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

Let L = −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 ∈ 1⁄2Q/2,∞Þ. Here, Q = 2n + 2 is the homogeneous dimension of Hn. Assume that fegt>0 is the heat semigroup generated by L . The semigroup maximal function related to the Schrödinger operator L is defined by T Lð f ÞðuÞ≔ supt>0je f ðuÞj. The Riesz transform associated with the operator L is defined by RL = ∇HnL, and the dual Riesz transform is defined by RL =L∇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 T L , RL , and R ∗ L acting on the Morrey spaces. In addition, it is shown that the Riesz transform RL = ∇HnL is of weak-type ð1, 1Þ. It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.


Introduction
1.1. The Heisenberg Group ℍ n . 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 in ℂ n+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][2][3]. We write ℕ = f1, 2, 3,⋯g for the set of natural numbers. The sets of real and complex numbers are denoted by ℝ and ℂ, respectively. Let ℍ n be a Heisenberg group of dimension 2n + 1, that is, a two-step nilpotent Lie group with underlying manifold ℂ n × ℝ. The group operation is given by where z = ðz 1 , z 2 , ⋯, z n Þ, w = ðw 1 , w 2 , ⋯, w n Þ ∈ ℂ n , and z• w ≔ ∑ n j=1 z j w j . Under this group operation, ℍ n becomes a nilpotent unimodular Lie group. It can easily be seen that the inverse element of u = ðz, tÞ ∈ ℍ n is u −1 = ð−z,−tÞ, and the identity element of this group is the origin 0 = ð0, 0Þ. The corresponding Lie algebra H n of left-invariant vector fields on ℍ n is spanned by All nontrivial commutation relations are given by Here, ½·, · is the usual Lie bracket. The sub-Laplacian Δ ℍ n and the gradient ∇ ℍ n are defined, respectively, by The Heisenberg group has a natural dilation structure which is consistent with the Lie group structure mentioned above. For each positive number a > 0, we define the dilation on ℍ n by δ a z, t ð Þ≔ az, Observe that δ a ða > 0Þ is an automorphism of the group ℍ n . For any given u = ðz, tÞ ∈ ℍ n , the homogeneous norm of u is given by the following form: Observe that |ðz, tÞ −1 | = |ðz, tÞ| and δ a z, t ð Þ j j= az j j 4 + a 2 t À Á 2 1/4 = a z, t ð Þ j j, a > 0: In addition, this norm j·j satisfies the triangle inequality and then leads to a left-invariant distance dðu, vÞ = ju −1 · vj for any u = ðz, tÞ, v = ðw, sÞ ∈ ℍ n . If r > 0 and u ∈ ℍ n , let Bðu, rÞ = fv ∈ ℍ n : dðu, vÞ < rg be the (open) ball with center u ∈ ℍ n and radius r ∈ ð0,∞Þ. Both left and right Haar measures on ℍ n coincide with the Lebesgue measure dzdt on ℂ n × ℝ. For any measurable set E ⊂ ℍ n , the Lebesgue measure of E is denoted by jEj. For ðu, rÞ ∈ ℍ n × ð0,∞Þ, it can be proved that the volume of Bðu, rÞ is where Q ≔ 2 n + 2 is the homogeneous dimension of ℍ n and |Bð0, 1Þ | is the volume of the unit ball in ℍ n . A simple calculation shows that Given a ball B = Bðu, rÞ ⊂ ℍ n and λ > 0, we adopt the notation λB to denote the ball with the same center u and radius λr. Clearly, by (8) For a radial function F on ℍ n , we have the following integration formula: ð where C n 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 Operator L. We recall some standard notation and definitions.
Definition 1. A nonnegative locally L q integrable function V on ℍ n is said to belong to the reverse Hölder class B q for some exponent 1 < q<∞, if there exists a positive constant C = Cðq ; VÞ such that the reverse Hölder inequality holds for every ball B in ℍ n .
In this article, we will always assume that 0≡V ∈ B q with q ∈ ½Q/2,∞Þ and Q = 2n + 2. We now consider the Schrödinger operator with the potential V ∈ B q on the Heisenberg group ℍ n (see [3]): 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 space ℝ n with nonnegative potentials that belong to the reverse Hölder class, see, for example, [6][7][8][9][10]. Concerning the weighted case, one can see [11][12][13][14][15][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][18][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 that V ∈ B q with q ∈ ½Q/2,∞Þ. For any given u ∈ ℍ n , the critical radius function ρðuÞ = ρðu ; VÞ is defined by 2 Journal of Function Spaces where Bðu, rÞ denotes the ball in ℍ n centered at u and with radius r.
It should be pointed out that the auxiliary function ρ ðu ; V Þ on the Euclidean space ℝ n 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 ρðuÞ determined by V ∈ B q satisfies for any given u ∈ ℍ n (see [3,22]). In particular, ρðuÞ = 1 with VðwÞ ≡ 1, and ρðuÞ ≈ 1/ð1 + jujÞ with VðwÞ = jwj 2 (Hermite operator).
It is easy to check that if Q/2 ≤ q 2 < q 1 , then B q 1 ⊂ B q 2 by the Hölder inequality. Furthermore, it can be shown that the B q class has a property of self-improvement. More precisely, if V ∈ B q , then V ∈ B q+ε for some ε > 0. By this fact, we know that the assumption q > Q/2 is equivalent to q ≥ Q/2.
When q ∈ ½Q/2, QÞ, we also write Let us give some elementary properties of the B q class. Assume that V ∈ B q with q ∈ ½Q/2,∞Þ. Lemma 3. The measure VðwÞdw satisfies the doubling condition; that is, there exists a constant C 0 > 0 such that for all balls Bðu 0 , rÞ in ℍ n .
for any u 0 ∈ ℍ n .
For more details, the reader may consult [3,22]. We also need the following technical lemma concerning the critical radius function (14).

