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

In this paper, assume that L = −ΔHn +V is a Schrödinger operator on the Heisenberg groupHn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the timefractional derivatives of semigroups fegt>0 and fe−t ffiffi L p gt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space H1,α L ðHnÞ associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.


Introduction
It is well-known that the Hardy spaces H p form a natural continuation of the Lebesgue spaces L p to the range 0 < p ≤ 1. Correspondingly, let I α and J α denote the classical Riesz potentials and Bessel potentials, respectively. The Hardy-Sobolev spaces I α ðH p Þ and J α ðH p Þ can be seen as natural generalizations of homogeneous and inhomogeneous Sobolev spaces. Compared with Hardy spaces, the elements of Hardy-Sobolev spaces are of regularities and have been widely used in the research of partial differential equations, potential theories, complex analysis and harmonic analysis, etc. In the last decades, the theory of Hardy-Sobolev spaces was investigated by many researchers extensively. In [1], Strichartz proved that I n/p ðH p Þ was an algebra and found equivalent norms for the Hardy-Sobolev space or, more generally, for the corresponding space with fractional smoothness and Lebesgue exponents in the range p > n/ðn + 1Þ. The trace properties of the space I α ðH p Þ were discussed by Torchinsky [2]. Miyachi [3] characterized the Hardy-Sobolev spaces in terms of maximal functions related to the mean oscillation of functions in cubes and obtained a counterpart of previous results of Calderón and of the general theory of De Vore and Sharpley [4]. For further information on Hardy-Sobolev spaces and their variants on ℝ d , or on subdomains, we refer the reader to [5][6][7][8][9][10][11][12].
The development of the theory of Hardy spaces with several real variables was initiated by Stein and Weiss. In [13], by use of square functions, Fefferman and Stein characterized the Hardy spaces H p ðℝ n Þ for 0 < p ≤ 1. From then on, such characterizations were extended to other settings, see [14][15][16] and the references therein. Since the 1990s, the theory of Hardy spaces associated with second-ordered differential operators on ℝ n attracts the attention of many researchers and has been investigated extensively, such as [15][16][17][18][19][20][21][22] and the references therein. In recent years, a lot of research has been done on the Hardy spaces associated with operators on the Heisenberg group and other settings, see [23][24][25].
Let L = −Δ ℍ n + V be a Schrödinger operator, where Δ H n is the sub-Laplacian on ℍ n and V belongs to the reverse Hölder class. Let fe −tL g t>0 be the heat semigroup generated by −L and denote by K L t ð·, · Þ the integral kernels. Since V is nonnegative, the Feynman-Kacformula asserts that Lin-Liu-Liu [25] introduced the Hardy space associated with L, which is defined as follows. Let M L denote the semigroup maximal function: M L ð f ÞðgÞ ≔ sup t>0 jT L t f ðgÞj, g ∈ ℍ n . The Hardy space H 1 L ðℍ n Þ associated with L is defined to be where k f k H 1 L = kM L ð f Þk L 1 . As an analogue of classical Hardy-Sobolev spaces, we introduce the following Hardy-Sobolev space associated with L on ℍ n : Definition 1. For α > 0, the Hardy-Sobolev space H 1,α L ðℍ n Þ is defined as the set of all functions f ∈ H 1 L ðℍ n Þ such that L α f ∈ H 1 L ðℍ n Þ with the norm Our motivation is inspired by the following square function characterization of H 1 L ðℍ n Þ. For k ∈ ℕ, let Define the square function associated with fQ k t g as In [16], Hoffmann et al. obtained the following square function characterization of H 1 L ðℍ n Þ: Proposition 2. Let k ∈ ℕ. A function f ∈ H 1 L ðℍ n Þ if and only if f ∈ L 1 ðℍ n Þ and the square function S k L ðf Þ ∈ L 1 ðℍ n Þ. Moreover, k f k H 1 L~k S k L ð f Þk L 1 + k f k L 1 . The goal of this paper is to characterize H 1,α L ðℍ n Þ by the square functions generated by semigroups associated with L. It can be seen from Definition 1 that the elements of H 1,α L ðℍ n Þ have the regularities of order α. Based on this observation, we introduced the following fractional square functions associated with semigroup generated by L. For α > 0, let ∂ α t K L t and ∂ α t P L t denote the time-fractional derivatives of the heat kernel and the Poisson kernel, respectively, (cf [26]), i.e., For α > 0, define the following family of operators: Similar to ( [27], Proposition 3.6), the regularities of the kernels of fQ L α,t g and fD L α,t g can be deduced from (6). In this paper, we apply a different method to derive the regularities. In Propositions 10 and 14, we estimate the regularities of ft α L α e −tL g and ft α L α/2 e −t ffiffi L p g, respectively. Then, by the functional calculus, we deduce the following relations: see Lemmas 15 and 11. Hence, the desired regularities of fQ L α,t g and fD L α,t g are corollaries of Propositions 10 and 14. Respect to Q L α,t , we introduce the following fractional square functions: In Section 3.1, we establish the characterizations of H 1 L ðℍ n Þ be the square function defined by (9), see Theorem 20. In Section 3.2, we introduce the fractional square functions as follows: Let For every f ∈ DðL α Þ and L α f ∈ L 2 ðℍ n Þ ∩ H 1 L ðℍ n Þ, we prove Journal of Function Spaces The above relations, together with Theorem 20, indicate that see Proposition 23. Finally, in Theorem 24, we obtain the desired characterizations of H 1,α L ðℍ n Þ via the fractional square functions defined in (10): for every f ∈ H 1,α L ðℍ n Þ, For the Poisson semigroup, via the operators fD L α,t g, we can also obtain the corresponding square function characterizations of H 1 L ðℍ n Þ and H 1,α L ðℍ n Þ, see Theorems 21 and 25 for the details.
(i) As far as the authors know, even on ℝ n , the regularities of the time-fractional derivatives of the heat kernels obtained in Section 2. The paper is organized as follows. In Section 2.1, we give some knowledge to be used throughout this paper. Sections 2.2 and 2.3 are devoted to the regularity estimates of fQ L α,t g and fD L α,t g, respectively. In Sections 3.1 and 3.2, we establish the fractional square function characterizations of H 1 L ðℍ n Þ and H 1,α L ðℍ n Þ. As an application, we deduce an equivalence of the norms of Hardy-Sobolev spaces associated with L.
1.1. Notations. Throughout this article, we will use c and C to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By B 1 B 2 , we mean that there exists a constant C > 1 such that 1/C ≤ B 1 /B 2 ≤ C.

