# Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Authors: Hua Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

LetL=−Δℍn+V be a Schrödinger operator on the Heisenberg group ℍn, where Δℍn is the sub-Laplacian on ℍn and the nonnegative potential V belongs to the reverse Hölder class Bq with q∈Q/2,∞. Here, Q=2n+2 is the homogeneous dimension of ℍn. Assume that e−tLt>0 is the heat semigroup generated by L. The semigroup maximal function related to the Schrödinger operator L is defined by TL∗fu≔supt>0e−tLfu. The Riesz transform associated with the operator L is defined by RL=∇ℍnL−1/2, and the dual Riesz transform is defined by RL∗=L−1/2∇ℍn, where ∇ℍn is the gradient operator on ℍn. In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍn. Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators TL∗, RL, and RL∗ acting on the Morrey spaces. In addition, it is shown that the Riesz transform RL=∇ℍnL−1/2 is of weak-type 1,1. It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.


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## Body

## 1. Introduction

### 1.1. The Heisenberg Groupℍn

The Heisenberg group is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as the representation theory, partial differential equations, several complex variables, and harmonic analysis. It is a remarkable fact that the Heisenberg group, an important example of the simply connected nilpotent Lie group, naturally arises in two fundamental but different settings in modern analysis. On the one hand, it can be naturally identified with the group of translations of the Siegel upper half-space inℂn+1 and plays an important role in our understanding of several problems in the complex function theory of the unit ball. On the other hand, it can also be realized as the group of unitary operators generated by the position and momentum operators in the context of quantum mechanics.We begin by recalling some notions and well-known results from [1–3]. We write ℕ=1,2,3,⋯ for the set of natural numbers. The sets of real and complex numbers are denoted by ℝ and ℂ, respectively. Let ℍn be a Heisenberg group of dimension 2n+1, that is, a two-step nilpotent Lie group with underlying manifold ℂn×ℝ. The group operation is given by
(1)z,t∙w,s≔z+w,t+s+2Imz∙w¯,where z=z1,z2,⋯,zn, w=w1,w2,⋯,wn∈ℂn, and z∙w¯≔∑j=1nzjwj¯.Under this group operation,ℍn becomes a nilpotent unimodular Lie group. It can easily be seen that the inverse element of u=z,t∈ℍn is u−1=−z,−t, and the identity element of this group is the origin 0=0,0. The corresponding Lie algebra Hn of left-invariant vector fields on ℍn is spanned by
(2)Xj≔∂∂xj+2yj∂∂t;Yj≔∂∂yj−2xj∂∂t;j=1,2,⋯,n,T≔∂∂t.All nontrivial commutation relations are given by(3)Xj,Yj=−4T,j=1,2,⋯,n.Here,·,· is the usual Lie bracket. The sub-LaplacianΔℍn and the gradient∇ℍn are defined, respectively, by
(4)Δℍn≔∑j=1nXj2+Yj2,∇ℍn≔X1,⋯,Xn,Y1,⋯,Yn.The Heisenberg group has a natural dilation structure which is consistent with the Lie group structure mentioned above. For each positive numbera>0, we define the dilation on ℍn by
(5)δaz,t≔az,a2t,z,t∈ℍn.Observe thatδaa>0 is an automorphism of the group ℍn. For any given u=z,t∈ℍn, the homogeneous norm of u is given by the following form:
(6)u=z,t≔z4+t21/4.Observe that∣z,t−1∣=∣z,t∣ and
(7)δaz,t=az4+a2t21/4=az,t,a>0.In addition, this norm· satisfies the triangle inequality and then leads to a left-invariant distance du,v=u−1·v for any u=z,t,v=w,s∈ℍn. If r>0 and u∈ℍn, let Bu,r=v∈ℍn:du,v<r be the (open) ball with center u∈ℍn and radius r∈0,∞. Both left and right Haar measures on ℍn coincide with the Lebesgue measure dzdt on ℂn×ℝ. For any measurable set E⊂ℍn, the Lebesgue measure of E is denoted by E. For u,r∈ℍn×0,∞, it can be proved that the volume of Bu,r is
(8)Bu,r=rQB0,1,where Q≔2n+2 is the homogeneous dimension of ℍn and |B0,1∣ is the volume of the unit ball in ℍn. A simple calculation shows that
(9)B0,1=2πn+1/2Γn/2n+1Γn+1/2.Given a ballB=Bu,r⊂ℍn and λ>0, we adopt the notation λB to denote the ball with the same center u and radius λr. Clearly, by (8), we have
(10)Bu,λr=λQBu,r,u,r∈ℍn×0,∞,λ∈0,∞.For a radial functionF on ℍn, we have the following integration formula:
(11)∫ℍnFudu=Cn∫0∞FϱϱQ−1dϱ,Q=2n+2,where Cn is a positive constant which is independent of F. For more information about harmonic analysis on the Heisenberg group, the reader is referred to [2, 4, 5] and the references therein.


### 1.2. The Schrödinger OperatorL

We recall some standard notation and definitions.Definition 1.
A nonnegative locallyLq integrable function V on ℍn is said to belong to the reverse Hölder class Bq for some exponent 1<q<∞, if there exists a positive constant C=Cq;V such that the reverse Hölder inequality
(12)1B∫BVwqdw1/q≤C1B∫Bwdw,holds for every ball B in ℍn.In this article, we will always assume that0≡V∈Bq with q∈Q/2,∞ and Q=2n+2. We now consider the Schrödinger operator with the potential V∈Bq on the Heisenberg group ℍn (see [3]):
(13)L≔−Δℍn+V.In recent years, there has been a lot of attention paid to the study of various function spaces associated with the Schrödinger operators, which has been an active research topic in harmonic analysis. For the investigation of Schrödinger operators on the Euclidean spaceℝn with nonnegative potentials that belong to the reverse Hölder class, see, for example, [6–10]. Concerning the weighted case, one can see [11–16] for more details. The extension to the setting of the Heisenberg group has been given by Lin and Liu in [3]. For further details, we refer the reader to [17–19], among others. Regarding the Schrödinger operators in a more general setting (such as the nilpotent Lie group), see, for example, [20, 21]. As in [3, 22], we introduce the following definition.Definition 2. Suppose thatV∈Bq with q∈Q/2,∞.
For any givenu∈ℍn, the critical radius function ρu=ρu;V is defined by
(14)ρu≔r∈0,∞:1rQ−2∫Bu,rVwdw≤1,where Bu,r denotes the ball in ℍn centered at u and with radius r.It should be pointed out that the auxiliary functionρu;V on the Euclidean space ℝn was introduced by Shen in [10]. Later, Li [21] defined it on the (simply connected) nilpotent Lie group. It is well known that the auxiliary function ρu determined by V∈Bq satisfies
(15)0<ρu<∞,for any given u∈ℍn (see [3, 22]). In particular, ρu=1 with Vw≡1, and ρu≈1/1+u with Vw=w2 (Hermite operator).It is easy to check that ifQ/2≤q2<q1, then Bq1⊂Bq2 by the Hölder inequality. Furthermore, it can be shown that the Bq class has a property of self-improvement. More precisely, if V∈Bq, then V∈Bq+ε for some ε>0. By this fact, we know that the assumption q>Q/2 is equivalent to q≥Q/2.Whenq∈Q/2,Q, we also write
(16)δ≔2−Qq∈0,1.Let us give some elementary properties of theBq class. Assume that V∈Bq with q∈Q/2,∞.Lemma 3.
The measureVwdw satisfies the doubling condition; that is, there exists a constant C0>0 such that
(17)∫Bu0,2rVwdw≤C0∫Bu0,rVwdw,for all balls Bu0,r in ℍn.Lemma 4.
Ifr=ρu0, then
(18)1rQ−2∫Bu0,rVwdw=1,for any u0∈ℍn.Lemma 5.
For0<r<R<∞, we have
(19)1rQ−2∫Bu0,rVwdw≤CrRδ1RQ−2∫Bu0,RVwdw,for any u0∈ℍn.For more details, the reader may consult [3, 22].We also need the following technical lemma concerning the critical radius function (14).Lemma 6.
Letρu be the auxiliary function determined by V. For any u and v in ℍn, there exist constants C1≥1 and N0>0 such that
(20)1C11+v−1uρu−N0≤ρvρu≤C11+v−1uρuN0/N0+1.Here, and in what follows,v−1·u is simply denoted by v−1u. Lemma 6 has been proved by Lu [22] (see also [3], Lemma 4). In the setting of ℝn, this result was first given by Shen in [10] (Lemma 1.4). As a direct consequence of (20), we can see that for each fixed k∈ℕ, the following estimate
(21)1+2krρu1+rρu−N0/N0+1≤C11+2krρv,holds for any v∈Bu,r with u∈ℍn and r∈0,∞, where C1 is the same as in (20). Let us verify (21). An application of (20) yields
(22)ρv≤C1·ρu1+v−1uρuN0/N0+1≤C1·ρu1+rρuN0/N0+1,which further implies that for each fixed k∈ℕ,
(23)2krρv≥1C1·2krρu1+rρu−N0/N0+1.Hence,(24)1+2krρv≥1+1C1·2krρu1+rρu−N0/N0+1≥1C11+rρu−N0/N0+1+1C1·2krρu1+rρu−N0/N0+1=1C11+2krρu1+rρu−N0/N0+1,which is the desired estimate. This estimate will often be used in the sequel.


