Herz-Type Hardy Spaces Associated with Operators

Chai, Yan; Han, Yaoyao; Zhao, Kai

doi:https://doi.org/10.1155/2018/1296837

Journal of Function Spaces

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

Research Article | Open Access

Volume 2018 | Article ID 1296837 | https://doi.org/10.1155/2018/1296837

Herz-Type Hardy Spaces Associated with Operators

Yan Chai,¹Yaoyao Han,¹and Kai Zhao¹

Academic Editor: Alberto Fiorenza

Received14 May 2018

Revised28 Jun 2018

Accepted08 Jul 2018

Published17 Jul 2018

Abstract

Suppose is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator . Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator . As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

1. Introduction

As we know, the theory of function spaces constitutes an important part of harmonic analysis and partial differential equations. Some results of the classical Hardy spaces can be found in [1–6], etc. Since there are some important situations in which the theory of classical Hardy spaces is not applicable, many authors begin to study Hardy spaces that are adapted to the differential operator . For example, Auscher, Duong, and McIntosh [7], then Duong and Yan [8, 9], introduced the Hardy and BMO spaces adapted to the operator which satisfies the Gaussian heat kernel upper bounds. Yang and his cooperators discussed new Orlicz-Hardy spaces associated with operators [10–13]. For more results, we refer to [14–19] and the references therein.

It is known that many classical function spaces and the Hardy type spaces associated with operators have the atomic decompositions and the molecular decompositions, and the atomic and molecular decompositions of function spaces make the linear operators acting on spaces very simple; see [20–29], etc. In fact, the characterizations of spaces of functions or distributions, including the atomic and molecular characterizations, have many important applications in harmonic analysis. In recent years, it has been proved that many results in the classical theory of Hardy spaces and singular integrals can transplant to the function spaces associated with operators, such as [30–36].

Suppose is a nonnegative, self-adjoint differential operator, and has -calculus on . The kernel of satisfies the Gaussian upper bound on . Motivated by [17, 20, 37], etc, in this paper, we use the area integral function associated with the operator to define the Herz-type Hardy space . In order to obtain the atomic and molecular decompositions of the Herz-type Hardy space, the -atom and the -molecule are introduced. By the method of the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we characterize spaces for atoms and molecules; that is, we prove the atomic and molecular decompositions of Herz-type Hardy spaces associated with the operator . Finally, as applications, we prove some singular integral operators are bounded from to Herz spaces and also bounded on .

Throughout the paper, we always use the letter to denote a positive constant, which may change from one to another and only depends on main parameters. We also use to denote the characteristic function of which is the subset of .

2. Preliminaries

For convenience, we recall the definitions of Herz and Herz-type Hardy spaces on . For details, we refer to [37, 38], etc.

Let , , and .

Definition 1 (see [38]). Let , , .
The homogeneous Herz space is defined by where The nonhomogeneous Herz space is defined by whereLet be the grand maximal function of defined as , where , .

Definition 2 (see [38]). Let .
The homogeneous Herz-Hardy space is defined by Moreover, The nonhomogeneous Herz-Hardy space is defined by Moreover,Now, we introduce the Herz-type Hardy spaces associated with operators.
Suppose that the differential operator satisfies the following two assumptions.

Assumption (A1). is a nonnegative self-adjoint operator on , and has a bounded -functional calculus in .

Assumption (A2). Each of the heat semigroup generated by has the kernel which satisfies the following Gaussian upper bounds; i.e., there exist constants such that

Obviously, the typical second-order elliptic or subelliptic differential operators are to satisfy these assumptions (see for instance, [39]).

Now we introduce the following lemmas which will be used in this paper.

Lemma 3 (see [40, 41]). Let be a nonnegative self-adjoint operator satisfying Assumptions (A1) and (A2). For every , there exist two positive constants such that the kernel of the operator satisfiesfor all and almost every .

Lemma 4 (see [19]). Let be even and . Suppose denotes the Fourier transform of . Then for each , kernel of satisfiesandfor all and

Lemma 5 (see [19]). For , we define Then for any nonzero function , .

Lemma 6 (see [19]). Let . Then, for any ,

Definition 7 (see [8]). For any , the area integral function associated with operators is defined bywhere , and satisfies Assumptions (A1) and (A2).

Thus, for any integer , , the kernel of satisfies , where is a positive decreasing function, satisfying , for any . Therefore, for convenience, in the following, we always set in (15).