Preliminaries
2.1. Heisenberg Groups and Hardy Spaces. The ð2n + 1Þ -dimensional Heisenberg group ℍ n is the Lie group with underlying manifold ℝ 2n × ℝ with the multiplication The Lie algebra of left-invariant vector fields on ℍ n is given by The sub-Laplacian Δ ℍ n is defined as Δ ℍ n = ∑ 2n j=1 X 2 j . The gradient ∇ ℍ n is defined by ∇ ℍ n = ðX 1 , ⋯, X 2n Þ. The leftinvariant distance is dðh, gÞ = jh −1 gj. The ball of radius r centered at g is denoted by Bðg, rÞ = fh ∈ ℍ n : jh −1 gj < rg whose volume is given by jBðg, rÞj = c n r Q , where c n denotes the volume of the unit ball in ℍ n and Q = 2n + 2 is the homogenous dimension of ℍ n . Let U n be the Siegel upper half-space in ℂ n+1 , i.e., Then, U n is holomorphically equivalent to the unit ball in ℂ n+1 . It is well known that the Heisenberg group ℍ n is a nilpotent subgroup of the automorphism group of U n , which consists of the translations of U n . The Heisenberg group ℍ n can be also identified with the boundary ∂U n via its action on the origin. We use the Heisenberg coordinates ðg, sÞ = ðx, t, sÞ to denote the points in U n , where A nonnegative locally L q -integrable function V on ℍ n is said to belong to the reverse Hölder class B q , 1 < q < ∞, if there exists C > 0 such that the reverse Hölder inequality holds for every ball B ∈ ℍ n : In the sequel, we always assume that 0≡V ∈ B Q/2 . The following auxiliary function ρðg, VÞ = ρðgÞ was first introduced by Shen [28] and widely used in the research of function spaces related to Schrödinger operators: The following atomic characterization of H 1 L ðℍ n Þ was obtained by Lin-Liu-Liu [25]. (ii) kak L ∞ ≤ jBðg 0 , rÞj 1/q−1 ; (iii) if r < ρðgÞ, then Ð Bðg,rÞ aðhÞdh = 0 The atomic norm of H 1 L ðℍ n Þ is defined by k f k L−atom,q ≔ inf f∑ | c j | g, where the infimum is taken over all decompositions f = Σc j a j , and a j are H q L -atoms.
Below, we give some results on the tent spaces introduced by Coifman-Meyer-Stein.
The following proposition is one of the main results of tent spaces. Proposition 8. Every element f ∈ T 1 2 ðU n Þ can be written as f = ∑ j λ j a j , where a j are T 1 2 -atoms, λ j ∈ ℂ, and ∑ j | a j | ≤C k f k T 1 2 . 2.2. Time-Fractional Derivatives of the Heat Semigroup. In this part, we estimate the time-fractional derivatives of the heat kernel associated with L. For k ∈ ℕ, define In ( [29], Proposition 2.9), the authors obtained the following estimates about the kernel Q L k,t ð·, · Þ.