Lemma 6.
Let ρðuÞ be the auxiliary function determined by V. For any u and v in ℍ n , there exist constants C 1 ≥ 1 and N 0 > 0 such that Here, and in what follows, v −1 · u is simply denoted by v −1 u. Lemma 6 has been proved by Lu [22] (see also [3], Lemma 4). In the setting of ℝ n , 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 holds for any v ∈ Bðu, rÞ with u ∈ ℍ n and r ∈ ð0,∞Þ, where C 1 is the same as in (20). Let us verify (21). An application of (20) yields which further implies that for each fixed k ∈ ℕ, Hence, which is the desired estimate. This estimate will often be used in the sequel.

Semigroup Maximal Functions and Riesz
Transforms. Let L = −Δ ℍ n + V be a Schrödinger operator on the Heisenberg group ℍ n , where Δ ℍ n is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q for q ∈ ½Q/2,∞Þ, and Q is the homogeneous dimension of ℍ n . Since V is nonnegative and belongs to L q loc ðℍ n Þ, L generates a ðC 0 Þ contraction semigroup fT L t g t>0 = fe −tL g t>0 . Let P t 3 Journal of Function Spaces ðu, vÞ denote the kernel of the semigroup fe −tL g t>0 .
We also denote by H t ðuÞ the convolution kernel of the heat semigroup fT t g t>0 = fe tΔ ℍ n g t>0 . Namely, For any u = ðz, sÞ ∈ ℍ n , it is well known that the heat kernel H t ðuÞ has the explicit expression: We consider the heat equation associated with the sub-Laplacian with the initial condition Fðu, 0Þ = f ðuÞ. In fact, the function H t ðuÞ stated the above exists as a solution to the heat equation. Moreover, by [23] (Theorem 2), we know that the heat kernel H t satisfies the Gaussian upper bound estimate: where the positive constants C and A are independent of t ∈ ð0,∞Þ and u ∈ ℍ n . By the Trotter product formula (see [24] for instance) and (29), one has 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 ℍ n , this estimate (30) can be significantly improved when V belongs to the reverse Hölder class B q for some q ∈ ½Q/2,∞Þ. The auxiliary function ρðuÞ arises naturally in the present situation.