By the definition (15) and Assumptions (A1) and (A2), it is easy to check that for any , ; there exist such that (or see, for example, [11]):

Definition 8. Suppose . Let satisfy Assumptions (A1) and (A2).
The homogeneous Herz-type Hardy space associated with the operator is defined by The norm of in is The nonhomogeneous Herz-type Hardy space associated with the operator is defined by The norm of in isSince , the Herz-type Hardy spaces associated with the operator introduced in Definition 8 are really the expansion of Hardy space associated with operators in [8].

3. The Decompositions of Herz-Type Hardy Spaces

In this section, we give the atomic decomposition and molecular decomposition of Herz-type Hardy spaces associated with the operator , respectively, which are the main results in this paper.

3.1. Atomic Decomposition of the Herz-Type Hardy Space

For the purpose of the atomic decomposition of the Herz-type Hardy space associated with operator, we first introduce the -atom.

Definition 9. Let , , . Set , where satisfies Assumptions (A1) and (A2).
A function is said to be an -atom, if there exists , such that(i);(ii);(iii);, .
Function is said to be a restrictive -atom, if there exists , satisfying (i), (ii), (iii), and

The main result of this subsection is the following atomic decomposition of the Herz-type Hardy spaces associated with the operator . The part of the idea is from [1].

Theorem 10. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of -atoms and a sequence of numbers such that can be represented in the following form: and the sum converges in the sense of -norm, . Moreover, where the infimum is taken over all of the decompositions of .

Proof. First, we prove the theorem for .
Necessity. Let and be the same as those in Lemma 4. Set . Then by -functional calculus (see for example, [42]), for every , there isSet , . denotes the collection of all dyadic cubes in . Let . Then, for any , there exists only one such that . Let ; i.e., denote the collection of maximal dyadic cubes in . Set where is the side length of .
Then, by (23), we have that where andWe will prove that, up to a normalization by a multiplicative constant, every is an -atom.
Obviously, by Lemma 4, we conclude that , .
For , and , then, by Hölder inequality together with Lemma 6, we have thatFurthermore, we prove the following estimate:Noting the definition of in (26) and Definition 8 for , it means that we should establish the following inequality:where is the characteristic function of .
It is sufficient to show thatIn fact, if , then , . Let denote the characteristic function of . Thus, by the definition of , we obtain that Therefore, Hence, the necessity is proved.
Sufficiency. Let , where every is an -atom. We will prove the sufficiency for two situations: and .
If , then, to prove , it is only need to show that, for any -atom , there exits a constant independent of such that In fact, then, we have Suppose that is an -atom with and for any () , . Then For , boundedness of and the size condition of atom tell us In order to estimate , we write For , noting that and , if , , then . Thus, by Lemma 3, we can have thatFor , noting that , if , , then . So that holds true. Therefore, we can obtain thatHence, combining (38) and (39), one can have If , then For , boundedness of and the Hölder inequality tell us For , similar to the estimate of , we can obtain the estimates of as (38) and (39). Thus, using the Hölder inequality, we have that The sufficiency is proved. Then the proof of Theorem 10 for is finished.
If , the proof that is exactly similar to the situation of , we only need to slightly modify some formulas above. We should set Inequalities (29) and (30) are replaced withandrespectively. To obtain inequality (46), there is Other details are omitted.

For the nonhomogeneous Herz-type Hardy space associated with the operator , there is the same result as follows.

Theorem 11. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of the restrictive -atoms and a sequence of numbers such that can be represented in the following form: and the sum converges in the sense of -norm, . Moreover, where the infimum is taken over all of the decompositions of .

3.2. Molecular Decomposition of the Herz-Type Hardy Space

In this subsection, we first introduce -molecule in the following; then we give the molecular decomposition of the Herz-type Hardy space associated with operator.

Definition 12. Let , , . Set , where satisfies Assumptions (A1) and (A2).
A function is said to be an -molecule, if there exists , such that(i);(ii) and , where , ; and Function is said to be a restrictive -molecule, if there exists , satisfying (i), (ii), and

It is not difficult to check that an -atom associated with ball is also an -molecule associated with the same ball .

The following molecular decomposition of the Herz-type Hardy spaces associated with the operator is the main result in this subsection.

Theorem 13. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of -molecules and a sequence of numbers such that can be represented in the following form: and the sum converges in the sense of -norm, . Moreover, where the infimum is taken over all of the decompositions of .