### 1.3. Semigroup Maximal Functions and Riesz Transforms

LetL=−Δℍn+V be a Schrödinger operator on the Heisenberg group ℍn, where Δℍn is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class Bq for q∈Q/2,∞, and Q is the homogeneous dimension of ℍn. Since V is nonnegative and belongs to Llocqℍn, L generates a C0 contraction semigroup TtLt>0=e−tLt>0. Let Ptu,v denote the kernel of the semigroup e−tLt>0.
(25)TtLfu≔e−tLfu=∫ℍnPtu,vfvdv,f∈L2ℍn,t>0.We also denote byHtu the convolution kernel of the heat semigroup Ttt>0=etΔℍnt>0. Namely,
(26)Ttfu≔etΔℍnfu=∫ℍnHtv−1ufvdv.For anyu=z,s∈ℍn, it is well known that the heat kernel Htu has the explicit expression:
(27)Htz,s=2π−14π−n∫ℝλsinhλtnexp−λz24cothλt−iλsdλ.We consider the heat equation associated with the sub-Laplacian(28)∂tFu,t=ΔℍnFu,t,u,t∈ℍn×0,∞,with the initial condition Fu,0=fu. In fact, the function Htu stated the above exists as a solution to the heat equation. Moreover, by [23] (Theorem 2), we know that the heat kernel Ht satisfies the Gaussian upper bound estimate:
(29)0<Htu≤C·t−Q/2exp−u2At,where the positive constants C and A are independent of t∈0,∞ and u∈ℍn. By the Trotter product formula (see [24] for instance) and (29), one has
(30)0≤Ptu,v≤Htv−1u≤C·t−Q/2exp−v−1u2At,where C and A are positive constants independent of u, v, and t. Furthermore, by using the estimates of the fundamental solution for the Schrödinger operator L on ℍn, this estimate (30) can be significantly improved when V belongs to the reverse Hölder class Bq for some q∈Q/2,∞. The auxiliary function ρu arises naturally in the present situation.Lemma 7.
Letρu be the auxiliary function determined by V. For every positive integer N∈ℕ, there exists a positive constant CN>0 such that, for any u and v in ℍn,
(31)0≤Ptu,v≤CN·t−Q/2exp−v−1u2At1+tρu+tρv−N,t>0.Remark 8.
This estimate ofPtu,v is much better than (30), which was given by Lin and Liu in [3] (Lemma 7). In the setting of ℝn, this result can be found in [9] (Proposition 2).In this article, we investigate the semigroup maximal function related to the Schrödinger operatorL, which is defined by (see [3])
(32)TL∗fu≔supt>0TtLfu=supt>0e−tLfu,u∈ℍn.We shall establish the strong-type and weak-type estimates of the operatorTL∗. Some other maximal functions will be discussed at the end of Section 3.Let us also consider the Riesz transformsRj and the dual Riesz transforms Rj∗ for the Schrödinger operator L, which are defined, respectively, by (see [3])
(33)Rj=XjL−1/2,Rj+1=YjL−1/2,j=1,2,⋯,n,Rj∗=L−1/2Xj,Rj+1∗=L−1/2Yj,j=1,2,⋯,n,where the Xj are left-invariant vector fields that generate the Lie algebra of ℍn. Let
(34)RL≔∇ℍnL−1/2=R1,R2,⋯,R2n,RL∗≔L−1/2∇ℍn=R1∗,R2∗,⋯,R2n∗.Here,∇ℍn is the gradient operator on ℍn. We shall be interested in the behavior of the (vector-valued) operators RL and RL∗ associated with the Schrödinger operator L on ℍn.For anyp∈1,∞, the Lebesgue space Lpℍn is defined to be the set of all measurable functions f on ℍn such that
(35)fLpℍn≔∫ℍnfupdu1/p<∞.The weak Lebesgue spaceWL1ℍn consists of all measurable functions f on ℍn such that
(36)fWL1ℍn≔supλ>0λ·u∈ℍn:fu>λ<∞.Recently, Lin and Liu ([3], Theorem 6 and Remark 3) established the strong-type and weak-type estimates of the operator TL∗ on the Lebesgue spaces.Theorem 9.
Let1≤p<∞. Then, the following statements are true:
(1)
Ifp>1, then the operator TL∗ is bounded on Lpℍn(2)
Ifp=1, then the operator TL∗ is bounded from L1ℍn into WL1ℍnRemark 10.
(i)
It was also shown by Lin and Liu that this operator is bounded onBMOLℍn(ii)
On the Euclidean spaceℝn, this maximal operator was studied by Dziubański et al. [9] (see also [11, 25])As for the (vector-valued) dual Riesz transformRL∗ defined above, we have the following estimate given in [21] (see also [19]).Theorem 11.
LetV∈Bq with q∈Q/2,Q, and let p0 be a number such that 1/p0=1/q−1/Q. Then, the (vector-valued) dual Riesz transform RL∗ is bounded on Lpℍn for p0′<p<∞.By duality, we could obtain the following result.Theorem 12.
LetV∈Bq with q∈Q/2,Q, and let p0 be a number such that 1/p0=1/q−1/Q. Then, the (vector-valued) Riesz transform RL is bounded on Lpℍn for 1<p<p0.Moreover, it will be proved in Section4 that RL is of weak-type 1,1 on the Heisenberg group. The case where q∈Q,∞ is also considered in Section 4.Remark 13.
(i)
It can be shown that the range ofp in the above theorems is optimal (see [3]). In this paper, the authors also proved that the dual Riesz transform RL∗ is bounded on BMOLℍn and gave the Fefferman-Stein-type decomposition of BMOL functions with respect to Rj∗, j=1,2,⋯,2n(ii)
It was shown in [26] that when V≡0, the operators R≔∇ℍn−Δℍn−1/2 and R∗=−Δℍn−1/2∇ℍn are uniformly bounded on Lpℍn with respect to n. More specifically, for every 1<p<∞, there exists a constant Cp>0 such that for every n∈ℕ(37)RfLpℍn+R∗fLpℍn≤CpfLpℍn.(iii)
Recall that in the setting ofℝn, the Riesz transform and its dual form were originally studied by Shen in [10]. It can be proved that the analog of Theorem 11 (also Theorem 12) on the Euclidean space is also true by the same argument (see [10, 27]). For the corresponding estimates for commutators generated by BMO functions, the reader is referred to [11, 12, 28] for more detailsThe paper is organized as follows. In Section2, we introduce the Morrey space and weak Morrey space associated with the Schrödinger operator L on ℍn and state our main results: Theorems 18, 19, 22, 23, and 24. Section 3 is devoted to Proofs of Theorems 18 and 19, which establish the strong-type and weak-type estimates for the semigroup maximal function TL∗ in the framework of Morrey spaces. The corresponding estimates for some other maximal functions are also proved in this section. Section 4 is devoted to proving the boundedness properties of the Riesz transform RL and its dual form RL∗. In Section 5, we extend the above results to the generalized Morrey spaces.Throughout this paper,C>0 denotes a universal constant which may change from line to line, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. The notation X≲Y means that X≤CY for some positive constant C. If X≲Y and Y≲X, then we write X≈Y to denote the equivalence of X and Y. For any p∈1,∞, the notation p′ denotes its conjugate number, namely, 1/p+1/p′=1 and 1′=∞.


## 1.1. The Heisenberg Groupℍn

The Heisenberg group is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as the representation theory, partial differential equations, several complex variables, and harmonic analysis. It is a remarkable fact that the Heisenberg group, an important example of the simply connected nilpotent Lie group, naturally arises in two fundamental but different settings in modern analysis. On the one hand, it can be naturally identified with the group of translations of the Siegel upper half-space inℂn+1 and plays an important role in our understanding of several problems in the complex function theory of the unit ball. On the other hand, it can also be realized as the group of unitary operators generated by the position and momentum operators in the context of quantum mechanics.We begin by recalling some notions and well-known results from [1–3]. We write ℕ=1,2,3,⋯ for the set of natural numbers. The sets of real and complex numbers are denoted by ℝ and ℂ, respectively. Let ℍn be a Heisenberg group of dimension 2n+1, that is, a two-step nilpotent Lie group with underlying manifold ℂn×ℝ. The group operation is given by
(1)z,t∙w,s≔z+w,t+s+2Imz∙w¯,where z=z1,z2,⋯,zn, w=w1,w2,⋯,wn∈ℂn, and z∙w¯≔∑j=1nzjwj¯.Under this group operation,ℍn becomes a nilpotent unimodular Lie group. It can easily be seen that the inverse element of u=z,t∈ℍn is u−1=−z,−t, and the identity element of this group is the origin 0=0,0. The corresponding Lie algebra Hn of left-invariant vector fields on ℍn is spanned by
(2)Xj≔∂∂xj+2yj∂∂t;Yj≔∂∂yj−2xj∂∂t;j=1,2,⋯,n,T≔∂∂t.All nontrivial commutation relations are given by(3)Xj,Yj=−4T,j=1,2,⋯,n.Here,·,· is the usual Lie bracket. The sub-LaplacianΔℍn and the gradient∇ℍn are defined, respectively, by
(4)Δℍn≔∑j=1nXj2+Yj2,∇ℍn≔X1,⋯,Xn,Y1,⋯,Yn.The Heisenberg group has a natural dilation structure which is consistent with the Lie group structure mentioned above. For each positive numbera>0, we define the dilation on ℍn by
(5)δaz,t≔az,a2t,z,t∈ℍn.Observe thatδaa>0 is an automorphism of the group ℍn. For any given u=z,t∈ℍn, the homogeneous norm of u is given by the following form:
(6)u=z,t≔z4+t21/4.Observe that∣z,t−1∣=∣z,t∣ and
(7)δaz,t=az4+a2t21/4=az,t,a>0.In addition, this norm· satisfies the triangle inequality and then leads to a left-invariant distance du,v=u−1·v for any u=z,t,v=w,s∈ℍn. If r>0 and u∈ℍn, let Bu,r=v∈ℍn:du,v<r be the (open) ball with center u∈ℍn and radius r∈0,∞. Both left and right Haar measures on ℍn coincide with the Lebesgue measure dzdt on ℂn×ℝ. For any measurable set E⊂ℍn, the Lebesgue measure of E is denoted by E. For u,r∈ℍn×0,∞, it can be proved that the volume of Bu,r is
(8)Bu,r=rQB0,1,where Q≔2n+2 is the homogeneous dimension of ℍn and |B0,1∣ is the volume of the unit ball in ℍn. A simple calculation shows that
(9)B0,1=2πn+1/2Γn/2n+1Γn+1/2.Given a ballB=Bu,r⊂ℍn and λ>0, we adopt the notation λB to denote the ball with the same center u and radius λr. Clearly, by (8), we have
(10)Bu,λr=λQBu,r,u,r∈ℍn×0,∞,λ∈0,∞.For a radial functionF on ℍn, we have the following integration formula:
(11)∫ℍnFudu=Cn∫0∞FϱϱQ−1dϱ,Q=2n+2,where Cn is a positive constant which is independent of F. For more information about harmonic analysis on the Heisenberg group, the reader is referred to [2, 4, 5] and the references therein.


