kφ2-jλ2k+dPkfxdλq1/qLp≤C∑j=-∞∞∫2-j2-j+1t-α∑kφt2k+dPkfxqdtt1/qLp≤C∫0∞t-α∑kφt2k+dPkfxqdtt1/qLp≤C∫0∞t-αφtHfxqdtt1/qLp.Step 2. We will show that the quasi-norm in (15) can be deduced from above by CfFpα,q. Let ρj∈φjx-∞∞. Then we have (13) with ρ instead of φ. Let ρm(x)=0,  -m∈N. For j∈Z, we get (31)2αjφjHfx=∑m=-∞∞2αjφjHρm+jHfx=∑m=-∞K⋯+∑m=K+1∞⋯.We give the estimate of the first sum in (31). Let (32)ρ~jx=2-jxα1ρjxj∈Z,α1>α.Then we have(33)∑m=-∞K2αjφjHρm+jHfx=∑m=-∞K2αj·∑kφ2-j2k+dρm+j2k+dPkfx=∑m=-∞K2αj∑kφ2-j2k+d2-m+j2k+dα1ρ~m+j2k+d·Pkfx=∑m=-∞K2αj+m2α1-αm·∑kΔlφ2-j2k+d2-j2k+dα1Akl-1Skl-1ρ~m+jHf·x.Thus(34)∑m=-∞K2αjφjHρm+jHfx≤∑m=-∞K2αj+m2α1-αm∑kΔlφ2-j2k+d2-j2k+dα1·Akl-1Skl-1ρ~m+jHfx≤C∑m=-∞K2αj+m2α1-αmMρ~m+jHfx,where j∈Z and l-1>(9d-8)/6. In addition, following the same estimate procedure of (27), we have (35)∑kΔlφ2-j2k+d2-j2k+dα1Akl-1≤C.
We useφ(tx),  2-j≤t≤2-j+1, instead of φj(x)=φ(2-jx) in (34); this gives(36)sup2-j≤t≤2-j+1⁡∑m=-∞K2αjφtHρm+jHfx≤C∑m=-∞K2αj+m2α1-αmMρ~m+jHfx.For the above inequality, we first apply the lq-quasi-norm with respect to j and then apply Lp-quasi-norm with respect to x. Note that α1>α. Then, we have(37)∑j=-∞∞sup2-j≤t≤2-j+1⁡∑m=-∞K2αjφtHρm+jHfxq1/qLp≤C∑j=-∞∞∑m=-∞K2αj+m2α1-αmMρ~m+jHfxq1/qLp≤C∑i=-∞∞2iαMρ~iHfxq1/qLp≤C∑i=-∞∞2iαρ~iHfxq1/qLp≤C∑i=-∞∞2iα∑k2-i2k+dα1ρi2k+dPkfxq1/qLp.
Letψ∈S,  supp⁡ψ⊂supp⁡ρ, and ψ(x)=1 if x∈supp⁡ρ. Then we have (38)∑i=-∞∞2iα∑k2-i2k+dα1ρi2k+dPkfxq1/qLp=∑i=-∞∞2iα∑k2-i2k+dα1ψ2-i2k+dρi2k+dPkfxq1/qLp=∑i=-∞∞2iα∑kΔl2-i2k+dα1ψ2-i2k+dAkl-1Skl-1ρiHfxq1/qLp≤C∑i=-∞∞2iαqρiHfxq1/qLp=CfFpα,q.In the proof of the above inequality, we use the following fact:(39)∑kΔl2-i2k+dα1ψ2-i2k+dAkl-1≤C∑k2-i+1l∑s=0lCls2-i2k+dα1l-s·ψs2-j2k+dkl-1≤C∑k22-i+1l·2-i2k+db1+2-i2k+dakl-1≤C∑k22-i+1l2a-bj·12i+2k+da-bkl-1.
Takinga-b=2l>l-1>0 yields (40)∑kΔl2-i2k+dα1ψ2-i2k+dAkl-1≤C.
Let us estimate the second sum in (31) now. Similar to the estimate of the first sum, we introduce (41)ρj′x=2-jxα0ρjxj∈Z,α0<α.Then we have(42)∑m=K+1∞2αjφjHρm+jHfx=∑m=K+1∞2αj+m2α0-αm·∑kφ2-j2k+d2-j2k+dα0ρm+j′2k+dPkfx≤∑m=K+1∞2αj+m2α0-αm∑kΔlφ2-j2k+d2-j2k+dα0·Akl-1Skl-1ρm+j′Hfx≤C∑m=K+1∞2αj+m2α0-αmMρm+j′Hfx,where j∈Z. Following the same estimate procedure of (27) gives (43)∑kΔlφ2-j2k+d2-j2k+dα0Akl-1≤C.