Lemma 7.
Let ρðuÞ be the auxiliary function determined by V. For every positive integer N ∈ ℕ, there exists a positive con-stant C N > 0 such that, for any u and v in ℍ n , Remark 8. This estimate of P t ðu, vÞ is much better than (30), which was given by Lin and Liu in [3] (Lemma 7). In the setting of ℝ n , this result can be found in [9] (Proposition 2).
In this article, we investigate the semigroup maximal function related to the Schrödinger operator L, which is defined by (see [3]) We shall establish the strong-type and weak-type estimates of the operator T * L . Some other maximal functions will be discussed at the end of Section 3.
Let us also consider the Riesz transforms R j and the dual Riesz transforms R * j for the Schrödinger operator L, which are defined, respectively, by (see [3]) where the X j are left-invariant vector fields that generate the Lie algebra of ℍ n . Let Here, ∇ ℍ n is the gradient operator on ℍ n . We shall be interested in the behavior of the (vector-valued) operators R L and R * L associated with the Schrödinger operator L on ℍ n .
For any p ∈ ½1,∞Þ, the Lebesgue space L p ðℍ n Þ is defined to be the set of all measurable functions f on ℍ n such that The weak Lebesgue space WL 1 ðℍ n Þ consists of all measurable functions f on ℍ n such that Recently, Lin and Liu ( [3], Theorem 6 and Remark 3) established the strong-type and weak-type estimates of the operator T * L on the Lebesgue spaces.

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Theorem 9. Let 1 ≤ p < ∞. Then, the following statements are true: (1) If p > 1, then the operator T * L is bounded on L p ðℍ n Þ (2) If p = 1, then the operator T * L is bounded from L 1 ðℍ n Þ into WL 1 ðℍ n Þ Remark 10.
(i) It was also shown by Lin and Liu that this operator is bounded on BMO L ðℍ n Þ (ii) On the Euclidean space ℝ n , this maximal operator was studied by Dziubański et al. [9] (see also [11,25]) As for the (vector-valued) dual Riesz transform R * L defined above, we have the following estimate given in [21] (see also [19]).

Theorem 11.
Let V ∈ B q with q ∈ ½Q/2, QÞ, and let p 0 be a number such that 1 By duality, we could obtain the following result.
Moreover, it will be proved in Section 4 that R L is of weak-type ð1, 1Þ on the Heisenberg group. The case where q ∈ ½Q,∞Þ is also considered in Section 4.

Remark 13.
(i) It can be shown that the range of p in the above theorems is optimal (see [3]). In this paper, the authors also proved that the dual Riesz transform R * L is bounded on BMO L ðℍ n Þ and gave the Fefferman-Stein-type decomposition of BMO L functions with respect to R * j , j = 1, 2, ⋯, 2n (ii) It was shown in [26] that when V ≡ 0, the operators R ≔ ∇ ℍ n ð−Δ ℍ n Þ −1/2 and R * = ð−Δ ℍ n Þ −1/2 ∇ ℍ n are uniformly bounded on L p ðℍ n Þ with respect to n. More specifically, for every 1 < p < ∞, there exists a constant C p > 0 such that for every n ∈ ℕ (iii) Recall that in the setting of ℝ n , 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 details The paper is organized as follows. In Section 2, we introduce the Morrey space and weak Morrey space associated with the Schrödinger operator L on ℍ n 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 strongtype and weak-type estimates for the semigroup maximal function T * L 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 R L and its dual form R * L . 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 ∈ ½1,∞Þ, the notation p′ denotes its conjugate number, namely, 1/p + 1/p ′ = 1 and 1 ′ = ∞.

Definitions and Main Theorems
A few historic remarks are in order. The classical Morrey space M p,λ ðℝ n Þ 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 M p,λ ðℝ n Þ the Morrey space, which consists of all p-locally integrable functions f on ℝ n such that where 1 ≤ p < ∞ and 0 ≤ λ ≤ n. It is known that M p,λ ðℝ n Þ is an extension of L p ðℝ n Þ in the sense that M p,0 ðℝ n Þ = L p ðℝ n Þ.
Note that M p,n ðℝ n Þ = L ∞ ðℝ n Þ by the Lebesgue differentiation theorem. If λ < 0 or λ > n, then M p,λ ðℝ n Þ = Θ, where Θ is the set of all functions equivalent to 0 on ℝ n . We also denote by WM 1,λ ðℝ n Þ the weak Morrey space, which consists of all measurable functions f on ℝ n such that