## 1.2. The Schrödinger OperatorL

We recall some standard notation and definitions.Definition 1.
A nonnegative locallyLq integrable function V on ℍn is said to belong to the reverse Hölder class Bq for some exponent 1<q<∞, if there exists a positive constant C=Cq;V such that the reverse Hölder inequality
(12)1B∫BVwqdw1/q≤C1B∫Bwdw,holds for every ball B in ℍn.In this article, we will always assume that0≡V∈Bq with q∈Q/2,∞ and Q=2n+2. We now consider the Schrödinger operator with the potential V∈Bq on the Heisenberg group ℍn (see [3]):
(13)L≔−Δℍn+V.In recent years, there has been a lot of attention paid to the study of various function spaces associated with the Schrödinger operators, which has been an active research topic in harmonic analysis. For the investigation of Schrödinger operators on the Euclidean spaceℝn with nonnegative potentials that belong to the reverse Hölder class, see, for example, [6–10]. Concerning the weighted case, one can see [11–16] for more details. The extension to the setting of the Heisenberg group has been given by Lin and Liu in [3]. For further details, we refer the reader to [17–19], among others. Regarding the Schrödinger operators in a more general setting (such as the nilpotent Lie group), see, for example, [20, 21]. As in [3, 22], we introduce the following definition.Definition 2. Suppose thatV∈Bq with q∈Q/2,∞.
For any givenu∈ℍn, the critical radius function ρu=ρu;V is defined by
(14)ρu≔r∈0,∞:1rQ−2∫Bu,rVwdw≤1,where Bu,r denotes the ball in ℍn centered at u and with radius r.It should be pointed out that the auxiliary functionρu;V on the Euclidean space ℝn was introduced by Shen in [10]. Later, Li [21] defined it on the (simply connected) nilpotent Lie group. It is well known that the auxiliary function ρu determined by V∈Bq satisfies
(15)0<ρu<∞,for any given u∈ℍn (see [3, 22]). In particular, ρu=1 with Vw≡1, and ρu≈1/1+u with Vw=w2 (Hermite operator).It is easy to check that ifQ/2≤q2<q1, then Bq1⊂Bq2 by the Hölder inequality. Furthermore, it can be shown that the Bq class has a property of self-improvement. More precisely, if V∈Bq, then V∈Bq+ε for some ε>0. By this fact, we know that the assumption q>Q/2 is equivalent to q≥Q/2.Whenq∈Q/2,Q, we also write
(16)δ≔2−Qq∈0,1.Let us give some elementary properties of theBq class. Assume that V∈Bq with q∈Q/2,∞.Lemma 3.
The measureVwdw satisfies the doubling condition; that is, there exists a constant C0>0 such that
(17)∫Bu0,2rVwdw≤C0∫Bu0,rVwdw,for all balls Bu0,r in ℍn.Lemma 4.
Ifr=ρu0, then
(18)1rQ−2∫Bu0,rVwdw=1,for any u0∈ℍn.Lemma 5.
For0<r<R<∞, we have
(19)1rQ−2∫Bu0,rVwdw≤CrRδ1RQ−2∫Bu0,RVwdw,for any u0∈ℍn.For more details, the reader may consult [3, 22].We also need the following technical lemma concerning the critical radius function (14).Lemma 6.
Letρu be the auxiliary function determined by V. For any u and v in ℍn, there exist constants C1≥1 and N0>0 such that
(20)1C11+v−1uρu−N0≤ρvρu≤C11+v−1uρuN0/N0+1.Here, and in what follows,v−1·u is simply denoted by v−1u. Lemma 6 has been proved by Lu [22] (see also [3], Lemma 4). In the setting of ℝn, this result was first given by Shen in [10] (Lemma 1.4). As a direct consequence of (20), we can see that for each fixed k∈ℕ, the following estimate
(21)1+2krρu1+rρu−N0/N0+1≤C11+2krρv,holds for any v∈Bu,r with u∈ℍn and r∈0,∞, where C1 is the same as in (20). Let us verify (21). An application of (20) yields
(22)ρv≤C1·ρu1+v−1uρuN0/N0+1≤C1·ρu1+rρuN0/N0+1,which further implies that for each fixed k∈ℕ,
(23)2krρv≥1C1·2krρu1+rρu−N0/N0+1.Hence,(24)1+2krρv≥1+1C1·2krρu1+rρu−N0/N0+1≥1C11+rρu−N0/N0+1+1C1·2krρu1+rρu−N0/N0+1=1C11+2krρu1+rρu−N0/N0+1,which is the desired estimate. This estimate will often be used in the sequel.


## 1.3. Semigroup Maximal Functions and Riesz Transforms

LetL=−Δℍn+V be a Schrödinger operator on the Heisenberg group ℍn, where Δℍn is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class Bq for q∈Q/2,∞, and Q is the homogeneous dimension of ℍn. Since V is nonnegative and belongs to Llocqℍn, L generates a C0 contraction semigroup TtLt>0=e−tLt>0. Let Ptu,v denote the kernel of the semigroup e−tLt>0.
(25)TtLfu≔e−tLfu=∫ℍnPtu,vfvdv,f∈L2ℍn,t>0.We also denote byHtu the convolution kernel of the heat semigroup Ttt>0=etΔℍnt>0. Namely,
(26)Ttfu≔etΔℍnfu=∫ℍnHtv−1ufvdv.For anyu=z,s∈ℍn, it is well known that the heat kernel Htu has the explicit expression:
(27)Htz,s=2π−14π−n∫ℝλsinhλtnexp−λz24cothλt−iλsdλ.We consider the heat equation associated with the sub-Laplacian(28)∂tFu,t=ΔℍnFu,t,u,t∈ℍn×0,∞,with the initial condition Fu,0=fu. In fact, the function Htu stated the above exists as a solution to the heat equation. Moreover, by [23] (Theorem 2), we know that the heat kernel Ht satisfies the Gaussian upper bound estimate:
(29)0<Htu≤C·t−Q/2exp−u2At,where the positive constants C and A are independent of t∈0,∞ and u∈ℍn. By the Trotter product formula (see [24] for instance) and (29), one has
(30)0≤Ptu,v≤Htv−1u≤C·t−Q/2exp−v−1u2At,where C and A are positive constants independent of u, v, and t. Furthermore, by using the estimates of the fundamental solution for the Schrödinger operator L on ℍn, this estimate (30) can be significantly improved when V belongs to the reverse Hölder class Bq for some q∈Q/2,∞. The auxiliary function ρu arises naturally in the present situation.Lemma 7.
Letρu be the auxiliary function determined by V. For every positive integer N∈ℕ, there exists a positive constant CN>0 such that, for any u and v in ℍn,
(31)0≤Ptu,v≤CN·t−Q/2exp−v−1u2At1+tρu+tρv−N,t>0.Remark 8.
This estimate ofPtu,v is much better than (30), which was given by Lin and Liu in [3] (Lemma 7). In the setting of ℝn, this result can be found in [9] (Proposition 2).In this article, we investigate the semigroup maximal function related to the Schrödinger operatorL, which is defined by (see [3])
(32)TL∗fu≔supt>0TtLfu=supt>0e−tLfu,u∈ℍn.We shall establish the strong-type and weak-type estimates of the operatorTL∗. Some other maximal functions will be discussed at the end of Section 3.Let us also consider the Riesz transformsRj and the dual Riesz transforms Rj∗ for the Schrödinger operator L, which are defined, respectively, by (see [3])
(33)Rj=XjL−1/2,Rj+1=YjL−1/2,j=1,2,⋯,n,Rj∗=L−1/2Xj,Rj+1∗=L−1/2Yj,j=1,2,⋯,n,where the Xj are left-invariant vector fields that generate the Lie algebra of ℍn. Let
(34)RL≔∇ℍnL−1/2=R1,R2,⋯,R2n,RL∗≔L−1/2∇ℍn=R1∗,R2∗,⋯,R2n∗.Here,∇ℍn is the gradient operator on ℍn. We shall be interested in the behavior of the (vector-valued) operators RL and RL∗ associated with the Schrödinger operator L on ℍn.For anyp∈1,∞, the Lebesgue space Lpℍn is defined to be the set of all measurable functions f on ℍn such that
(35)fLpℍn≔∫ℍnfupdu1/p<∞.The weak Lebesgue spaceWL1ℍn consists of all measurable functions f on ℍn such that
(36)fWL1ℍn≔supλ>0λ·u∈ℍn:fu>λ<∞.Recently, Lin and Liu ([3], Theorem 6 and Remark 3) established the strong-type and weak-type estimates of the operator TL∗ on the Lebesgue spaces.Theorem 9.
Let1≤p<∞. Then, the following statements are true:
(1)
Ifp>1, then the operator TL∗ is bounded on Lpℍn(2)
Ifp=1, then the operator TL∗ is bounded from L1ℍn into WL1ℍnRemark 10.
(i)
It was also shown by Lin and Liu that this operator is bounded onBMOLℍn(ii)
On the Euclidean spaceℝn, this maximal operator was studied by Dziubański et al. [9] (see also [11, 25])As for the (vector-valued) dual Riesz transformRL∗ defined above, we have the following estimate given in [21] (see also [19]).Theorem 11.
LetV∈Bq with q∈Q/2,Q, and let p0 be a number such that 1/p0=1/q−1/Q. Then, the (vector-valued) dual Riesz transform RL∗ is bounded on Lpℍn for p0′<p<∞.By duality, we could obtain the following result.Theorem 12.
LetV∈Bq with q∈Q/2,Q, and let p0 be a number such that 1/p0=1/q−1/Q. Then, the (vector-valued) Riesz transform RL is bounded on Lpℍn for 1<p<p0.Moreover, it will be proved in Section4 that RL is of weak-type 1,1 on the Heisenberg group. The case where q∈Q,∞ is also considered in Section 4.Remark 13.
(i)
It can be shown that the range ofp in the above theorems is optimal (see [3]). In this paper, the authors also proved that the dual Riesz transform RL∗ is bounded on BMOLℍn and gave the Fefferman-Stein-type decomposition of BMOL functions with respect to Rj∗, j=1,2,⋯,2n(ii)
It was shown in [26] that when V≡0, the operators R≔∇ℍn−Δℍn−1/2 and R∗=−Δℍn−1/2∇ℍn are uniformly bounded on Lpℍn with respect to n. More specifically, for every 1<p<∞, there exists a constant Cp>0 such that for every n∈ℕ(37)RfLpℍn+R∗fLpℍn≤CpfLpℍn.(iii)
Recall that in the setting ofℝn, the Riesz transform and its dual form were originally studied by Shen in [10]. It can be proved that the analog of Theorem 11 (also Theorem 12) on the Euclidean space is also true by the same argument (see [10, 27]). For the corresponding estimates for commutators generated by BMO functions, the reader is referred to [11, 12, 28] for more detailsThe paper is organized as follows. In Section2, we introduce the Morrey space and weak Morrey space associated with the Schrödinger operator L on ℍn and state our main results: Theorems 18, 19, 22, 23, and 24. Section 3 is devoted to Proofs of Theorems 18 and 19, which establish the strong-type and weak-type estimates for the semigroup maximal function TL∗ in the framework of Morrey spaces. The corresponding estimates for some other maximal functions are also proved in this section. Section 4 is devoted to proving the boundedness properties of the Riesz transform RL and its dual form RL∗. In Section 5, we extend the above results to the generalized Morrey spaces.Throughout this paper,C>0 denotes a universal constant which may change from line to line, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. The notation X≲Y means that X≤CY for some positive constant C. If X≲Y and Y≲X, then we write X≈Y to denote the equivalence of X and Y. For any p∈1,∞, the notation p′ denotes its conjugate number, namely, 1/p+1/p′=1 and 1′=∞.