Similarly, we useφ(tx), 2-j≤t≤2-j+1, instead of φj(x)=φ(2-jx) in (42). Then, we have (44)sup2-j≤t≤2-j+1⁡∑m=K+1∞2αjφtHρm+jHfx≤C∑m=K+1∞2αj+m2α0-αmMρ~m+jHfx.Since α0<α, similar to the estimate of the first sum and the estimate of (38), we arrive at (45)∑m=K+1∞sup2-j≤t≤2-j+1⁡∑m=K+1∞2αjφtHρm+jHfxq1/qLp≤C∑i=-∞∞2iα∑k2-i2k+dα0ρi2k+dPkfxq1/qLp≤C∑i=-∞∞2iαqρiHfxq1/qLp=CfFpα,q,which is even stronger than the desires estimate.
The proof is complete.The Poisson semigroupPtt>0, associated with H, is given by(46)Ptf=e-tH1/2f=∑k=0∞e-t2k+d1/2Pkf,f∈L2.We refer the reader to Thangavelu’s monograph [3] for a detailed description of the Poisson semigroup Ptt>0 associated with H. In [9], the authors studied the g-function associated with the Poisson semigroup: (47)gfx=∫0∞∂tft,x2tdt1/2,where f∈Lp, 1≤p<∞, and f(t,x) is the Poisson integral of f that is given by the convergent series (48)ft,x=∑k=0∞e-t2k+d1/2∑α=kf,hαhαx.They obtained that, for 1<p<∞ and ω∈Ap, (49)gfLωp≤CfLωp,f∈Lωp,which implies (50)Cp-1fLp≤gfLp≤CpfLp,1<p<∞.That is a consequent of a general result for symmetric contraction semigroups. This general result is a refinement done by Coifman et al. [10], of the Littlewood-Paley theory for symmetric contraction semigroups satisfying, also, positivity and conservation properties which was developed by Stein [11]; see also [12]. Inspired by these results, we give the harmonic and thermic characterization of the Triebel-Lizorkin spaces associated with the Hermite expansions by the Poisson semigroup related to the Hermite expansion, which shows Fp0,2=Lp, for 1<p<∞.Theorem 5.
Letα∈R, 1<p<∞, and 1<q≤∞. If m is a sufficiently large natural number, then (51)∫0∞tm-αq∂m∂tmPtfxqdtt1/qLPRd(with usual modification when q=∞) is equivalent quasi-norm in Fpα,q.Proof.
We use Theorem1 to prove that (51) is an equivalent quasi-norm in Fpα,q. Let φ(x)=-xm/2e-x1/2. Obviously, φ∈S; we get(52)φt2Hfx=-tmHm/2e-tH1/2fx=tm∂m∂tmPtfx.It means that (51) is an equivalent quasi-norm in Fpα,q. Here, we substituted t by t2.


## 3. Endpoint Estimates

From Theorem5, we know that Fp0,2=Lp, for 1<p<∞. However, this conclusion does not hold when p=1. So, the purpose of this section is to discuss the characterization of the space F10,2. We first present the definition of the local Hardy space below.Definition 6.
Letφ∈S(Rd), φ(0)=1, and φt(x)=φ(tx). The local Hardy space associated with the Hermite operator is defined by the quasi-norm (53)fhn1=sup0<t<2-n⁡φtHf·L1Rd.We now give some properties about the local Hardy space.Proposition 7.
Letm(x) be an infinitely differentiable complex-valued function in Rd such that(54)supx∈Rd⁡1+xαDαmx<∞for every multi-index α. If p=1, then there exists a constant C such that(55)mHf·hn1Rd≤Cfhn1Rdholds for all f∈hn1(Rd).In order to prove Proposition7, we recall some results from the theory of local Hardy spaces [13]; compare also [14]. A function a is an atom for the local Hardy space hn1 if there is a ball B(x0,r), r≤21-n/2, such that(56)supp⁡a⊂Bx0,r,aL∞≤Bx0,r-1,ifr≤2-1-n/2,then∫axdx=0.The atomic norm in hn1 is defined by (57)fha,n1=inf⁡∑jcj,where the infimum is taken over all decompositions f=∑‍cjaj; aj are hn1 atoms.Theorem GO.The norms·hn1and·ha,n1are equivalent with constants independent ofn∈Z.Moreover, if
f
∈
h
n
1
,
supp
⁡
f
⊂
B
(
x
,
2
1
-
n
/
2
)
, then there are
h
n
1
atoms
a
j
such that
supp
⁡
a
j
∈
B
(
x
,
2
2
-
n
/
2
)
and
(58)
f
=
∑
j
c
j
a
j
,
∑
j
c
j
≤
f
h
n
1
with a constant
C
independent of
n.Proof of Proposition7.
