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

Wang, Zhiyong; Sun, Chuanhong; Li, Pengtao

doi:https://doi.org/10.1155/2020/8851287

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8851287 | https://doi.org/10.1155/2020/8851287

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

Zhiyong Wang,¹Chuanhong Sun,¹and Pengtao Li¹

Academic Editor: Yoshihiro Sawano

Received19 Aug 2020

Accepted22 Sept 2020

Published10 Oct 2020

Abstract

In this paper, assume that is a Schrödinger operator on the Heisenberg group , where the nonnegative potential belongs to the reverse Hölder class . By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups and , respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space associated with . Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with .

1. Introduction

It is well-known that the Hardy spaces form a natural continuation of the Lebesgue spaces to the range . Correspondingly, let and denote the classical Riesz potentials and Bessel potentials, respectively. The Hardy-Sobolev spaces and 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 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 . The trace properties of the space 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 , or on subdomains, we refer the reader to [5–12].

The development of the theory of Hardy spaces with several real variables was initiated by Stein and Weiss. In [13], by use of square functions, Fefferman and Stein characterized the Hardy spaces for . From then on, such characterizations were extended to other settings, see [14–16] and the references therein. Since the 1990s, the theory of Hardy spaces associated with second-ordered differential operators on attracts the attention of many researchers and has been investigated extensively, such as [15–22] and the references therein. In recent years, a lot of research has been done on the Hardy spaces associated with operators on the Heisenberg group and other settings, see [23–25].

Let be a Schrödinger operator, where is the sub-Laplacian on and belongs to the reverse Hölder class. Let be the heat semigroup generated by and denote by the integral kernels. Since is nonnegative, the Feynman-Kacformula asserts that

Lin-Liu-Liu [25] introduced the Hardy space associated with , which is defined as follows. Let denote the semigroup maximal function: . The Hardy space associated with is defined to be where .

As an analogue of classical Hardy-Sobolev spaces, we introduce the following Hardy-Sobolev space associated with on :

Definition 1. For , the Hardy-Sobolev space is defined as the set of all functions such that with the norm Our motivation is inspired by the following square function characterization of . For , let Define the square function associated with as

In [16], Hoffmann et al. obtained the following square function characterization of :

Proposition 2. Let . A function if and only if and the square function . Moreover, .
The goal of this paper is to characterize by the square functions generated by semigroups associated with . It can be seen from Definition 1 that the elements of have the regularities of order . Based on this observation, we introduced the following fractional square functions associated with semigroup generated by . For , let and denote the time-fractional derivatives of the heat kernel and the Poisson kernel, respectively, (cf [26]), i.e., For , define the following family of operators: Similar to ([27], Proposition 3.6), the regularities of the kernels of and 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 and , respectively. Then, by the functional calculus, we deduce the following relations: see Lemmas 15 and 11. Hence, the desired regularities of and are corollaries of Propositions 10 and 14.
Respect to , we introduce the following fractional square functions: In Section 3.1, we establish the characterizations of 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 and , we prove The above relations, together with Theorem 20, indicate that see Proposition 23. Finally, in Theorem 24, we obtain the desired characterizations of via the fractional square functions defined in (10): for every , For the Poisson semigroup, via the operators , we can also obtain the corresponding square function characterizations of and , see Theorems 21 and 25 for the details.

Remark 3. (i)As far as the authors know, even on , the regularities of the time-fractional derivatives of the heat kernels obtained in Section 2.2 are new. The results obtained in Section 2.3 generalize those of [27] to the setting of Heisenberg groups. Moreover, all results in Sections 2.2 and 2.3 apply to some other operators, for example, the degenerate Schrödinger operators, the Schrödinger operators on stratified Lie groups, and so on(ii)Lemma 22 implies that the operators and can be expressed by the spectrum integral of Schrödinger operator. In the sequel, sometime, we formally denote by and by and , respectively

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 and , respectively. In Sections 3.1 and 3.2, we establish the fractional square function characterizations of and . As an application, we deduce an equivalence of the norms of Hardy-Sobolev spaces associated with .

1.1. Notations

Throughout this article, we will use and to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By , we mean that there exists a constant such that .