## 2. Definitions and Main Theorems

A few historic remarks are in order. The classical Morrey spaceMp,λℝn was originally introduced and studied by Morrey in [29] to deal with the local behavior of solutions to second-order elliptic partial differential equations. Since then, this space was systematically developed by a number of authors. Nowadays, this space has been investigated extensively and widely used in analysis, geometry, mathematical physics, and other related fields. We denote by Mp,λℝn the Morrey space, which consists of all p-locally integrable functions f on ℝn such that
(38)fMp,λℝn≔supx∈ℝn,r>0r−λ/pfLpBx,r=supx∈ℝn,r>0r−λ/p∫Bx,rfypdy1/p<∞,where 1≤p<∞ and 0≤λ≤n. It is known that Mp,λℝn is an extension of Lpℝn in the sense that Mp,0ℝn=Lpℝn. Note that Mp,nℝn=L∞ℝn by the Lebesgue differentiation theorem. If λ<0 or λ>n, then Mp,λℝn=Θ, where Θ is the set of all functions equivalent to 0 on ℝn. We also denote by WM1,λℝn the weak Morrey space, which consists of all measurable functions f on ℝn such that
(39)fWM1,λℝn≔supx∈ℝn,r>0r−λfWL1Bx,r=supx∈ℝn,r>0r−λsupσ>0σy∈Bx,r:fy>σ<∞.For the properties of classical Morrey spaces, we refer the readers to [30–34] and the references therein. Moreover, the Morrey spaces were found to have many important applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients, and the potential analysis (one can see [30, 35–38]).In this section, we introduce some types of Morrey spaces associated with the Schrödinger operatorL on ℍn (see [39]) and then give our main results.Definition 14.
Letρ be the auxiliary function determined by V∈Bq with q∈Q/2,∞. Let 1≤p<∞ and 0≤κ<1. For each given 0<θ<∞, the Morrey space Lρ,θp,κℍn is defined to be the set of all p-locally integrable functions f on ℍn such that
(40)1Bu0,rκ∫Bu0,rfupdu1/p≤C·1+rρu0θ,holds for every ball Bu0,r in ℍn, where u0 and r denote the center and radius of Bu0,r, respectively. The smallest constant appearing in (40) is called the norm of f, which is denoted by fLρ,θp,κℍn. It is a Banach space with respect to the norm ·Lρ,θp,κℍn. Define
(41)Lρ,∞p,κℍn≔⋃0<θ<∞Lρ,θp,κℍn.Definition 15.
Letρ be the auxiliary function determined by V∈Bq with q∈Q/2,∞. Let p=1 and 0≤κ<1. For each given 0<θ<∞, the weak Morrey space WLρ,θ1,κℍn is defined to be the set of all measurable functions f on ℍn such that
(42)1Bu0,rκsupλ>0λ·u∈Bu0,r:fu>λ≤C·1+rρu0θ,holds for every ball Bu0,r in ℍn. The smallest constant appearing in (42) is called the (quasi-)norm of f, which is denoted by fWLρ,θ1,κℍn. It is a (quasi-)Banach space with respect to the (quasi-)norm ·WLρ,θ1,κℍn. Correspondingly, we define
(43)WLρ,∞1,κℍn≔⋃0<θ<∞WLρ,θ1,κℍn.Remark 16.
(i)
Obviously, if we takeθ=0 or V≡0, then this Morrey space Lρ,θp,κℍn (or weak Morrey space WLρ,θ1,κℍn) is just the Morrey space Lp,κℍn (or weak Morrey space WL1,κℍn), which was defined and studied by Guliyev et al. [40](ii)
According to the above definitions, one has(44)Lp,κℍn⊂Lρ,θ1p,κℍn⊂Lρ,θ2p,κℍn,(45)WL1,κℍn⊂WLρ,θ11,κℍn⊂WLρ,θ21,κℍn,whenever 0<θ1<θ2<∞. Hence,
(46)Lp,κℍn⊂Lρ,∞p,κℍn,WL1,κℍn⊂WLρ,∞1,κℍn,for all p,κ∈1,∞×0,1. When κ=0, the spaces Lp,κℍn and WL1,κℍn reduce to Lpℍn and WL1ℍn, respectively.
(iii)
It follows directly from Chebyshev’s inequality thatfWLρ,θ1,κℍn≤fLρ,θ1,κℍn, and hence,(47)Lρ,θ1,κℍn⊂WLρ,θ1,κℍn,Lρ,∞1,κℍn⊂WLρ,∞1,κℍn.
Moreover, the inclusion is strict.Remark 17.
(i)
We can define a norm on the spaceLρ,∞p,κℍn (see [39]), which makes it into a Banach space. In view of (44), for any given f∈Lρ,∞p,κℍn, let(48)θ∗≔infθ>0:f∈Lρ,θp,κℍn.
Now define the functional·å by
(49)f⋆=fLρ,∞p,κℍn≔fLρ,θ∗p,κℍn.
It is easy to check that the functional·⋆ defined by (49) is indeed a norm on Lρ,∞p,κℍn provided p,κ∈1,∞×0,1; i.e., it satisfies the following conditions:
(a)
It is positive definite:f⋆≥0, and f⋆=0 if and only if f=0(b)
It is multiplicative:λf⋆=λf⋆, for any λ∈ℝ(c)
It satisfies the triangle inequality:f+g⋆≤f⋆+g⋆, for any f,g∈Lρ,∞p,κℍn(ii)
In view of (45), for any given f∈WLρ,∞1,κℍn, let(50)θ∗∗≔infθ>0:f∈WLρ,θ1,κℍn.
Similarly, we define the functional·⋆⋆ by
(51)f⋆⋆=fWLρ,∞1,κℍn≔fWLρ,θ∗∗1,κℍn.
It is easily checked that the functional·⋆⋆ defined by (51) is a (quasi-)norm on WLρ,∞1,κℍn for all 0<κ<1; i.e., it satisfies the following conditions:
(a)
It is positive definite:f⋆⋆≥0, and f⋆⋆=0 if and only if f=0(b)
It is multiplicative:λf⋆⋆=∣λ∣f⋆⋆, for any λ∈ℝ(c)
It satisfies the inequality:f+g⋆⋆≤2f⋆⋆+g⋆⋆, for any f,g∈WLρ,∞1,κℍn
The spaceWLρ,∞1,κℍn is a (quasi-)Banach space with respect to the (quasi-)norm ·⋆⋆.Since Morrey spaceLρ,θp,κℍn (or weak Morrey space WLρ,θ1,κℍn) could be viewed as an extension of the Lebesgue space (or the weak Lebesgue space) on ℍn (when κ=θ=0, or κ=0,V≡0), it is natural to study the boundedness properties of the operators TL∗, RL, and RL∗ in the context of Morrey spaces. In this paper, we will extend Theorems 9, 11, and 12 to the Morrey spaces on ℍn. Let ρ be the same as before. Now let us formulate our main results as follows.Theorem 18.
Let1<p<∞, 0<κ<1, and 0<θ<∞. If V∈Bq with q∈Q/2,∞, then the semigroup maximal function TL∗ is a bounded sublinear operator on Lρ,θp,κℍn.Theorem 19.
Letp=1, 0<κ<1, and 0<θ<∞. If V∈Bq with q∈Q/2,∞, then the semigroup maximal function TL∗ is a bounded sublinear operator from Lρ,θ1,κℍn into WLρ,θ1,κℍn.As an immediate consequence of Theorems18 and 19 and Remark 17, we have the following results.Corollary 20.
Let1<p<∞ and 0<κ<1. If V∈Bq with q∈Q/2,∞, then the semigroup maximal function TL∗ is a bounded sublinear operator on Lρ,∞p,κℍn.Corollary 21.
Letp=1 and 0<κ<1. If V∈Bq with q∈Q/2,∞, then the semigroup maximal function TL∗ is a bounded sublinear operator from Lρ,∞1,κℍn into WLρ,∞1,κℍn.Theorem 22.
Let0<κ<1 and 0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the dual Riesz transform RL∗ is a bounded linear operator on Lρ,θp,κℍn provided that p0′<p<∞.Theorem 23.
Let0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the Riesz transform RL is a bounded linear operator on Lρ,θp,κℍn provided that 1<p<p0 and 0<κ<1/s′ with s≔p0/p.It is worth pointing out that we cannot use Theorem22 to prove Theorem 23 in a direct way by duality, since the predual to Lρ,θp,κℍn is unknown. Motivated by the ideas in [30, 31], it is an interesting and natural problem to investigate the dual theory for the Morrey space Lρ,θp,κℍn through a geometric analysis of the Hausdorff capacity and Choquet integrals, which will be treated in a subsequent paper. In addition, we will prove that the operator RL is of weak-type 1,1. Based on this result, we can further prove the following.Theorem 24.
Let0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the Riesz transform RL is a bounded linear operator from Lρ,θ1,κℍn into WLρ,θ1,κℍn provided that 0<κ<1/p0′.As a straightforward consequence of Theorems22–24 and Remark 17, we obtain the following estimates.Corollary 25.
Let0<κ<1. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the dual Riesz transform RL∗ is a bounded linear operator on Lρ,∞p,κℍn provided that p0′<p<∞.Corollary 26.
IfV∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the Riesz transform RL is a bounded linear operator on Lρ,∞p,κℍn provided that 1<p<p0 and 0<κ<1/s′ with s≔p0/p.Corollary 27.
IfV∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the Riesz transform RL is a bounded linear operator from Lρ,∞1,κℍn into WLρ,∞1,κℍn provided that 0<κ<1/p0′.