Proposition 9.
(i) For M > 0, there exists a constant C M > 0 such that (ii) Assume that 0 < δ′ ≤ min f1, δg. For any M > 0, there exists a constant C M > 0 such that, for all |ω | < ffiffi t p (iii) For any M > 0, there exists a constant C M > 0 such that Denote byQ L α,t ð·, · Þ the kernel of t α L α e −tL . In the following proposition, we investigate the regularities ofQ Proof.
(i) The proof of (i) is divided into the following two cases.
Case 1. α ∈ ð0, 1Þ. For this case, it follows from functional calculus that By (i) of Proposition 9, we obtaiñ On the one hand, a direct computation gives On the other hand, because the heat kernel decays rapidly, we can get Since m = ½α + 1, we can get It can be deduced from (i) of Proposition 9 that Similarly, an application of (i) of Proposition 9 again yields (ii) We first consider the case α ∈ ð0, 1Þ. By (ii) of Proposition 9, we obtain 5 Journal of Function Spaces Changing the order of integration, we obtaiñ Alternatively, we can also get For α ≥ 1, by (ii) of Proposition 9, we can get Similar to the case α ∈ ð0, 1Þ, the rest of the proof can be finished by applying change of order of integration. We omit the details.
(iii) For α ∈ ð0, 1Þ, by (iii) of Proposition 9, we change the order of integration to obtain If For α ≥ 1, we have If The following lemma can be deduced from the functional calculus immediately.
2.3. Time-Fractional Derivatives of the Poisson Semigroup. In this part, our aim is to give some regularity estimates of the Poisson kernel associated with ffiffiffi hÞ. In ( [29], Proposition 2.12), the authors obtained the following estimates about the kernel D L k,t ð·, · Þ.
Denote byD L α,t ð·, · Þ the kernel t α L α/2 P L t ð·, · Þ. Similar to Proposition 10, we have Proof. Let us prove (i) first. The following two cases are considered. Case 1. α ∈ ð0, 1Þ. By the functional calculus, we obtain which, together with Proposition 13, implies that One the one hand, we use the change of order of integration to get One the other hand, for α ∈ ð0, 1Þ, Setting β = α − ½α, we obtain Since m = ½α + 1, It follows from Proposition 13 that Journal of Function Spaces Also, noticing that α < m, we obtaiñ (ii) We first consider the case α ∈ ð0, 1Þ. Sincẽ we apply (ii) of Proposition 13 to obtaiñ One the one hand, we havẽ On the other hand, since α < 1, it holdsD For α ≥ 1, noticing we can use (ii) of Proposition 13 to get The rest of the proof can be completed by the procedure of the case α > 1 in (i), so we omit the details.