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For the properties of classical Morrey spaces, we refer the readers to [30][31][32][33][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][36][37][38]).
In this section, we introduce some types of Morrey spaces associated with the Schrödinger operator L on ℍ n (see [39]) and then give our main results.
Definition 14. Let ρ be the auxiliary function determined by V ∈ B q with q ∈ ½Q/2,∞Þ. Let 1 ≤ p < ∞ and 0 ≤ κ < 1. For each given 0 < θ < ∞, the Morrey space L p,κ ρ,θ ðℍ n Þ is defined to be the set of all p-locally integrable functions f on ℍ n such that holds for every ball Bðu 0 , rÞ in ℍ n , where u 0 and r denote the center and radius of Bðu 0 , rÞ, respectively. The smallest constant appearing in (40) is called the norm of f , which is denoted by kf k L p,κ ρ,θ ðℍ n Þ . It is a Banach space with respect to the norm k·k L p,κ ρ,θ ðℍ n Þ . Define Definition 15. Let ρ be the auxiliary function determined by V ∈ B q with q ∈ ½Q/2,∞Þ. Let p = 1 and 0 ≤ κ < 1. For each given 0 < θ < ∞, the weak Morrey space WL 1,κ ρ,θ ðℍ n Þ is defined to be the set of all measurable functions f on ℍ n such that holds for every ball Bðu 0 , rÞ in ℍ n . The smallest constant appearing in (42) Remark 16.
(iii) It follows directly from Chebyshev's inequality that Moreover, the inclusion is strict.

Remark 17.
(i) We can define a norm on the space L p,κ ρ,∞ ðℍ n Þ (see [39]), which makes it into a Banach space. In view of (44), for any given f ∈ L p,κ ρ,∞ ðℍ n Þ, let Now define the functional k·k å by It is easy to check that the functional k·k ⋆ defined by (49) is indeed a norm on L p,κ ρ,∞ ðℍ n Þ provided ðp, κÞ ∈ ½1,∞Þ × ð0, 1Þ; i.e., it satisfies the following conditions: (45), for any given f ∈ WL 1,κ ρ,∞ ðℍ n Þ, let Journal of Function Spaces Similarly, we define the functional k·k ⋆⋆ by It is easily checked that the functional k·k ⋆⋆ defined by (51) is a (quasi-)norm on WL 1,κ ρ,∞ ðℍ n Þ for all 0 < κ < 1; i.e., it satisfies the following conditions: ρ,θ ðℍ n Þ) could be viewed as an extension of the Lebesgue space (or the weak Lebesgue space) on ℍ n (when κ = θ = 0, or κ = 0, V ≡ 0), it is natural to study the boundedness properties of the operators T * L , R L , and R * L in the context of Morrey spaces. In this paper, we will extend Theorems 9, 11, and 12 to the Morrey spaces on ℍ n . Let ρ be the same as before. Now let us formulate our main results as follows. Theorem 19. Let p = 1, 0 < κ < 1, and 0 < θ < ∞. If V ∈ B q with q ∈ ½Q/2,∞Þ, then the semigroup maximal function T * L is a bounded sublinear operator from L 1,κ ρ,θ ðℍ n Þ into WL 1,κ ρ,θ ðℍ n Þ.
As an immediate consequence of Theorems 18 and 19 and Remark 17, we have the following results.
It is worth pointing out that we cannot use Theorem 22 to prove Theorem 23 in a direct way by duality, since the predual to L p,κ ρ,θ ðℍ n Þ 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 L p,κ ρ,θ ðℍ n Þ 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 R L is of weaktype ð1, 1Þ. Based on this result, we can further prove the following.
As a straightforward consequence of Theorems 22-24 and Remark 17, we obtain the following estimates.

