# Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

Authors: Zhiyong Wang; Chuanhong Sun; Pengtao Li
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, assume thatL=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.


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## Body

## 1. Introduction

It is well-known that the Hardy spacesHp form a natural continuation of the Lebesgue spaces Lp to the range 0<p≤1. Correspondingly, let Iα and Jα denote the classical Riesz potentials and Bessel potentials, respectively. The Hardy-Sobolev spaces IαHp and JαHp can be seen as natural generalizations of homogeneous and inhomogeneous Sobolev spaces. Compared with Hardy spaces, the elements of Hardy-Sobolev spaces are of regularities and have been widely used in the research of partial differential equations, potential theories, complex analysis and harmonic analysis, etc. In the last decades, the theory of Hardy-Sobolev spaces was investigated by many researchers extensively. In [1], Strichartz proved that In/pHp was an algebra and found equivalent norms for the Hardy-Sobolev space or, more generally, for the corresponding space with fractional smoothness and Lebesgue exponents in the range p>n/n+1. The trace properties of the space IαHp were discussed by Torchinsky [2]. Miyachi [3] characterized the Hardy-Sobolev spaces in terms of maximal functions related to the mean oscillation of functions in cubes and obtained a counterpart of previous results of Calderón and of the general theory of De Vore and Sharpley [4]. For further information on Hardy-Sobolev spaces and their variants on ℝd, or on subdomains, we refer the reader to [5–12].The development of the theory of Hardy spaces with several real variables was initiated by Stein and Weiss. In [13], by use of square functions, Fefferman and Stein characterized the Hardy spaces Hpℝn for 0<p≤1. From then on, such characterizations were extended to other settings, see [14–16] and the references therein. Since the 1990s, the theory of Hardy spaces associated with second-ordered differential operators on ℝn attracts the attention of many researchers and has been investigated extensively, such as [15–22] and the references therein. In recent years, a lot of research has been done on the Hardy spaces associated with operators on the Heisenberg group and other settings, see [23–25].LetL=−Δℍn+V be a Schrödinger operator, where ΔHn is the sub-Laplacian on ℍn and V belongs to the reverse Hölder class. Let e−tLt>0 be the heat semigroup generated by −L and denote by KtL·,· the integral kernels. Since V is nonnegative, the Feynman-Kacformula asserts that
(1)0<KtLg,h≤T~tg,h≔4πt−Q/2e−g−1h2/4t.Lin-Liu-Liu [25] introduced the Hardy space associated with L, which is defined as follows. Let ML denote the semigroup maximal function: MLfg≔supt>0TtLfg,g∈ℍn. The Hardy space HL1ℍn associated with L is defined to be
(2)HL1ℍn=f∈L1ℍn:MLf∈L1ℍn,where fHL1=MLfL1.As an analogue of classical Hardy-Sobolev spaces, we introduce the following Hardy-Sobolev space associated withL on ℍn:Definition 1.
Forα>0, the Hardy-Sobolev space HL1,αℍn is defined as the set of all functions f∈HL1ℍn such that Lαf∈HL1ℍn with the norm
(3)fHL1,α≔LαfHL1+fHL1<∞.
Our motivation is inspired by the following square function characterization ofHL1ℍn. For k∈ℕ, let
(4)Qtkfg≔t2k∂skTsLs=t2fg.
Define the square function associated withQtk as
(5)SkLfg≔∫0∞∫∣g−1h∣<tQtkfh2dhdttQ+11/2.In [16], Hoffmann et al. obtained the following square function characterization of HL1ℍn:Proposition 2.
Letk∈ℕ. A function f∈HL1ℍn if and only if f∈L1ℍn and the square function SLkf∈L1ℍn. Moreover, fHL1~SLkfL1+fL1.
The goal of this paper is to characterizeHL1,αℍn by the square functions generated by semigroups associated with L. It can be seen from Definition 1 that the elements of HL1,αℍn have the regularities of order α. Based on this observation, we introduced the following fractional square functions associated with semigroup generated by L. For α>0, let ∂tαKtL and ∂tαPtL denote the time-fractional derivatives of the heat kernel and the Poisson kernel, respectively, (cf [26]), i.e.,
(6)∂tαKtLg,h≔eiπm−αΓm−α∫0∞∂tmKt+sLg,hsm−αdss,m=α+1;∂tαPtLg,h≔eiπm−αΓm−α∫0∞∂tmPt+sLg,hsm−αdss,m=α+1,
Forα>0, define the following family of operators:
(7)Qα,tLf≔t2α∂sαe−sLs=t2f,t>0;Dα,tLf≔tα∂tαe−tLf,t>0.
Similar to ([27], Proposition 3.6), the regularities of the kernels of Qα,tL and Dα,tL can be deduced from (6). In this paper, we apply a different method to derive the regularities. In Propositions 10 and 14, we estimate the regularities of tαLαe−tL and tαLα/2e−tL, respectively. Then, by the functional calculus, we deduce the following relations:
(8)tαLαe−tL·,·=tα∂tαe−tL·,·;tαLα/2e−tL·,·=tα∂tαe−tL·,·,
see Lemmas15 and 11. Hence, the desired regularities of Qα,tL and Dα,tL are corollaries of Propositions 10 and 14.
Respect toQα,tL, we introduce the following fractional square functions:
(9)gH,αfg≔∫0∞Qα,tLfh2dtt1/2;SH,αfg≔∫0∞∫Bg,tQα,tLfh2dhdttQ+11/2;gH,α,λ∗fg≔∫0∞∫ℍntt+∣g−1h∣2λQα,tLfh2dhdttQ+11/2.
In Section3.1, we establish the characterizations of HL1ℍn be the square function defined by (9), see Theorem 20. In Section 3.2, we introduce the fractional square functions as follows:
(10)gk,αHf≔∫0∞t2k−2α∂ke−sL∂sks=t2f2dtt1/2,k≥α>0;Sk,αHf≔∫0∞∫Bg,tt2k−2α∂ke−sL∂sks=t2f2dhdttQ+11/2,k≥α>0;gk,α,λH,∗f≔∫0∞∫ℍntt+∣g−1h∣2λt2k−2α∂ke−sL∂sks=t2f2dhdttQ+11/2,k≥α>0.
Let(11)DML=f∈L2ℍn:∫0∞Mλ2dELλf,f<∞.
For everyf∈DLα and Lαf∈L2ℍn∩HL1ℍn, we prove
(12)gk−αHLαf=gk,αHf,Sk−αHLαf=Sk,αHf,gk−α,λH,∗Lαf=gk,α,λH,∗f,
The above relations, together with Theorem20, indicate that
(13)LαfHL1~gk,αHfL1~Sk,αHfL1~gk,α,λH,∗fL1,see Proposition 23. Finally, in Theorem 24, we obtain the desired characterizations of HL1,αℍn via the fractional square functions defined in (10): for every f∈HL1,αℍn,
(14)fHL1,α~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.
For the Poisson semigroup, via the operatorsDα,tL, we can also obtain the corresponding square function characterizations of HL1ℍn and HL1,αℍn, see Theorems 21 and 25 for the details.Remark 3.
(i)
As far as the authors know, even onℝn, the regularities of the time-fractional derivatives of the heat kernels obtained in Section 2.2 are new. The results obtained in Section 2.3 generalize those of [27] to the setting of Heisenberg groups. Moreover, all results in Sections 2.2 and 2.3 apply to some other operators, for example, the degenerate Schrödinger operators, the Schrödinger operators on stratified Lie groups, and so on(ii)
Lemma22 implies that the operators Qα,tL and Dα,tL can be expressed by the spectrum integral of Schrödinger operator. In the sequel, sometime, we formally denote by tα∂tαe−tL and tα∂tαe−tL by Qα,tL and Dα,tL, respectivelyThe paper is organized as follows. In Section2.1, we give some knowledge to be used throughout this paper. Sections 2.2 and 2.3 are devoted to the regularity estimates of Qα,tL and Dα,tL, respectively. In Sections 3.1 and 3.2, we establish the fractional square function characterizations of HL1ℍn and HL1,αℍn. As an application, we deduce an equivalence of the norms of Hardy-Sobolev spaces associated with L.


### 1.1. Notations

Throughout this article, we will usec and C to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By B1~B2, we mean that there exists a constant C>1 such that 1/C≤B1/B2≤C.


## 1.1. Notations

Throughout this article, we will usec and C to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By B1~B2, we mean that there exists a constant C>1 such that 1/C≤B1/B2≤C.


## 2. Preliminaries

### 2.1. Heisenberg Groups and Hardy Spaces

The2n+1-dimensional Heisenberg group ℍn is the Lie group with underlying manifold ℝ2n×ℝ with the multiplication
(15)x,ty,s=x+y,t+s+2∑j=1nxn+jyj−xjyn+j.The Lie algebra of left-invariant vector fields onℍn is given by
(16)X2n+1=∂∂t,Xj=∂∂xj+2xn+j∂∂t,Xn+j=∂∂xn+j+2xj∂∂t,j=1,2,⋯,n.The sub-LaplacianΔℍn is defined as Δℍn=∑j=12nXj2. The gradient ∇ℍn is defined by ∇ℍn=X1,⋯,X2n. The left-invariant distance is dh,g=h−1g. The ball of radius r centered at g is denoted by Bg,r=h∈ℍn:h−1g<r whose volume is given by Bg,r=cnrQ, where cn denotes the volume of the unit ball in ℍn and Q=2n+2 is the homogenous dimension of ℍn. Let Un be the Siegel upper half-space in ℂn+1, i.e.,
(17)Un=z∈ℂn+1:Imzn+1>∑j=1nzj2.Then,Un is holomorphically equivalent to the unit ball in ℂn+1. It is well known that the Heisenberg group ℍn is a nilpotent subgroup of the automorphism group of Un, which consists of the translations of Un. The Heisenberg group ℍn can be also identified with the boundary ∂Un via its action on the origin. We use the Heisenberg coordinates g,s=x,t,s to denote the points in Un, where
(18)xj+ixn+j=zj,j=1,⋯,n;t=Rezn+1;s=Imzn+1−∑j=1nzj2.A nonnegative locallyLq-integrable function V on ℍn is said to belong to the reverse Hölder class Bq,1<q<∞, if there exists C>0 such that the reverse Hölder inequality
(19)1∣B∣∫BVqhdh1/q≤C∣B∣∫BVhdhholds for every ball B∈ℍn. In the sequel, we always assume that 0≡V∈BQ/2.The following auxiliary functionρg,V=ρg was first introduced by Shen [28] and widely used in the research of function spaces related to Schrödinger operators:Definition 4.
The auxiliary functionρ· is defined by
(20)ρg≔supr>0r:1rQ−2∫Bg,rVhdh≤1,g∈ℍn.The following atomic characterization ofHL1ℍn was obtained by Lin-Liu-Liu [25].Definition 5.
A functiona is called a 1,q-atom of the Hardy space HL1ℍn related with a ball Bg0,r if
(i)
suppa⊂Bg0,r;(ii)
aL∞≤Bg0,r1/q−1;(iii)
ifr<ρg, then ∫Bg,rahdh=0
The atomic norm ofHL1ℍn is defined by fL−atom,q≔inf∑∣cj∣, where the infimum is taken over all decompositions f=Σcjaj, and aj are HLq-atoms.Proposition 6.
Let1≤q≤∞. The norms fL−atom,q and fHL1 are equivalent, that is, there exists a constant C>0 such that C−1fHL1≤fL−atom,q≤CfHL1.Below, we give some results on the tent spaces introduced by Coifman-Meyer-Stein.Definition 7.
Assume that0<p,q<∞. The tent space Tqpℍn is defined as the set of all functions f·,· on ℍn satisfying Aqf·∈Lpℍn, where
(21)Aqf≔∫∫Γgfh,tqdhdttQ+11/q,Γg=h,t:h−1g<t.
Coifman, Meyer, and Stein established the following atomic decomposition ofT21Un. A function a·,· is called a T21-atom if (i) a is supported in B^ for some ball B⊂ℍn; (ii) ∫∫B^ag,t2dgdt/t≤1/∣B∣.
The following proposition is one of the main results of tent spaces.Proposition 8.
Every elementf∈T21Un can be written as f=∑jλjaj, where aj are T21-atoms, λj∈ℂ, and ∑j∣aj∣≤CfT21.