## 3. Boundedness of the Semigroup Maximal Functions

In this section, we will prove the conclusions of Theorems18 and 19. Let Ptu,v denote the integral kernel of e−tL related to the Schrödinger operator L (see [3]). Then, we can write TL∗ as follows:
(52)TL∗fu=supt>0e−tLfu=supt>0∫ℍnPtu,vfvdv,u∈ℍn.Proof of Theorem 18.
For any givenf∈Lρ,θp,κℍn with p,κ∈1,∞×0,1 and θ∈0,∞, by definition, we only need to show that for any given ball B=Bu0,r of ℍn, the following inequality
(53)1Bu0,rκ∫Bu0,rTL∗fupdu1/p≲1+rρu0θ,holds true. By a standard argument, we decompose the function f as
(54)f=f1+f2∈Lρ,θp,κℍn,f1=f·χ2B,f2=f·χ2B∁,where 2B denotes the open ball centered at u0 of radius 2r, χE denotes the characteristic function of the set E, and 2B∁=ℍn\2B denotes its complement. Then, by the sublinearity of TL∗, we write
(55)1Bu0,rκ∫Bu0,rTL∗fupdu1/p≤1Bu0,rκ∫Bu0,rTL∗f1updu1/p+1Bu0,rκ∫Bu0,rTL∗f2updu1/p≔I1+I2.
Let us consider the first termI1. Making use of the first part of Theorem 9, we have
(56)I1=1Bu0,rκ∫Bu0,rTL∗fupdu1/p≤C·1Bκ/p∫ℍnf1updu1/p=C·1Bκ/p∫2Bfupdu1/p≤CfLρ,θp,κℍn·2Bκ/pBκ/p·1+2rρu0θ.
Moreover, note that for any fixedθ∈0,∞,
(57)1≤1+2rρu0θ≤2θ1+rρu0θ.
This, combined with (10), yields
(58)I1≤Cθ,nfLρ,θp,κℍn1+rρu0θ.
We now turn to estimate the second termI2. We first assert that the following inequality
(59)TL∗f2u≤CN,A∫2B∁1+v−1uρu−N1v−1uQ∙fvdv,holds for any u∈Bu0,r, where N and A are given as in Lemma 7. Indeed, this can be done by considering the following two cases: 0<t≤v−1u2 and v−1u2<t<∞. It follows directly from Lemma 7 that
(60)TL∗f2u=supt>0∫ℍnPtu,vf2vdv≤supt>0∫ℍnCNtQ/2·exp−v−1u2At1+tρu−Nf2vdv≤supv−1u2<t<∞∫ℍnCNtQ/2·exp−v−1u2At1+tρu−Nf2vdv+sup0<t≤v−1u2∫ℍnCNtQ/2·exp−v−1u2At1+tρu−Nf2vdv≔T∞∗f2u+T0∗f2u.
Whent>v−1u2, then t>∣v−1u∣, and hence,
(61)T∞∗f2u≤supv−1u2<t<∞∫ℍnCN,AtQ/2·v−1u2t−Q/21+tρu−N∣f2v∣dv=supv−1u2<t<∞∫ℍnCN,Av−1uQ·1+tρu−Nf2vdv≤CN,A∫2B∁1+∣v−1u∣ρu−N1v−1uQ·∣fv∣dv.
On the other hand, we can easily see that(62)T0∗f2u≤sup0<t≤v−1u2∫ℍnCN,AtQ/2·v−1u2t−Q/2+N/21+tρu−Nf2vdv=sup0<t≤v−1u2∫ℍnCN,Av−1uQ·t∣v−1u∣N1+tρu−Nf2vdv.
When0<t≤v−1u2, then t≤v−1u. In this case, it is easy to check that for any N∈ℕ,
(63)tv−1uN≤t+ρuv−1u+ρuN.
From this, it follows immediately that(64)T0∗f2u≤sup0<t≤v−1u2∫ℍnCN,Av−1uQ·t+ρuv−1u+ρuNt+ρuρu−Nf2vdv≤CN,A∫2B∁1+∣v−1u∣ρu−N1v−1uQ·∣fv∣dv.
Putting all together produces the required inequality (59). Notice that for any u∈Bu0,r and v∈2B∁, one has
(65)v−1u=v−1u0∙u0−1u≤v−1u0+u0−1u,v−1u=v−1u0∙u0−1u≥v−1u0−u0−1u.
Thus,(66)12v−1u0≤v−1u≤32v−1u0.
That is,v−1u≈v−1u0. Combining this fact with (59) yields that for any u∈Bu0,r and any positive integer N∈ℕ,
(67)TL∗f2u≤CN,A,n∫2B∁1+∣v−1u0∣ρu−N1v−1u0Q·fvdv=CN,A,n∑k=1∞∫2kr≤v−1u0<2k+1r1+v−1u0ρu−N1v−1u0Q·fvdv≤CN,A,n∑k=1∞1Bu0,2k+1r∫v−1u0<2k+1r1+2krρu−Nfvdv.
Furthermore, in view of (21) and (57), we can see that the above expression (67) does not exceed
(68)C∑k=1∞1Bu0,2k+1r∫Bu0,2k+1r1+rρu0N·N0/N0+11+2krρu0−Nfvdv≤C∑k=1∞1+2krρu0−N·1/N0+11Bu0,2k+1r∫Bu0,2k+1rfvdv≤C∑k=1∞1+2k+1rρu0−N·1/N0+11Bu0,2k+1r∫Bu0,2k+1rfvdv.
By using the Hölder inequality, we obtain that for each fixedk∈ℕ,
(69)1Bu0,2k+1r∫Bu0,2k+1rfvdv≤1Bu0,2k+1r∫Bu0,2k+1rfvpdv1/p∫Bu0,2k+1r1dv1/p′≤CfLρ,θp,κℍn·Bu0,2k+1rκ/pBu0,2k+1r1/p1+2k+1rρu0θ.
Substituting the above inequality into formula (68), we conclude that
(70)I2≤CfLρ,θp,κℍn∑k=1∞Bu0,r1−κ/pBu0,2k+1r1−κ/p1+2k+1rρu0−N·1/N0+1+θ.
By choosing some sufficiently large numberN such that N>N0+1θ, then we have
(71)I2≤CfLρ,θp,κℍn∑k=1∞Bu0,rBu0,2k+1r1−κ/p≤CfLρ,θp,κℍn≤CfLρ,θp,κℍn1+rρu0θ,where the last inequality follows from the fact that 1−κ>0 and θ>0. Combining the above estimates for I1 and I2, we obtain the desired inequality (53). This ends Proof of Theorem 18.Proof of Theorem 19.
Letf∈Lρ,θ1,κℍn with κ∈0,1 and θ∈0,∞. Fix u0,r∈ℍn×0,∞. Our aim is to prove, by definition, that for each given ball B=Bu0,r of ℍn, the following estimate
(72)1Bu0,rκsupλ>0λ·u∈Bu0,rTL∗fu>λ≲1+rρu0θ,holds true. To this end, we decompose the function f as
(73)f=f1+f2∈Lρ,θ1,κℍn,f1=f·χ2B,f2=f·χ2B∁.
Then, for any fixedλ>0, we can write
(74)1Bu0,rκλ·u∈Bu0,r:TL∗fu>λ≤1Bu0,rκλ·u∈Bu0,r:TL∗f1u>λ2+1Bu0,rκλ·u∈Bu0,r:TL∗f2u>λ2≔J1+J2.
We first give the estimate for the termJ1. By the second part of Theorem 9, we get
(75)J1=1Bu0,rκλ·u∈Bu0,r:TL∗f1u>λ2≤C·1Bκ∫ℍnf1udu=C·1Bκ∫2Bfudu≤CfLρ,θ1,κℍn·2BκBκ·1+2rρu0θ.
Therefore, in view of (57) and (10), we have
(76)J1≤Cθ,nfLρ,θ1,κℍn1+rρu0θ.
To estimate the second termJ2, by using the pointwise inequality (68) and Chebyshev’s inequality, we can deduce that
(77)J2≤2Bu0,rκ∫Bu0,rTL∗f2udu≤C·Bu0,rBu0,rκ∫2B∁1+v−1uρu−N1v−1uQ·fvdv≤C·Bu0,rBu0,rκ∑k=1∞1+2k+1rρu0−N·1/N0+11Bu0,2k+1r∫Bu0,2k+1rfvdv.
Moreover, for each fixedk∈ℕ, we compute
(78)1Bu0,2k+1r∫Bu0,2k+1rfvdv≤CfLρ,θ1,κℍn·Bu0,2k+1rκBu0,2k+1r1+2k+1rρu0θ.
Consequently, substituting this inequality into formula (77),
(79)J2≤CfLρ,θ1,κℍn∑k=1∞Bu0,r1−κBu0,2k+1r1−κ1+2k+1rρu0−N·1/N0+1+θ.
By selecting some large enoughN such that N>N0+1θ, we thus have
(80)J2≤CfLρ,θ1,κℍn∑k=1∞Bu0,rBu0,2k+1r1−κ≤CfLρ,θ1,κℍn≤CfLρ,θ1,κℍn1+rρu0θ,where the last step is again due to the fact that 0<κ<1 and θ>0. Summing up the above estimates for J1 and J2, and then taking the supremum over all λ>0, we obtain our desired result (72). This completes Proof of Theorem 19.We also consider the maximal function with respect to the Poisson semigroupe−tLt>0, which is defined by
(81)TL∗fu≔supt>0e−tLfu,u∈ℍn.We now begin to prove that under the conditions of Theorems18 and 19, the same results also hold for the operator TL∗ related to L. Recall the subordination formula (see [41], p. 6)
(82)e−β=1π∫0∞e−vv·e−β2/4vdv.Thus, from the heat semigroup, we can define the Poisson semigroupe−tLt>0 by
(83)e−tLfu=1π∫0∞e−vv·e−t2/4vLfudv.From this, it follows immediately that for allu∈ℍn,
(84)TL∗fu≤supt>01π∫0∞e−vv·e−t2/4vLfudv≤1π∫0∞e−vv·sups>0e−sLfudv=TL∗fu·1π∫0∞e−vvdv=TL∗fu.Hence, as an immediate consequence of Theorems18 and 19 and Corollaries 20 and 21, we have the following results. Let ρ be the same as in (14).Theorem 28.
Let1<p<∞, 0<κ<1, and 0<θ<∞. If V∈Bq with q∈Q/2,∞, then the operator TL∗ is bounded on Lρ,θp,κℍn and hence bounded on Lρ,∞p,κℍn.Theorem 29.
Letp=1, 0<κ<1, and 0<θ<∞. If V∈Bq with q∈Q/2,∞, then the operator TL∗ is bounded from Lρ,θ1,κℍn into WLρ,θ1,κℍn and hence bounded from Lρ,∞1,κℍn into WLρ,∞1,κℍn.Remark 30.
(i)
A slightly more general point of view is as follows. Motivated by the work in [9, 25], we introduce the semigroup nontangential maximal function related to L, which is given as follows:(85)TL∗∗fu≔supt>0supv∈Bu,te−tLfv,u∈ℍn.
It is interesting to investigate the boundedness of the operatorTL∗∗ related to L. Following the same arguments as in Proofs of Theorems 4 and 6 in [3], we are able to prove that the operator TL∗∗ is bounded on Lpℍn for all 1<p<∞ and bounded from L1ℍn into WL1ℍn. Based on this result, we can further prove that the corresponding estimates for the operator TL∗∗ remain valid in the context of Morrey spaces. The proof needs appropriate but minor modifications, and we leave this to the interested reader.
(ii)
In view of the above results, we consider here the nontangential maximal function with respect to the Poisson semigroupe−tLt>0, which is defined by(86)TL∗∗fu≔supt>0supv∈Bu,te−tLfv,u∈ℍn.
For the same reason as above, it can be shown that this new maximal operatorTL∗∗ is dominated by TL∗∗ in some sense. Therefore, all the results mentioned above hold as well for the maximal operator in this more general situation.