Corollary 27.
If V ∈ B q with q ∈ ½Q/2, QÞ, and p 0 is a number such that 1/p 0 = 1/q − 1/Q, then the Riesz transform R L is a bounded linear operator from

Boundedness of the Semigroup Maximal Functions
In this section, we will prove the conclusions of Theorems 18 and 19. Let P t ðu, vÞ denote the integral kernel of e −tL related to the Schrödinger operator L (see [3]). Then, we can write T * L as follows: Proof of Theorem 18. For any given f ∈ L p,κ ρ,θ ðℍ n Þ with ðp, κÞ ∈ ð1,∞Þ × ð0, 1Þ and θ ∈ ð0,∞Þ, by definition, we only need to show that for any given ball B = Bðu 0 , rÞ of ℍ n , the 7 Journal of Function Spaces following inequality holds true. By a standard argument, we decompose the function f as where 2B denotes the open ball centered at u 0 of radius 2r, χ E denotes the characteristic function of the set E, and ð2BÞ ∁ = ℍ n \ ð2BÞ denotes its complement. Then, by the sublinearity of T * L , we write Let us consider the first term I 1 . Making use of the first part of Theorem 9, we have Moreover, note that for any fixed θ ∈ ð0,∞Þ, This, combined with (10), yields We now turn to estimate the second term I 2 . We first assert that the following inequality holds for any u ∈ Bðu 0 , rÞ, where N and A are given as in Lemma 7. Indeed, this can be done by considering the following two cases: When t > jv −1 uj 2 , then ffiffi ffi t p > |v −1 u | , and hence,

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On the other hand, we can easily see that When 0 < t ≤ jv −1 uj 2 , then ffiffi ffi t p ≤ jv −1 uj. In this case, it is easy to check that for any N ∈ ℕ, From this, it follows immediately that Putting all together produces the required inequality (59). Notice that for any u ∈ Bðu 0 , rÞ and v ∈ ð2BÞ ∁ , one has Thus, That is, jv −1 uj ≈ jv −1 u 0 j. Combining this fact with (59) yields that for any u ∈ Bðu 0 , rÞ and any positive integer N ∈ ℕ, Furthermore, in view of (21) and (57), we can see that the above expression (67) does not exceed By using the Hölder inequality, we obtain that for each fixed k ∈ ℕ, Substituting the above inequality into formula (68), we conclude that

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By choosing some sufficiently large number N such that N > ðN 0 + 1Þθ, then we have where the last inequality follows from the fact that 1 − κ > 0 and θ > 0. Combining the above estimates for I 1 and I 2 , we obtain the desired inequality (53). This ends Proof of Theorem 18.
Proof of Theorem 19. Let f ∈ L 1,κ ρ,θ ðℍ n Þ with κ ∈ ð0, 1Þ and θ ∈ ð0,∞Þ. Fix ðu 0 , rÞ ∈ ℍ n × ð0,∞Þ. Our aim is to prove, by definition, that for each given ball B = Bðu 0 , rÞ of ℍ n , the following estimate holds true. To this end, we decompose the function f as Then, for any fixed λ > 0, we can write We first give the estimate for the term J 1 . By the second part of Theorem 9, we get Therefore, in view of (57) and (10), we have To estimate the second term J 2 , by using the pointwise inequality (68) and Chebyshev's inequality, we can deduce that Moreover, for each fixed k ∈ ℕ, we compute Consequently, substituting this inequality into formula (77), By selecting some large enough N such that N > ðN 0 + 1Þθ, we thus have where the last step is again due to the fact that 0 < κ < 1 and θ > 0. Summing up the above estimates for J 1 and J 2 , 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 semigroup fe −t ffiffiffi ffi L p g t>0 , which is defined by

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: It is interesting to investigate the boundedness of the operator T * * L 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 T * * L is bounded on L p ðℍ n Þ for all 1 < p < ∞ and bounded from L 1 ðℍ n Þ into WL 1 ðℍ n Þ. Based on this result, we can further prove that the corresponding estimates for the operator T * * L 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 semigroup fe −t ffiffiffi ffi L p g t>0 , which is defined by For the same reason as above, it can be shown that this new maximal operator T * * ffiffiffi ffi L p is dominated by T * * L in some sense. Therefore, all the results mentioned above hold as well for the maximal operator in this more general situation.

Boundedness of the Riesz Transforms
This section is concerned with Proofs of Theorems 22, 23, and 24. Recall that the operators R L and R * L have singular kernels with values in ℝ 2n that will be denoted by Kðu, vÞ and K * ðu, vÞ, respectively (see [3]). Obviously, The next lemma plays a crucial role in our Proofs of Theorems 22-24.

