Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators

Yang, Li; Li, Pengtao

doi:https://doi.org/10.1155/2023/8001131

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Research Article | Open Access

Volume 2023 | Article ID 8001131 | https://doi.org/10.1155/2023/8001131

Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators

Li Yang¹and Pengtao Li¹

Academic Editor: Andrea Scapellato

Received19 Dec 2022

Revised08 Mar 2023

Accepted10 Mar 2023

Published24 Mar 2023

Abstract

Let be a Schrödinger operator on , where denotes the Laplace operator and is a nonnegative potential belonging to a certain reverse Hölder class with . In this paper, by the regularity estimate of the fractional heat kernel related with , we establish the quantitative weighted boundedness of Littlewood-Paley functions generated by fractional heat semigroups related with the Schrödinger operators.

1. Introduction

Let be a Schrödinger operator, where denotes the Laplace operator and the potential . The Schrödinger operators originate from the famous Schrödinger equations which are used to describe the microparticle state in quantum mechanics. Due to the profound background in mathematic physics, the theory of Schrödinger operators has attracted the attention of a lot of mathematicians (see Simon [1] for a summary on the development of Schrödinger operators). Since 1990s, by the aid of the technology of PDEs and the functional analysis, there has been a considerable development on the theory of harmonic analysis associated with Schrödinger operators. In 1995, under the assumption that the potential , Shen [2] gave the estimates of the fundamental solution for . Here, a locally -integrable function is said to belong to on , , if there exists such that the reverse Hölder inequality holds. The analogue for magnetic Schrödinger operators and the generalized Schrödinger operators are established by Shen in [3, 4], respectively. In 1999, via the semigroup maximal function, Dziubański and Zienkiewicz [5] introduced the Hardy space related with , where . By the aid of the local Hardy spaces, the atomic characterization and the Riesz transform characterization of were obtained in [5] (see also [6] for the theory of ). As the dual space of , the bounded mean oscillation space related with denoted by was introduced by Dziubański et al. in [7]. For further progress on this topic, we refer the reader to Bongioanni et al. [8, 9], Duong et al. [10], Duong et al. [11], Guo et al. [12], Li [13], Lin and Liu [14], Ma et al. [15], Tang [16], Yang et al. [17] and the references therein.

In recent years, the quantitative weighted bounds for operators in the Schrödinger settings have been investigated by many researchers. Li et al. [18] established the quantitative weighted boundedness of maximal functions, maximal heat semigroups, and fractional integral operators related to . In , Zhang and Yang [19] showed that the quantitative weighted boundedness for Littlewood-Paley functions in the Schrödinger setting. Bui et al. [20] investigated the quantitative boundedness for square functions with new class of weights. In [21], Bui et al. studied the quantitative estimates for some singular integrals associated with critical functions. Wen and Wu [22] obtained the quantitative weighted strong and weak-type estimates for variation operators related to heat semigroups associated with .

Inspired by the results in [18, 19, 22], in this paper, we investigate the quantitative weighted boundedness of Littlewood-Paley functions generated by fractional heat semigroups associated with . Let denote the integral kernel of the heat semigroup . For , the subordinative formula (cf. [23]) indicates that there exists a continuous function on such that the fractional heat kernel related with can be represented as

Let , , be the fractional heat semigroup associated with , i.e.

For the special case , is exactly the fractional heat kernel related with the Laplace operator. Denote by the Fourier transform of . The fractional heat semigroup is defined via the Fourier multiplier: with the integral kernel:

In the literature, the fractional heat semigroups related with second-order differential operators have been widely used in the study of partial differential equations, harmonic analysis, potential theory, and modern probability theory. For example, the semigroup is usually applied to construct the linear part of solutions to fluid equations in the mathematic physics, e.g., the generalized Navier-Stokes equation, the quasigeostrophic equation, and the generalized MHD equations. In the field of probability theory, the researchers use to describe some kind of Markov processes with jumps.

Let , , , and . Via the higher-order derivatives of in the time variable , the Littlewood-Paley type functions associated with are defined, respectively, as

Specially, if in (7), we write as . If and in (6) and (7), then and related to are the classical Littlewood-Paley type functions.

Let . Similarly, we can introduce the Littlewood-Paley type functions associated with the spatial gradient of as follows:

If in (10), we write as . If and in (9) and (10), then and related to are the classical Littlewood-Paley-type functions.