Fractional Square Functions Characterizations of
In this section, we will characterize the Hardy space H 1 L ðℍ n Þ by the fractional square functions defined by (9) and (84). Now, we first prove the following reproducing formulas.
(i) The operator Q L α,t defines an isometry from L 2 ðℍ n Þ into L 2 ðU n , dgdt/tÞ. Moreover, in the sense of L 2 ðℍ n Þ, (ii) The operator D L α,t defines an isometry from L 2 ðℍ n Þ into L 2 ðU n , dgdt/tÞ. Moreover, in the sense of L 2 ðℍ n Þ, Proof. The proofs of (i) and (ii) are standard and can be deduced from the spectral techniques. For completeness, we give the proof of (i) and omit the details of the proof of (ii). Since e −t 2 L = Ð ∞ 0 e −t 2 λ dEðλÞ, we have Thus, for all f ∈ L 2 ðℍ n Þ, we get For the second part, it suffices to show that, for every pair of sequences Indeed, if (89) holds, we can find h ∈ L 2 ðℍ n Þ such that lim k→∞ Ð n k ε k ðQ L α,t f Þ 2 ðdt/tÞ = h. Therefore, it follows from a polarized version of the first part that for g ∈ L 2 ðℍ n Þ, which implies h = C α f . To prove (89), we use again the functional calculus to deduce that Computing the integral inside one yields k dE f ,f ðλÞasn k → ∞, which by dominated convergence tends to 0. Observe that the last step makes use of the fact that 0 is not an eigenvalue of L because VðgÞ > 0 for almost every g, and hLf , f i ≥ hV f , f i > 0 unless f ≡ 0. One proceeds similarly when ε k → 0.
The following boundedness of square functions can be deduced from the spectral theorem immediately.
The operators g P,α , G P,α and g * P,α,λ are bounded on L 2 ðℍ n Þ. Moreover, there exist constants C, C 1 and 11 Journal of Function Spaces Proof. We only prove (i), and (ii) can be done similarly. For g H,α , using the reproducing formula on L 2 ðℍ n Þ, we can get For G H,α , we have For g * H,α,λ , the relation: g * H,α,λ f ðgÞ ≤ CG H,α ð f ÞðgÞ indicates that kg * H,α,λ f k L 2 ≤ C 2 k f k L 2 : Proposition 19. Let α > 0 and λ > Q/2.
(i) There exists a constant C such that for any function f which is a linear combination of H 1 L -atoms (ii) There exists a constant C such that for any function f which is a linear combination of H 1 L -atoms Proof. We only prove (i), and (ii) can be dealt with similarly. Firstly, by Lemma 18, we can get kg H,α ð f Þk L 2 = Ck f k L 2 . For f ∈ H 1 L ðℍ n Þ, it holds an atomic decomposition: f = ∑ j c j a j . Then, So we only need to verify that G H,α ðaÞ is in L 1 ðℍ n Þ for any H 1 L -atom a uniformly. By Lemma 18, For the estimate of B, the following two cases are considered.
Moreover, for every f ∈ H 1 L ðℍ n Þ, Proof. This theorem can be proved similarly as the proof of Theorem 20, so we omit it.

Fractional Square Functions
Characterizations of H 1,α L ðℍ n Þ. In this part, we will give the characterizations of Hardy-Sobolev space H 1,α L ðℍ n Þ by fractional square functions. Firstly, we give the following Lemma, which will be used in the sequel. Similar to ([31], Proposition 2.4), we can express the operators ∂ α t e −tL and ∂ α t e −t ffiffi L p as follows.
(i) For every f ∈ L 2 ðℍ n Þ, (ii) For every f ∈ L 2 ðℍ n Þ, Proof. Let EðλÞ denote a resolution of the identity. It follows from the spectral decomposition: By (6) and (129), we have where k is the smallest integer satisfying k > α. Then, the integral is absolutely convergent. By the fact that k∂ α t e −tL f k L p ≤ C α k f k L p /t α , the integral in (6) is absolutely convergent in L 2 ðℍ n Þ. Hence, by (130), we can get for g ∈ L 2 ðℍ n Þ, which implies (i). The assertion (ii) can be obtained by the aid of functional calculus similarly.
The following result can be deduced from Lemma 22 immediately.
Using Lemma 22, we can get ð136Þ Using Theorem 20, we can get Using Proposition 23, g H k,α , S H k,α , and g H, * k,α,λ can be extended to H 1,α L ðℍ n Þ as bounded operators from H 1,α L ðℍ n Þ to L 1 ðℍ n Þ. Let f g H k,α be the extension of g H k,α to H 1,α L ðℍ n Þ as a bounded operator from H 1,α L ðℍ n Þ to L 1 ðℍ n Þ. Then, there exists C > 0 such that for f ∈ H 1,α L ðℍ n Þ, Below, we give the square function characterizations of the Hardy-Sobolev space H 1,α L ðℍ n Þ as follows.
Theorem 24. Let α ≥ 1/2, k ∈ ℕf0g, and λ > Q. Then, the following assertions are equivalent: Moreover, for every f ∈ H 1,α L ðℍ n Þ, Proof. We first prove k f k H 1 Then, By the definition of H 1 L ðℍ n Þ, we conclude that the operator where the positive constant C is independent of N ∈ ℕ. Letting N → ∞ yields Since G α,L is dense in H 1,α L ðℍ n Þ, for f ∈ H 1,α L ðℍ n Þ, we obtain The proofs for S H k,α and g H, * k,α,λ are similar, and so is omitted.
For the reverse, we only deal with the case of g H k,α for simplicity.
Step I. We prove For m ∈ ℕ and m > α, by (141), we obtain Therefore, (147) follows from the definition of H 1 L ðℍ n Þ.
For the Poisson semigroup fP L t g t>0 , we define the fractional square functions as follows: Similar to the proof of Theorem 24, we can apply (ii) of Proposition 23 to establish the following characterization of H 1,α L ðℍ n Þ via the fractional square functions related to the Poisson semigroup. We omit the proof. Theorem 25. Let α ≥ 1/2, k ∈ ℕ \ f0g and λ > Q. Then, the following assertions are equivalent: (i) f ∈ H 1,α L ðℍ n Þ (ii) f ∈ H 1 L ðℍ n Þ and g P k,α ðf Þ ∈ L 1 ðℍ n Þ for k > α (iii) f ∈ H 1 L ðℍ n Þ and S P k,α ðf Þ ∈ L 1 ðℍ n Þ for α < k − ðQ + 1Þ/2 (iv) f ∈ H 1 L ðℍ n Þ and g P, * k,α,λ ð f Þ ∈ L 1 ðℍ n Þ for α < k − ðQ + 1Þ/2 Moreover, for every f ∈ H 1,α L ðℍ n Þ, The purpose of this section is to characterize H 1,α L ðℍ n Þ by the fractional square functions defined by (10) and (156), respectively. As an application, it follows from the fractional square function characterizations of H 1,α L ðℍ n Þ and H 1,α L ðℍ n Þ that the two Hardy-Sobolev spaces are equivalent.
Let E L be the spectral decomposition of the operator L. For a bounded function M on ð0, ∞Þ, the spectral multiplier MðLÞ is defined by where DðMðLÞÞ denotes the domain, i.e., 18

Journal of Function Spaces
We have the following version of spectral multiplier theorems.

Data Availability
The data used to support the findings of this study have not been made available because this is a mathematical article, which is pure theoretical proof and derivation, no specific data information.

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