Lemma 31.
Let V ∈ B q with q ∈ ½Q/2, QÞ, and let ρð·Þ be the auxiliary function determined by V. For every positive integer N ∈ ℕ, there exists a positive constant C N > 0 such that, for any u and v in ℍ n , where K * ðu, vÞ denotes the (vector-valued) kernel of the operator R * L . Moreover, the above inequality also holds with ρðuÞ replaced by ρðvÞ.
Lemma 31 was proved by Li [21] in a more general setting (connected nilpotent Lie group) (see also [19] for ℍ n ). For 11 Journal of Function Spaces such kernels, we also give the following result, which establishes the Lipschitz regularity of K * ðu, vÞ. Lemma 32. Let V ∈ B q with q ∈ ½Q/2, QÞ, and let ρð·Þ be the auxiliary function determined by V. For every positive integer N ∈ ℕ, there exists a positive constant C N > 0 such that, for any u and v in ℍ n , and for some fixed δ ′ ∈ ð0, δÞ, δ is given as in (16), whenever jhj ≤ jv −1 uj/4/. Moreover, the above inequality also holds with ρðuÞ replaced by ρðvÞ.
The above kernel estimate in Lemma 32 was obtained by Pengtao and Lizhong in [19], which will be used to prove that the operator R L is of weak-type ð1, 1Þ. 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. Let f ∈ L p,κ ρ,θ ðℍ n Þ with ðp, κÞ ∈ ðp ′ 0 ,∞Þ × ð0, 1Þ and θ ∈ ð0,∞Þ, where 1/p 0 = 1/q − 1/Q. Fix ðu 0 , rÞ ∈ ℍ n × ð0,∞Þ. By definition, we only need to show that for any given ball B = Bðu 0 , rÞ of ℍ n , the following inequality holds true. Using the standard technique, we decompose the function f as Then, by using the linearity of R * L , we write Let us consider the first term K 1 . Making use of (10), (57), and Theorem 11, we have Now let us turn to estimate the second term K 2 . By using Lemma 31, we obtain that for any u ∈ Bðu 0 , rÞ, where Hence, K 2 can be written as follows: Arguing as in Proof of Theorem 18, we can also obtain We only have to deal with the term K 4 . As mentioned in the previous proof, one has Journal of Function Spaces whenever u ∈ Bðu 0 , rÞ and v ∈ ð2BÞ ∁ . Thus, for any positive integer N ∈ ℕ, It is not difficult to check that when w ∈ Bðv, jv −1 uj/4Þ and v ∈ Bðu 0 , 2 k+1 rÞ, one has w ∈ Bðu 0 , 2 k+2 rÞ, which implies where I 1 stands for the fractional integral operator of order one defined by For this operator, a classical result of Folland and Stein [43] states that I 1 is bounded from L q ðℍ n Þ into L p 0 ðℍ n Þ for 1 < q < Q and 1/p 0 = 1/q − 1/Q (see also [39,44]). Namely, there exists a constant C > 0 such that for any g ∈ L q ðℍ n Þ, Since ðp 0 Þ′ < p < ∞, we can choose a number s > 0 such that 1/p + 1/p 0 + 1/s = 1. A combination of the Hölder inequality and (103) gives where the last inequality holds by our assumption V ∈ B q . We now claim that the following inequality holds. For any N 1 > log 2 C 0 (C 0 is the doubling constant in Lemma 3), there exists a constant C > 0 such that for any u 0 ∈ ℍ n and τ ∈ ð0,∞Þ, Taking this claim momentarily for granted, then we have 13 Journal of Function Spaces where in the last step we have invoked (57) and (10). In addition, it follows immediately from (21) and (57) that A trivial computation shows that Therefore, in view of (108) and (107), we conclude that Consequently, By choosing some sufficiently large number N such that N > ðN 0 + 1ÞðN 1 + θÞ, then we have where the last inequality follows again from the fact that 1 − κ > 0 and θ > 0. Combining the above estimates for K 1 , K 3 , and K 4 produces the desired inequality (91).
Finally, let us verify (105). Suppose that V ∈ B q with Q/ 2 ≤ q < Q. Using the same method as in Proof of Lemma 1 in [42], for any u 0 ∈ ℍ n and τ ∈ ð0,∞Þ, there must exist an integer j such that 2 j τ ≤ ρðu 0 Þ < 2 j+1 τ. Two cases are considered below.
Case 1. j < 0. In this case, one has ρðu 0 Þ < τ. This fact together with Lemmas 3 and 4 yields Since N 1 > log 2 C 0 , it is easy to see that Case 2. j ≥ 0. In this case, one has τ ≤ ρðu 0 Þ. This fact, along with Lemmas 5 and 4, implies that Thus, in both cases, (105) holds. This completes Proof of Theorem 22.
In order to prove Theorem 24, let us first set up the following result, which is based on a version of the Calderón-Zygmund decomposition on ℍ n and Lemma 32.
Proof. For any given f ∈ L 1 ðℍ n Þ 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 = ∑ i b i such that (1) |gðuÞ | ≤C · σ, for almost every u ∈ ℍ n and Journal of Function Spaces (2) each b i is supported in the ball B i = Bðu i , r i Þ, and we denote the center and the radius of B i by u i and r i , respectively ð (3) the sets B i are finitely overlapping and From this construction, we have that for any fixed σ > 0, The part of the argument involving the function g proceeds as follows. By using the L 2 ðℍ n Þ boundedness of R L (see Theorem 12 with 1 < 2 < p 0 ), we obtain Setting E = S i 4B i = S i Bðu i , 4r i Þ, we split II into two parts as follows: It is obvious that Therefore, in order to complete our proof, we need only to show that An application of Chebyshev's inequality yields We observe that whenever v ∈ B i and u ∈ ð4B i Þ ∁ . Then, we apply Lemma 32, (88), and the cancelation condition of b i to get Obviously, the first term on the right-hand side of (125) is bounded by Using this estimate together with (11), we can deduce that On the other hand, the latter term on the right-hand side of (125) is controlled by