We point out that the quantitative weighted boundedness of Littlewood-Paley functions generated by is not a simple analogue of [19] which deals with the case . For , it is well known that the heat kernel of the Laplace operator has a good decay properties. Precisely, , which indicates that it can be dominated as follows.

The arbitrariness of ensures that the square functions generated with can be dominated by the intrinsic Lusin area function: for (see [19], Lemma 3.1).

For the case , Miao et al. [24] obtained the following regularity estimate for : (see ([24], Remark 2.1). By this estimate, it can be only obtained that , which indicates that (13) does not hold.

In Lemmas 11 and 12, adopting the subordinate formula (2), we obtain the following revised decay estimate for : there exists a constant such that

The estimates (14) enable us to establish the quantitative weighted bounds for , , and with (see Theorem 25). By the aid of the quantitative version of the extrapolation theorem in [19], we prove the quantitative weighted boundedness of function under the assumption that , see Theorem 3.12.

The structure of the article is as follows. In Section 2, we give some symbol notations and estimates of kernels which will be used in the sequel. In Section 3, we investigate quantitative weighted bounds for Littlewood-Paley-type functions.

Some notations are as follows: In this article, denotes the set of all infinitely differential functions with compact supports. is a ball centered at and with radius . Given and , we will write . Unless otherwise indicated, we will denote by a positive constant, which is different from range to range and depends on the main parameters.

For any constant , denote as the conjugate of , i.e., . By , we mean that there is a such that . means that there exist that satisfy . We write , where . For any set on , we denote and its characteristic function.

2. Preliminaries

2.1. Some Notations

In this article, a weight denotes a nonnegative locally integrable function. Given a Lebesgue measurable set and a weight , the symbol denotes the Lebesgue measure of and

For , the weighted Lebesgue space is defined as the set of all measurable functions satisfying

In [25], to estimate the fundamental solution of Schrödinger operators, Shen first introduced the following auxiliary function .

Definition 1. Assume that for some . The function is defined by Obviously, if .

The following properties of were obtained by Shen [2].

Lemma 2 (see [2], Lemma 1.4). Assume that for some . There exist constants , , and such that In particular, if .

Lemma 3 (see [2], page 196). Assume that , . For any and , there exists a constant such that In [8], Bongioanni et al. introduced a new class of weights related with .

Definition 4. Let , , and . is defined as the set of all weights satisfying any ball : where .

Definition 5. The classical Hardy-Littlewood maximal operator is defined by setting, for any and : where the supremum is taken over all balls of containing .

Definition 6. Let . The Hardy-Littlewood-type maximal operator related to is defined by setting, for any and : where the supremum is taken over all balls of containing .

In [18], Li et al. obtained the quantitative estimate for the maximal operator .

Lemma 7 (see [18], Theorem 1.3). Let , , and . Then there exists a positive constant such that for any with and :

Specially, if , the following result can be deduced from Lemma 7.

Corollary 8 (see [19]). Let and . There exists a positive constant such that for any and Zhang and Yang [19] established the quantitative version of the extrapolation theorem for weights.

Lemma 9 (see [19], Lemma 2.6). Let , , and be an operator defined on . Suppose that there exist positive constants and such that for any and Then for any , there exists a positive constant such that for any and

By the aid of Lemma 2.9, in [19], Zhang and Yang obtained the following conclusion.

Corollary 10 (see [19], Remark 2.7). Let be a given family of pairs of nonnegative measurable functions on . Suppose that there exist positive constants and such that for some fixed and for any with Then there exists a positive constant such that for any and with satisfying (26)

2.2. Estimations of Fractional Heat Kernels

In this subsection, we list some estimations of kernels used in the proof. Denote by the integral kernel of fractional heat semigroups with . For the special case , , we denote by the integral kernel of fractional heat semigroups with . In ([24], Lemma 2.1), Miao et al. proved that the kernel satisfies the following pointwise estimate.

We denote by the integral kernel of heat semigroup . Specially, when and , we write as the integral kernel of heat semigroup . The Feynman-Kac formula implies that

Now we give the regularity estimate of the fractional heat kernel .

Lemma 11. Let , , and for some . For all , there exists a positive constant such that for

Proof. Since , , is a analytic semigroup, then the estimate for can be deduced by that of and the Cauchy integral formula. We omit the details.