### 2.2. Time-Fractional Derivatives of the Heat Semigroup

In this part, we estimate the time-fractional derivatives of the heat kernel associated withL. For k∈ℕ, define
(22)Qk,tLg,h≔t2k∂skKsLg,hs=t2.In ([29], Proposition 2.9), the authors obtained the following estimates about the kernel Qk,tL·,·.Proposition 9.
(i)
ForM>0, there exists a constant CM>0 such that(23)Qk,tLg,h≤CMt−Qe−cg−1h2/t21+tρg+tρh−M.(ii)
Assume that0<δ′≤min1,δ. For any M>0, there exists a constant CM>0 such that, for all ∣ω∣<t(24)Qk,tLgω,h−Qtkg,h≤CM∣ω∣tδ′t−Qe−cg−1h2/t21+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(25)∫ℍnQk,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Denote byQ~α,tL·,· the kernel of tαLαe−tL. In the following proposition, we investigate the regularities of Q~α,tL·,·.Proposition 10.
Letα>0.
(i)
ForM>0, there exists a constant CM>0 such that(26)Q~α,tLg,h≤CMmin1tQ/2,tα/2g−1hQ+α1+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0, there exists a constant CM>0 such that for all ∣ω∣≤t(27)Q~α,tLgω,h−Q~α,tLg,h≤CM∣ω∣tδ′min1tQ/2,tα/2g−1hQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(28)∫ℍnQ~α,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Proof.
(i)
The proof of (i) is divided into the following two cases.
Case 1. α∈0,1. For this case, it follows from functional calculus that
(29)tαLαe−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α.
By (i) of Proposition9, we obtain
(30)Q~α,tLg,h≤tα∫0∞∫0sQt+rLg,hdrt+rdss1+α≤tα∫0∞∫0se−g−1h2/t+rt+rQ/21+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
On the one hand, a direct computation gives(31)Q~α,tLg,h≤tα∫0∞∫0s1t+rQ/2t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαρgMρhM∫0∞∫r∞dss1+α1t+rQ/2+M+1dr≤t−Q/2tρg−Mtρh−M.
On the other hand, because the heat kernel decays rapidly, we can get(32)Q~α,tLg,h≤tαρgMρhM∫0∞∫0s1t+rQ/2+M+1g−1h2t+s−Q+α/2drdss1+α≤tαρgMρhMg−1hQ+α∫0∞∫r∞dss1+αt+r−M−1+α/2dr≤tαρgMρhMg−1hQ+α∫0∞r−αt+r−M−1+α/2dr≤tα/2g−1hQ+α1+tρg−M1+tρh−M.
Case 2. α>1. Let α−α=β. Write
(33)tαLαe−tL=tαLαLβe−tL=tαLα∫0∞∫0s∂re−t+rLdrdss1+β.
Sincem=α+1, we can get
(34)tαLαe−tL=tαLα∫0∞∫0s−Le−t+rLdrdss1+α−α=tα∫0∞∫0s−Lme−t+rLdrdss2+α−m=tα∫0∞∫0sQt+r,mLg,hdrt+rmdss2+α−m.
It can be deduced from (i) of Proposition9 that
(35)Q~α,tLg,h≤tα∫0∞∫0st+r−Q/2t+rρg−Mt+rρh−Mdrt+rmdss2+α−m≤tαρgMρhM∫0∞∫0st+r−Q/2−M−mdrdss2+α−m≤tαρgMρhM∫0∞∫r∞dss2+α−mt+r−Q/2−M−mdr≤t−Q/21+tρg−M1+tρh−M.
Similarly, an application of (i) of Proposition9 again yields
(36)Q~α,tLg,h≤tαρgMρhM∫0∞∫0st+r−Q/2−M−mg−1h2t+s−Q+α/2drdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞∫r∞dss2+α−mt+r−M−m+α/2dr≤tαρgMρhMg−1hQ+α∫0∞rm−α−1t+r−M−m+α/2dr≤tα/2g−1hQ+α1+tρg−M1+tρh−M.(ii)
We first consider the caseα∈0,1. By (ii) of Proposition 9, we obtain(37)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0sQt+rLgω,h−Qt+rLg,hdrt+rdss1+α≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′e−g−1h2/t+r×1+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
Changing the order of integration, we obtain(38)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαωδ′ρgMρhM∫0∞t+r−Q/2−δ′/2−M−1∫r∞dss1+αdr≤∣ω∣tδ′tρg−Mtρh−Mt−Q/2.
Alternatively, we can also get(39)Q~α,tLgω,h−Q~α,tLg,h≤tαωδ′ρgMρhM∫0∞∫0st+r−Q/2−δ′/2−M−1g−1h2t+r−Q+α/2drdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞t+r−δ′/2−M−1+α/2∫r∞dss1+αdr≤∣ω∣tδ′tρg−Mtρh−Mtα/2g−1hα+Q.
Forα≥1, by (ii) of Proposition 9, we can get
(40)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0sQt+rLgω,h−Qt+rLg,hdrt+rmdss2+α−m≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′e−g−1h2/t+r×1+t+rρg−M1+t+rρh−Mdrt+rmdss2+α−m.
Similar to the caseα∈0,1, the rest of the proof can be finished by applying change of order of integration. We omit the details.
(iii)
Forα∈0,1, by (iii) of Proposition 9, we change the order of integration to obtain(41)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0s∫ℍnQt+rLg,hdhdrt+rdss1+α≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rdss1+α.
Ift>ρg, then
(42)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rdss1+α≤tαρgM−δ′∫0∞∫r∞dss1+αdrt+rM−δ′/2+1≤t/ρgδ′1+t/ρgM.
Ift≤ρg, then
(43)∫ℍnQ~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrt+rrα≤tαρg−δ′∫0∞t+rδ′/2−1drrα≤t/ρgδ′1+t/ρgM.
Forα≥1, we have
(44)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+rLg,hdhdrt+rmdss2+α−m≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rmdss2+α−m.
Ift>ρg, then
(45)∫ℍnQ~α,tLg,hdh≤tαρgM−δ′∫0∞t+rδ′−M/2−m∫r∞dss2+α−mdr≤tαρgM−δ′∫0∞t+rδ′−M/2−mrm−α−1dr≤t/ρgδ′1+t/ρgM.
Ift≤ρg, then
(46)∫ℍnQ~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrt+rmrα+1−m≤tαρg−δ′∫0∞t+rδ′/2−1drrα≤t/ρgδ′1+t/ρgM.The following lemma can be deduced from the functional calculus immediately.Lemma 11.
Letα>0. The operators tα∂tαe−tL and tαLαe−tL are equivalent.Proof.
Forα∈0,1, we have
(47)tαLαe−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α=tα∫0∞∫r∞dss1+α∂re−t+rLdr=tα∫0∞r−α∂re−t+rLdr=tα∂tαe−tL.
Forα>1, let α−α=β. Since m=α+1, it holds
(48)tαLαe−tL=tαLαLβe−tL=tαLα∫0∞−Le−s+tLs1−βdss=tα∫0∞−Lme−t+sLsm−αdss=tα∂tαe−tL.Denote byQα,tL·,· the integral kernel of Qα,tL. By Proposition 10 and Lemma 11, we have the following result.Corollary 12.
Letα>0.
(i)
ForM>0, there exists a constant C>0 such that(49)Qα,tLg,h≤Ctαg−1h+tQ+α1+tρg+tρh−M.(ii)
Let0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant C>0 such that, for all ∣ω∣≤t(50)Qα,tLgω,h−Qtαg,h≤C∣ω∣tδ′tαg−1h+tQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant C>0 such that(51)∫ℍnQα,tLg,hdh≤Ct/ρgδ′1+t/ρgM.


### 2.3. Time-Fractional Derivatives of the Poisson Semigroup

In this part, our aim is to give some regularity estimates of the Poisson kernel associated withL. For k∈ℕ, define Dk,tLg,h≔tk∂tkPtLg,h. In ([29], Proposition 2.12), the authors obtained the following estimates about the kernel Dk,tL·,·.Proposition 13 (see [29], Proposition 2.12).
(i)
ForM>0, there exists a constant CM>0 such that(52)Dk,tLg,h≤CMtkt2+g−1h2Q+k/21+tρg+tρh−M.(ii)
Assume that0<δ′≤min1,δ. For any M>0 there exists a constant CM>0 such that, for all ∣ω∣<t(53)Dk,tLgω,h−Dk,tLg,h≤CM∣ω∣tδ′tkt2+g−1h2Q+k/21+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(54)∫ℍnDk,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Denote byD~α,tL·,· the kernel tαLα/2PtL·,·. Similar to Proposition 10, we haveProposition 14.
Letα>0.
(i)
For everyM, there is a constant CM such that(55)D~α,tLg,h≤CMmin1tQ,tαg−1hQ+α1+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant C>0 such that for all ∣ω∣≤t(56)D~α,tLgω,h−D~α,tLg,h≤CM∣ω∣tδ′min1tQ,tαg−1hQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(57)∫0∞D~α,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Proof.
Let us prove (i) first. The following two cases are considered.
Case 1. α∈0,1. By the functional calculus, we obtain
(58)tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α,which, together with Proposition 13, implies that
(59)D~α,tLg,h=tα∫0∞∫0sDt+r,1Lg,hdrt+rdss1+α≤tα∫0∞∫0st+rt+r2+g−1h2Q+1/21+t+rρg+t+rρh−Mdrt+rdss1+α.
One the one hand, we use the change of order of integration to get(60)D~α,tLg,h≤tα∫0∞∫0st+r−Qt+rρg−Mt+rρh−Mdrt+rdss1+α≤tαρgMρhM∫0∞∫r∞dss1+αt+r−Q−2M−1dr≤tαρgMρhM∫0∞r−αt+r−Q−2M−1dr≤t−Qtρg−Mtρh−M.
One the other hand, forα∈0,1,
(61)D~α,tLg,h≤tα∫0∞∫0st+r−Q−11+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrdss1+α≤tαρgMρhMg−1hQ+α∫0∞∫0st+rα−2M−1drdss1+α≤tαρgMρhMg−1hQ+α∫0∞r−αt+rα−2M−1dr≤tαg−1h−Q−αtρg−Mtρh−M.
Case 2. α≥1. Since, for α∈0,1, tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrds/s1+α. We can get
(62)tαLα/2e−tL=tαLα/2Lα−α/2e−tL.
Settingβ=α−α, we obtain
(63)tαLα/2e−tL=tαLα/2Lβ/2e−tL=tαLα/2∫0∞∫0s∂re−t+rLdrdss1+β.
Sincem=α+1,
(64)tαLα/2e−tL=tαLα/2∫0∞∫0s−L1/2e−t+rLdrdss1+α−α=tα∫0∞∫0s−Lm/2e−t+rLdrdss2+α−m=tα∫0∞∫0sDt+r,mLg,hdrt+rmdss2+α−m.
It follows from Proposition13 that
(65)D~α,tLg,h≤tα∫0∞∫0st+r−Qt+rρg−Mt+rρh−Mdrt+rmdss2+α−m≤tαρgMρhM∫0∞∫0st+r−Q−2M−mdrdss2+α−m≤tαρgMρhM∫0∞∫r∞dss2+α−mt+r−Q−2M−mdr≤t−Q1+tρg−M1+tρh−M.
Also, noticing thatα<m, we obtain
(66)D~α,tLg,h≤tα∫0∞∫0st+r−Q−m1+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞∫0st+rα−2M−mdrdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞rm−α−1t+rα−2M−mdr≤tαg−1hQ+α1+tρg−M1+tρh−M.(ii)
We first consider the caseα∈0,1. Since(67)D~α,tLg,h=tα∫0∞∫0sDt+r,1Lg,hdrt+rdss1+α,we apply (ii) of Proposition 13 to obtain
(68)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rt+r2+g−1h2Q+1/2×1+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
One the one hand, we have(69)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rρg−Mt+rρh−Mdrt+rQ+1dss1+α≤tαωδ′ρgMρhM∫0∞∫r∞dss1+αt+r−δ′−Q−2M−1dr≤t−Q∣ω∣tδ′1+tρg−M1+tρh−M.
On the other hand, sinceα<1, it holds
(70)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+r−Q1+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞∫0st+r−δ′+α−2M−1drdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞r−αt+r−δ′+α−2M−1dr≤tαg−1hQ+α∣ω∣tδ′1+tρg−M1+tρh−M.
Forα≥1, noticing
(71)D~α,tLgω,h−D~α,tLg,h=tα∫0∞∫0sDt+r,mLgω,h−Dt+r,mLg,hdrt+rmdss2+α−m,we can use (ii) of Proposition 13 to get
(72)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rmt+r2+g−1h2Q+m/2×1+t+rρg−M1+t+rρh−Mdrt+rmdss2+α−m.
The rest of the proof can be completed by the procedure of the caseα>1 in (i), so we omit the details.
(iii)
Forα∈0,1, it follows from (iii) of Proposition 13 that(73)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+r,1Lg,hdhdrt+rdss1+α≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rdss1+α.
Ift/ρg≥1, then
(74)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rdss1+α≤tαρgM−δ′∫0∞∫r∞dss1+αt+rδ′−M−1dr≤t/ρgδ′t/ρgM≤Ct/ρgδ′1+t/ρgM.
Ift/ρg<1, then
(75)∫ℍnD~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrrαt+r≤tαρg−δ′∫0∞t+rδ′−1r−αdr≤tρgδ′≤Ct/ρgδ′1+t/ρgM.
Forα≥1, using (iii) of Proposition 13 again, we have
(76)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+r,mLg,hdhdrt+rmdss2+α−m≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rmdss2+α−m.
Ift/ρg≥1, we obtain
(77)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rmdss2+α−m≤tαρgM−δ′∫0∞∫r∞dss2+α−mt+rδ′−M−mdr≤t/ρgδ′t/ρgM≤Ct/ρgδ′1+t/ρgM.
Ift/ρg<1, similarly, we can get
(78)∫ℍnD~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrrα+1−mt+rm≤tαρg−δ′∫0∞t+rδ′−mrm−α−1dr≤tρgδ′≤Ct/ρgδ′1+t/ρgM.The following result can be obtained similar to Lemma11.Lemma 15.
Letα>0. The operators tα∂tαe−tL and tαLα/2e−tL are equivalent.Proof.
Forα∈0,1, we have
(79)tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α=tα∫0∞∫r∞dss1+α∂re−t+rLdr=tα∫0∞r−α∂re−t+rLdr=tα∂tαe−tL.
Forα>1, let α−α=β. Noticing m=α+1, we obtain
(80)tαLα/2e−tL=tαLα/2Lβ/2e−tL=tαLα/2∫0∞−Le−s+tLs1−βdss=tα∫0∞−Lα+1e−t+sLs1−α+αdss=tα∫0∞−Lme−t+sLsm−αdss=tα∂tαe−tL.
Define an operatorDα,tLf=tα∂tαPtL. Denote by Dα,tL·,· the integral kernel of Dα,tL. The following estimates are immediate corollaries of Proposition 14 and Lemma 15.Corollary 16.
Letα>0.
(i)
For everyM, there is a constant CM such that(81)Dα,tLg,h≤CMtαt2+g−1h2Q+α/21+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant CM>0 such that for all ∣ω∣≤t(82)Dα,tLgω,h−Dα,tLPtLg,h≤CMtαt2+g−1h2Q+α/2∣ω∣tδ′1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(83)∫0∞Dα,tLg,hdh≤CMt/ρgδ′1+t/ρgM.