2. Preliminaries

2.1. Heisenberg Groups and Hardy Spaces

The -dimensional Heisenberg group is the Lie group with underlying manifold with the multiplication

The Lie algebra of left-invariant vector fields on is given by

The sub-Laplacian is defined as . The gradient is defined by . The left-invariant distance is . The ball of radius centered at is denoted by whose volume is given by , where denotes the volume of the unit ball in and is the homogenous dimension of . Let be the Siegel upper half-space in , i.e.,

Then, is holomorphically equivalent to the unit ball in . It is well known that the Heisenberg group is a nilpotent subgroup of the automorphism group of , which consists of the translations of . The Heisenberg group can be also identified with the boundary via its action on the origin. We use the Heisenberg coordinates to denote the points in , where

A nonnegative locally -integrable function on is said to belong to the reverse Hölder class , if there exists such that the reverse Hölder inequality holds for every ball In the sequel, we always assume that .

The following auxiliary function was first introduced by Shen [28] and widely used in the research of function spaces related to Schrödinger operators:

Definition 4. The auxiliary function is defined by

The following atomic characterization of was obtained by Lin-Liu-Liu [25].

Definition 5. A function is called a -atom of the Hardy space related with a ball if (i);(ii);(iii)if , then The atomic norm of is defined by , where the infimum is taken over all decompositions , and are -atoms.

Proposition 6. Let . The norms and are equivalent, that is, there exists a constant such that .

Below, we give some results on the tent spaces introduced by Coifman-Meyer-Stein.

Definition 7. Assume that . The tent space is defined as the set of all functions on satisfying , where Coifman, Meyer, and Stein established the following atomic decomposition of . A function is called a -atom if (i) is supported in for some ball ; (ii) .
The following proposition is one of the main results of tent spaces.

Proposition 8. Every element can be written as , where are -atoms, , and .

2.2. Time-Fractional Derivatives of the Heat Semigroup

In this part, we estimate the time-fractional derivatives of the heat kernel associated with . For , define

In ([29], Proposition 2.9), the authors obtained the following estimates about the kernel .

Proposition 9. (i)For , there exists a constant such that(ii)Assume that . For any , there exists a constant such that, for all (iii)For any , there exists a constant such that

Denote by the kernel of . In the following proposition, we investigate the regularities of .

Proposition 10. Let . (i)For , there exists a constant such that(ii)Assume that with . For any , there exists a constant such that for all (iii)For any , there exists a constant such that

Proof. (i)The proof of (i) is divided into the following two cases.Case 1. . For this case, it follows from functional calculus that By (i) of Proposition 9, we obtain On the one hand, a direct computation gives On the other hand, because the heat kernel decays rapidly, we can get Case 2. . Let . Write Since , 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 . By (ii) of Proposition 9, we obtainChanging the order of integration, we obtain Alternatively, we can also get For , by (ii) of Proposition 9, we can get Similar to the case , the rest of the proof can be finished by applying change of order of integration. We omit the details. (iii)For , by (iii) of Proposition 9, we change the order of integration to obtainIf , then If , then For , we have If , then If , then

The following lemma can be deduced from the functional calculus immediately.

Lemma 11. Let . The operators and are equivalent.

Proof. For , we have For , let . Since , it holds

Denote by the integral kernel of . By Proposition 10 and Lemma 11, we have the following result.

Corollary 12. Let . (i)For , there exists a constant such that(ii)Let with . For any there exists a constant such that, for all (iii)For any , there exists a constant such that

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 . For , define . In ([29], Proposition 2.12), the authors obtained the following estimates about the kernel .

Proposition 13 (see [29], Proposition 2.12). (i)For , there exists a constant such that(ii)Assume that . For any there exists a constant such that, for all (iii)For any , there exists a constant such that

Denote by the kernel . Similar to Proposition 10, we have

Proposition 14. Let . (i)For every , there is a constant such that(ii)Assume that with . For any there exists a constant such that for all (iii)For any , there exists a constant such that

Proof. Let us prove (i) first. The following two cases are considered.
Case 1. . 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 , Case 2. . Since, for , We can get Setting , we obtain Since , It follows from Proposition 13 that Also, noticing that , we obtain (ii)We first consider the case . Sincewe apply (ii) of Proposition 13 to obtain One the one hand, we have On the other hand, since , it holds For , noticing we can use (ii) of Proposition 13 to get The rest of the proof can be completed by the procedure of the case in (i), so we omit the details. (iii)For , it follows from (iii) of Proposition 13 thatIf , then If , then For , using (iii) of Proposition 13 again, we have If , we obtain If , similarly, we can get

The following result can be obtained similar to Lemma 11.