Journal of Function Spaces
When w ∈ Bðu,|u −1 i u|/4Þ, by the triangle inequality, one has From this, it follows that the above expression is bounded by Consequently, where I 1 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, where the last inequality is obtained by the hypothesis V ∈ B q . Moreover, in view of (10), (105), and (57), we can see that the above expression (132) is bounded by where the last step is due to the fact that 1/Q + 1/p 0 − 1/q = 0. Therefore, by selecting some large enough N such that N > N 1 , we thus have Collecting all these estimates and then taking the supremum over all σ > 0, we conclude the proof of Theorem 33.
Proof of Theorem 24. Let p 0 be a positive number such that 1/p 0 = 1/q − 1/Q. To prove Theorem 24, it is enough to prove that for each given ball B = Bðu 0 , rÞ of ℍ n , the following estimate holds true for any given f ∈ L 1,κ ρ,θ ðℍ n Þ with some θ ∈ ð0,∞Þ and κ ∈ ð0, 1/p ′ 0 Þ. Using the standard technique, we decompose the function f as Journal of Function Spaces Then, for any fixed σ > 0, we can write Let us estimate the first term L 1 . By Theorem 33, (57), and (10), we get As for the second term L 2 , from (88) and Lemma 31, it follows that for any u ∈ Bðu 0 , rÞ, where Thus, by (139) and Chebyshev's inequality, L 2 can be written as follows: We now proceed exactly as we did in Proof of Theorem 19 and have the following estimate as well: Let us analyze the latter term L 4 . In order to do this, we first observe that whenever u ∈ Bðu 0 , rÞ and v ∈ ð2BÞ ∁ . Hence, for any positive integer N ∈ ℕ, It is easy to verify that when w ∈ Bðu, jv −1 uj/4Þ and v ∈ Bðu 0 , 2 k+1 rÞ, one has w ∈ Bðu 0 , 2 k+2 rÞ. This fact together with (107) implies that for any u ∈ Bðu 0 , rÞ,