## 2.1. Heisenberg Groups and Hardy Spaces

The2n+1-dimensional Heisenberg group ℍn is the Lie group with underlying manifold ℝ2n×ℝ with the multiplication
(15)x,ty,s=x+y,t+s+2∑j=1nxn+jyj−xjyn+j.The Lie algebra of left-invariant vector fields onℍn is given by
(16)X2n+1=∂∂t,Xj=∂∂xj+2xn+j∂∂t,Xn+j=∂∂xn+j+2xj∂∂t,j=1,2,⋯,n.The sub-LaplacianΔℍn is defined as Δℍn=∑j=12nXj2. The gradient ∇ℍn is defined by ∇ℍn=X1,⋯,X2n. The left-invariant distance is dh,g=h−1g. The ball of radius r centered at g is denoted by Bg,r=h∈ℍn:h−1g<r whose volume is given by Bg,r=cnrQ, where cn denotes the volume of the unit ball in ℍn and Q=2n+2 is the homogenous dimension of ℍn. Let Un be the Siegel upper half-space in ℂn+1, i.e.,
(17)Un=z∈ℂn+1:Imzn+1>∑j=1nzj2.Then,Un is holomorphically equivalent to the unit ball in ℂn+1. It is well known that the Heisenberg group ℍn is a nilpotent subgroup of the automorphism group of Un, which consists of the translations of Un. The Heisenberg group ℍn can be also identified with the boundary ∂Un via its action on the origin. We use the Heisenberg coordinates g,s=x,t,s to denote the points in Un, where
(18)xj+ixn+j=zj,j=1,⋯,n;t=Rezn+1;s=Imzn+1−∑j=1nzj2.A nonnegative locallyLq-integrable function V on ℍn is said to belong to the reverse Hölder class Bq,1<q<∞, if there exists C>0 such that the reverse Hölder inequality
(19)1∣B∣∫BVqhdh1/q≤C∣B∣∫BVhdhholds for every ball B∈ℍn. In the sequel, we always assume that 0≡V∈BQ/2.The following auxiliary functionρg,V=ρg was first introduced by Shen [28] and widely used in the research of function spaces related to Schrödinger operators:Definition 4.
The auxiliary functionρ· is defined by
(20)ρg≔supr>0r:1rQ−2∫Bg,rVhdh≤1,g∈ℍn.The following atomic characterization ofHL1ℍn was obtained by Lin-Liu-Liu [25].Definition 5.
A functiona is called a 1,q-atom of the Hardy space HL1ℍn related with a ball Bg0,r if
(i)
suppa⊂Bg0,r;(ii)
aL∞≤Bg0,r1/q−1;(iii)
ifr<ρg, then ∫Bg,rahdh=0
The atomic norm ofHL1ℍn is defined by fL−atom,q≔inf∑∣cj∣, where the infimum is taken over all decompositions f=Σcjaj, and aj are HLq-atoms.Proposition 6.
Let1≤q≤∞. The norms fL−atom,q and fHL1 are equivalent, that is, there exists a constant C>0 such that C−1fHL1≤fL−atom,q≤CfHL1.Below, we give some results on the tent spaces introduced by Coifman-Meyer-Stein.Definition 7.
Assume that0<p,q<∞. The tent space Tqpℍn is defined as the set of all functions f·,· on ℍn satisfying Aqf·∈Lpℍn, where
(21)Aqf≔∫∫Γgfh,tqdhdttQ+11/q,Γg=h,t:h−1g<t.
Coifman, Meyer, and Stein established the following atomic decomposition ofT21Un. A function a·,· is called a T21-atom if (i) a is supported in B^ for some ball B⊂ℍn; (ii) ∫∫B^ag,t2dgdt/t≤1/∣B∣.
The following proposition is one of the main results of tent spaces.Proposition 8.
Every elementf∈T21Un can be written as f=∑jλjaj, where aj are T21-atoms, λj∈ℂ, and ∑j∣aj∣≤CfT21.


## 2.2. Time-Fractional Derivatives of the Heat Semigroup

In this part, we estimate the time-fractional derivatives of the heat kernel associated withL. For k∈ℕ, define
(22)Qk,tLg,h≔t2k∂skKsLg,hs=t2.In ([29], Proposition 2.9), the authors obtained the following estimates about the kernel Qk,tL·,·.Proposition 9.
(i)
ForM>0, there exists a constant CM>0 such that(23)Qk,tLg,h≤CMt−Qe−cg−1h2/t21+tρg+tρh−M.(ii)
Assume that0<δ′≤min1,δ. For any M>0, there exists a constant CM>0 such that, for all ∣ω∣<t(24)Qk,tLgω,h−Qtkg,h≤CM∣ω∣tδ′t−Qe−cg−1h2/t21+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(25)∫ℍnQk,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Denote byQ~α,tL·,· the kernel of tαLαe−tL. In the following proposition, we investigate the regularities of Q~α,tL·,·.Proposition 10.
Letα>0.
(i)
ForM>0, there exists a constant CM>0 such that(26)Q~α,tLg,h≤CMmin1tQ/2,tα/2g−1hQ+α1+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0, there exists a constant CM>0 such that for all ∣ω∣≤t(27)Q~α,tLgω,h−Q~α,tLg,h≤CM∣ω∣tδ′min1tQ/2,tα/2g−1hQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(28)∫ℍnQ~α,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Proof.
(i)
The proof of (i) is divided into the following two cases.
Case 1. α∈0,1. For this case, it follows from functional calculus that
(29)tαLαe−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α.
By (i) of Proposition9, we obtain
(30)Q~α,tLg,h≤tα∫0∞∫0sQt+rLg,hdrt+rdss1+α≤tα∫0∞∫0se−g−1h2/t+rt+rQ/21+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
On the one hand, a direct computation gives(31)Q~α,tLg,h≤tα∫0∞∫0s1t+rQ/2t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαρgMρhM∫0∞∫r∞dss1+α1t+rQ/2+M+1dr≤t−Q/2tρg−Mtρh−M.
On the other hand, because the heat kernel decays rapidly, we can get(32)Q~α,tLg,h≤tαρgMρhM∫0∞∫0s1t+rQ/2+M+1g−1h2t+s−Q+α/2drdss1+α≤tαρgMρhMg−1hQ+α∫0∞∫r∞dss1+αt+r−M−1+α/2dr≤tαρgMρhMg−1hQ+α∫0∞r−αt+r−M−1+α/2dr≤tα/2g−1hQ+α1+tρg−M1+tρh−M.
Case 2. α>1. Let α−α=β. Write
(33)tαLαe−tL=tαLαLβe−tL=tαLα∫0∞∫0s∂re−t+rLdrdss1+β.
Sincem=α+1, we can get
(34)tαLαe−tL=tαLα∫0∞∫0s−Le−t+rLdrdss1+α−α=tα∫0∞∫0s−Lme−t+rLdrdss2+α−m=tα∫0∞∫0sQt+r,mLg,hdrt+rmdss2+α−m.
It can be deduced from (i) of Proposition9 that
(35)Q~α,tLg,h≤tα∫0∞∫0st+r−Q/2t+rρg−Mt+rρh−Mdrt+rmdss2+α−m≤tαρgMρhM∫0∞∫0st+r−Q/2−M−mdrdss2+α−m≤tαρgMρhM∫0∞∫r∞dss2+α−mt+r−Q/2−M−mdr≤t−Q/21+tρg−M1+tρh−M.
Similarly, an application of (i) of Proposition9 again yields
(36)Q~α,tLg,h≤tαρgMρhM∫0∞∫0st+r−Q/2−M−mg−1h2t+s−Q+α/2drdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞∫r∞dss2+α−mt+r−M−m+α/2dr≤tαρgMρhMg−1hQ+α∫0∞rm−α−1t+r−M−m+α/2dr≤tα/2g−1hQ+α1+tρg−M1+tρh−M.(ii)
We first consider the caseα∈0,1. By (ii) of Proposition 9, we obtain(37)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0sQt+rLgω,h−Qt+rLg,hdrt+rdss1+α≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′e−g−1h2/t+r×1+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
Changing the order of integration, we obtain(38)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαωδ′ρgMρhM∫0∞t+r−Q/2−δ′/2−M−1∫r∞dss1+αdr≤∣ω∣tδ′tρg−Mtρh−Mt−Q/2.
Alternatively, we can also get(39)Q~α,tLgω,h−Q~α,tLg,h≤tαωδ′ρgMρhM∫0∞∫0st+r−Q/2−δ′/2−M−1g−1h2t+r−Q+α/2drdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞t+r−δ′/2−M−1+α/2∫r∞dss1+αdr≤∣ω∣tδ′tρg−Mtρh−Mtα/2g−1hα+Q.
Forα≥1, by (ii) of Proposition 9, we can get
(40)Q~α,tLgω,h−Q~α,tLg,h≤tα∫0∞∫0sQt+rLgω,h−Qt+rLg,hdrt+rmdss2+α−m≤tα∫0∞∫0st+r−Q/2∣ω∣t+rδ′e−g−1h2/t+r×1+t+rρg−M1+t+rρh−Mdrt+rmdss2+α−m.
Similar to the caseα∈0,1, the rest of the proof can be finished by applying change of order of integration. We omit the details.
(iii)
Forα∈0,1, by (iii) of Proposition 9, we change the order of integration to obtain(41)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0s∫ℍnQt+rLg,hdhdrt+rdss1+α≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rdss1+α.
Ift>ρg, then
(42)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rdss1+α≤tαρgM−δ′∫0∞∫r∞dss1+αdrt+rM−δ′/2+1≤t/ρgδ′1+t/ρgM.
Ift≤ρg, then
(43)∫ℍnQ~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrt+rrα≤tαρg−δ′∫0∞t+rδ′/2−1drrα≤t/ρgδ′1+t/ρgM.
Forα≥1, we have
(44)∫ℍnQ~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+rLg,hdhdrt+rmdss2+α−m≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rmdss2+α−m.
Ift>ρg, then
(45)∫ℍnQ~α,tLg,hdh≤tαρgM−δ′∫0∞t+rδ′−M/2−m∫r∞dss2+α−mdr≤tαρgM−δ′∫0∞t+rδ′−M/2−mrm−α−1dr≤t/ρgδ′1+t/ρgM.
Ift≤ρg, then
(46)∫ℍnQ~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrt+rmrα+1−m≤tαρg−δ′∫0∞t+rδ′/2−1drrα≤t/ρgδ′1+t/ρgM.The following lemma can be deduced from the functional calculus immediately.Lemma 11.
Letα>0. The operators tα∂tαe−tL and tαLαe−tL are equivalent.Proof.
Forα∈0,1, we have
(47)tαLαe−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α=tα∫0∞∫r∞dss1+α∂re−t+rLdr=tα∫0∞r−α∂re−t+rLdr=tα∂tαe−tL.
Forα>1, let α−α=β. Since m=α+1, it holds
(48)tαLαe−tL=tαLαLβe−tL=tαLα∫0∞−Le−s+tLs1−βdss=tα∫0∞−Lme−t+sLsm−αdss=tα∂tαe−tL.Denote byQα,tL·,· the integral kernel of Qα,tL. By Proposition 10 and Lemma 11, we have the following result.Corollary 12.
Letα>0.
(i)
ForM>0, there exists a constant C>0 such that(49)Qα,tLg,h≤Ctαg−1h+tQ+α1+tρg+tρh−M.(ii)
Let0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant C>0 such that, for all ∣ω∣≤t(50)Qα,tLgω,h−Qtαg,h≤C∣ω∣tδ′tαg−1h+tQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant C>0 such that(51)∫ℍnQα,tLg,hdh≤Ct/ρgδ′1+t/ρgM.