We only need to show that there exists a constantC>0 such that (59)∫Rdsup0<t≤2-n⁡φtHmHaxdx≤C,where a is any hn1 atom associated with a ball B(x0,r). Let B∗=B∗(x0,2r). Clearly, this ball is concentric with B. Then(60)∫B∗sup0<t≤2-n⁡φtHmHaxdx≤CB∗1/2∫B∗sup0<t≤2-n⁡∑kφt2k+dm2k+dPkax2dx1/2≤CB∗1/2∫B∗sup0<t≤2-n⁡∑kΔlφt2k+dm2k+dAkl-1Skl-1ax2dx1/2≤CB∗1/2∫B∗Max2dx1/2≤CB∗1/2∫Bax2dx1/2≤CB∗1/2B1/2B-1≤C.In the above inequality, we use the fact that (61)sup0<t≤2-n⁡∑kΔlφt2k+dm2k+dAkl-1≤C.
Ifx∉B∗, we use the representation (62)φtHmHax=∫Ktx,yaydy,where(63)Ktx,y=∑kφt2k+dm2k+dΦkx,y=∑kΔlφt2k+dm2k+dAkl-1Skl-1x,y,and Φk(x,y) is the kernel of the projection operator Pk; see [3] for detailed description. From Lemma 3, we conclude that(64)Ktx,y≤C∑k11+t2k+da11+2k+dbkl-1kd/2·11+x-yl-3d-2/6+11+x+yl-3d-2/6.We choose l-1>(3d-2)/6; then we have (65)∫B∗Csup0<t≤2-n⁡φtHmHaxdx≤C∫B∗Csup0<t≤2-n⁡∫Ktx,yaydydx≤C∫B∗C∫11+x-yl-3d-2/6+11+x+yl-3d-2/6aydydx≤C∫11+xl-3d-2/6dx≤C.Then we complete our proof of Proposition 7.Ifη has the same properties as the function φ (including ηt(x)=η(tx)), then the l2-valued counterpart of fhn1η is given by(66)fjhn1Rd,l2η=sup0<t≤2-n⁡∑j=-∞∞ηtHfj·21/2L1Rd.In the sequel we omit the index η in (66). Then the l2-valued counterpart of (54), (55) reads as follows.Proposition 8.
Letmj,lxj,l=-∞∞ be a sequence of infinitely differentiable complex-valued functions in Rd such that(67)supx∈Rd⁡1+xα∑j,l=0∞Dαmj,lx21/2<∞for every multi-index α. Then there exists a constant C such that(68)∑l=-∞∞mj,lHfl·hn1Rd,l2≤Cfjhn1Rd,l2.After giving the above theoretical preparations, we now state our next result.Theorem 9.
Letα=0, q=2, and p=1. Then (69)F10,2Rd=hn1Rdwhich are equivalent quasi-norms.Proof.
Consider the following.
Step 1. The functions f(x)∈S(Rd) with compact support are dense in F10,2(Rd) and dense in hn1(Rd); compare [13], so we assume f(x)∈S(Rd). First, we prove that(70)fhn1Rd~φjHfhn1Rd,l2with φj∈φjx-∞∞ (equivalent quasi-norms).
Letf∈Lp(Rd), φj∈φjx-∞∞; and let mj,0=φj and mj,l=0 for j,l∈Z, l≠0. Substituting f0=0 and fl=0 (l∈Z, l≠0) into (68) yields(71)φjHfhn1Rd,l2=∑l=-∞∞mj,lHfl·hn1Rd,l2≤Cfjhn1Rd,l2=fhn1Rd.Hence f∈hn1(Rd,l2). In order to prove the converse assertion, we assume that f∈hn1(Rd,l2). Let(72)ψjx=∑r=-11φj+rxwithφ-1=0.If x∈supp⁡φj, it is easy to get ψj=1. Let m0,l=ψl and mj,l=0 if j∈Z, l∈Z, and l≠0. We apply (68) with fj=φj(H)f. Then (73)∑l=-∞∞mj,lHfl·hn1Rd,l2=sup0<t≤2-n⁡∑j=-∞∞ηtH∑l=-∞∞mj,lHfl·21/2L1Rd=sup0<t≤2-n⁡ηtH∑l=-∞∞ψlHfl·L1Rd=sup0<t≤2-n⁡ηtH∑l=-∞∞φlHf·L1Rd=sup0<t≤2-n⁡ηtHf·L1Rd≤Cfjhn1Rd,l2=φjHfhn1Rd,l2.Therefore (74)fhn1Rd≤CφjHfhn1Rd,l2.Thus, (70) holds.