Journal of Function Spaces
Consequently, Applying the Hölder inequality along with (103), we can compute the above integral as follows: where the last inequality holds since V ∈ B q . Moreover, by (105), (57), and (10), we can see that the above expression is bounded by Taking into account the fact that 1/Q + 1/q′ = 1/p′ 0 , then we have where a large enough N is chosen satisfying N > ðN 0 + 1Þ ðθ + N 1 Þ. By the choice of κ, it guarantees that the exponent −ð1/p ′ 0 − κÞ of the last summation is negative, and hence, it is convergent. Therefore, we conclude that Combining these estimates for L 1 , L 3 , and L 4 , 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 Theorem 22, we shall only indicate the necessary modifications. As before, it is enough for us to show that for an arbitrary fixed ball Bðu 0 , rÞ in ℍ n , the following estimate holds true for any given f ∈ L p,κ ρ,θ ðℍ n Þ with some θ ∈ ð0,∞Þ, p ∈ ð1, p 0 Þ, and κ ∈ ð0, 1/s ′ Þ, where s = p 0 /p and 1/p 0 = 1/p − 1/Q. To this end, we split f = f 1 + f 2 through f 1 = f · χ 2B and f 2 = f · χ ð2BÞ ∁ . Then, the left-hand side of (151) will be divided into two parts given below.
Since the Riesz transform R L is bounded on L p ðℍ n Þ for 1 < p < p 0 (see Theorem 12), we can deal with M 1 in the same manner as in Proof of Theorem 22 and obtain On the other hand, from (139), it follows that Lemma 34. Let V ∈ B q with q ∈ ½Q,∞Þ, and let ρð·Þ be the auxiliary function determined by V. For every positive integer N ∈ ℕ, there exists a positive constant C N > 0 such that, for any u and v in ℍ n , We remark that in the Euclidean case, this lemma was already obtained by Shen in [10]. Moreover, Shen [10] actually showed that R L and its dual form R * L are standard Calderón-Zygmund singular integral operators in ℝ d , and hence, these two operators R L and R * L are all bounded on L p ðℝ d Þ for 1 < p < ∞ and are of weak-type ð1, 1Þ, when V ∈ B q 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 R L and R * L are bounded on L p ðℍ n Þ for all 1 < p < ∞ and are bounded from L 1 ðℍ n Þ into WL 1 ðℍ n Þ 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 R L and R * L on ℍ n .
We recall the relation 1/p 0 = 1/p − 1/Q. Since p 0 tends to ∞ as q → Q, so we have the following: ðp 0 Þ′ tends to 1, and s ′ tends to 1 with s = p 0 /p. Hence, the above theorems can be regarded as the limiting case of the results of Theorems 22, 23, and 24.

Generalized Morrey Spaces
In the last section, let us give the definitions of the generalized Morrey spaces related to the nonnegative potential V on ℍ n . Let Φ = ΦðrÞ, r > 0, be a growth function, that is, a positive increasing function on ð0, ∞Þ, and satisfy the following doubling condition: for all r > 0, where D = DðΦÞ ≥ 1 is a doubling constant independent of r.
Definition 37. Let ρð·Þ be the auxiliary function determined by V ∈ B q with q ∈ ½Q/2,∞Þ. Let 1 ≤ p < ∞ and Φ be a growth function. For any given 0 < θ < ∞, the generalized Morrey space L p,Φ ρ,θ ðℍ n Þ is defined to be the set of all p -locally integrable functions f on ℍ n such that holds for every ball Bðu 0 , rÞ in ℍ n , and we denote the smallest constant C satisfying (167) by ∥f ∥ L p,Φ ρ,θ ðℍ n Þ . It is easy to see that the functional ∥·∥ L p,Φ ρ,θ ðℍ n Þ is a norm on the linear space L p,Φ ρ,θ ðℍ n Þ that makes it into a Banach space under this norm. Define Journal of Function Spaces Definition 38. Let ρð·Þ be the auxiliary function determined by V ∈ B q with q ∈ ½Q/2,∞Þ. Let p = 1 and Φ be a growth function. For any given 0 < θ < ∞, the generalized weak Morrey space WL 1,Φ ρ,θ ðℍ n Þ is defined to be the set of all measurable functions f on ℍ n such that holds for every ball Bðu 0 , rÞ in ℍ n , and we denote the smallest constant C satisfying (169) by ∥f ∥ WL 1,Φ ρ,θ ðℍ n Þ . Correspondingly, we define Remark 39.
We point out that the same conclusions also hold for T * L and other maximal functions (T * ffiffiffi ffi L p , T * * L , and T * * ffiffiffi ffi L p ) discussed in Section 3.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.