## 2.3. Time-Fractional Derivatives of the Poisson Semigroup

In this part, our aim is to give some regularity estimates of the Poisson kernel associated withL. For k∈ℕ, define Dk,tLg,h≔tk∂tkPtLg,h. In ([29], Proposition 2.12), the authors obtained the following estimates about the kernel Dk,tL·,·.Proposition 13 (see [29], Proposition 2.12).
(i)
ForM>0, there exists a constant CM>0 such that(52)Dk,tLg,h≤CMtkt2+g−1h2Q+k/21+tρg+tρh−M.(ii)
Assume that0<δ′≤min1,δ. For any M>0 there exists a constant CM>0 such that, for all ∣ω∣<t(53)Dk,tLgω,h−Dk,tLg,h≤CM∣ω∣tδ′tkt2+g−1h2Q+k/21+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(54)∫ℍnDk,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Denote byD~α,tL·,· the kernel tαLα/2PtL·,·. Similar to Proposition 10, we haveProposition 14.
Letα>0.
(i)
For everyM, there is a constant CM such that(55)D~α,tLg,h≤CMmin1tQ,tαg−1hQ+α1+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant C>0 such that for all ∣ω∣≤t(56)D~α,tLgω,h−D~α,tLg,h≤CM∣ω∣tδ′min1tQ,tαg−1hQ+α1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(57)∫0∞D~α,tLg,hdh≤CMt/ρgδ′1+t/ρgM.Proof.
Let us prove (i) first. The following two cases are considered.
Case 1. α∈0,1. By the functional calculus, we obtain
(58)tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α,which, together with Proposition 13, implies that
(59)D~α,tLg,h=tα∫0∞∫0sDt+r,1Lg,hdrt+rdss1+α≤tα∫0∞∫0st+rt+r2+g−1h2Q+1/21+t+rρg+t+rρh−Mdrt+rdss1+α.
One the one hand, we use the change of order of integration to get(60)D~α,tLg,h≤tα∫0∞∫0st+r−Qt+rρg−Mt+rρh−Mdrt+rdss1+α≤tαρgMρhM∫0∞∫r∞dss1+αt+r−Q−2M−1dr≤tαρgMρhM∫0∞r−αt+r−Q−2M−1dr≤t−Qtρg−Mtρh−M.
One the other hand, forα∈0,1,
(61)D~α,tLg,h≤tα∫0∞∫0st+r−Q−11+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrdss1+α≤tαρgMρhMg−1hQ+α∫0∞∫0st+rα−2M−1drdss1+α≤tαρgMρhMg−1hQ+α∫0∞r−αt+rα−2M−1dr≤tαg−1h−Q−αtρg−Mtρh−M.
Case 2. α≥1. Since, for α∈0,1, tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrds/s1+α. We can get
(62)tαLα/2e−tL=tαLα/2Lα−α/2e−tL.
Settingβ=α−α, we obtain
(63)tαLα/2e−tL=tαLα/2Lβ/2e−tL=tαLα/2∫0∞∫0s∂re−t+rLdrdss1+β.
Sincem=α+1,
(64)tαLα/2e−tL=tαLα/2∫0∞∫0s−L1/2e−t+rLdrdss1+α−α=tα∫0∞∫0s−Lm/2e−t+rLdrdss2+α−m=tα∫0∞∫0sDt+r,mLg,hdrt+rmdss2+α−m.
It follows from Proposition13 that
(65)D~α,tLg,h≤tα∫0∞∫0st+r−Qt+rρg−Mt+rρh−Mdrt+rmdss2+α−m≤tαρgMρhM∫0∞∫0st+r−Q−2M−mdrdss2+α−m≤tαρgMρhM∫0∞∫r∞dss2+α−mt+r−Q−2M−mdr≤t−Q1+tρg−M1+tρh−M.
Also, noticing thatα<m, we obtain
(66)D~α,tLg,h≤tα∫0∞∫0st+r−Q−m1+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞∫0st+rα−2M−mdrdss2+α−m≤tαρgMρhMg−1hQ+α∫0∞rm−α−1t+rα−2M−mdr≤tαg−1hQ+α1+tρg−M1+tρh−M.(ii)
We first consider the caseα∈0,1. Since(67)D~α,tLg,h=tα∫0∞∫0sDt+r,1Lg,hdrt+rdss1+α,we apply (ii) of Proposition 13 to obtain
(68)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rt+r2+g−1h2Q+1/2×1+t+rρg−M1+t+rρh−Mdrt+rdss1+α.
One the one hand, we have(69)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rρg−Mt+rρh−Mdrt+rQ+1dss1+α≤tαωδ′ρgMρhM∫0∞∫r∞dss1+αt+r−δ′−Q−2M−1dr≤t−Q∣ω∣tδ′1+tρg−M1+tρh−M.
On the other hand, sinceα<1, it holds
(70)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+r−Q1+g−1h2/t+r2Q+α/2t+rρg−Mt+rρh−Mdrt+rdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞∫0st+r−δ′+α−2M−1drdss1+α≤tαωδ′ρgMρhMg−1hQ+α∫0∞r−αt+r−δ′+α−2M−1dr≤tαg−1hQ+α∣ω∣tδ′1+tρg−M1+tρh−M.
Forα≥1, noticing
(71)D~α,tLgω,h−D~α,tLg,h=tα∫0∞∫0sDt+r,mLgω,h−Dt+r,mLg,hdrt+rmdss2+α−m,we can use (ii) of Proposition 13 to get
(72)D~α,tLgω,h−D~α,tLg,h≤tα∫0∞∫0s∣ω∣t+rδ′t+rmt+r2+g−1h2Q+m/2×1+t+rρg−M1+t+rρh−Mdrt+rmdss2+α−m.
The rest of the proof can be completed by the procedure of the caseα>1 in (i), so we omit the details.
(iii)
Forα∈0,1, it follows from (iii) of Proposition 13 that(73)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+r,1Lg,hdhdrt+rdss1+α≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rdss1+α.
Ift/ρg≥1, then
(74)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rdss1+α≤tαρgM−δ′∫0∞∫r∞dss1+αt+rδ′−M−1dr≤t/ρgδ′t/ρgM≤Ct/ρgδ′1+t/ρgM.
Ift/ρg<1, then
(75)∫ℍnD~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrrαt+r≤tαρg−δ′∫0∞t+rδ′−1r−αdr≤tρgδ′≤Ct/ρgδ′1+t/ρgM.
Forα≥1, using (iii) of Proposition 13 again, we have
(76)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0s∫ℍnDt+r,mLg,hdhdrt+rmdss2+α−m≤tα∫0∞∫0st+rρgδ′11+t+r/ρgMdrt+rmdss2+α−m.
Ift/ρg≥1, we obtain
(77)∫ℍnD~α,tLg,hdh≤tα∫0∞∫0st+rρgδ′−Mdrt+rmdss2+α−m≤tαρgM−δ′∫0∞∫r∞dss2+α−mt+rδ′−M−mdr≤t/ρgδ′t/ρgM≤Ct/ρgδ′1+t/ρgM.
Ift/ρg<1, similarly, we can get
(78)∫ℍnD~α,tLg,hdh≤tα∫0∞t+rρgδ′11+t+r/ρgMdrrα+1−mt+rm≤tαρg−δ′∫0∞t+rδ′−mrm−α−1dr≤tρgδ′≤Ct/ρgδ′1+t/ρgM.The following result can be obtained similar to Lemma11.Lemma 15.
Letα>0. The operators tα∂tαe−tL and tαLα/2e−tL are equivalent.Proof.
Forα∈0,1, we have
(79)tαLα/2e−tL=tα∫0∞∫0s∂re−t+rLdrdss1+α=tα∫0∞∫r∞dss1+α∂re−t+rLdr=tα∫0∞r−α∂re−t+rLdr=tα∂tαe−tL.
Forα>1, let α−α=β. Noticing m=α+1, we obtain
(80)tαLα/2e−tL=tαLα/2Lβ/2e−tL=tαLα/2∫0∞−Le−s+tLs1−βdss=tα∫0∞−Lα+1e−t+sLs1−α+αdss=tα∫0∞−Lme−t+sLsm−αdss=tα∂tαe−tL.
Define an operatorDα,tLf=tα∂tαPtL. Denote by Dα,tL·,· the integral kernel of Dα,tL. The following estimates are immediate corollaries of Proposition 14 and Lemma 15.Corollary 16.
Letα>0.
(i)
For everyM, there is a constant CM such that(81)Dα,tLg,h≤CMtαt2+g−1h2Q+α/21+tρg+tρh−M.(ii)
Assume that0<δ′≤δ with 0<δ′<α. For any M>0 there exists a constant CM>0 such that for all ∣ω∣≤t(82)Dα,tLgω,h−Dα,tLPtLg,h≤CMtαt2+g−1h2Q+α/2∣ω∣tδ′1+tρg+tρh−M.(iii)
For anyM>0, there exists a constant CM>0 such that(83)∫0∞Dα,tLg,hdh≤CMt/ρgδ′1+t/ρgM.