Lemma 15. Let . The operators and are equivalent.

Proof. For , we have For , let . Noticing , we obtain Define an operator Denote by the integral kernel of . The following estimates are immediate corollaries of Proposition 14 and Lemma 15.

Corollary 16. Let . (i)For every , there is a constant such that(ii)Assume that with . For any there exists a constant such that for all (iii)For any , there exists a constant such that

3. Square Function Characterizations of Hardy-Sobolev Type Spaces

3.1. Fractional Square Functions Characterizations of

Define

In this section, we will characterize the Hardy space by the fractional square functions defined by (9) and (84). Now, we first prove the following reproducing formulas.

Lemma 17. Let . (i)The operator defines an isometry from into . Moreover, in the sense of ,(ii)The operator defines an isometry from into . Moreover, in the sense of ,

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 , we have Thus, for all , we get For the second part, it suffices to show that, for every pair of sequences , Indeed, if (89) holds, we can find such that . Therefore, it follows from a polarized version of the first part that for , which implies . To prove (89), we use again the functional calculus to deduce that Computing the integral inside one yields , which by dominated convergence tends to . Observe that the last step makes use of the fact that is not an eigenvalue of because for almost every g, and unless . One proceeds similarly when .

The following boundedness of square functions can be deduced from the spectral theorem immediately.

Lemma 18. Let and . (i)The operators , and are bounded on . Moreover, there exist constants , and such that (ii)The operators , and are bounded on . Moreover, there exist constants , and such that

Proof. We only prove (i), and (ii) can be done similarly. For , using the reproducing formula on , we can get For , we have For , the relation: indicates that

Proposition 19. Let and . (i)There exists a constant such that for any function which is a linear combination of -atoms(ii)There exists a constant such that for any function which is a linear combination of -atoms

Proof. We only prove (i), and (ii) can be dealt with similarly. Firstly, by Lemma 18, we can get . For , it holds an atomic decomposition: . Then, So we only need to verify that is in for any -atom uniformly. By Lemma 18, Write where and For , it is clear that

For the estimate of , the following two cases are considered.

Case 1. . By the cancelation property of the atom , we have where For , since and , we can get . For and , we have . Using (ii) of Corollary 12 and the symmetry, we can get The above estimate for implies that Next, we estimate . Since , the estimate implies that

Case 2. . In this case, we write , where We first estimate the term . Since , and , . For , . Using the triangle inequality, we apply (i) of Corollary 12 to estimate as follows. For , since , , then . We have At last, we estimate . For , we have . Then, we can get The above estimates for indicate that

Now, we give the following characterizations of .

Theorem 20. Let and . The following assertions are equivalent: (i);(ii)and;(iii)and;(iv)andMoreover, for every ,

Proof. By Proposition 19, for , we know that , , and , respectively.
For the reverse, we first show that for , . Assume that . When , we can see that which implies that , where . By Proposition 8, , where are -atoms and . Assume that the atom is supported on . By Lemma 17, where . For simplicity, we denote by for . Write where . For , we use Hölder’s inequality to deduce that which gives .
Now, we deal with . For , by functional calculus and Proposition 2.9, we have When and , we have , and Finally, we get When , let be the bounded extension of from to . Since is dense in , there exists a sequence such that as in . By Corollary 12, we conclude that as . By the definition of , we know that as . Therefore, for , which gives For the Littlewood-Paley -function, it is sufficient to prove . For , we define by Similarly, we can prove that if and only if and . Moreover, .
Let and . Then . Therefore, When , we have . Hence, Let . Then, . Therefore, , where can be seen as a vector-valued Hardy space (cf. [30]). This shows that , where We can assume that . Then, the identity (6) gives When , we get . Via integration by substitution, we can change the orders of integration to obtain which implies , and therefore, . Since in the cone , we have This completes the proof of Theorem 20.