## 4. Boundedness of the Riesz Transforms

This section is concerned with Proofs of Theorems22, 23, and 24. Recall that the operators RL and RL∗ have singular kernels with values in ℝ2n that will be denoted by Ku,v and K∗u,v, respectively (see [3]).
(87)RLfu=∇ℍnL−1/2fu=∫ℍnKu,vfvdv,RL∗fu=L−1/2∇ℍnfu=∫ℍnK∗u,vfvdv.Obviously,(88)K∗u,v=−Kv,uK∗u,v=Kv,u.The next lemma plays a crucial role in our Proofs of Theorems22–24.Lemma 31.
LetV∈Bq with q∈Q/2,Q, and let ρ· be the auxiliary function determined by V. For every positive integer N∈ℕ, there exists a positive constant CN>0 such that, for any u and vinℍn,
(89)K∗u,v≤CN1+v−1uρu−N1v−1uQ−1×∫Bv,v−1u/4Vwv−1wQ−1dw+1v−1u,where K∗u,v denotes the (vector-valued) kernel of the operator RL∗. Moreover, the above inequality also holds with ρu replaced by ρv.Lemma31 was proved by Li [21] in a more general setting (connected nilpotent Lie group) (see also [19] for ℍn). For such kernels, we also give the following result, which establishes the Lipschitz regularity of K∗u,v.Lemma 32.
LetV∈Bq with q∈Q/2,Q, and let ρ· be the auxiliary function determined by V. For every positive integer N∈ℕ, there exists a positive constant CN>0 such that, for any u and v in ℍn, and for some fixed δ′∈0,δ, δ is given as in (16),
(90)K∗uh,v−K∗u,v≤CN1+v−1uρu−Nhδ′v−1uQ−1+δ′×∫Bv,v−1u/4Vwv−1wQ−1dw+1v−1u,whenever h≤v−1u/4/. Moreover, the above inequality also holds with ρu replaced by ρv.The above kernel estimate in Lemma32 was obtained by Pengtao and Lizhong in [19], which will be used to prove that the operator RL is of weak-type 1,1. Recall that in the Euclidean setting, the kernel estimates (89) and (90) were proved in [10, 42].We are now in a position to give the proofs of our main theorems.Proof of Theorem 22.
Letf∈Lρ,θp,κℍn with p,κ∈p′0,∞×0,1 and θ∈0,∞, where 1/p0=1/q−1/Q. Fix u0,r∈ℍn×0,∞. By definition, we only need to show that for any given ball B=Bu0,r of ℍn, the following inequality
(91)1Bu0,rκ∫Bu0,rRL∗fupdu1/p≲1+rρu0θ,holds true. Using the standard technique, we decompose the function f as
(92)f=f1+f2∈Lρ,θp,κℍn,f1=f·χ2B,f2=f·χ2B∁.
Then, by using the linearity ofRL∗, we write
(93)1Bu0,rκ∫Bu0,rRL∗fupdu1/p≤1Bu0,rκ∫Bu0,rRL∗f1updu1/p+1Bu0,rκ∫Bu0,rRL∗f2updu1/p≔K1+K2.
Let us consider the first termK1. Making use of (10), (57), and Theorem 11, we have
(94)K1≤C·1Bκ/p∫ℍnf1updu1/p=C·1Bκ/p∫2Bfupdu1/p≤CfLρ,θp,κℍn·2Bκ/pBκ/p·1+2rρu0θ≤Cθ,nfLρ,θp,κℍn1+rρu0θ.
Now let us turn to estimate the second termK2. By using Lemma 31, we obtain that for any u∈Bu0,r,
(95)RL∗f2u=∫2B∁K∗u,vfvdv≤CNIu+IIu,where
(96)Iu≔∫2B∁1+v−1uρu−N1v−1uQ·fvdv,IIu≔∫2B∁1+v−1uρu−N1v−1uQ−1∫Bv,v−1u/4Vwv−1wQ−1dw·fvdv.
Hence,K2 can be written as follows:
(97)K2=1Bu0,rκ∫Bu0,rRL∗f2updu1/p≲1Bu0,rκ∫Bu0,rIupdu1/p+1Bu0,rκ∫Bu0,rIIupdu1/p≔K3+K4.
Arguing as in Proof of Theorem18, we can also obtain
(98)K3≤CfLρ,θp,κℍn1+rρu0θ.
We only have to deal with the termK4. As mentioned in the previous proof, one has
(99)12v−1u0≤v−1u≤32v−1u0,whenever u∈Bu0,r and v∈2B∁. Thus, for any positive integer N∈ℕ,
(100)∣IIu∣≲∫2B∁1+v−1u0ρu−N1v−1u0Q−1∫Bv,∣v−1u∣/4Vwv−1wQ−1dw·fvdv=∑k=1∞∫2kr≤v−1u0<2k+1r1+v−1u0ρu−N1v−1u0Q−1×∫Bv,v−1u/4Vwv−1wQ−1dw·fvdv≲∑k=1∞1Bu0,2k+1r1−1/Q∫Bu0,2k+1r1+2krρu−N×∫Bv,v−1u/4Vwv−1wQ−1dw·fvdv.
It is not difficult to check that whenw∈Bv,v−1u/4 and v∈Bu0,2k+1r, one has w∈Bu0,2k+2r, which implies
(101)IIu≲∑k=1∞1Bu0,2k+1r1−1/Q1+2krρu−N×∫Bu0,2k+1rI1VχBu0,2k+2rv·fvdv,where I1 stands for the fractional integral operator of order one defined by
(102)I1gv≔∫ℍngwv−1wQ−1dw.
For this operator, a classical result of Folland and Stein [43] states that I1 is bounded from Lqℍn into Lp0ℍn for 1<q<Q and 1/p0=1/q−1/Q (see also [39, 44]). Namely, there exists a constant C>0 such that for any g∈Lqℍn,
(103)I1gLp0ℍn≤CgLqℍn.
Sincep0′<p<∞, we can choose a number s>0 such that 1/p+1/p0+1/s=1. A combination of the Hölder inequality and (103) gives
(104)∫Bu0,2k+1rI1VχBu0,2k+2rv·∣fv∣dv≤∫Bu0,2k+1rfvpdv1/p∫Bu0,2k+1r1dv1/s∫Bu0,2k+1rI1VχBu0,2k+2rvp0dv1/p0≤C∫Bu0,2k+1rfvpdv1/pBu0,2k+1r1/s∫Bu0,2k+2rVwqdw1/q≤C∫Bu0,2k+1rfvpdv1/pBu0,2k+1r1/s1Bu0,2k+2r1/q′∫Bu0,2k+2rVwdw,where the last inequality holds by our assumption V∈Bq. We now claim that the following inequality holds. For any N1>log2C0 (C0 is the doubling constant in Lemma 3), there exists a constant C>0 such that for any u0∈ℍn and τ∈0,∞,
(105)∫Bu0,τVwdw≤C·τQ−21+τρu0N1.
Taking this claim momentarily for granted, then we have(106)∫Bu0,2k+1rI1VχBu0,2k+2rv·fvdv≤C∫Bu0,2k+1rfvpdv1/pBu0,2k+1r1/s·2k+2rQ−2Bu0,2k+2r1/q′1+2k+2rρu0N1≤C∫Bu0,2k+1rfvpdv1/p·1Bu0,2k+1r1/q′−1/s−1+2/Q1+2k+1rρu0N1,where in the last step we have invoked (57) and (10). In addition, it follows immediately from (21) and (57) that
(107)1+2krρu−N≲1+rρu0N·N0/N0+11+2krρu0−N≤1+2krρu0−N·1/N0+1≤1+2k+1rρu0−N·1/N0+1.
A trivial computation shows that(108)1−1Q+1q′−1s−1+2Q=1Q+1q′−1s=1p.
Therefore, in view of (108) and (107), we conclude that
(109)IIu≲∑k=1∞1Bu0,2k+1r1/p1+2k+1rρu0−N·1/N0+1+N1×∫Bu0,2k+1rfvpdv1/p≲fLρ,θp,κℍn∑k=1∞Bu0,2k+1rκ/pBu0,2k+1r1/p1+2k+1rρu0−N·1/N0+1+N1+θ.
Consequently,(110)K4≤CfLρ,θp,κℍn∑k=1∞Bu0,r1−κ/pBu0,2k+1r1−κ/p1+2k+1rρu0−N·1/N0+1+N1+θ.
By choosing some sufficiently large numberN such that N>N0+1N1+θ, then we have
(111)K4≤CfLρ,θp,κℍn∑k=1∞Bu0,rBu0,2k+1r1−κ/p≤CfLρ,θp,κℍn≤CfLρ,θp,κℍn1+rρu0θ,where the last inequality follows again from the fact that 1−κ>0 and θ>0. Combining the above estimates for K1, K3, and K4 produces the desired inequality (91).Finally, let us verify (105). Suppose that V∈Bq with Q/2≤q<Q. Using the same method as in Proof of Lemma 1 in [42], for any u0∈ℍn and τ∈0,∞, there must exist an integer j such that 2jτ≤ρu0<2j+1τ. Two cases are considered below.Case 1.
j<0. In this case, one has ρu0<τ. This fact together with Lemmas 3 and 4 yields
(112)1+τρu0−N1∫Bu0,τVwdw≤2−j−N1∫Bu0,2−jρu0Vwdw≤2−j−N1C0−j∫Bu0,ρu0Vwdw≤2−j−N1C0−jτQ−2.
SinceN1>log2C0, it is easy to see that
(113)1+τρu0−N1∫Bu0,τVwdw≤τQ−2.Case 2.
j≥0. In this case, one has τ≤ρu0. This fact, along with Lemmas 5 and 4, implies that
(114)1+τρu0−N1∫Bu0,τVwdw≤∫Bu0,τVwdw=τQ−2·1τQ−2∫Bu0,τVwdw≤C·τQ−2·1ρu0Q−2∫Bu0,ρu0Vwdw=C·τQ−2.Thus, in both cases, (105) holds. This completes Proof of Theorem 22.In order to prove Theorem24, let us first set up the following result, which is based on a version of the Calderón-Zygmund decomposition on ℍn and Lemma 32.Theorem 33.
LetV∈Bq with q∈Q/2,Q. Then, the Riesz transform RL is bounded from L1ℍn into WL1ℍn.Proof.
For any givenf∈L1ℍn and σ>0, making use of the Calderón-Zygmund decomposition of f at height σ (see [43]), we have the decomposition f=g+b with b=∑ibi such that
(1)
∣gu∣≤C·σ, for almost every u∈ℍn and(115)gL1ℍn≤CfL1ℍn,(2)
eachbi is supported in the ball Bi=Bui,ri, and we denote the center and the radius of Bi by ui and ri, respectively(116)∫Bibiudu=0,biL1≤CσBi,(3)
the setsBi are finitely overlapping and(117)∑i∣Bi∣≤CσfL1ℍn.
From this construction, we have that for any fixedσ>0,
(118)u∈ℍn:RLfu>σ≤u∈ℍn:RLgu>σ2+u∈ℍn:RLbu>σ/2≔I+II.
The part of the argument involving the functiong proceeds as follows. By using the L2ℍn boundedness of RL (see Theorem 12 with 1<2<p0), we obtain
(119)I≤Cσ2fL2ℍn2≤CσgL1ℍn≤CσfL1ℍn.
SettingE=⋃i4Bi=⋃iBui,4ri, we split II into two parts as follows:
(120)II≤u∈E:RLbu>σ2+u∈E∁:RLbu>σ2≔III+IV.
It is obvious that(121)III≤⋃i4Bi≤C∑i∣Bi∣≤CσfL1ℍn.
Therefore, in order to complete our proof, we need only to show that(122)IV≤CσfL1ℍn.
An application of Chebyshev’s inequality yields(123)IV≤2σ∫E∁RLbudu≤2σ∑i∫4Bi∁RLbiudu.
We observe that(124)ui−1v<ri<ui−1u4,whenever v∈Bi and u∈4Bi∁. Then, we apply Lemma 32, (88), and the cancelation condition of bi to get
(125)RLbiu=∫ℍnKu,vbivdv=∫BiKu,v−Ku,uibivdv≤∫BiK∗v,u−K∗ui,ubivdv≲∫Biui−1vδ′ui−1uQ+δ′bivdv+∫Bi1+ui−1uρui−N∫Bu,ui−1u/4Vwu−1wQ−1dwui−1vδ′ui−1uQ−1+δ′bivdv.
Obviously, the first term on the right-hand side of (125) is bounded by
(126)∫Biriδ′ui−1uQ+δ′bivdv≤riδ′biL1·1ui−1uQ+δ′.
Using this estimate together with (11), we can deduce that
(127)IV1≔1σ∑i∫4Bi∁∫Biui−1vδ′ui−1uQ+δ′bivdvdu≤1σ∑iriδ′biL1∫ui−1u>4riduui−1uQ+δ′=1σ∑iriδ′biL1∫4ri∞1ρQ+δ′·ρQ−1dρ≤Cσ∑iσBi≤CσfL1ℍn.
On the other hand, the latter term on the right-hand side of (125) is controlled by
(128)∫Bi1+ui−1uρui−N∫Bu,ui−1u/4Vwu−1wQ−1dwriδ′ui−1uQ−1+δ′bivdv≤riδ′biL1·1ui−1uQ−1+δ′1+∣ui−1u∣ρui−N∫Bu,ui−1u/4Vwu−1wQ−1dw.
Whenw∈Bu,∣ui−1u∣/4, by the triangle inequality, one has
(129)ui−1w≤ui−1u+u−1w≤2ui−1u.
From this, it follows that the above expression is bounded by(130)riδ′biL1·1ui−1uQ−1+δ′1+ui−1uρui−N∫Bui,2ui−1uVwu−1wQ−1dw.
Consequently,(131)IV2≔1σ∑i∫4Bi∁∫Bi1+∣ui−1u∣ρui−N×∫Bu,ui−1u/4Vwu−1wQ−1dwui−1vδ′ui−1uQ−1+δ′bivdvdu≤1σ∑i∫4Bi∁riδ′biL1·1ui−1uQ−1+δ′1+ui−1uρui−N×∫Bui,2ui−1uVwu−1wQ−1dwdu≤1σ∑iriδ′biL1∑k=1∞1+2k+1riρui−N×∫2k+1ri≤ui−1u<2k+2ri1ui−1uQ−1+δ′·I1VχBui,2k+3riudu,where I1 is the fractional integral operator of order one given in (102). A combination of the Hölder inequality and (103) implies that for each fixed k∈ℕ,