## 3. Square Function Characterizations of Hardy-Sobolev Type Spaces

### 3.1. Fractional Square Functions Characterizations ofHL1ℍn

Define(84)gP,αfg≔∫0∞Dα,tLfg2dtt1/2;SP,αfg≔∫0∞∫Bg,tDα,tLfg2dhdttQ+11/2;gP,α,λ∗fg≔∫0∞∫ℍntt+∣g−1h∣2λDα,tLfg2dhdttQ+11/2.In this section, we will characterize the Hardy spaceHL1ℍn by the fractional square functions defined by (9) and (84). Now, we first prove the following reproducing formulas.Lemma 17.
Letα>0.
(i)
The operatorQα,tL defines an isometry from L2ℍn into L2Un,dgdt/t. Moreover, in the sense of L2ℍn,(85)f=Cαlimε→0limN→∞∫εNQα,tL2fdtt.(ii)
The operatorDα,tL defines an isometry from L2ℍn into L2Un,dgdt/t. Moreover, in the sense of L2ℍn,(86)f=Cαlimε→0limN→∞∫εNDα,tL2fdtt.Proof.
The proofs of (i) and (ii) are standard and can be deduced from the spectral techniques. For completeness, we give the proof of (i) and omit the details of the proof of (ii). Sincee−t2L=∫0∞e−t2λdEλ, we have
(87)t2ddse−sLs=t2=−t2Le−t2L=−∫0∞t2λe−t2λdEλ.
Thus, for allf∈L2ℍn, we get
(88)gH,αf22=∫ℍn∫0∞Qα,tLfg2dtdgt=∫0∞Qα,tL2f,fdtt=∫0∞∫0∞t4αλ2αe−2t2λdttdEf,fλ=Cαf22.
For the second part, it suffices to show that, for every pair of sequencesnk→∞&εk→0,
(89)limk→∞∫nknk+mQα,tLf2dtt=limk→∞∫εk+mεkQα,tLf2dtt=0∀m≥1.
Indeed, if (89) holds, we can find h∈L2ℍn such that limk→∞∫εknkQα,tLf2dt/t=h. Therefore, it follows from a polarized version of the first part that for g∈L2ℍn,
(90)h,g=limk→∞∫εknkQα,tLf,Qα,tLgdtt=∫0∞Qα,tLf,Qα,tLgdtt=Cαf,g,which implies h=Cαf. To prove (89), we use again the functional calculus to deduce that
(91)∫nknk+mQα,tLf2dtt22≤∫0∞∫nknk+mt4αλ2αe−2t2λdtt2dEf,fλ.
Computing the integral inside one yields∫0∞1+2λnk2e−2λnk2dEf,fλasnk→∞, which by dominated convergence tends to 0. Observe that the last step makes use of the fact that 0 is not an eigenvalue of L because Vg>0 for almost every g, and Lf,f≥Vf,f>0 unless f≡0. One proceeds similarly when εk→0.The following boundedness of square functions can be deduced from the spectral theorem immediately.Lemma 18.
Letα>0 and λ>Q/2.
(i)
The operatorsgH,α, GH,α and gH,α,λ∗ are bounded on L2ℍn. Moreover, there exist constants C, C1 and C2 such that gH,αfL2=CfL2,GH,αfL2≤C1fL2,gH,α,λ∗fL2≤C2fL2(ii)
The operatorsgP,α, GP,α and gP,α,λ∗ are bounded on L2ℍn. Moreover, there exist constants C, C1 and C2 such that gP,αfL2=CfL2,GP,αfL2≤C1fL2,gP,α,λ∗fL2≤C2fL2Proof.
We only prove (i), and (ii) can be done similarly. ForgH,α, using the reproducing formula on L2ℍn, we can get
(92)gH,αfL22=∫0∞Qα,tLf,Qtαfdtt=∫0∞Qα,tL2f,fdtt=∫0∞e−iπα∫0∞t4αλ2αe−2t2λdttdEf,fλ=CfL22.
ForGH,α, we have
(93)GH,αfL22≤∫0∞∫ℍn1tQ∫ℍnχΓgh,tdgQα,tLfg2dhdtt≤∫0∞∫ℍnQα,tLfg2dhdtt=gH,αfL22≤C1fL22.
ForgH,α,λ∗, the relation: gH,α,λ∗fg≤CGH,αfg indicates that gH,α,λ∗fL2≤C2fL2.Proposition 19.
Letα>0 and λ>Q/2.
(i)
There exists a constantC such that for any function f which is a linear combination of HL1-atoms(94)GH,αfL1≤CfHL1,gH,αfL1≤CfHL1,gH,α,λ∗L1≤CfHL1.(ii)
There exists a constantC such that for any function f which is a linear combination of HL1-atoms(95)GP,αfL1≤CfHL1,gP,αfL1≤CfHL1,gP,α,λ∗L1≤CfHL1.Proof.
We only prove (i), and (ii) can be dealt with similarly. Firstly, by Lemma18, we can get gH,αfL2=CfL2. For f∈HL1ℍn, it holds an atomic decomposition: f=∑jcjaj. Then,
(96)GH,αfg=∫0∞∫Bg,t∑jcjQα,tLajh2dhdttQ+11/2≤∑j∣cj∣GH,αajg.
So we only need to verify thatGH,αa is in L1ℍn for any HL1-atom a uniformly. By Lemma 18,
(97)GH,αaL22≤∫0∞∫ℍnQα,tLah2dhdtt=gH,αfL22≤CaL22≤CBg0,r−1.
WriteGH,αaL1=A+B, where A=∫Bg0,4rGH,αagdg and B=∫Bcg0,4rGH,αagdg. For A, it is clear that
(98)A≤Bg0,4r1/2∫Bg0,4rGH,αag2dg1/2≤Bg0,4r1/2CBg0,r−1≤C.For the estimate ofB, the following two cases are considered.Case 1.
r<ρg0/4. By the cancelation property of the atom a, we have GH,αag≤B1+B2, where
(99)B1≔∫0∣g−1g0∣/2∫∣g−1h∣<t∫Bg0,rQα,tLh,z−Qα,tLh,g0azdz2dhdttQ+11/2;B2≔∫∣g−1g0∣/2∞∫∣g−1h∣<t∫Bg0,rQα,tLh,z−Qα,tLh,g0azdz2dhdttQ+11/2.
ForB1, since 0<t<g−1g0/2 and g−1h<t, we can get h−1g0~g−1g0. For z∈Bg0,r and g∈Bcg0,4r, we have g0−1z<r≤Cg0−1h/4. Using (ii) of Corollary 12 and the symmetry, we can get
(100)B1≤∫0∣g−1g0∣/2∫∣g−1h∣<t∫Bg0,rCMtαt+∣g0−1h∣Q+α∣z−1g0∣tδ′∣az∣dz2dhdttQ+11/2≤CM∫0∣g−1g0∣/2∫∣g−1h∣<tt2αt+∣g0−1g∣2Q+αrt2δ′dhdttQ+11/2≤CMrδ′g0−1gQ+α∫0∣g−1g0∣/21t2δ′−2α+1dt1/2≤CMrδ′g0−1gQ+δ′.
The above estimate forB1 implies that
(101)∫Bcg0,4rB1dg≤CM∑k=2∞∫2kr≤∣g−1g0∣<2k+1rrδ′dgg−1g0Q+δ′≤CM∑k=2∞rδ′2k+1rQ2krQ+δ′≤C.
Next, we estimateB2. Since z−1g0≤r<g−1g0/2≤t, the estimate
(102)B2≤CM∫∣g−1g0∣/2∞∫∣g−1h∣<t∫Bg0,rtαt+∣g0−1h∣Q+α∣z−1g0∣tδ′∣az∣dz2dhdttQ+11/2≤CM∫∣g−1g0∣/2∞∫∣g−1h∣<trt2δ′t2αt+∣g0−1h∣2Q+αdhdttQ+11/2≤CMg−1g0Qrδ′g−1g0δ′implies that
(103)∫Bcg0,4rB2dg≤CM∫∣g−1g0∣≥4rrδ′g−1g0Q+δ′dg≤C.Case 2.
ρg0/4≤r<ρg0. In this case, we write GH,αag2=D1+D2+D3, where
(104)D1≔∫0r/2∫∣g−1h∣<tQα,tLah2dhdttQ+1;D2≔∫r/2∣g−1g0∣/4∫∣g−1h∣<tQα,tLah2dhdttQ+1;D3≔∫∣g−1g0∣/4∞∫∣g−1h∣<tQα,tLah2dhdttQ+1.
We first estimate the termD1. Since g−1g0>4r, g0−1z<r and g−1h<t<r/2, h−1g0>7r/2. For z∈Bg0,r, z−1g0<r<g−1g0/4. Using the triangle inequality, we apply (i) of Corollary 12 to estimate D1 as follows.
(105)D1≤CM∫0r/2∫∣g−1h∣<t∫Bg0,rtαt+∣g−1h∣Q+α∣az∣dz2dhdttQ+1≤CM∫0r/2∫∣g−1h∣<tt2αt+∣g−1g0∣2Q+αdhdttQ+1≤CMg−1g02Q+2α∫0r/2t2α−1dt≤CMr2αg−1g02Q+2α.
ForD2, since z∈Bg0,r, ∣z−1g0∣<r<ρg0, then ρz~ρg0~r. We have
(106)D2≤CM∫r/2∣g−1g0∣/4∫∣g−1h∣<t∫Bg0,rtαt+∣g−1h∣Q+α∣az∣dz1+t/ρh+t/ρzM2dhdttQ+1,≤CM∫r/2∣g−1g0∣/4∫∣g−1h∣<tt2αt+∣g−1h∣2Q+α1+tρg0−2MdhdttQ+1≤CM∫r/2∣g−1g0∣/4rt2Mt2α−1g−1g02Q+2αdt≤CMr2Mg−1g02Q+2M.
At last, we estimateD3. For ∣z−1g0∣<r<ρg0, we have ρg0~ρz. Then, we can get
(107)D3≤CM∫∣g−1g0∣/4∞∫∣g−1h∣<t∫Bg0,rtαt+∣h−1z∣Q+α∣az∣dz1+t/ρh+t/ρzM2dhdttQ+1≤CM∫∣g−1g0∣/4∞∫∣g−1h∣<t∫Bg0,r1+tρg0−M1t+∣h−1z∣Q∣az∣dz2dhdttQ+1≤CMr2Mg−1g02Q∫∣g−1g0∣/4∞1t2M+1dt≤CMr2Mg−1g02Q+2M.
The above estimates forDi,i=1,2,3, indicate that
(108)∫BcGH,αagdg≤∑k=2∞∫2kr≤∣g−1g0∣<2k+1rD11/2g+D21/2g+D31/2gdg≤C.Now, we give the following characterizations ofHL1ℍn.Theorem 20.
Letα≥1/2 and λ>Q/2. The following assertions are equivalent:
(i)
f∈HL1ℍn;(ii)
f∈L1ℍnandgH,αf∈L1ℍn;(iii)
f∈L1ℍnandGH,αf∈L1ℍn;(iv)
f∈L1ℍnandgH,α,λ∗f∈L1ℍn
Moreover, for everyf∈HL1ℍn,
(109)fHL1~f1+gH,αf1~f1+GH,αf1~f1+gH,α,λ∗f1.Proof.
By Proposition19, for f∈HL1ℍn, we know that gH,αf∈L1ℍn, GH,αf∈L1ℍn, and gH,α,λ∗f∈L1ℍn, respectively.
For the reverse, we first show that forGH,αf∈L1ℍn, f∈HL1ℍn. Assume that f∈L1ℍn∩L2ℍn. When GH,αf∈L1ℍn, we can see that
(110)∫ℍnGH,αfgdg=∫ℍn∫0∞∫Bg,tQα,tLfh2dhdttQ+11/2dg,which implies that Qα,tLfg∈T21, where Qα,tLfg≔∫ℍnQα,tLg,hfhdh. By Proposition 8, Qα,tLfg=Σkλkakg,t, where ak·,· are T21-atoms and Σk∣λk∣<∞. Assume that the atom a·,· is supported on B^g0,r. By Lemma 17,
(111)fg=C∫0∞Qtα∑k=1∞λkakg,tdtt≔∑k=1∞λkTkg,where Tkg=∫0∞Qα,tLakg,tdt/t. For simplicity, we denote Tkg by Tg for k=1,2,⋯. Write
(112)supt>0e−tLTgL1≤supt>0e−tLTgχB∗L1+supt>0e−tLTgχB∗cL1=I1+I2,where B∗=Bg0,2r. For I1, we use Hölder’s inequality to deduce that
(113)TL2=sup∥ϕ∥2≤1∫ℍn∫0∞Qα,tLag,tdttϕ¯gdg≤sup∥ϕ∥2≤1∫0∞∫ℍnag,t2dgdtt1/2∫0∞∫ℍnQα,tLϕ¯g2dgdtt1/2≤sup∥ϕ∥2≤1B−1/2ϕ2≤B−1/2,which gives I1≤B∗1/2B−1/2≤C.
Now, we deal withI2. For s>0, by functional calculus and Proposition 2.9, we have
(114)e−sL∫0∞Qα,tLag,tdtt=∫0∞∫0∞t2α∂sKs+t+λLs=t2ag,tλ1−αdλλdtt=∫0∞t2α∂sαKs+tLs=t2ag,tdtt≤C∫0∞tαah,ts+t2+∣g−1h∣Q+αdhdtt.
Whenh∈Bg0,r and g∈B∗c, we have ∣g−1h∣~∣g−1g0∣, and
(115)e−sL∫0∞Qα,tLag,tdtt≤Cg−1g0−Q+α/2∫0r∫Bt2α−1dhdt1/2∫0r∫Bah,t2dhdtt1/2≤CB−1/2g−1g0−Q+α∫0r∫Bt2α−1dhdt1/2≤Crαg−1g0−Q+α.
Finally, we get(116)I2≤∫Bcg0,rrαg−1g0Q+αdg≤C.
Whenf∈HL1ℍn, let G~H,α be the bounded extension of GH,αf from L2∩HL1ℍn to HL1ℍn. Since L2∩HL1ℍn is dense in HL1ℍn, there exists a sequence fn⊂L2∩HL1ℍn such that fn→f as n→∞ in HL1ℍn. By Corollary 12, we conclude that GH,αfn→GH,αf as n→∞. By the definition of G~H,α, we know that GH,αfn→G~H,αf as n→∞. Therefore, GH,αf=G~H,αf for f∈HL1ℍn, which gives
(117)fHL1=limn→∞fnHL1≤limn→∞GH,αfnL1=G~H,αfL1=GH,αfL1.
For the Littlewood-Paleyg-function, it is sufficient to prove GH,αfL1≤CgH,αfL1. For β>0, we define G~H,βf by
(118)G~H,βfg=∫0∞∫∣g−1h∣<βtQα,tLfh2dhdttQ+11/2.
Similarly, we can prove thatf∈HL1ℍn if and only if G~H,βf∈L1ℍn and f∈L1ℍn. Moreover, fHL1~G~H,βfL1.
LetFgt≔∂sαe−sLs=t2fg and Vg,s≔e−sLFg. Then Vg,st=∂rαe−s+rLr=t2fg. Therefore,
(119)∫0∞Vg,st2dtt1−4α=∫0∞∂rαe−s+rLr=t2fg2dtt1−4α=∫s∞∂rαe−rLr=t2fg2tdtt2−s1−2α.
Whenα≥1/2, we have t2−s2α−1≤t4α−1. Hence,
(120)sups>0∫0∞Vg,st2t4α−1dt≤∫0∞∂rαe−rLr=t2fg2t4α−1dt=gH,αfg2.
LetX=L20,∞,t4α−1dt. Then, sups>0e−sLFgX≤gH,αfg∈L1ℍn. Therefore, F∈HX1ℍn, where HX1ℍn can be seen as a vector-valued Hardy space (cf. [30]). This shows that G~2XFg∈L1ℍn, where
(121)G~2XFg=∫0∞∫∣g−1h∣<2tQα,tLFhX2dhdttQ+11/2.
We can assume that1/2≤α<1. Then, the identity (6) gives
(122)∂tαKtLt=s2∂sαKsLs=t2=C∫0∞∫0∞∂aKa+tLt=s2∂bKs+bLs=t2a−αb−αdadb=C∫0∞∫0∞∂a2Ka+b+s+tLs=t2,t=s2a−αb−αdadb=C∫0∞∂λ2Kλ+s+tLs=t2,t=s2λ1−2αdλ.
Whenα≥1/2, we get ∂tαKtLt=s2∂sαKsLs=t2=∂t2αKs+tLs=t2,t=s2. Via integration by substitution, we can change the orders of integration to obtain
(123)G~2XFg2=∫0∞∫∣g−1h∣<2t∫0∞t2α∂s2αe−sLs=t2Fhs2s4α−1dsdhdttQ+1≥∫0∞∫03t/2∫∣g−1h∣<2t2−s2∂s2αe−sLs=t2fh2ts4α−1dhdtdst2−s21+Q/2−2α≥∫0∞∫03t/2∫∣g−1h∣<t∂s2αe−sLs=t2fh2t4α−1−Qs4α−1dhdsdt=C∫0∞∫∣g−1h∣<tt4α∂s2αe−sLs=t2fh2dhdttQ+1=CG~L1fg2,which implies GH,αf∈L1ℍn, and therefore, f∈HL1ℍn. Since t/∣g−1h∣+t2λ>2−2λ in the cone Γg=h,t:∣g−1h∣<t, we have
(124)GH,αfg≤∫Γg22λt∣g−1h∣+t2λQtαfh2dhdttQ+11/2≤2λgH,α,λ∗fg.
This completes the proof of Theorem20.Theorem 21.
Letα≥1/2 and λ>Q/2. The following assertions are equivalent:
(i)
f∈HL1ℍn(ii)
f∈L1ℍn and gP,αf∈L1ℍn(iii)
f∈L1ℍn and GP,αf∈L1ℍn(iv)
f∈L1ℍn and gP,α,λ∗f∈L1ℍn
Moreover, for everyf∈HL1ℍn,
(125)fHL1~f1+gP,αf1~f1+GP,αf1~f1+gP,α,λ∗f1.Proof.
This theorem can be proved similarly as the proof of Theorem20, so we omit it.


### 3.2. Fractional Square Functions Characterizations ofHL1,αℍn

In this part, we will give the characterizations of Hardy-Sobolev spaceHL1,αℍn by fractional square functions. Firstly, we give the following Lemma, which will be used in the sequel. Similar to ([31], Proposition 2.4), we can express the operators ∂tαe−tL and ∂tαe−tL as follows.Lemma 22.
Letα>0.
(i)
For everyf∈L2ℍn,(126)∂tαe−tLf=eiπα∫0∞λαe−tλdELλf,t>0.(ii)
For everyf∈L2ℍn,(127)∂tαe−tLf=eiπα∫0∞λα/2e−tλdELλf,t>0.Proof.
LetEλ denote a resolution of the identity. It follows from the spectral decomposition:
(128)e−tLf=∫0∞e−λtdEfλ∀f∈L2ℍnthat
(129)∂tke−tLf=e−iπk∫0∞λke−λtdEfλ,k=1,2,⋯.
By (6) and (129), we have
(130)∂tαe−tLf=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEfλsk−α−1ds,where k is the smallest integer satisfying k>α. Then, the integral
(131)∫0∞∫0∞λke−t+sλ∣dEfλ∣sk−α−1dsis absolutely convergent. By the fact that ∂tαe−tLfLp≤CαfLp/tα, the integral in (6) is absolutely convergent in L2ℍn. Hence, by (130), we can get for g∈L2ℍn,
(132)∂tαe−tLf,g=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEfλsk−α−1ds,g=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEf,gλsk−α−1ds=e−iπαΓk−α∫0∞∫0∞λke−t+sλsk−α−1dsdEf,g=e−iπα∫0∞λαe−tλdEfλ,g,which implies (i). The assertion (ii) can be obtained by the aid of functional calculus similarly.The following result can be deduced from Lemma22 immediately.Proposition 23.
(i)
Let0<α<k, k∈ℕ and λ>Q. Iff∈DLα∩HL1ℍn and Lαf∈L2ℍn∩HL1ℍn. Then,(133)LαfHL1~gk,αHfL1~Sk,αHfL1~gk,α,λH,∗fL1.(ii)
Let0<α<k, k∈ℕ and λ>Q. If f∈DLα∩HL1ℍn and Lαf∈L2ℍn∩HL1ℍn. Then,(134)Lα/2fHL1~gk,αPfL1~Sk,αPfL1~gk,α,λP,∗fL1.Proof.
We only prove (i), and (ii) can be dealt with similarly. Using Lemma22, we can get
(135)∂sk−αe−sLs=t2Lαf=Lk−αe−t2LLαf=Lke−t2Lf=∂skKsLs=t2f,therefore,
(136)gk−αHLαf=gk,αHf,Sk−αHLαf=Sk,αHf,gk−α,λH,∗Lαf=gk,α,λH,∗f.
Using Theorem20, we can get
(137)LαfHL1~gk,αHfL1~Sk,αHfL1~gk,α,λH,∗fL1.
LetGα,L=f∈HL1ℍn:Lαf∈Cc∞ℍn. Since Cc∞ℍn is dense in HL1ℍn, Gα,L is dense in HL1,αℍn. Note that Gα,L⊂DLα∩HL1ℍn, and
(138)LαGα,L=Cc∞ℍn⊂L2ℍn∩HL1ℍn.
Using Proposition23, gk,αH, Sk,αH, and gk,α,λH,∗ can be extended to HL1,αℍn as bounded operators from HL1,αℍn to L1ℍn. Let gk,αH~ be the extension of gk,αH to HL1,αℍn as a bounded operator from HL1,αℍn to L1ℍn. Then, there exists C>0 such that for f∈HL1,αℍn,
(139)fHL1+gk,αH~f1≤CfHL1,α.
Below, we give the square function characterizations of the Hardy-Sobolev spaceHL1,αℍn as follows.Theorem 24.
Letα≥1/2, k∈ℕ0, and λ>Q. Then, the following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αHf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αHf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λH,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(140)fHL1,α~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Proof.
We first provefHL1+gk,αHfL1≤CfHL1,α. By (139), it is sufficient to prove gk,αHf=gk,αH~f. For N∈ℕ and h∈HL1ℍn, by the subordination formula, we obtain
(141)∂k∂tkKtLhg≤Ct−2ksupt>0Tt2/2Lhg.
Then,(142)∫1/N∞t2k−2α∂k∂skKsLs=t2hg2dtt1/2≤C∫1/N∞t−1−4αdt1/2supt>0Tt2/2Lhg≤CN2αsupt>0Tt2/2Lhg.
By the definition ofHL1ℍn, we conclude that the operator
(143)h→∫1/N∞t2k−2α∂k∂skKsLs=t2hg2dtt1/2is bounded from HL1ℍn to L1ℍn. Therefore, if f=L−αh, where h∈HL1ℍn∩L2ℍn, we have
(144)∫1/N∞t2k−2α∂k∂skKsLs=t2L−αhg2dtt1/2L1=∫1/N∞t2k−2α∂k−α∂sk−αKsLs=t2hg2dtt1/2L1≤ChHL1,where the positive constant C is independent of N∈ℕ. Letting N→∞ yields
(145)gk,αH~fL1=∫0∞t2k−2α∂k∂skKsLs=t2fg2dtt1/2L1≤CfHL1,α.
SinceGα,L is dense in HL1,αℍn, for f∈HL1,αℍn, we obtain
(146)gk,αHfL1=∫0∞t2k−2α∂k∂skKsLs=t2fg2dtt1/2L1=gk,αH~fL1≤CfHL1,α.
The proofs forSk,αH and gk,α,λH,∗ are similar, and so is omitted.
For the reverse, we only deal with the case ofgk,αH for simplicity.
Step I. We prove(147)∂sαKsLs=t2fHL1≤Ct−2αfHL1.
Form∈ℕ and m>α, by (141), we obtain
(148)supβ>0TβL∂sαKsLs=t2fg≤supβ>0TβL∫0∞um−α−1∂m∂umKu+sLs=t2fgds≤Csupβ>0∫0∞sm−α−1Tt2+u/2+βLfgt2+s−mds≤Csupt>0Tt2/2Lfg∫0∞sm−α−1t2+smds≤Ct−2αsupt>0Tt2/2Lfg.
Therefore, (147) follows from the definition of HL1ℍn.
Step II. Assume thatf∈HL1ℍn and gk,αHf∈L1ℍn. Let fn be a sequence in Cc∞ℍn such that limn→∞fn=f in HL1ℍn. For fixed t>0, set ut,·≔e−tLf· and unt,·≔e−tLfn·, n∈ℕ. Then, ut,· and unt,· belong to HL1ℍn. By Lemma 22 and (147), we have
(149)∂tαuns,·s=t2=Lαunt2,·∈HL1ℍn,which implies that unt,·∈HL1,αℍn with unt2,·HL1,α=unt2,·HL1+∂tαuns,·s=t2HL1. By (147) again,
(150)limn→∞∂tαuns,·s=t2−∂tαus,·s=t2HL1=0.
This indicates thatunt2,· is a Cauchy sequence in HL1,αℍn. Therefore, there exists vt,·∈HL1,αℍn such that unt2,·−vt,·HL1,α→0 as n→∞. Hence, unt2,·−vt,·HL1→0 as n→∞, which yields ut2,·=vt,·∈HL1,αℍn and unt2,·→ut2,·HL1,α→0 as n→∞.
Step III. Noting thatunt2,·∈L2ℍn∩HL1ℍn and Lαunt2,·∈L2ℍn∩HL1ℍn, by Proposition 23, we get
(151)unt2,·HL1+gk,αHunt2,·1~unt2,·HL1,α.
Lettingn→∞, we have ut2,·HL1,α≤Cut2,·HL1+gk,αHut2,·1. Since
(152)gk,αHftg=∫0∞s2k−2α∂k∂tkKtLt=s2ut2,·g2dss1/2=∫0∞e−t2Ls2k−2α∂k∂tkKtLt=s2fg2dss1/2≤e−t2L∫0∞s2k−2α∂tkKtLt=s2f·2dss1/2g,we get gk,αHut,·1≤gk,αHf1. Furthermore, this gives
(153)ut2,·HL1,α≤Cut2,·HL1+gk,αHf1,where C>0 is independent of t. By (153), we know ut2,· are uniformly bounded in HL1,αℍn, i.e., Lαut2,· are uniformly bounded in HL1ℍn. Since HL1ℍn is a Banach space, we can find g∈HL1ℍn such that Lαujt2,·→g as j→∞, where ujt2,· is a subsequence of ut2,·. Since HL1ℍn is the dual space of VMOLℍn and Cc∞ℍn is dense in VMOLℍn with norm of VMOLℍn (cf. [32]), we get limj→∞Lαujt2,·,ϕ=g,ϕ,ϕ∈Cc∞ℍn. Let h=L−αg. Then, h∈HL1,αℍn and limj→∞ujt2,·,ϕ=h,ϕ,ϕ∈Cc∞ℍn. By the arguments analogous to ([33] page 776), which relay on the decay of the kernel of e−tL, we can get
(154)limt→0ut2,·,ϕ=f,ϕ,ϕ∈Cc∞ℍn.
It follows thatf=h and
(155)fHL1,α≤CfHL1+gk,αHfL1.
This completes the proof of Theorem24.For the Poisson semigroupPtLt>0, we define the fractional square functions as follows:
(156)gk,αPf≔∫0∞tk−α∂kPtL∂tkf2dtt1/2,k≥α>0;Sk,αPf≔∫0∞∫Bg,ttk−α∂kPtL∂tkf2dhdttQ+11/2,k≥α>0;gk,α,λP,∗f≔∫0∞∫ℍntt+g−1h2λtk−α∂kPtL∂tkf2dhdttQ+112,k≥α>0.Similar to the proof of Theorem24, we can apply (ii) of Proposition 23 to establish the following characterization of HL1,αℍn via the fractional square functions related to the Poisson semigroup. We omit the proof.Theorem 25.
Letα≥1/2, k∈ℕ\0 and λ>Q. Then, the following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αPf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αPf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λP,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(157)fHL1,α~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.