Theorem 21. Let and . The following assertions are equivalent: (i)(ii) and (iii) and (iv) and Moreover, for every ,

Proof. This theorem can be proved similarly as the proof of Theorem 20, so we omit it.

3.2. Fractional Square Functions Characterizations of

In this part, we will give the characterizations of Hardy-Sobolev space 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 and as follows.

Lemma 22. Let . (i)For every ,(ii)For every ,

Proof. Let denote a resolution of the identity. It follows from the spectral decomposition: that By (6) and (129), we have where is the smallest integer satisfying . Then, the integral is absolutely convergent. By the fact that , the integral in (6) is absolutely convergent in . Hence, by (130), we can get for , 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.

Proposition 23. (i)Let , and . If and . Then,(ii)Let , and . If and . Then,

Proof. We only prove (i), and (ii) can be dealt with similarly. Using Lemma 22, we can get therefore, Using Theorem 20, we can get Let . Since is dense in , is dense in . Note that , and Using Proposition 23, , , and can be extended to as bounded operators from to . Let be the extension of to as a bounded operator from to . Then, there exists such that for , Below, we give the square function characterizations of the Hardy-Sobolev space as follows.

Theorem 24. Let , , and . Then, the following assertions are equivalent: (i)(ii) and for (iii) and for (iv) and for Moreover, for every ,

Proof. We first prove . By (139), it is sufficient to prove . For and , by the subordination formula, we obtain Then, By the definition of , we conclude that the operator is bounded from to . Therefore, if , where , we have where the positive constant is independent of . Letting yields Since is dense in , for , we obtain The proofs for and are similar, and so is omitted.
For the reverse, we only deal with the case of for simplicity.
Step I. We prove For and , by (141), we obtain Therefore, (147) follows from the definition of .
Step II. Assume that and . Let be a sequence in such that in . For fixed , set and , . Then, and belong to . By Lemma 22 and (147), we have which implies that with . By (147) again, This indicates that is a Cauchy sequence in . Therefore, there exists such that as . Hence, as , which yields and as .
Step III. Noting that and , by Proposition 23, we get Letting , we have . Since we get . Furthermore, this gives where is independent of . By (153), we know are uniformly bounded in , i.e., are uniformly bounded in . Since is a Banach space, we can find such that as where is a subsequence of . Since is the dual space of and is dense in with norm of (cf. [32]), we get Let . Then, and . By the arguments analogous to ([33] page 776), which relay on the decay of the kernel of , we can get It follows that and This completes the proof of Theorem 24.

For the Poisson semigroup , 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 via the fractional square functions related to the Poisson semigroup. We omit the proof.

Theorem 25. Let , and . Then, the following assertions are equivalent: (i)(ii) and for (iii) and for (iv) and for Moreover, for every ,

3.3. Equivalent Norms of Hardy-Sobolev Spaces

We define the following Hardy-Sobolev space as the set of all functions such that , with the norm

The purpose of this section is to characterize by the fractional square functions defined by (10) and (156), respectively. As an application, it follows from the fractional square function characterizations of and that the two Hardy-Sobolev spaces are equivalent.

Let be the spectral decomposition of the operator . For a bounded function on , the spectral multiplier is defined by where denotes the domain, i.e.,

We say that a function on belongs to the space , , if

We have the following version of spectral multiplier theorems.

Proposition 26 (see [34], Theorem 1.11). Let be a bounded continuous function on . If for some and a nonzero function , there exists a constant such that for every , then the operator is bounded on .
Let . For , define

Then, it is clear that , are smooth and bounded on . It follows from Proposition 26 that

Proposition 27. Let . The operators , can be extended to bounded operators on .

Theorem 28. Let , and . If and ,

Proof. We give the proof of . The proofs for the cases of and are similar. By Proposition 27, we know that the operators and are bounded on . Then, following from Proposition 23, we obtain For the reverse, we take the function . For any , Letting , we get . By Proposition 27 again, we obtain and Theorem 28 follows from Proposition 23.

Similar to Theorem 28, we also can obtain

Theorem 29. Let, and. Ifand, Let Since is dense in , is dense in . Note that , and Using Theorem 28, , , and can be extended to as bounded operators from to . Let be the extension of to as a bounded operator from to . Then, there exists such that for ,
Similar to Theorems 24 and 25, we will give the following characterizations of the Hardy-Sobolev space as follows. We omit the proof.