Proof. (i) The proof of necessity is a direct consequence of the necessity in Theorem 10, since an -atom is also an -molecule for all .
(ii) The proof of sufficiency is similar to that of the sufficiency in Theorem 10. The main difference is that the support of -molecule is not the ball . However, we can overcome this difficulty by decomposing into annuli associated with the ball , then using the same argument as in Theorem 10 to get sufficiency. We omit the details here.

Similarly, for the nonhomogeneous Herz-type Hardy space associated with operator , there is the same result as follows.

Theorem 14. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of the restrictive -molecules and a sequence of numbers such that can be represented in the following form: and the sum converges in the sense of -norm, . Moreover, where the infimum is taken over all of the decompositions of .

4. Boundedness of Singular Integral Operators

This section is based on the decompositions of in the previous section; as applications, we give some boundedness of sublinear operators satisfying certain conditions on Herz-type Hardy spaces associated with operator.

Theorem 15. Let , , . If a sublinear operator satisfies that(i) is bounded on ;(ii) satisfies the size conditionfor suitable function with .
Then is bounded from to , that is,

Proof. Suppose . By Theorem 10, we have where each is a -atom with , and , , .
Thus, First, we estimate . By the boundedness of in , we can infer that Therefore, if , then, by the Jensen inequality, we have If , let . Then we can obtain Hence, .
Second, we estimate . By the Hölder inequality and (55), we obtain Thus, Therefore, if , by the Jensen inequality, we have If , by the Hölder inequality, we obtain Hence,

Theorem 16. Let , , . Suppose that is a sublinear operator as Theorem 15 and and are commutative. Then is bounded on .

Proof. Suppose is an -atom. According to Definition 9, there exists , such that(i);(ii);(iii) being an -molecule only needs to be prove, such that(1);(2) and In fact, it is enough to check . For any and , we have Thus, we complete the proof of Theorem 16.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (11471176).