### 3.3. Equivalent Norms of Hardy-Sobolev Spaces

We define the following Hardy-Sobolev spaceHL1,αℍn as the set of all functions f∈HL1ℍn such that I+Lαf∈HL1ℍn, with the norm
(158)fHL1,α=I+LαfHL1+fHL1.The purpose of this section is to characterizeHL1,αℍn by the fractional square functions defined by (10) and (156), respectively. As an application, it follows from the fractional square function characterizations of HL1,αℍn and HL1,αℍn that the two Hardy-Sobolev spaces are equivalent.LetEL be the spectral decomposition of the operator L. For a bounded function M on 0,∞, the spectral multiplier ML is defined by
(159)MLf=∫0∞MλdELλf,f∈DML,where DML denotes the domain, i.e.,
(160)DML=f∈L2ℍn:∫0∞Mλ2dELλf,f<∞.We say that a functionM on −∞,+∞ belongs to the space Cs, s>0, if
(161)MCs≔∑k=0ssupMkλ<∞,s∈ℤ;MsLips−s+∑k=0ssupMkλ<∞,s∉ℤ.We have the following version of spectral multiplier theorems.Proposition 26 (see [34], Theorem 1.11).
LetM be a bounded continuous function on 0,∞. If for some ε>0 and a nonzero function ϕ∈Cc∞0,∞, there exists a constant C>0 such that for every t>0,
(162)ϕ·Mt·CQ/2+ε≤C,then the operator ML is bounded on HL1ℍn.
Letα,β>0. For λ>0, define
(163)M1λ=λα1+λα,M2λ=1+λα1+λα,M3λ=β+λ−α.Then, it is clear thatMi,i=1,2,3, are smooth and bounded on 0,∞. It follows from Proposition 26 thatProposition 27.
Letα,β>0. The operators MiL,i=1,2,3, can be extended to bounded operators on HL1ℍn.Theorem 28.
Let0<α<k, k∈ℕ and λ>Q. Iff∈DI+Lα∩HL1ℍn and I+Lαf∈L2ℍn∩HL1ℍn,
(164)I+LαfHL1~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Proof.
We give the proof ofI+LαfHL1~fHL1+gk,αHfL1. The proofs for the cases of Sk,αHf and gk,α,λH,∗f are similar. By Proposition 27, we know that the operators LαI+L−α and I+LαI+Lα−1 are bounded on HL1ℍn. Then, following from Proposition 23, we obtain
(165)I+LαfHL1=I+LαI+Lα−1I+LαfHL1≤I+LαfHL1≤CfHL1+LαfHL1≤CfHL1+gk,αHfL1.
For the reverse, we take the functionM1λ=λα1+λ−α,λ>0. For any r∈0,∞,
(166)∫0rλαdELλf=∫0rλα1+λα1+λαdELλf=M1L∫0r1+λαdELλf.
Lettingr→∞, we get Lαf=M1λI+Lαf. By Proposition 27 again, we obtain LαfHL1≤CI+LαfHL1, and
(167)fHL1=I+L−αI+LαfHL1≤CI+LαfHL1.
Theorem28 follows from Proposition 23.Similar to Theorem28, we also can obtainTheorem 29.
Let0<α<k, k∈ℕandλ>Q. Iff∈DI+Lα/2∩HL1ℍnandI+Lα/2f∈L2ℍn∩HL1ℍn,
(168)I+Lα/2fHL1~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.
Let(169)Gα,L=f∈HL1ℍn:I+Lαf∈Cc∞ℍn.
SinceCc∞ℍn is dense in HL1ℍn, Gα,L is dense in HL1,αℍn. Note that Gα,L⊂DI+Lα∩HL1ℍn, and
(170)I+LαGα,L=Cc∞ℍn⊂L2ℍn∩HL1ℍn.
Using Theorem28, gk,αH, Sk,αH, and gk,α,λH,∗ can be extended to HL1,αℍn as bounded operators from HL1,αℍn to L1ℍn. Let gk,αH~ be the extension of gk,αH to HL1,αℍn as a bounded operator from HL1,αℍn to L1ℍn. Then, there exists C>0 such that for f∈HL1,αℍn, fHL1+gk,αH~f1≤CfHL1,α.
Similar to Theorems24 and 25, we will give the following characterizations of the Hardy-Sobolev space HL1,αℍn as follows. We omit the proof.Theorem 30.
Letα≥1/2, k∈ℕ\0 and λ>Q. The following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αHf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αHf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λH,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(171)fHL1,α~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Theorem 31.
Letα≥1/2, k∈ℕ0 and λ>Q. The following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αPf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αPf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λP,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(172)fHL1,α~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.Theorems24, 25, 30, and 31 indicate the following equivalence relation:Corollary 32.
Letα≥1/2. HL1,αℍn=HL1,αℍn.