Step 2. We recall that η(0)=1. It follows from (66) with fj=φj(H)f and (70) that(75)fjhn1Rd,l2=φjHfhn1Rd,l2=sup0<t≤2-n⁡∑j=-∞∞ηtHφjHf·21/2L1Rd≥C∑j=-∞∞φjHf·21/2L1Rd=fF10,2Rd.Step 3. We prove the converse assertion of (75). From (66) and (75) we have(76)fhn1Rd≤Csup0<t≤2-n⁡∑j=-∞∞ηtHφjHf·21/2L1Rd.Since η(x) has compact support, ηt(2k+d)φj(2k+d)≠0 implies k≈2j. So, we have (77)sup0<t≤2-n⁡∑j=-∞∞ηtHφjHfx=sup0<t≤2-n⁡∑j=-∞∞∑kηt2k+dPkφjHfx≤Csup0<t≤2-n⁡∫RdKtx,yφjHfydy,where (78)Ktx,y=∑kηt2k+dΦkx,y=∑kΔlηt2k+dAkl-1Skl-1x,y≤C∑k11+t2k+dakl-1kd/2·11+2j/2x-yl-3d-2/6+11+2j/2x+yl-3d-2/6.If φj,s∗(f)(x) is the maximal function in (11), where s=l-(3d-2)/6>d/min⁡(p,q), then by Theorem GS we get(79)fhn1Rd≤C∑j=-∞∞φj,s∗fx21/2L1Rd(80)≤CfF10,2Rd.
In summary, (75) and (79) tell us that the quasi-norms fhn1(Rd) and fF10,2(Rd) are equivalent on the functions f∈S(Rd) with compact support. Theorem 9 follows by completion.


## 4. The Boundedness of the Riesz Transform onFpα,q

This section presents the proof of the boundedness of the Riesz transforms associated with Hermite expansions on the spaceFpα,q.Since(81)L=-12∇+x·∇-x+∇-x·∇+x,it is reasonable to define the Riesz transforms Rj±, j=1,2,…,d, by (82)Rj±=∂∂xj±xjL-1/2.This definition was suggested by Thangavelu [15]; compare also [3]. The definition fits a general framework and, in dimension one, matches the notion of conjugacy for the Hermite function expansions investigated by Gosselin and Stempak [16]. In [17], Stempak and Torrea have already proved that the kernels of the Riesz transforms satisfy the Calderón-Zygmund standard conditions. Their results are as below.Theorem KJ.The Riesz operatorsRj±, initially defined onL2, are Calderón-Zygmund operators associated with the (Calderón-Zygmund) kernelsRj±(x,y), which satisfy(83)Rj±x,y≤Cx-yd,∇xRj±x,y+∇yRj±x,y≤Cx-yd+1.We now state the last result of this paper.Theorem 10.
Letα∈R, 0<p, and q<∞. Then the Riesz transform associated with H is a bounded operator on fFpα,q.Proof.
Using the equivalent quasi-norm ofFpα,q defined in (51), we yield that (84)TRj±=Rj±T,where T is the operator f→T(f) that is defined by(85)Tfx=∂m∂tmPtfxt>0.We obtain (86)Rj±Fpα,q=∫0∞tm-αTRj±fxqdtt1/qLpRd=∫0∞tm-αRj±Tfxqdtt1/qLpRd,where (87)∫0∞tm-αRj±Tfxqdtt1/q≤∫Rd∫0∞tm-αRj±x,yTfyqtdt1/qdy≤C∫Rd1x-yd∫0∞tm-αTfyqtdt1/qdy≤C∫Rd1x-ydp′dy1/p′·∫Rn∫0∞tm-αTfyqtdtp/qdy1/p≤C∫Rd1x-ydp′dy1/p′fFpα,q.
Applying theLp-norm with respect to x in (87) gives (88)Rj±fFpα,q≤CfFpα,q.The results presented in Theorem10 can be further enlarged to the boundedness of the g-functions of Littlewood-Paley defined with either the heat semigroup or the Poisson semigroup. We use a theorem to describe this bounded result.Theorem 11.
Ifg is commutative with the operator T defined by (85) and if the kernel of g satisfies the Calderón-Zygmund size condition, then g is bounded on Fpα,q.Since the proof of Theorem11 is very similar to the proof of Theorem 10, we omit the proof here in order to make the paper concise.



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*Source: 104897-2015-05-28.xml*