Lemma 12. Let and for some . For all , there exists a constant such that for all and

Proof. The subordinate formula gives where satisfies (cf. [23]) A direct computation gives On the one hand, by (35) and the change of variables: and , we obtain On the other hand, we can get Thus, it is easy to see that Case 1: . We have . Moreover, it leads to Case 2: . It holds

Lemma 13 (see [26], Proposition 3.3). Let , , and for some . For any and all , there exists a positive constant such that for

Lemma 14 (see [26], Proposition 3.6). Let , for some and . For any and all , there exists a constant such that

Lemma 15 (see [27], Proposition 7). Let , , and . For all , there exist constants and with such that

Lemma 16 (see [27], Proposition 6). Let and . For all , there exist constants and with such that

3. Quantitative Boundedness

3.1. Localized Weights and Operators

Bongioanni et al. [8] introduced the following -localized weights.

Definition 17. Let and . The local weight class is defined as the set of all nonnegative locally integrable functions satisfying any ball where .

Lemma 18 (see [5], Lemma 3). There exists a sequence of points in such that the family of balls satisfies that For any , there exist positive constants and such that for any , .

We list some propositions about -localized weights.

Lemma 19 (see [8], Proposition 3, Corollary 1 and Lemma 1). Let and .
For any , with .
Let and . Then it is easy to see that , and there exists a constant such that for any , .
Let be a ball in . Assume that . Then has an extension on such that for any , and , where the implicit positive equivalence constants are independent of and .
The radial maximal function and the nontangential maximal function related to are defined, respectively, by setting, for any and where , , and denotes the cone of aperture with vertex .
For any , let . For any , , and , we define the following -localized , , , , , and , respectively, by setting, for any

Proposition 20. Let , , , and . There exists such that for any and

Proof. We first prove (51). Let be the family of balls given in Lemma 18. Let with positive constants and same as in Lemma 2. For any , let and . Via Lemma 2, for any , we have that is, (cf. ([19], Lemma 3.6). For any , has an extension on denoted by such that , for any , and where the implicit positive constants are independent of . We have It follows from (30) that For any and , combining with Corollary 8, Lemma 18, and (53), we obtain Next, we consider (50). Note that The rest of the proof of (50) is similar to that of (51), so we omit the details.
Wilson [28] introduced the following intrinsic Littlewood-Paley functions. For any , let be the set of all function defined on , such that For any , , and , define where with . In particular, if , .
For any , , and , let be the set of all functions defined on such that for any For any , , , and , define where with . In particular, if , .
Lerner [29] investigated the sharp bounds for and .

Lemma 21 (see [29], Theorem 1.1). Let and . There exists a constant such that for any and

Lemma 22 (see [28], Theorem 6.3). Let and . There is a constant such that, for all having pointwise.

Proposition 23. Let , , , , and . There exists such that for any and

Proof. For , we denote with . Thus where . On the one hand on the other hand where . Furthermore, it follows from Lemma 11 that

For any , we divide into two cases.

Case 1: if , we obtain

Case 2: if , we consider . Then , , and . Hence

When , for any , we have . Moreover

Therefore, we obtain for any , , and . Combining with Lemma 22 and the fact (from [30], Exercise 6.4) that , we have , . Via Lemma 21, similar to the proof of Proposition 20, we can get (66).

For any and where . Note that for any and

It follows from Lemma 22 that for any

Using Lemma 21, we obtain for any and

Similar to the proof of Proposition 20, we can get (67).

Proposition 24. Let , , , and . There exists such that for any and

Proof. For , we denote with . Thus where . On the one hand, for On the other hand where . Furthermore, it follows from Lemma 12 that Similar to the proof of Proposition 24, we have for any , , and . Following the process of the proof of Proposition 23, for , we can get (79) and (80).

3.2. Main Results

Theorem 25. Let , , and . Assume that with and , where , and , where and satisfies (18). There exists a constant such that for any and

Proof. We split the proof into three steps.
Step : for any , and Firstly, for , , , and , we divide To deal with , via Lemma 13, we obtain where , and .
Next, we estimate . Write where For , we have . To deal with , it follows from Lemma 15 that for with which indicates For where . It holds where . For the term , we can get where . Thus, we obtain Furthermore, it holds Let . By Lemmas 7 and 19 and Propositions 20 and 23, we can get for any , , and The (86) can be deduced from Lemma 9 and the density of in .
Step : for any , and For , , and , we have To deal with , let and . Split Firstly, we further split as , where Via Lemma 13, we obtain Since and , it holds By the fact that and Lemma 2, we can get for any where is the same constant in Lemma 2. Hence, it leads to where , . Next, we estimate . We consider two cases.
Case : . For this case, . It follows from Lemma 13 and (105) that where and .
Case : . Note that . By Lemma 13 and (105), it holds where . Furthermore, we can get where and . Hence, it is easy to see that Next, we estimate . Split where For , it can be seen that . To deal with , using Lemma 15, we obtain where Since , , we have . For , we can get where . For , it holds Hence, we obtain where . For , note that . Via Lemma 13 and (105), we can get where . Furthermore, it is easy to see that Thus, for and Let and . By the fact that , where and satisfies (18), we can get Via Lemma 9 and the fact that is dense in , we obtain (99).
Step : for any , and Since Combining with (99), we can get (123). This completes the proof of Theorem 25.