## 3.1. Fractional Square Functions Characterizations ofHL1ℍn

Define(84)gP,αfg≔∫0∞Dα,tLfg2dtt1/2;SP,αfg≔∫0∞∫Bg,tDα,tLfg2dhdttQ+11/2;gP,α,λ∗fg≔∫0∞∫ℍntt+∣g−1h∣2λDα,tLfg2dhdttQ+11/2.In this section, we will characterize the Hardy spaceHL1ℍn by the fractional square functions defined by (9) and (84). Now, we first prove the following reproducing formulas.Lemma 17.
Letα>0.
(i)
The operatorQα,tL defines an isometry from L2ℍn into L2Un,dgdt/t. Moreover, in the sense of L2ℍn,(85)f=Cαlimε→0limN→∞∫εNQα,tL2fdtt.(ii)
The operatorDα,tL defines an isometry from L2ℍn into L2Un,dgdt/t. Moreover, in the sense of L2ℍn,(86)f=Cαlimε→0limN→∞∫εNDα,tL2fdtt.Proof.
The proofs of (i) and (ii) are standard and can be deduced from the spectral techniques. For completeness, we give the proof of (i) and omit the details of the proof of (ii). Sincee−t2L=∫0∞e−t2λdEλ, we have
(87)t2ddse−sLs=t2=−t2Le−t2L=−∫0∞t2λe−t2λdEλ.
Thus, for allf∈L2ℍn, we get
(88)gH,αf22=∫ℍn∫0∞Qα,tLfg2dtdgt=∫0∞Qα,tL2f,fdtt=∫0∞∫0∞t4αλ2αe−2t2λdttdEf,fλ=Cαf22.
For the second part, it suffices to show that, for every pair of sequencesnk→∞&εk→0,
(89)limk→∞∫nknk+mQα,tLf2dtt=limk→∞∫εk+mεkQα,tLf2dtt=0∀m≥1.
Indeed, if (89) holds, we can find h∈L2ℍn such that limk→∞∫εknkQα,tLf2dt/t=h. Therefore, it follows from a polarized version of the first part that for g∈L2ℍn,
(90)h,g=limk→∞∫εknkQα,tLf,Qα,tLgdtt=∫0∞Qα,tLf,Qα,tLgdtt=Cαf,g,which implies h=Cαf. To prove (89), we use again the functional calculus to deduce that
(91)∫nknk+mQα,tLf2dtt22≤∫0∞∫nknk+mt4αλ2αe−2t2λdtt2dEf,fλ.
Computing the integral inside one yields∫0∞1+2λnk2e−2λnk2dEf,fλasnk→∞, which by dominated convergence tends to 0. Observe that the last step makes use of the fact that 0 is not an eigenvalue of L because Vg>0 for almost every g, and Lf,f≥Vf,f>0 unless f≡0. One proceeds similarly when εk→0.The following boundedness of square functions can be deduced from the spectral theorem immediately.Lemma 18.
Letα>0 and λ>Q/2.
(i)
The operatorsgH,α, GH,α and gH,α,λ∗ are bounded on L2ℍn. Moreover, there exist constants C, C1 and C2 such that gH,αfL2=CfL2,GH,αfL2≤C1fL2,gH,α,λ∗fL2≤C2fL2(ii)
The operatorsgP,α, GP,α and gP,α,λ∗ are bounded on L2ℍn. Moreover, there exist constants C, C1 and C2 such that gP,αfL2=CfL2,GP,αfL2≤C1fL2,gP,α,λ∗fL2≤C2fL2Proof.
We only prove (i), and (ii) can be done similarly. ForgH,α, using the reproducing formula on L2ℍn, we can get
(92)gH,αfL22=∫0∞Qα,tLf,Qtαfdtt=∫0∞Qα,tL2f,fdtt=∫0∞e−iπα∫0∞t4αλ2αe−2t2λdttdEf,fλ=CfL22.
ForGH,α, we have
(93)GH,αfL22≤∫0∞∫ℍn1tQ∫ℍnχΓgh,tdgQα,tLfg2dhdtt≤∫0∞∫ℍnQα,tLfg2dhdtt=gH,αfL22≤C1fL22.
ForgH,α,λ∗, the relation: gH,α,λ∗fg≤CGH,αfg indicates that gH,α,λ∗fL2≤C2fL2.Proposition 19.
Letα>0 and λ>Q/2.
(i)
There exists a constantC such that for any function f which is a linear combination of HL1-atoms(94)GH,αfL1≤CfHL1,gH,αfL1≤CfHL1,gH,α,λ∗L1≤CfHL1.(ii)
There exists a constantC such that for any function f which is a linear combination of HL1-atoms(95)GP,αfL1≤CfHL1,gP,αfL1≤CfHL1,gP,α,λ∗L1≤CfHL1.Proof.
We only prove (i), and (ii) can be dealt with similarly. Firstly, by Lemma18, we can get gH,αfL2=CfL2. For f∈HL1ℍn, it holds an atomic decomposition: f=∑jcjaj. Then,
(96)GH,αfg=∫0∞∫Bg,t∑jcjQα,tLajh2dhdttQ+11/2≤∑j∣cj∣GH,αajg.
So we only need to verify thatGH,αa is in L1ℍn for any HL1-atom a uniformly. By Lemma 18,
(97)GH,αaL22≤∫0∞∫ℍnQα,tLah2dhdtt=gH,αfL22≤CaL22≤CBg0,r−1.
WriteGH,αaL1=A+B, where A=∫Bg0,4rGH,αagdg and B=∫Bcg0,4rGH,αagdg. For A, it is clear that
(98)A≤Bg0,4r1/2∫Bg0,4rGH,αag2dg1/2≤Bg0,4r1/2CBg0,r−1≤C.For the estimate ofB, the following two cases are considered.Case 1.
r<ρg0/4. By the cancelation property of the atom a, we have GH,αag≤B1+B2, where
(99)B1≔∫0∣g−1g0∣/2∫∣g−1h∣<t∫Bg0,rQα,tLh,z−Qα,tLh,g0azdz2dhdttQ+11/2;B2≔∫∣g−1g0∣/2∞∫∣g−1h∣<t∫Bg0,rQα,tLh,z−Qα,tLh,g0azdz2dhdttQ+11/2.
ForB1, since 0<t<g−1g0/2 and g−1h<t, we can get h−1g0~g−1g0. For z∈Bg0,r and g∈Bcg0,4r, we have g0−1z<r≤Cg0−1h/4. Using (ii) of Corollary 12 and the symmetry, we can get
(100)B1≤∫0∣g−1g0∣/2∫∣g−1h∣<t∫Bg0,rCMtαt+∣g0−1h∣Q+α∣z−1g0∣tδ′∣az∣dz2dhdttQ+11/2≤CM∫0∣g−1g0∣/2∫∣g−1h∣<tt2αt+∣g0−1g∣2Q+αrt2δ′dhdttQ+11/2≤CMrδ′g0−1gQ+α∫0∣g−1g0∣/21t2δ′−2α+1dt1/2≤CMrδ′g0−1gQ+δ′.
The above estimate forB1 implies that
(101)∫Bcg0,4rB1dg≤CM∑k=2∞∫2kr≤∣g−1g0∣<2k+1rrδ′dgg−1g0Q+δ′≤CM∑k=2∞rδ′2k+1rQ2krQ+δ′≤C.
Next, we estimateB2. Since z−1g0≤r<g−1g0/2≤t, the estimate
(102)B2≤CM∫∣g−1g0∣/2∞∫∣g−1h∣<t∫Bg0,rtαt+∣g0−1h∣Q+α∣z−1g0∣tδ′∣az∣dz2dhdttQ+11/2≤CM∫∣g−1g0∣/2∞∫∣g−1h∣<trt2δ′t2αt+∣g0−1h∣2Q+αdhdttQ+11/2≤CMg−1g0Qrδ′g−1g0δ′implies that
(103)∫Bcg0,4rB2dg≤CM∫∣g−1g0∣≥4rrδ′g−1g0Q+δ′dg≤C.Case 2.
ρg0/4≤r<ρg0. In this case, we write GH,αag2=D1+D2+D3, where
(104)D1≔∫0r/2∫∣g−1h∣<tQα,tLah2dhdttQ+1;D2≔∫r/2∣g−1g0∣/4∫∣g−1h∣<tQα,tLah2dhdttQ+1;D3≔∫∣g−1g0∣/4∞∫∣g−1h∣<tQα,tLah2dhdttQ+1.
We first estimate the termD1. Since g−1g0>4r, g0−1z<r and g−1h<t<r/2, h−1g0>7r/2. For z∈Bg0,r, z−1g0<r<g−1g0/4. Using the triangle inequality, we apply (i) of Corollary 12 to estimate D1 as follows.
(105)D1≤CM∫0r/2∫∣g−1h∣<t∫Bg0,rtαt+∣g−1h∣Q+α∣az∣dz2dhdttQ+1≤CM∫0r/2∫∣g−1h∣<tt2αt+∣g−1g0∣2Q+αdhdttQ+1≤CMg−1g02Q+2α∫0r/2t2α−1dt≤CMr2αg−1g02Q+2α.
ForD2, since z∈Bg0,r, ∣z−1g0∣<r<ρg0, then ρz~ρg0~r. We have
(106)D2≤CM∫r/2∣g−1g0∣/4∫∣g−1h∣<t∫Bg0,rtαt+∣g−1h∣Q+α∣az∣dz1+t/ρh+t/ρzM2dhdttQ+1,≤CM∫r/2∣g−1g0∣/4∫∣g−1h∣<tt2αt+∣g−1h∣2Q+α1+tρg0−2MdhdttQ+1≤CM∫r/2∣g−1g0∣/4rt2Mt2α−1g−1g02Q+2αdt≤CMr2Mg−1g02Q+2M.
At last, we estimateD3. For ∣z−1g0∣<r<ρg0, we have ρg0~ρz. Then, we can get
(107)D3≤CM∫∣g−1g0∣/4∞∫∣g−1h∣<t∫Bg0,rtαt+∣h−1z∣Q+α∣az∣dz1+t/ρh+t/ρzM2dhdttQ+1≤CM∫∣g−1g0∣/4∞∫∣g−1h∣<t∫Bg0,r1+tρg0−M1t+∣h−1z∣Q∣az∣dz2dhdttQ+1≤CMr2Mg−1g02Q∫∣g−1g0∣/4∞1t2M+1dt≤CMr2Mg−1g02Q+2M.
The above estimates forDi,i=1,2,3, indicate that
(108)∫BcGH,αagdg≤∑k=2∞∫2kr≤∣g−1g0∣<2k+1rD11/2g+D21/2g+D31/2gdg≤C.Now, we give the following characterizations ofHL1ℍn.Theorem 20.
Letα≥1/2 and λ>Q/2. The following assertions are equivalent:
(i)
f∈HL1ℍn;(ii)
f∈L1ℍnandgH,αf∈L1ℍn;(iii)
f∈L1ℍnandGH,αf∈L1ℍn;(iv)
f∈L1ℍnandgH,α,λ∗f∈L1ℍn
Moreover, for everyf∈HL1ℍn,
(109)fHL1~f1+gH,αf1~f1+GH,αf1~f1+gH,α,λ∗f1.Proof.
By Proposition19, for f∈HL1ℍn, we know that gH,αf∈L1ℍn, GH,αf∈L1ℍn, and gH,α,λ∗f∈L1ℍn, respectively.
For the reverse, we first show that forGH,αf∈L1ℍn, f∈HL1ℍn. Assume that f∈L1ℍn∩L2ℍn. When GH,αf∈L1ℍn, we can see that
(110)∫ℍnGH,αfgdg=∫ℍn∫0∞∫Bg,tQα,tLfh2dhdttQ+11/2dg,which implies that Qα,tLfg∈T21, where Qα,tLfg≔∫ℍnQα,tLg,hfhdh. By Proposition 8, Qα,tLfg=Σkλkakg,t, where ak·,· are T21-atoms and Σk∣λk∣<∞. Assume that the atom a·,· is supported on B^g0,r. By Lemma 17,
(111)fg=C∫0∞Qtα∑k=1∞λkakg,tdtt≔∑k=1∞λkTkg,where Tkg=∫0∞Qα,tLakg,tdt/t. For simplicity, we denote Tkg by Tg for k=1,2,⋯. Write
(112)supt>0e−tLTgL1≤supt>0e−tLTgχB∗L1+supt>0e−tLTgχB∗cL1=I1+I2,where B∗=Bg0,2r. For I1, we use Hölder’s inequality to deduce that
(113)TL2=sup∥ϕ∥2≤1∫ℍn∫0∞Qα,tLag,tdttϕ¯gdg≤sup∥ϕ∥2≤1∫0∞∫ℍnag,t2dgdtt1/2∫0∞∫ℍnQα,tLϕ¯g2dgdtt1/2≤sup∥ϕ∥2≤1B−1/2ϕ2≤B−1/2,which gives I1≤B∗1/2B−1/2≤C.
Now, we deal withI2. For s>0, by functional calculus and Proposition 2.9, we have
(114)e−sL∫0∞Qα,tLag,tdtt=∫0∞∫0∞t2α∂sKs+t+λLs=t2ag,tλ1−αdλλdtt=∫0∞t2α∂sαKs+tLs=t2ag,tdtt≤C∫0∞tαah,ts+t2+∣g−1h∣Q+αdhdtt.
Whenh∈Bg0,r and g∈B∗c, we have ∣g−1h∣~∣g−1g0∣, and
(115)e−sL∫0∞Qα,tLag,tdtt≤Cg−1g0−Q+α/2∫0r∫Bt2α−1dhdt1/2∫0r∫Bah,t2dhdtt1/2≤CB−1/2g−1g0−Q+α∫0r∫Bt2α−1dhdt1/2≤Crαg−1g0−Q+α.
Finally, we get(116)I2≤∫Bcg0,rrαg−1g0Q+αdg≤C.
Whenf∈HL1ℍn, let G~H,α be the bounded extension of GH,αf from L2∩HL1ℍn to HL1ℍn. Since L2∩HL1ℍn is dense in HL1ℍn, there exists a sequence fn⊂L2∩HL1ℍn such that fn→f as n→∞ in HL1ℍn. By Corollary 12, we conclude that GH,αfn→GH,αf as n→∞. By the definition of G~H,α, we know that GH,αfn→G~H,αf as n→∞. Therefore, GH,αf=G~H,αf for f∈HL1ℍn, which gives
(117)fHL1=limn→∞fnHL1≤limn→∞GH,αfnL1=G~H,αfL1=GH,αfL1.
For the Littlewood-Paleyg-function, it is sufficient to prove GH,αfL1≤CgH,αfL1. For β>0, we define G~H,βf by
(118)G~H,βfg=∫0∞∫∣g−1h∣<βtQα,tLfh2dhdttQ+11/2.
Similarly, we can prove thatf∈HL1ℍn if and only if G~H,βf∈L1ℍn and f∈L1ℍn. Moreover, fHL1~G~H,βfL1.
LetFgt≔∂sαe−sLs=t2fg and Vg,s≔e−sLFg. Then Vg,st=∂rαe−s+rLr=t2fg. Therefore,
(119)∫0∞Vg,st2dtt1−4α=∫0∞∂rαe−s+rLr=t2fg2dtt1−4α=∫s∞∂rαe−rLr=t2fg2tdtt2−s1−2α.
Whenα≥1/2, we have t2−s2α−1≤t4α−1. Hence,
(120)sups>0∫0∞Vg,st2t4α−1dt≤∫0∞∂rαe−rLr=t2fg2t4α−1dt=gH,αfg2.
LetX=L20,∞,t4α−1dt. Then, sups>0e−sLFgX≤gH,αfg∈L1ℍn. Therefore, F∈HX1ℍn, where HX1ℍn can be seen as a vector-valued Hardy space (cf. [30]). This shows that G~2XFg∈L1ℍn, where
(121)G~2XFg=∫0∞∫∣g−1h∣<2tQα,tLFhX2dhdttQ+11/2.
We can assume that1/2≤α<1. Then, the identity (6) gives
(122)∂tαKtLt=s2∂sαKsLs=t2=C∫0∞∫0∞∂aKa+tLt=s2∂bKs+bLs=t2a−αb−αdadb=C∫0∞∫0∞∂a2Ka+b+s+tLs=t2,t=s2a−αb−αdadb=C∫0∞∂λ2Kλ+s+tLs=t2,t=s2λ1−2αdλ.
Whenα≥1/2, we get ∂tαKtLt=s2∂sαKsLs=t2=∂t2αKs+tLs=t2,t=s2. Via integration by substitution, we can change the orders of integration to obtain
(123)G~2XFg2=∫0∞∫∣g−1h∣<2t∫0∞t2α∂s2αe−sLs=t2Fhs2s4α−1dsdhdttQ+1≥∫0∞∫03t/2∫∣g−1h∣<2t2−s2∂s2αe−sLs=t2fh2ts4α−1dhdtdst2−s21+Q/2−2α≥∫0∞∫03t/2∫∣g−1h∣<t∂s2αe−sLs=t2fh2t4α−1−Qs4α−1dhdsdt=C∫0∞∫∣g−1h∣<tt4α∂s2αe−sLs=t2fh2dhdttQ+1=CG~L1fg2,which implies GH,αf∈L1ℍn, and therefore, f∈HL1ℍn. Since t/∣g−1h∣+t2λ>2−2λ in the cone Γg=h,t:∣g−1h∣<t, we have
(124)GH,αfg≤∫Γg22λt∣g−1h∣+t2λQtαfh2dhdttQ+11/2≤2λgH,α,λ∗fg.
This completes the proof of Theorem20.Theorem 21.
Letα≥1/2 and λ>Q/2. The following assertions are equivalent:
(i)
f∈HL1ℍn(ii)
f∈L1ℍn and gP,αf∈L1ℍn(iii)
f∈L1ℍn and GP,αf∈L1ℍn(iv)
f∈L1ℍn and gP,α,λ∗f∈L1ℍn
Moreover, for everyf∈HL1ℍn,
(125)fHL1~f1+gP,αf1~f1+GP,αf1~f1+gP,α,λ∗f1.Proof.
This theorem can be proved similarly as the proof of Theorem20, so we omit it.