References

S.-Y. A. Chang and R. Fefferman, “A continuous version of the duality of H¹ and BMO on the bidisk,” Annals of Mathematics, vol. 112, no. 1, pp. 179–201, 1980.
View at: Publisher Site | Google Scholar | MathSciNet
C. Fefferman and E. M. Stein, “Hp spaces of several variables,” Acta Mathematica, vol. 129, no. 1, pp. 137–193, 1972.
View at: Publisher Site | Google Scholar | MathSciNet
J. García-Cuerva, “Weighted Hp spaces,” Dissertationes Mathematicae, vol. 162, pp. 1–63, 1979.
View at: Google Scholar
J. García-Cuerva and J. Rubio De Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amestrdam, Netherlands, 1985.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.
View at: MathSciNet
E. M. Stein and G. Weiss, “On the theory of harmonic functions of several variables. I. The theory of H^p-spaces,” Acta Mathematica, vol. 103, pp. 25–62, 1960.
View at: Publisher Site | Google Scholar | MathSciNet
P. Auscher, X. T. Duong, and A. McIntosh, “Boundedness of Banach space valued singular integral operators and Hardy spaces,” Unpublished preprint, 2005.
View at: 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 | MathSciNet
X. T. Duong and L. Yan, “New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications,” Communications on Pure and Applied Mathematics, vol. 58, no. 10, pp. 1375–1420, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
T. A. Bui, J. Cao, L. D. Ky, D. Yang, and S. Yang, “Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates,” Analysis and Geometry in Metric Spaces, vol. 1, pp. 69–129, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
R. Jiang and D. Yang, “New Orlicz-Hardy spaces associated with divergence form elliptic operators,” Journal of Functional Analysis, vol. 258, no. 4, pp. 1167–1224, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
R. Jiang and D. Yang, “Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates,” Communications in Contemporary Mathematics, vol. 13, no. 2, pp. 331–373, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
D. Yang and S. Yang, “Musielak-Orlicz-Hardy spaces associated with operators and their applications,” The Journal of Geometric Analysis, vol. 24, no. 1, pp. 495–570, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
P. Auscher, A. McIntosh, and E. Russ, “Hardy spaces of differential forms on Riemannian manifolds,” The Journal of Geometric Analysis, vol. 18, no. 1, pp. 192–248, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
P. Auscher and E. Russ, “Hardy spaces and divergence operators on strongly Lipschitz domains of Rⁿ,” Journal of Functional Analysis, vol. 201, no. 1, pp. 148–184, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
J. Cao and D. Yang, “Hardy spaces associated with operators satisfying k-Davies-Gaffney estimates,” Science China Mathematics, vol. 55, pp. 1403–1440, 2012.
View at: Google Scholar
S. Hofmann, G. Z. Lu, D. Mitrea, M. Mitrea, and L. X. 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 | MathSciNet
S. Hofmann and S. Mayboroda, “Hardy and BMO spaces associated to divergence form elliptic operators,” Mathematische Annalen, vol. 344, no. 1, pp. 37–116, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
S. Liu, K. Zhao, and S. Zhou, “Weighted Hardy spaces associated to self-adjoint operators and BMO_L,w,” Taiwanese Journal of Mathematics, vol. 18, no. 5, pp. 1663–1678, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
T. Anh, J. Cao, L. D. Ky, D. Yang, and S. Yang, “Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates,” Taiwanese Journal of Mathematics, vol. 17, no. 4, pp. 1127–1166, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
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 | MathSciNet
L. Deng, B. Ma, and S. Liu, “A Marcinkiewicz criterion for L^p -multipliers related to Schrödinger operators with constant magnetic fields,” Science China Mathematics, vol. 58, no. 2, pp. 389–404, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
X. Fu, H. Lin, D. Yang, and D. Yang, “Hardy spaces H^p over non-homogeneous metric measure spaces and their applications,” Science China Mathematics, vol. 58, no. 2, pp. 309–388, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
R. Gong, J. Li, and L. Yan, “A local version of Hardy spaces associated with operators on metric spaces,” Science China Mathematics, vol. 56, no. 2, pp. 315–330, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
H. Lin and D. Yang, “Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces,” Science China Mathematics, vol. 57, no. 1, pp. 123–144, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
A. McIntosh, “Operators which have an H∞-calculus, Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986),” in Proceedings of the Centre for Mathematical Analysis, vol. 14, pp. 210–231, ANU, Canberra, Australia, 1986.
View at: Google Scholar | MathSciNet
Y. Meyer and R. Coifman, Wavelets, Calderón-Zygmund and Multilinear Operators, Cambridge University Press, Cambridge, UK, 1997.
View at: MathSciNet
D. Yang and S. Yang, “Local Hardy spaces of Musielak-Orlicz type and their applications,” Science China Mathematics, vol. 55, no. 8, pp. 1677–1720, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
K. Zhao and Y. Han, “Boundedness of operators on Hardy spaces,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 319–327, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Jianfeng Dong, Jizheng Huang, and Heping Liu, “Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators,” Journal of Function Spaces, vol. 2015, pp. 1–11, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Guo, P. Li, and L. Peng, “L^p Boundedness of commutator of riesz transform associated to Schrödinger operator,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 421–432, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Liang, D. Yang, and S. Yang, “Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates,” Science China Mathematics, vol. 54, no. 11, pp. 2395–2426, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Liu, J. Huang, and J. Dong, “Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators,” Science China Mathematics, vol. 56, no. 9, pp. 1895–1913, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. Liu and K. Zhao, “Various characterizations of product Hardy spaces associated to Schrödinger operators,” Science China Mathematics, vol. 58, no. 12, pp. 2549–2564, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
L. Song, C. Tan, and L. Yan, “An atomic decompostion for Hardy spaces associated to Schrödinger operator,” Journal of the Australian Mathematical Society, vol. 91, no. 1, pp. 125–144, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Liang Song and Chaoqiang Tan, “Hardy Spaces Associated to Schrödinger Operators on Product Spaces,” journal of function spaces and applications, vol. 2012, pp. 1–17, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Lu, D. Yang, and G. Hu, Herz type Spaces and Their Applications, Science Press, Beijing, China, 2008.
S. Lu, “Herz type spaces,” Advances in Mathematics (China), vol. 33, pp. 257–272, 2004.
View at: Google Scholar | MathSciNet
X. Thinh Duong, E. M. Ouhabaz, and A. Sikora, “Plancherel-type estimates and sharp spectral multipliers,” Journal of Functional Analysis, vol. 196, no. 2, pp. 443–485, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
E. B. Davies, Heat Kernels and Spectral Theory, vol. 92, Cambridge University Press, Cambridge, Cambridge, UK, 1989.
View at: Publisher Site | MathSciNet
E. M. Ouhabaz, Analysis of Heat Equations on Domains, vol. 31 of London Mathematical Society Monographs, Princeton University Press, Princeton, NJ, USA, 2005.
View at: MathSciNet
L. Song and L. X. Yan, “Riesz transforms associated to Schrödinger operators on weighted Hardy spaces,” Journal of Functional Analysis, vol. 259, no. 6, pp. 1466–1490, 2010.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2018 Yan Chai 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.

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