Theorem 26. Let , , and . Assume that with and , where , and , where and satisfies (18). There exists a constant such that for any and

Proof. Similar to the proof of Theorem 25, we can prove this theorem by the aid of Lemmas 14 and 16 and Proposition 24. Thus, we omit the details.

Lemma 27 (see [19], Lemma 4.1). Let , , , . Assume that . There exists a constant such that for any and ball of

Theorem 28. Let , , and . Assume that and , where , with . There exists a constant such that for any and

Proof. Let . We first introduce an auxiliary function , which is defined by setting, for any and It is easy to see that , where For , noting that , we have . Thus For , it follows from Lemma 13 that where For , similar to the proof of Theorem 25, we have where . To deal with , similarly, for , and , we obtain where . Thus, and Using (67), Lemma 9 and Theorem 25, we can get that for , , where and with , , and Next, we claim that for any , , , and Indeed, for any and , using Lemma 27, we have Hence Choose to be the family of all pairs . Since it can be deduced from Corollary 10 that for which implies (137) holds. Hence Furthermore, for any , , , and which completes the proof of Theorem 28.

Theorem 29. Let , , and . Assume that with and , where , and . There exists a constant such that for any and ,

Proof. Similar to the proof of Theorem 28, we can prove this theorem via Lemma 14 and Theorem 26. Hence, we omit the details.

Data Availability

The findings in this research do not make use of data.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was in part supported by the National Science Foundation of China (Nos. 11871293, 12071272) and Natural Science Foundation of Shandong Province (No. ZR2020MA004).