## 3.2. Fractional Square Functions Characterizations ofHL1,αℍn

In this part, we will give the characterizations of Hardy-Sobolev spaceHL1,αℍn by fractional square functions. Firstly, we give the following Lemma, which will be used in the sequel. Similar to ([31], Proposition 2.4), we can express the operators ∂tαe−tL and ∂tαe−tL as follows.Lemma 22.
Letα>0.
(i)
For everyf∈L2ℍn,(126)∂tαe−tLf=eiπα∫0∞λαe−tλdELλf,t>0.(ii)
For everyf∈L2ℍn,(127)∂tαe−tLf=eiπα∫0∞λα/2e−tλdELλf,t>0.Proof.
LetEλ denote a resolution of the identity. It follows from the spectral decomposition:
(128)e−tLf=∫0∞e−λtdEfλ∀f∈L2ℍnthat
(129)∂tke−tLf=e−iπk∫0∞λke−λtdEfλ,k=1,2,⋯.
By (6) and (129), we have
(130)∂tαe−tLf=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEfλsk−α−1ds,where k is the smallest integer satisfying k>α. Then, the integral
(131)∫0∞∫0∞λke−t+sλ∣dEfλ∣sk−α−1dsis absolutely convergent. By the fact that ∂tαe−tLfLp≤CαfLp/tα, the integral in (6) is absolutely convergent in L2ℍn. Hence, by (130), we can get for g∈L2ℍn,
(132)∂tαe−tLf,g=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEfλsk−α−1ds,g=e−iπαΓk−α∫0∞∫0∞λke−t+sλdEf,gλsk−α−1ds=e−iπαΓk−α∫0∞∫0∞λke−t+sλsk−α−1dsdEf,g=e−iπα∫0∞λαe−tλdEfλ,g,which implies (i). The assertion (ii) can be obtained by the aid of functional calculus similarly.The following result can be deduced from Lemma22 immediately.Proposition 23.
(i)
Let0<α<k, k∈ℕ and λ>Q. Iff∈DLα∩HL1ℍn and Lαf∈L2ℍn∩HL1ℍn. Then,(133)LαfHL1~gk,αHfL1~Sk,αHfL1~gk,α,λH,∗fL1.(ii)
Let0<α<k, k∈ℕ and λ>Q. If f∈DLα∩HL1ℍn and Lαf∈L2ℍn∩HL1ℍn. Then,(134)Lα/2fHL1~gk,αPfL1~Sk,αPfL1~gk,α,λP,∗fL1.Proof.
We only prove (i), and (ii) can be dealt with similarly. Using Lemma22, we can get
(135)∂sk−αe−sLs=t2Lαf=Lk−αe−t2LLαf=Lke−t2Lf=∂skKsLs=t2f,therefore,
(136)gk−αHLαf=gk,αHf,Sk−αHLαf=Sk,αHf,gk−α,λH,∗Lαf=gk,α,λH,∗f.
Using Theorem20, we can get
(137)LαfHL1~gk,αHfL1~Sk,αHfL1~gk,α,λH,∗fL1.
LetGα,L=f∈HL1ℍn:Lαf∈Cc∞ℍn. Since Cc∞ℍn is dense in HL1ℍn, Gα,L is dense in HL1,αℍn. Note that Gα,L⊂DLα∩HL1ℍn, and
(138)LαGα,L=Cc∞ℍn⊂L2ℍn∩HL1ℍn.
Using Proposition23, gk,αH, Sk,αH, and gk,α,λH,∗ can be extended to HL1,αℍn as bounded operators from HL1,αℍn to L1ℍn. Let gk,αH~ be the extension of gk,αH to HL1,αℍn as a bounded operator from HL1,αℍn to L1ℍn. Then, there exists C>0 such that for f∈HL1,αℍn,
(139)fHL1+gk,αH~f1≤CfHL1,α.
Below, we give the square function characterizations of the Hardy-Sobolev spaceHL1,αℍn as follows.Theorem 24.
Letα≥1/2, k∈ℕ0, and λ>Q. Then, the following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αHf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αHf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λH,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(140)fHL1,α~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Proof.
We first provefHL1+gk,αHfL1≤CfHL1,α. By (139), it is sufficient to prove gk,αHf=gk,αH~f. For N∈ℕ and h∈HL1ℍn, by the subordination formula, we obtain
(141)∂k∂tkKtLhg≤Ct−2ksupt>0Tt2/2Lhg.
Then,(142)∫1/N∞t2k−2α∂k∂skKsLs=t2hg2dtt1/2≤C∫1/N∞t−1−4αdt1/2supt>0Tt2/2Lhg≤CN2αsupt>0Tt2/2Lhg.
By the definition ofHL1ℍn, we conclude that the operator
(143)h→∫1/N∞t2k−2α∂k∂skKsLs=t2hg2dtt1/2is bounded from HL1ℍn to L1ℍn. Therefore, if f=L−αh, where h∈HL1ℍn∩L2ℍn, we have
(144)∫1/N∞t2k−2α∂k∂skKsLs=t2L−αhg2dtt1/2L1=∫1/N∞t2k−2α∂k−α∂sk−αKsLs=t2hg2dtt1/2L1≤ChHL1,where the positive constant C is independent of N∈ℕ. Letting N→∞ yields
(145)gk,αH~fL1=∫0∞t2k−2α∂k∂skKsLs=t2fg2dtt1/2L1≤CfHL1,α.
SinceGα,L is dense in HL1,αℍn, for f∈HL1,αℍn, we obtain
(146)gk,αHfL1=∫0∞t2k−2α∂k∂skKsLs=t2fg2dtt1/2L1=gk,αH~fL1≤CfHL1,α.
The proofs forSk,αH and gk,α,λH,∗ are similar, and so is omitted.
For the reverse, we only deal with the case ofgk,αH for simplicity.
Step I. We prove(147)∂sαKsLs=t2fHL1≤Ct−2αfHL1.
Form∈ℕ and m>α, by (141), we obtain
(148)supβ>0TβL∂sαKsLs=t2fg≤supβ>0TβL∫0∞um−α−1∂m∂umKu+sLs=t2fgds≤Csupβ>0∫0∞sm−α−1Tt2+u/2+βLfgt2+s−mds≤Csupt>0Tt2/2Lfg∫0∞sm−α−1t2+smds≤Ct−2αsupt>0Tt2/2Lfg.
Therefore, (147) follows from the definition of HL1ℍn.
Step II. Assume thatf∈HL1ℍn and gk,αHf∈L1ℍn. Let fn be a sequence in Cc∞ℍn such that limn→∞fn=f in HL1ℍn. For fixed t>0, set ut,·≔e−tLf· and unt,·≔e−tLfn·, n∈ℕ. Then, ut,· and unt,· belong to HL1ℍn. By Lemma 22 and (147), we have
(149)∂tαuns,·s=t2=Lαunt2,·∈HL1ℍn,which implies that unt,·∈HL1,αℍn with unt2,·HL1,α=unt2,·HL1+∂tαuns,·s=t2HL1. By (147) again,
(150)limn→∞∂tαuns,·s=t2−∂tαus,·s=t2HL1=0.
This indicates thatunt2,· is a Cauchy sequence in HL1,αℍn. Therefore, there exists vt,·∈HL1,αℍn such that unt2,·−vt,·HL1,α→0 as n→∞. Hence, unt2,·−vt,·HL1→0 as n→∞, which yields ut2,·=vt,·∈HL1,αℍn and unt2,·→ut2,·HL1,α→0 as n→∞.
Step III. Noting thatunt2,·∈L2ℍn∩HL1ℍn and Lαunt2,·∈L2ℍn∩HL1ℍn, by Proposition 23, we get
(151)unt2,·HL1+gk,αHunt2,·1~unt2,·HL1,α.
Lettingn→∞, we have ut2,·HL1,α≤Cut2,·HL1+gk,αHut2,·1. Since
(152)gk,αHftg=∫0∞s2k−2α∂k∂tkKtLt=s2ut2,·g2dss1/2=∫0∞e−t2Ls2k−2α∂k∂tkKtLt=s2fg2dss1/2≤e−t2L∫0∞s2k−2α∂tkKtLt=s2f·2dss1/2g,we get gk,αHut,·1≤gk,αHf1. Furthermore, this gives
(153)ut2,·HL1,α≤Cut2,·HL1+gk,αHf1,where C>0 is independent of t. By (153), we know ut2,· are uniformly bounded in HL1,αℍn, i.e., Lαut2,· are uniformly bounded in HL1ℍn. Since HL1ℍn is a Banach space, we can find g∈HL1ℍn such that Lαujt2,·→g as j→∞, where ujt2,· is a subsequence of ut2,·. Since HL1ℍn is the dual space of VMOLℍn and Cc∞ℍn is dense in VMOLℍn with norm of VMOLℍn (cf. [32]), we get limj→∞Lαujt2,·,ϕ=g,ϕ,ϕ∈Cc∞ℍn. Let h=L−αg. Then, h∈HL1,αℍn and limj→∞ujt2,·,ϕ=h,ϕ,ϕ∈Cc∞ℍn. By the arguments analogous to ([33] page 776), which relay on the decay of the kernel of e−tL, we can get
(154)limt→0ut2,·,ϕ=f,ϕ,ϕ∈Cc∞ℍn.
It follows thatf=h and
(155)fHL1,α≤CfHL1+gk,αHfL1.
This completes the proof of Theorem24.For the Poisson semigroupPtLt>0, we define the fractional square functions as follows:
(156)gk,αPf≔∫0∞tk−α∂kPtL∂tkf2dtt1/2,k≥α>0;Sk,αPf≔∫0∞∫Bg,ttk−α∂kPtL∂tkf2dhdttQ+11/2,k≥α>0;gk,α,λP,∗f≔∫0∞∫ℍntt+g−1h2λtk−α∂kPtL∂tkf2dhdttQ+112,k≥α>0.Similar to the proof of Theorem24, we can apply (ii) of Proposition 23 to establish the following characterization of HL1,αℍn via the fractional square functions related to the Poisson semigroup. We omit the proof.Theorem 25.
Letα≥1/2, k∈ℕ\0 and λ>Q. Then, the following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αPf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αPf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λP,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(157)fHL1,α~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.


## 3.3. Equivalent Norms of Hardy-Sobolev Spaces

We define the following Hardy-Sobolev spaceHL1,αℍn as the set of all functions f∈HL1ℍn such that I+Lαf∈HL1ℍn, with the norm
(158)fHL1,α=I+LαfHL1+fHL1.The purpose of this section is to characterizeHL1,αℍn by the fractional square functions defined by (10) and (156), respectively. As an application, it follows from the fractional square function characterizations of HL1,αℍn and HL1,αℍn that the two Hardy-Sobolev spaces are equivalent.LetEL be the spectral decomposition of the operator L. For a bounded function M on 0,∞, the spectral multiplier ML is defined by
(159)MLf=∫0∞MλdELλf,f∈DML,where DML denotes the domain, i.e.,
(160)DML=f∈L2ℍn:∫0∞Mλ2dELλf,f<∞.We say that a functionM on −∞,+∞ belongs to the space Cs, s>0, if
(161)MCs≔∑k=0ssupMkλ<∞,s∈ℤ;MsLips−s+∑k=0ssupMkλ<∞,s∉ℤ.We have the following version of spectral multiplier theorems.Proposition 26 (see [34], Theorem 1.11).
LetM be a bounded continuous function on 0,∞. If for some ε>0 and a nonzero function ϕ∈Cc∞0,∞, there exists a constant C>0 such that for every t>0,
(162)ϕ·Mt·CQ/2+ε≤C,then the operator ML is bounded on HL1ℍn.
Letα,β>0. For λ>0, define
(163)M1λ=λα1+λα,M2λ=1+λα1+λα,M3λ=β+λ−α.Then, it is clear thatMi,i=1,2,3, are smooth and bounded on 0,∞. It follows from Proposition 26 thatProposition 27.
Letα,β>0. The operators MiL,i=1,2,3, can be extended to bounded operators on HL1ℍn.Theorem 28.
Let0<α<k, k∈ℕ and λ>Q. Iff∈DI+Lα∩HL1ℍn and I+Lαf∈L2ℍn∩HL1ℍn,
(164)I+LαfHL1~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Proof.
We give the proof ofI+LαfHL1~fHL1+gk,αHfL1. The proofs for the cases of Sk,αHf and gk,α,λH,∗f are similar. By Proposition 27, we know that the operators LαI+L−α and I+LαI+Lα−1 are bounded on HL1ℍn. Then, following from Proposition 23, we obtain
(165)I+LαfHL1=I+LαI+Lα−1I+LαfHL1≤I+LαfHL1≤CfHL1+LαfHL1≤CfHL1+gk,αHfL1.
For the reverse, we take the functionM1λ=λα1+λ−α,λ>0. For any r∈0,∞,
(166)∫0rλαdELλf=∫0rλα1+λα1+λαdELλf=M1L∫0r1+λαdELλf.
Lettingr→∞, we get Lαf=M1λI+Lαf. By Proposition 27 again, we obtain LαfHL1≤CI+LαfHL1, and
(167)fHL1=I+L−αI+LαfHL1≤CI+LαfHL1.
Theorem28 follows from Proposition 23.Similar to Theorem28, we also can obtainTheorem 29.
Let0<α<k, k∈ℕandλ>Q. Iff∈DI+Lα/2∩HL1ℍnandI+Lα/2f∈L2ℍn∩HL1ℍn,
(168)I+Lα/2fHL1~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.
Let(169)Gα,L=f∈HL1ℍn:I+Lαf∈Cc∞ℍn.
SinceCc∞ℍn is dense in HL1ℍn, Gα,L is dense in HL1,αℍn. Note that Gα,L⊂DI+Lα∩HL1ℍn, and
(170)I+LαGα,L=Cc∞ℍn⊂L2ℍn∩HL1ℍn.
Using Theorem28, gk,αH, Sk,αH, and gk,α,λH,∗ can be extended to HL1,αℍn as bounded operators from HL1,αℍn to L1ℍn. Let gk,αH~ be the extension of gk,αH to HL1,αℍn as a bounded operator from HL1,αℍn to L1ℍn. Then, there exists C>0 such that for f∈HL1,αℍn, fHL1+gk,αH~f1≤CfHL1,α.
Similar to Theorems24 and 25, we will give the following characterizations of the Hardy-Sobolev space HL1,αℍn as follows. We omit the proof.Theorem 30.
Letα≥1/2, k∈ℕ\0 and λ>Q. The following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αHf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αHf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λH,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(171)fHL1,α~fHL1+gk,αHfL1~fHL1+Sk,αHfL1~fHL1+gk,α,λH,∗fL1.Theorem 31.
Letα≥1/2, k∈ℕ0 and λ>Q. The following assertions are equivalent:
(i)
f∈HL1,αℍn(ii)
f∈HL1ℍn and gk,αPf∈L1ℍn for k>α(iii)
f∈HL1ℍn and Sk,αPf∈L1ℍn for α<k−Q+1/2(iv)
f∈HL1ℍn and gk,α,λP,∗f∈L1ℍn for α<k−Q+1/2
Moreover, for everyf∈HL1,αℍn,
(172)fHL1,α~fHL1+gk,αPfL1~fHL1+Sk,αPfL1~fHL1+gk,α,λP,∗fL1.Theorems24, 25, 30, and 31 indicate the following equivalence relation:Corollary 32.
Letα≥1/2. HL1,αℍn=HL1,αℍn.



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