(132)∫2k+1ri≤ui−1u<2k+2ri1ui−1uQ−1+δ′·I1VχBui,2k+3riudu≤∫2k+1ri≤ui−1u<2k+2ri1ui−1uQ−1+δ′p′0du1/p′0×∫2k+1ri≤ui−1u<2k+2riI1VχBui,2k+3riup0du1/p0≤∫2k+1ri≤ui−1u<2k+2ri1ui−1uQ−1+δ′p′0du1/p′0I1VχBui,2k+3riLp0≤C·Bui,2k+2ri1/p′02k+1riQ−1+δ′∫Bui,2k+3riVwqdw1/q≤C·Bui,2k+2ri1/p′02k+1riQ−1+δ′1Bui,2k+3ri1/q′∫Bui,2k+3riVwdw,where the last inequality is obtained by the hypothesis V∈Bq. Moreover, in view of (10), (105), and (57), we can see that the above expression (132) is bounded by
(133)Bui,2k+2ri1/p′02k+1ri1+δ′·1Bui,2k+3ri1/q′1+2k+3riρuiN1≤C·12k+1riδ′Bui,2k+1ri1/p′0Bui,2k+1ri1/Q·1Bui,2k+1ri1/q′1+2k+1riρuiN1=C·12k+1riδ′1+2k+1riρuiN1,where the last step is due to the fact that 1/Q+1/p0−1/q=0. Therefore, by selecting some large enough N such that N>N1, we thus have
(134)IV2≤Cσ∑iriδ′biL1∑k=1∞12k+1riδ′1+2k+1riρui−N+N1≤Cσ∑ibiL1∑k=1∞12k+1δ′≤Cσ∑iσBi≤CσfL1ℍn.
Collecting all these estimates and then taking the supremum over allσ>0, we conclude the proof of Theorem 33.Proof of Theorem 24.
Letp0 be a positive number such that 1/p0=1/q−1/Q. To prove Theorem 24, it is enough to prove that for each given ball B=Bu0,r of ℍn, the following estimate
(135)1Bu0,rκsupσ>0σ·u∈Bu0,r:RLfu>σ≲1+rρu0θ,holds true for any given f∈Lρ,θ1,κℍn with some θ∈0,∞ and κ∈0,1/p′0. Using the standard technique, we decompose the function f as
(136)f=f1+f2∈Lρ,θ1,κℍn,f1=f·χ2B,f2=f·χ2B∁.
Then, for any fixedσ>0, we can write
(137)1Bu0,rκσ·u∈Bu0,r:RLfu>σ≤1Bu0,rκσ·u∈Bu0,r:RLf1u>σ2+1Bu0,rκσ·u∈Bu0,r:RLf2u>σ2≔L1+L2.
Let us estimate the first termL1. By Theorem 33, (57), and (10), we get
(138)L1≤C·1Bκ∫ℍnf1udu=C·1Bκ∫2Bfudu≤CfLρ,θ1,κℍn·2BκBκ1+2rρu0θ≤Cθ,nfLρ,θ1,κℍn1+rρu0θ.
As for the second termL2, from (88) and Lemma 31, it follows that for any u∈Bu0,r,
(139)RLf2u=∫2B∁Ku,vfvdv=∫2B∁K∗v,ufvdv≤CNIIIu+IVu,where
(140)IIIu≔∫2B∁1+v−1uρu−N1v−1uQ·fvdv,IVu≔∫2B∁1+∣v−1u∣ρu−N1v−1uQ−1∫Bu,∣v−1u∣/4Vwu−1wQ−1dw·fvdv.
Thus, by (139) and Chebyshev’s inequality, L2 can be written as follows:
(141)L2≤2Bu0,rκ∫Bu0,rRLf2udu≲1Bu0,rκ∫Bu0,rIIIudu+1Bu0,rκ∫Bu0,rIVudu≔L3+L4.
We now proceed exactly as we did in Proof of Theorem19 and have the following estimate as well:
(142)L3≤CfLρ,θ1,κℍn1+rρu0θ.
Let us analyze the latter termL4. In order to do this, we first observe that
(143)12v−1u0≤v−1u≤32v−1u0,whenever u∈Bu0,r and v∈2B∁. Hence, for any positive integer N∈ℕ,
(144)IVu≲∫2B∁1+v−1u0ρu−N1v−1u0Q−1∫Bu,∣v−1u∣/4Vwu−1wQ−1dw·fvdv=∑k=1∞∫2kr≤v−1u0<2k+1r1+v−1u0ρu−N1v−1u0Q−1×∫Bu,∣v−1u∣/4Vwu−1wQ−1dw·fvdv≲∑k=1∞1Bu0,2k+1r1−1/Q∫Bu0,2k+1r1+2krρu−N×∫Bu,v−1u/4Vwu−1wQ−1dw·fvdv.
It is easy to verify that whenw∈Bu,v−1u/4 and v∈Bu0,2k+1r, one has w∈Bu0,2k+2r. This fact together with (107) implies that for any u∈Bu0,r,
(145)IVu≲∑k=1∞1Bu0,2k+1r1−1/Q1+2krρu−N×∫Bu0,2k+1rI1VχBu0,2k+2ru·fvdv≲∑k=1∞1Bu0,2k+1r1−1/Q1+2k+1rρu0−N·1/N0+1∫Bu0,2k+1rfvdv×I1VχBu0,2k+2ru≲∑k=1∞Bu0,2k+1rκBu0,2k+1r1−1/Q1+2k+1rρu0−N·1/N0+1+θfLρ,θ1,κℍn×I1VχBu0,2k+2ru.
Consequently,(146)L4≤CfLρ,θ1,κℍn1Bu0,rk∑k=1∞Bu0,2k+1rκBu0,2k+1r1−1/Q1+2k+1rρu0−N·1/N0+1+θ×∫Bu0,rI1VχBu0,2k+2rudu.
Applying the Hölder inequality along with (103), we can compute the above integral as follows:
(147)∫Bu0,rI1VχBu0,2k+2rudu≤∫Bu0,r1du1/p′0∫Bu0,rI1VχBu0,2k+2rup0du1/p0≤Bu0,r1/p′0I1VχBu0,2k+2rLp0≤CBu0,r1/p′0∫Bu0,2k+2rVwqdw1/q≤CBu0,r1/p′01Bu0,2k+2r1/q′∫Bu0,2k+2rVwdw,where the last inequality holds since V∈Bq. Moreover, by (105), (57), and (10), we can see that the above expression is bounded by
(148)CBu0,r1/p′0·2k+2rQ−2Bu0,2k+2r1/q′1+2k+2rρu0N1≤CBu0,r1/p′0·2k+1rQ−2Bu0,2k+1r1/q′1+2k+1rρu0N1.
Taking into account the fact that1/Q+1/q′=1/p′0, then we have
(149)L4≤CfLρ,θ1,κℍnBu0,r1/p′0−κ∑k=1∞Bu0,2k+1rκBu0,2k+1r1−1/Q+1/q′−1+2/Q1+2k+1rρu0−N·1/N0+1+θ+N1=CfLρ,θ1,κℍn∑k=1∞Bu0,r1/p′0−κBu0,2k+1r1/p′0−κ1+2k+1rρu0−N·1/N0+1+θ+N1≤CfLρ,θ1,κℍn∑k=1∞Bu0,r1/p′0−κBu0,2k+1r1/p′0−κ,where a large enough N is chosen satisfying N>N0+1θ+N1. By the choice of κ, it guarantees that the exponent −1/p′0−κ of the last summation is negative, and hence, it is convergent. Therefore, we conclude that
(150)L4≤CfLρ,θ1,κℍn≤CfLρ,θ1,κℍn1+rρu0θ.
Combining these estimates forL1, L3, and L4, and then taking the supremum over all σ>0, we get the desired estimate (135). This finishes Proof of Theorem 24.Proof of Theorem 23.
Since the proof is similar to that of Theorem22, we shall only indicate the necessary modifications. As before, it is enough for us to show that for an arbitrary fixed ball Bu0,r in ℍn, the following estimate
(151)1Bu0,rκ∫Bu0,rRLfupdu1/p≲1+rρu0θ,holds true for any given f∈Lρ,θp,κℍn with some θ∈0,∞, p∈1,p0, and κ∈0,1/s′, where s=p0/p and 1/p0=1/p−1/Q. To this end, we split f=f1+f2 through f1=f·χ2B and f2=f·χ2B∁. Then, the left-hand side of (151) will be divided into two parts given below.
(152)1Bu0,rκ∫Bu0,rRLfupdu1/p≤1Bu0,rκ∫Bu0,rRLf1updu1/p+1Bu0,rκ∫Bu0,rRLf2updu1/p≔M1+M2.
Since the Riesz transformRL is bounded on Lpℍn for 1<p<p0 (see Theorem 12), we can deal with M1 in the same manner as in Proof of Theorem 22 and obtain
(153)M1≤CfLρ,θp,κℍn1+rρu0θ.
On the other hand, from (139), it follows that
(154)M2≲1Bu0,rκ∫Bu0,rIIIupdu1/p+1Bu0,rκ∫Bu0,rIVupdu1/p≔M3+M4.
Here(155)IIIu=∫2B∁1+v−1uρu−N1v−1uQ·fvdv,IVu=∫2B∁1+∣v−1u∣ρu−N1v−1uQ−1∫Bu,v−1u/4Vwu−1wQ−1dw·fvdv.
We follow the same arguments as in Proof of Theorem18 and obtain the following estimate as well:
(156)M3≤CfLρ,θp,κℍn1+rρu0θ.
It remains to check that (151) holds for the last term M4. As it was shown in Theorem 24, it holds that for any u∈Bu0,r,
(157)IVu≲∑k=1∞1Bu0,2k+1r1−1/Q1+2krρu−N×∫Bu0,2k+1rI1VχBu0,2k+2ru·∣fv∣dv≲∑k=1∞1Bu0,2k+1r1−1/Q1+2k+1rρu0−N·1/N0+1∫Bu0,2k+1rfvdv×I1VχBu0,2k+2ru.
Moreover, by using the Hölder inequality, we obtain that for each fixedk∈ℕ,
(158)∫Bu0,2k+1rfvdv≤∫Bu0,2k+1rfvpdv1/p∫Bu0,2k+1r1dv1/p′≤CfLρ,θp,κℍnBu0,2k+1rκ/p1+2k+1rρu0θ·Bu0,2k+1r1/p′,which in turn gives
(159)M4≤CfLρ,θp,κℍn1Bu0,rκ/p∑k=1∞Bu0,2k+1rκ/pBu0,2k+1r1/p−1/Q1+2k+1rρu0−N·1/N0+1+θ×∫Bu0,rI1VχBu0,2k+2rupdu1/p.
For the latter integral, we use the Hölder inequality with exponents=p0/p>1 and (103) to derive
(160)∫Bu0,rI1VχBu0,2k+2rupdu1/p≤∫Bu0,r1du1/ps′∫Bu0,rI1VχBu0,2k+2rup0du1/p0≤Bu0,r1/ps′I1VχBu0,2k+2rLp0≤CBu0,r1/ps′∫Bu0,2k+2rVwqdw1/q≤CBu0,r1/ps′1Bu0,2k+2r1/q′∫Bu0,2k+2rVwdw,where the last inequality is obtained by the fact that V∈Bq. Furthermore, in view of (105), (57), and (10), we can see that the above expression is controlled by
(161)CBu0,r1/ps′·2k+2rQ−2Bu0,2k+2r1/q′1+2k+2rρu0N1≤CBu0,r1/ps′·2k+1rQ−2Bu0,2k+1r1/q′1+2k+1rρu0N1.
A trivial computation leads to(162)1p−1Q+1q′−1+2Q=1p−1p0=1ps′.
Taking into account this fact, then we have(163)M4≤CfLρ,θp,κℍnBu0,r1/ps′−κ/p×∑k=1∞Bu0,2k+1rκ/pBu0,2k+1r1/p−1/Q+1/q′−1+2/Q1+2k+1rρu0−N·1/N0+1+θ+N1=CfLρ,θp,κℍn∑k=1∞Bu0,r1/s′−κ/pBu0,2k+1r1/s′−κ/p1+2k+1rρu0−N·1/N0+1+θ+N1≤CfLρ,θp,κℍn∑k=1∞Bu0,r1/s′−κ/pBu0,2k+1r1/s′−κ/p,where a large enough N is chosen such that N>N0+1θ+N1. By the hypothesis, we know that the exponent in the last summation is negative, and hence, it is convergent. Therefore, we conclude that
(164)M4≤CfLρ,θp,κℍn≤CfLρ,θp,κℍn1+rρu0θ,as desired. This finishes Proof of Theorem 23.If one has a slightly stronger assumption on the potentialV, then we have the following improved estimates for the kernels Ku,v and K∗u,v.Lemma 34.
LetV∈Bq with q∈Q,∞, and let ρ· be the auxiliary function determined by V. For every positive integer N∈ℕ, there exists a positive constant CN>0 such that, for any u and v in ℍn,
(165)Ku,v≤CN1+v−1uρu−N1v−1uQ,K∗u,v≤CN1+∣v−1u∣ρu−N1v−1uQ.We remark that in the Euclidean case, this lemma was already obtained by Shen in [10]. Moreover, Shen [10] actually showed that RL and its dual form RL∗ are standard Calderón-Zygmund singular integral operators in ℝd, and hence, these two operators RL and RL∗ are all bounded on Lpℝd for 1<p<∞ and are of weak-type 1,1, when V∈Bq with d≤q<∞. We adapt the arguments used in [10, 42] (see also [19, 21]) to our present situation and prove Lemma 34 similarly. Furthermore, by adopting the same method given in [3, 10], we can also prove that the operators RL and RL∗ are bounded on Lpℍn for all 1<p<∞ and are bounded from L1ℍn into WL1ℍn in such a situation. Repeating the arguments above, we are able to show that under the same assumptions as in Theorems 18 and 19, the corresponding results also hold for the operators RL and RL∗ on ℍn.Theorem 35.
Let0<κ<1 and 0<θ<∞. If V∈Bq with q∈Q,∞, then the operators RL∗ and RL are bounded linear operators on Lρ,θp,κℍn for all 1<p<∞ and hence bounded on Lρ,∞p,κℍn.Theorem 36.
Let0<κ<1 and 0<θ<∞. If V∈Bq with q∈Q,∞, then the operators RL∗ and RL are bounded linear operators from Lρ,θ1,κℍn into WLρ,θ1,κℍn and hence bounded from Lρ,∞1,κℍn into WLρ,∞1,κℍn.We recall the relation1/p0=1/p−1/Q. Since p0 tends to ∞ as q→Q, so we have the following: p0′ tends to 1, and s′ tends to 1 with s=p0/p. Hence, the above theorems can be regarded as the limiting case of the results of Theorems 22, 23, and 24.