References

B. Simon, “Schrödinger operators in the twentieth century,” Journal of Mathematical Physics, vol. 41, no. 6, pp. 3523–3555, 2000.
View at: Publisher Site | Google Scholar
Z. Shen, “ estimates for Schrödinger operators with certain potentials,” Annales de l’institut Fourier, vol. 45, pp. 513–546, 1995.
View at: Publisher Site | Google Scholar
Z. Shen, “Estimates in for magnetic Schrödinger operators,” Indiana University Mathematics Journal, vol. 45, pp. 817–841, 1996.
View at: Google Scholar
Z. Shen, “On fundamental solutions of generalized Schrodinger operators,” Journal of Functional Analysis, vol. 167, no. 2, pp. 521–564, 1999.
View at: Publisher Site | Google Scholar
J. Dziubański and J. Zienkiewicz, “Hardy space associated to Schrödinger operator with potential satisfying reverse Hölder inequality,” Revista Matemática Iberoamericana, vol. 15, pp. 279–296, 1999.
View at: Google Scholar
J. Dziubański and J. Zienkiewicz, “ spaces associated with Schrödinger operators with potentials from reverse Hölder classes,” Colloquium Mathematicum, vol. 98, pp. 5–38, 2003.
View at: Publisher Site | Google Scholar
J. Dziubański, G. Garrigós, T. Martínez, L. Torrea, and J. Zienkiewicz, “BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality,” Mathematische Zeitschrift, vol. 249, pp. 329–356, 2005.
View at: Publisher Site | Google Scholar
B. Bongioanni, E. Harboure, and O. Salinas, “Classes of weights related to Schrodinger operators,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 563–579, 2011.
View at: Publisher Site | Google Scholar
B. Bongioanni, E. Harboure, and O. Salinas, “Commutators of Riesz transforms related to Schrödinger operators,” Journal of Fourier Analysis and Applications, vol. 17, no. 1, pp. 115–134, 2011.
View at: Publisher Site | Google Scholar
X. Duong, E. Ouhabaz, and L. Yan, “Endpoint estimates for Riesz transforms of magnetic Schrödinger operators,” Arkiv för Matematik, vol. 44, no. 2, pp. 261–275, 2006.
View at: Publisher Site | Google Scholar
X. Duong, L. Yan, and C. Zhang, “On characterization of Poisson integrals of Schrodinger operators with BMO traces,” Journal of Functional Analysis, vol. 266, no. 4, pp. 2053–2085, 2014.
View at: Publisher Site | Google Scholar
Z. Guo, P. Li, and L. Peng, “ boundedness of commutators of Riesz transforms associated to Schrödinger operator,” Journal of Mathematical Analysis and Applications, vol. 341, pp. 421–432, 2008.
View at: Publisher Site | Google Scholar
H.-Q. Li, “Estimations des opérateurs de Schrödinger sur les groupes nilpotents,” Journal of Functional Analysis, vol. 161, pp. 152–218, 1999.
View at: Publisher Site | Google Scholar
C. Lin and H. Liu, “ spaces and Carleson measures for Schrödinger operators,” Advances in Mathematics, vol. 228, pp. 1631–1688, 2011.
View at: Publisher Site | Google Scholar
T. Ma, P. Stinga, J. Torrea, and C. Zhang, “Regularity properties of Schrodinger operators,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 817–837, 2012.
View at: Publisher Site | Google Scholar
L. Tang, “Weighted norm inequalities for Schrödinger type operators,” Forum Mathematicum, vol. 27, no. 4, pp. 2491–2532, 2015.
View at: Publisher Site | Google Scholar
D. Yang, D. Yang, and Y. Zhou, “Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators,” Nagoya Mathematical Journal, vol. 198, pp. 77–119, 2010.
View at: Publisher Site | Google Scholar
J. Li, R. Rahm, and B. Wick, “ weights and quantitative estimates in the Schrödinger setting,” Mathematische Zeitschrift, vol. 293, pp. 259–283, 2019.
View at: Publisher Site | Google Scholar
J. Zhang and D. Yang, “Quantitative boundedness of Littlewood-Paley functions on weighted Lebesgue spaces in the Schrodinger setting,” Journal of Mathematical Analysis and Applications, vol. 484, no. 2, article 123731, 2020.
View at: Publisher Site | Google Scholar
T. Bui, T. Bui, and X. Duong, “Quantitative estimates for square functions with new class of weights,” Potential Analysis, vol. 57, no. 4, pp. 545–569, 2022.
View at: Publisher Site | Google Scholar
T. Bui, T. Bui, and X. Duong, “Quantitative weighted estimates for some singular integrals related to critical functions,” Journal of Geometric Analysis, vol. 31, no. 10, pp. 10215–10245, 2021.
View at: Publisher Site | Google Scholar
Y. Wen and H. Wu, “Quantitative weighted bounds for variation operators associated with heat semigroups in the Schrödinger setting,” Rocky Mountain Journal of Mathematics, 2022.
View at: Google Scholar
A. Grigor’yan, “Heat kernels and function theory on metric measure spaces,” Contemporary Mathematics, vol. 338, pp. 143–172, 2003.
View at: Publisher Site | Google Scholar
C. Miao, B. Yuan, and B. Zhang, “Well-posedness of the Cauchy problem for the fractional power dissipative equations,” Nonlinear Analysis, vol. 68, no. 3, pp. 461–484, 2008.
View at: Publisher Site | Google Scholar
Z. Shen, “On the Neumann problem for Schrödinger operators in Lipschitz domains,” Indiana University Mathematics Journal, vol. 43, no. 1, pp. 143–176, 1994.
View at: Publisher Site | Google Scholar
P. Li, Z. Wang, T. Qian, and C. Zhang, “Regularity of fractional heat semigroup associated with Schrödinger operators,” Fractal and Fractional, vol. 6, no. 2, p. 112, 2022.
View at: Publisher Site | Google Scholar
Z. Wang, P. Li, and C. Zhang, “Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via theorem,” Banach Journal of Mathematical Analysis, vol. 15, no. 4, pp. 1–37, 2021.
View at: Publisher Site | Google Scholar
J. Wilson, “The intrinsic square function,” Revista Matemática Iberoamericana, vol. 23, pp. 771–791, 2007.
View at: Google Scholar
A. Lerner, “Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals,” Advances in Mathematics, vol. 226, no. 5, pp. 3912–3926, 2011.
View at: Publisher Site | Google Scholar
J. Wilson, Weighted Littlewood-Paley Theory and Exponential-Square Integrability, vol. 1924 of Lecture Notes in Mathematics, Springer, Berlin, 2008.

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Copyright © 2023 Li Yang and Pengtao Li. 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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