Theorem 30. Let , and . The following assertions are equivalent: (i)(ii) and for (iii) and for (iv) and for Moreover, for every ,

Theorem 31. Let , and . The following assertions are equivalent: (i)(ii) and for (iii) and for (iv) and for Moreover, for every ,

Theorems 24, 25, 30, and 31 indicate the following equivalence relation:

Corollary 32. Let . .

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.

Acknowledgments

The authors thank Professor Yu Liu and Professor Jizheng Huang for their illuminating discussion on this topic. P.T. Li was supported by the Shandong Natural Science Foundation of China (No. ZR2017JL008), the National Natural Science Foundation of China (No. 11871293), and the University Science and Technology Projects of Shandong Province (No. J15LI15).

References

R. Strichartz, “H^p Sobolev spaces,” Colloquium Mathematicum, vol. 60-61, no. 1, pp. 129–139, 1990.
View at: Google Scholar
A. Torchinsky, “Restrictions and extensions of potentials of H^p distributions,” Journal of Functional Analysis, vol. 31, no. 1, pp. 24–41, 1979.
View at: Publisher Site | Google Scholar
A. Miyachi, “Hardy-Sobolev spaces and maximal functions,” Journal of the Mathematical Society of Japan, vol. 42, no. 1, pp. 73–90, 1990.
View at: Publisher Site | Google Scholar
R. A. DeVore and R. C. Sharpley, “Maximal functions measuring smoothness,” Memoirs of the American Mathematical Society, vol. 47, no. 293, p. 0, 1984.
View at: Publisher Site | Google Scholar
P. Auscher, E. Russ, and P. Tchamitchian, “Hardy Sobolev spaces on strongly Lipschitz domains of ,” Journal of Functional Analysis, vol. 218, no. 1, pp. 54–109, 2005.
View at: Publisher Site | Google Scholar
N. Badr and F. Bernicot, “Abstract Hardy-Sobolev spaces and interpolation,” Journal of Functional Analysis, vol. 259, no. 5, pp. 1169–1208, 2010.
View at: Publisher Site | Google Scholar
N. Badr and G. Dafni, “Maximal characterization of Hardy-Sobolev spaces on manifolds,” Contemporary Mathematics, vol. 545, pp. 13–21, 2011.
View at: Publisher Site | Google Scholar
Y.-K. Cho and J. Kim, “Atomic decomposition on Hardy-Sobolev spaces,” Studia Mathematica, vol. 177, no. 1, pp. 25–42, 2006.
View at: Publisher Site | Google Scholar
A. Gatto, C. Segovia, and J. Jiménez, “On the solution of the equation for ,” in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I,II, pp. 394–415, Chicago, III., 1981.
View at: Google Scholar
S. Janson, “On functions with derivatives in H¹,” in Harmonic Analysis and Partial Differential Equations. Lecture Notes in Mathematics, vol 1384, J. García-Cuerva, Ed., pp. 193–201, Springer, Berlin, Heidelberg, 1989.
View at: Publisher Site | Google Scholar
Z. Lou and S. Yang, “An atomic decomposition for the HARDY-SOBOLEV space,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1167–1176, 2007.
View at: Publisher Site | Google Scholar
J. Orobitg, “Spectral synthesis in spaces of functions with derivatives in ,” in Harmonic Analysis and Partial Differential Equations. Lecture Notes in Mathematics, vol 1384, J. García-Cuerva, Ed., pp. 202–206, Springer, Berlin, Heidelberg.
View at: Publisher Site | Google Scholar
C. Fefferman and E. M. Stein, “H^p spaces of several variables,” Acta Mathematica, vol. 129, pp. 137–193, 1972.
View at: Publisher Site | Google Scholar
R. R. Coifman, Y. Meyer, and E. M. Stein, “Some new function spaces and their applications to harmonic analysis,” Journal of Functional Analysis, vol. 62, no. 2, pp. 304–335, 1985.
View at: Publisher Site | Google Scholar
X. T. Duong and L. Yan, “Duality of Hardy and BMO spaces associated with operators with heat kernel bounds,” Journal of the American Mathematical Society, vol. 18, no. 4, pp. 943–973, 2005.
View at: Publisher Site | Google Scholar
S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, and L. Yan, “Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates,” Memoirs of the American Mathematical Society, vol. 214, no. 1007, 2011.