## 5. Generalized Morrey Spaces

In the last section, let us give the definitions of the generalized Morrey spaces related to the nonnegative potentialV on ℍn. Let Φ=Φr, r>0, be a growth function, that is, a positive increasing function on 0,∞, and satisfy the following doubling condition:
(166)Φ2r≤DΦr,for all r>0, where D=DΦ≥1 is a doubling constant independent of r.Definition 37.
Letρ· be the auxiliary function determined by V∈Bq with q∈Q/2,∞. Let 1≤p<∞ and Φ be a growth function. For any given 0<θ<∞, the generalized Morrey space Lρ,θp,Φℍn is defined to be the set of all p-locally integrable functions f on ℍn such that
(167)1Φ∣Bu0,r∣∫Bu0,rfupdu1/p≤C·1+rρu0θ,holds for every ball Bu0,r in ℍn, and we denote the smallest constant C satisfying (167) by ∥f∥Lρ,θp,Φℍn. It is easy to see that the functional ∥·∥Lρ,θp,Φℍn is a norm on the linear space Lρ,θp,Φℍn that makes it into a Banach space under this norm. Define
(168)Lρ,∞p,Φℍn≔⋃0<θ<∞Lρ,θp,Φℍn.Definition 38.
Letρ· be the auxiliary function determined by V∈Bq with q∈Q/2,∞. Let p=1 and Φ be a growth function. For any given 0<θ<∞, the generalized weak Morrey space WLρ,θ1,Φℍn is defined to be the set of all measurable functions f on ℍn such that
(169)1Φ∣Bu0,r∣supλ>0λ·u∈Bu0,r:fu>λ≤C·1+rρu0θ,holds for every ball Bu0,r in ℍn, and we denote the smallest constant C satisfying (169) by ∥f∥WLρ,θ1,Φℍn. Correspondingly, we define
(170)WLρ,∞1,Φℍn≔⋃0<θ<∞WLρ,θ1,Φℍn.Remark 39.
(i)
As in Section2 (Remark 17), we can also define a norm and a (quasi-)norm on the linear spaces Lρ,∞p,Φℍn and WLρ,∞1,Φℍn, respectively(ii)
According to this definition, we recover the spacesLρ,θp,κℍn and WLρ,θ1,κℍn under the choice Φt=tκ, for all p,κ∈1,∞×0,1(iii)
In the Euclidean setting, whenθ=0 or V≡0, the classes Lρ,θp,Φ and WLρ,θ1,Φ reduce to the classes Lp,Φ and WL1,Φ, which were introduced and studied by Mizuhara in [45]. We refer the reader to [46, 47] for further detailsBy using the same procedure as in the proofs of our main results, we have the following theorems. We leave the details to the reader. Letρ be as in (14).Theorem 40.
Let1<p<∞, 1≤DΦ<2, and 0<θ<∞. If V∈Bq with q∈Q,∞, then the operators RL∗ and RL are bounded on Lρ,θp,Φℍn and hence bounded on Lρ,∞p,Φℍn.Theorem 41.
Letp=1, 1≤DΦ<2, and 0<θ<∞. If V∈Bq with q∈Q,∞, then the operators RL∗ and RL are bounded from Lρ,θ1,Φℍn into WLρ,θ1,Φℍn and hence bounded from Lρ,∞1,Φℍn into WLρ,∞1,Φℍn.We point out that the same conclusions also hold forTL∗ and other maximal functions (TL∗, TL∗∗, and TL∗∗) discussed in Section 3.Theorem 42.
Let1≤DΦ<2 and 0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the operator RL∗ is bounded on Lρ,θp,Φℍn and hence bounded on Lρ,∞p,Φℍn provided that p0′<p<∞.Theorem 43.
Let0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the operator RL is bounded on Lρ,θp,Φℍn and hence bounded on Lρ,∞p,Φℍn provided that 1<p<p0 and 1≤DΦ<21/s′ with s≔p0/p.Theorem 44.
Let0<θ<∞. If V∈Bq with q∈Q/2,Q, and p0 is a number such that 1/p0=1/q−1/Q, then the operator RL is bounded from Lρ,θ1,Φℍn into WLρ,θ1,Φℍn and hence bounded from Lρ,∞1,Φℍn into WLρ,∞1,Φℍn provided that 1≤DΦ<21/p0′.



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*Source: 8839785-2020-11-21.xml*