View at: Publisher Site | Google Scholar
H.-Q. Bui, X. T. Duong, and L. Yan, “Calderón reproducing formulas and new Besov spaces associated with operators,” Advances in Mathematics, vol. 229, no. 4, pp. 2449–2502, 2012.
View at: Publisher Site | Google Scholar
J. Dziubański, “Note on H¹ spaces related to degenerate Schrödinger operators,” Illinois Journal of Mathematics, vol. 49, no. 4, pp. 1271–1297, 2005.
View at: Publisher Site | Google Scholar
J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea, and J. Zienkiewicz, “BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality,” Mathematische Zeitschrift, vol. 249, no. 2, pp. 329–356, 2005.
View at: Publisher Site | Google Scholar
J. Dziubański and J. Zienkiewicz, “Hardy space H¹ associated to Schrödinger operator with potential satisfying reverse Hölder inequality,” Revista Matemática Iberoamericana, vol. 15, pp. 279–296, 1999.
View at: Publisher Site | Google Scholar
J. Dziubański and J. Zienkiewicz, “H^p spaces for Schrödinger operators,” in Fourier Analysis and Related Topics, Banach Center Publications, Volume 56, pp. 45–53, Institute of Mathematics Polish Academy of Sciences Warszawa, 2002.
View at: Google Scholar
L. Wu and L. Yan, “Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators,” Journal of Functional Analysis, vol. 270, no. 10, pp. 3709–3749, 2016.
View at: Publisher Site | Google Scholar
J. Cao and D. Yang, “Hardy spaces associated with operators satisfying k-Davies-Gaffney estimates,” Science China Mathematics, vol. 55, no. 7, pp. 1403–1440, 2012.
View at: Publisher Site | Google Scholar
J. Huang, P. Li, and Y. Liu, “Poisson semigroup, area function, and the characterization of Hardy space associated to degenerate Schrödinger operators,” Banach Journal of Mathematical Analysis, vol. 10, no. 4, pp. 727–749, 2016.
View at: Publisher Site | Google Scholar
C. Lin, C. Liu, and Y. Liu, “Hardy spaces associated with Schrödinger operators on the Heisenberg group,” 2011, http://arxiv.org/abs/1106.4960.
View at: Google Scholar
C. Segovia and R. Wheeden, “On certain fractional area integrals,” Journal of Mathematics and Mechanics, vol. 19, pp. 247–262, 1969/1970.
View at: Google Scholar
T. Ma, P. R. Stinga, J. L. Torrea, and C. Zhang, “Regularity properties of Schrödinger operators,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 817–837, 2012.
View at: Publisher Site | Google Scholar
Z. Shen, “L^p estimates for Schrödinger operators with certain potentials,” Annales de l’institut Fourier, vol. 45, no. 2, pp. 513–546, 1995.
View at: Publisher Site | Google Scholar
Y. Wang, Y. Liu, C. Sun, and P. Li, “Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups,” Forum Mathematicum, vol. 32, no. 5, pp. 1337–1373, 2020.
View at: Publisher Site | Google Scholar
G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, 1982.
J. Torrea and C. Zhang, “Fractional vector-valued Littlewood-Paley-Stein theory for semigroups,” Proceedings of the Royal Society of Edinburgh, vol. 144A, pp. 637–667, 2014.
View at: Google Scholar
P. Auscher and P. Tchamitchian, “Calcul fonctionnel précisé pour dos opéralcurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux),” Annales de l'Institut Fourier, vol. 45, pp. 721–778, 1995.
View at: Google Scholar
R. Jiang and D. Yang, “Predual spaces of Banach completions of Orlicz-Hardy spaces associated with operators,” Journal of Fourier Analysis and Applications, vol. 17, no. 1, pp. 1–35, 2011.
View at: Publisher Site | Google Scholar
J. Dziubański, “Spectral multiplier theorem for H¹ spaces associated with some Schrödinger operators,” Proceedings of the American Mathematical Society, vol. 127, no. 12, pp. 3605–3613, 1999.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Zhiyong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies