A Remark Related Pseudo-Differential Operators on Multi-Parameter Besov-Lipschitz and Triebel-Lizorkin Spaces

Xu, Chengdan

doi:https://doi.org/10.1155/2022/7953457

Journal of Function Spaces

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Research Article | Open Access

Volume 2022 | Article ID 7953457 | https://doi.org/10.1155/2022/7953457

A Remark Related Pseudo-Differential Operators on Multi-Parameter Besov-Lipschitz and Triebel-Lizorkin Spaces

Chengdan Xu¹

Academic Editor: Mitsuru Sugimoto

Received23 Sept 2022

Accepted17 Oct 2022

Published07 Nov 2022

Abstract

Let a symbol belongs to the class with and . This paper mainly establish the boundedness of the class of all bi-parameter pseudo-differential operators associated with the symbol on the multi-parameter non-homogeneous Besov-Lipschitz and Triebel-Lizorkin spaces.

1. Introduction

In this paper, we mainly consider the bi-parameter pseudo-differential operator for , where the symbol with , , , and , , for .

The pseudo-differential operator, which is formalised in Kohn and Nirenberg [1] and Hörmander [2] in 1960s, have been developed as a popular mathematical tool and widely applied in various fields of mathematics. Over the last few decades, the study of pseudo-differential operator have attracted many related mathematicians and many authors have been working on proving boundednesses of the one-parameter pseudo-differential operators on various classical spaces. It’s worth noting that the boundedness for the class was first treated by Calderón-Vaillancourt in [3], which means that the boundedness of all the derivatives of a symbol ensures the -boundedness of the corresponding operator. Further, Laptev [4] proved that any pseudo-differential operator with symbols in is a standard Calderón-Zygmund operator, hence the -boundedness is obvious. Hörmander [5] pointed out that the pseudo-differential operators with symbols in are bounded on if and only if , where , and (see also Fefferman [6] and Stein [7], Chapter VII, Section 5.12). Sugimoto [8] showed the pseudo-differential operators with symbols in are bounded on Besov spaces when , where , , and . Furthermore, Jun Park [9] studied the boundedness of pseudo-differential operators with symbols in on Triebel-Lizorkin spaces and Besov spaces when , where , and . Some weighted results about boundednesses of pseudo-differential operators also have been verified, such as in [10–12] and so on.

On the other hand, it’s well-known that there are two different directions being extended to developing the Calderón-Zygmund singular integral theory. One of the two extensions is non-convolution operators with a kernel, such as the Cauchy-operator on Lipschitz curves and the interested reader can see [13–16] for more reference. The other extension is due to R. Fefferman and E. M. Stein, they extended the classical singular integrals into the product situation in [17]. Later, Journé have unified these two different directions into a certain extent in [18], and studied a class of singular integral operators defined on a product of Euclidean spaces, the reader can see [16, 19–23] for further information on this topic. For the theory of multi-parameter singular integrals and function spaces plays an important role in many aspects of harmonic analysis, many authors have studied the pure product theory (which is also called multi-parameter harmonic analysis) over the last decades. For instance, the estimates for multi-parameter Coifman-Meyer Fourier multipliers in [24]; the estimates and Calderón-Vaillancourt type theorem for multi-parameter and multi-linear Fourier multipliers and pseudo-differential operators in [25, 26]. In particular, Sugimoto [27] showed that any pseudo-differential operators with symbols in Besov space implies the -boundedness. Furthermore, Tomita [28] generalized the results in [27] and studied the -boundedness of pseudo-differential operators with non-regular symbols, where . As some applications, Tomita [28] also obtained the -boundedness for symbols in Besov spaces and modulation spaces, respectively. Recently, the properties of multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces and the boundedness of multi-parameter singular integral operators on them have been established in [29–32].

Generally speaking, the pseudo-differential operators can be regarded as an extension for Fourier multipliers. On the other hand, the -modulation spaces , which were introduced by Gröbner’s PH.D. thesis [33], function as a kind of intermediate space between modulation spaces and the well-known Besov spaces. Here the parameter works as the “adjuster”, modulation spaces are the special -modulation spaces when and Besov spaces can be treated as the limit case of -modulation spaces when the parameter tends to . The difference between modulation spaces and Besov spaces can be referred to Tomita [28]. We also remark that the boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces have been obtained in [34]. The author in this paper and others have introduced the bi-parameter -modulation spaces with and bi-parameter modulation spaces in [35], and established the boundedness of bi-parameter pseudo-differential operators on these spaces. While not much is known about the boundedness of pseudo-differential operators on multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces:

The main purpose of this paper is to investigate the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces. Let us recall the definition of the bi-parameter pseudo-differential operators as the following.

For , be real pairs, where with . A function is said to belong to if there exists a positive constant such that the following inequality holds for all multi-indices , where , .

Denote the class of all bi-parameter pseudo-differential operators with by , for , where , .

Next we give the main results about the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces.

Theorem 1. Let , and with and , . Then the bi-parameter pseudo-differential operators defined in the form of (3) with symbols is bounded on the bi-parameter non-homogeneous Besov-Lipschitz . That is, there exists a positive constant such that holds for all and .

Theorem 2. Let , and with and , . Then the bi-parameter pseudo-differential operators defined in the form of (3) with symbols is bounded on the bi-parameter non-homogeneous Triebel-Lizorkin spaces . That is, there exists a positive constant such that holds for all and .

The contents of this paper are organized as follows: In Section 2, we state some preliminary notation that will be used throughout this paper and introduce the bi-parameter non-homogeneous Besov-Lipschitz spaces and the bi-parameter non-homogeneous Triebel-Lizorkin spaces. Moveover, we give some necessary lemmas which will serve to prove the main theorems in Section 3. Since it is of importance for its own sake, we will give some self-contained proofs here. We give the detailed proof for Theorem 1 in Section 4 and the detailed proof for Theorem 2 in Section 5.

As it was pointed out by one of the referees our work seems to have overlapping with our previous work [35]. The mainly thought to prove the main theorems maybe seem similar. However, we also give some new results, such as some norm estimates related to the bi-parameter pseudo-differential operators with different symbols (see lemma 16, lemma 17 and lemma 18). In particular, the method to prove Theorem 2 is slightly different from the idea to deal with Theorem 1, where we have applied the bi-parameter -multiplier estimates (see lemma 19). These new lemmas are key tools in our proofs.

2. Preliminary Notation and Definition

In this section, we fix notations and provide some preliminary definitions that will be used in this paper.

General notation. We will denote universal positive constants by which can be different line to line. If and are two quantities (typically nonnegative), we use to denote the statement for some absolute constant . Also, we use to denote the statement and .

Let and denote the Schwartz space of rapidly decreasing smooth functions and its dual space of tempered distributions, respectively, with their usual topologies. Write , for any multi-index , denote and .

For , the bi-parameter Lebesgue space is equipped with the (quasi)-norm

When ,

For , we denote by the set of all complex number sequences such that with the usual modification for .

Through out the following paper, we denote and , and . We use denote the Fourier transform and denote the inverse Fourier transform. That is, for any .

We also use denote the partial Fourier transform of functions with respect to the -variable and the -variable, and denote the inverse partial Fourier transform of functions with respect to the -variable and the -variable. That is, for any .

A class of Fourier multiplier operators can be stated as follow:

In order to introduce the definition of bi-parameter non-homogeneous Besov-Lipschitz spaces and non-homogeneous Triebel-Lizorkin spaces, we first recall the definition of the test function which is defined on .

Let denote the space of Schwartz function and be its dual. The test function defined on can be given by where , . Set

Definition 3. Let be the collection of all the systems that satisfy the following conditions: (i)supp , supp for , ;(ii) for all , ;(iii)There exists a positive number such that the inequality holds for all non-negative integer and every multi-index and , .

Definition 4. Let , . The bi-parameter non-homogeneous Besov-Lipschitz space is defined by where with a slight modification in the case of and . Here are non-negative integers.

Definition 5. Let , . The bi-parameter non-homogeneous Triebel-Lizorkin space is defined by where with a slight modification in the case of and . Here are non-negative integers.

It should pointed out that the definition of bi-parameter non-homogeneous Besov-Lipschitz space and the definition of bi-parameter non-homogeneous Triebel-Lizorkin space are independent with the choosing of and . Definition 4 and Definition 5 are identical to definitions of weighted Besov-Lipschitz and Triebel-Lizorkin spaces defined in Sugimoto [7, 27] with the weight . We also refer the reader to [29, 32] for the bi-parameter homogeneous cases.

For simplicity and convenience, we will choose and denote the inverse Fourier transform of by , that is, . Denote and for all , , . Hence we can deduce , for , and where denotes the identity operator. When the integer , we denote by for convenience. Hence we have

For short, the bi-parameter non-homogeneous Besov-Lipschitz space will be called as the bi-parameter Besov space and the bi-parameter non-homogeneous Triebel-Lizorkin space called as the bi-parameter Triebel space in the next. According to the above, the bi-parameter Besov space and the bi-parameter Triebel space can also be defined as follows:

Definition 6. Let , . The bi-parameter Besov space is defined by where

Definition 7. Let , . The bi-parameter Triebel space is defined by where

According to (18) and (20), the symbol can be decomposed as where the sequence and are the decomposition constituting bi-parameter Besov spaces and Triebel space, and by the support condition of in Definition 3 for . By using Definition 6 and Definition 7, we have where

3. Necessary Lemmas

In this section, we show some necessary lemmas that will be used to prove the main theorem in this paper. Let be a compact subset of and be a sequence of compact subsets of . Write where for .

The following lemmas (Lemma 8 to Lemma 10) are requirements to prove the main theorem and they have been considered in our previous work [35], we will omit the detailed proof here.

Lemma 8. Let , and . Then we have the estimates holds for all , where the implicit constant is independent of and .

Lemma 9. (see [36]). Let , then there exists a constant such that for all sequences of locally integrable functions on , where denotes the strong maximal operator. The strong maximal operator is defined by for any , where , and is a locally integrable function on .

Lemma 10. Let , for any and any function whose Fourier transform is supported in , assume that is an integrable function. Then there exists two constants and (which depend only on ) such that where denotes the strong maximal operator.

By Lemma 9 and Lemma 10, we can deduce the following Lemma 11.

Lemma 11. Let , and . If, then we have holds for all , where the implicit constant is independent of and .

Lemma 12. Let . If , then we have the estimates holds for all .

Proof. We only need to prove by Lemma 11. Since By the definition of the sequence , we get Let and for in Lemma 11, hence we have .
By Lemma 12, we can obtain the following lemma due to Triebel [37].

Lemma 13. Let and . If , then we have the estimates holds for all .

For any and with , the bi-parameter Bessel potential can be denoted as . By adopting the main idea for Besov spaces and Triebel spaces in Triebel [37], we give the following lemma for the bi-parameter case.

Lemma 14. Let , , with , . Then (i)The mapping is isomorphic(ii)The mapping is isomorphic

To prove Theorem 1 and Theorem 2, we will change the formulation of slightly as follows. Choosing a function and satisfy on , and besides with .

Set

Then, we have on the support of . Thus, we can deduce where

Lemma 15. Let with , . Suppose . Then the estimates holds for any , where the implicit constant is independent of and

Proof. Firstly, we deal with . For any and any multi-index with , according to the Leibniz rule, we get By the construction of , , we can deduce If in the above inequality, then Similarly, if we have
Now, we start to estimate . By using the Fubini-Tonelli theorem, we can deduce Hence, we have Obviously, when , it gives hat Therefore, On the other hand, by using the times integration by parts with respect to the -variable and using the times integration by parts with respect to the -variable, respectively, we can infer that for any .
According to above, we can obtain holds for any and with . Thus, our desired result can be immediately obtained.
By applying Lemma 15, we have the following lemma.

Lemma 16. Let , with , . Suppose . Then we have the estimates holds for any .

Proof. Taking and , respectively. By Lemma 15 and the Fubini-Tonelli theorem, we have Set It remains to estimate the integration expression of . Next we will deal with by dividing it into the following four parts Now we need to consider and , respectively.

Case 1. When and , then

Case 2. When and , by transformation of spherical polar coordinates, we get

Case 3. When and , by symmetry with Case , we obtain

Case 4. When and , by transformation of spherical polar coordinates, we have

Combining above four cases together, we have

Taking the (quasi) norm to the both sides and using Lemma 12, we can get the desired result.

Lemma 17. Let and , with , . Suppose . If , then we have the estimates holds for any .

Proof. By the estimates in the proof of Lemma 16, we get Taking the (quasi) norm on and and (quasi) norm on and both sides, respectively. We can obtain the desired result by using Lemma 13.
At last, we need the following lemma for the preparation of proving Theorem 1.

Lemma 18. Let and for . Then we have where the implicit constant is independent of , .

Proof. When , by the Young inequality, we get When , by the decomposition of the symbol and the Fubini-Tonelli theorem, we can deduce According to the facts that It follows that Besides, by the support of defined as in the definition of the bi-parameter Besov space, we have According to the above, if for any non-negative integer , we can get the following four cases:
Case 1:
Case 2:
Case 3:
Case 4:
Therefore, the estimates always vanishes unless the above four cases. From above four cases, it implies that Let and set Then, by Lemma 8, we have Hence, it follows that According to above, we can get the desired result.

4. Proof of Theorem 1

In this section, we will give a detailed proof for Theorem 1.

Note that for all , we have

Now we give the detailed proof of Theorem 1.

Proof of Theorem 1. Set . If all pseudo-differential operators with symbols are bounded on , by using Lemma 14, we can deduce On the other hand, if all are pseudo-differential operators with symbols , then the operators are pseudo-differential operators with symbols . Hence, we only need to prove the case that . That is, we need to prove By Definition 6, we obtian For any , by Lemma 16 and Lemma 18, we have Taking -th power and summing the both sides over and , we get Choosing and such that Hence, we can obtain Then the proof of Theorem 1 is accomplished.

5. Proof of Theorem 2

If , the bi-parameter Bessel-potential space is defined by where .

Next, we show the bi-parameter -multiplier estimates, which will be used to prove Theorem 2. The main idea of proving this estimate is enlightened by Triebel’s book [37], we will omit the proof here.

Lemma 19 (Bi-parameter -multiplier estimates). Let and , , and be a sequence of compact subsets of . If for Then there exists a constant such that holds for all systems and all function spaces

Finally, we give the detailed proof of Theorem 2.

Proof of Theorem 2. Similar to the analysis in the proof of Theorem 1, we only need to prove the case for . That is, it is enough to prove By Definition 7, for any , By Lemma 19 and Lemma 17, one can obtain Choosing and , hence we have Then the proof of Theorem 2 is accomplished.

Data Availability

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The research was supported by Research Start-Up Fund (NO. 11432832611915).

References

J. J. Kohn and L. Nirenberg, “An algebra of pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 18, no. 1-2, pp. 269–305, 1965.
View at: Publisher Site | Google Scholar
L. Hörmander, “Pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 18, no. 3, pp. 501–517, 1965.
View at: Publisher Site | Google Scholar
A. P. Calderón and R. Vaillancourt, “A class of bounded pseudo-differential operators,” Proceedings of the National Academy of Sciences of the United States of America, vol. 69, no. 5, pp. 1185–1187, 1972.
View at: Publisher Site | Google Scholar
A. Laptev, “Spectral asymptotics of a class of Fourier integral operators,” Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 43, pp. 92–115, 1981.
View at: Google Scholar
L. Hörmander, “Pseudo-differential operators and hypoelliptic equations,” Proceedings of Symposia in Pure Mathematics, vol. 10, pp. 138–183, 1967.
View at: Publisher Site | Google Scholar
C. Fefferman, “ bounds for pseudo-differential operators,” Israel Journal of Mathematics, vol. 14, no. 4, pp. 413–417, 1973.
View at: Publisher Site | Google Scholar
M. Sugimoto, “-boundedness of pseudo-differential operators satisfying Besov estimates II,” Journal of the Faculty of Science, University of Tokyo, vol. 35, pp. 149–162, 1988.
View at: Google Scholar
M. Sugimoto, “Pseudo-differential operators on Besov spaces,” Tsukuba Journal of Mathematics, vol. 12, no. 1, pp. 43–63, 1988.
View at: Publisher Site | Google Scholar
B. J. Park, “On the -boundedness of pseudo-differential operators on Triebe-Lizorkin ans Besov spaces,” Journal of Mathematical Analysis and Applications, vol. 461, pp. 544–576, 2018.
View at: Google Scholar
Y. Lin, Z. Liu, C. Xu, and Z. Ren, “Weighted estimates for Toeplitz operators related to pseudodifferential operators,” Journal of Function Spaces, vol. 2016, Article ID 1084859, 19 pages, 2016.
View at: Publisher Site | Google Scholar
N. Miller, “Weighted sobolev spaces and pseudodifferential operators with smooth symbols,” Transactions of the American Mathematical Society, vol. 269, no. 1, pp. 91–109, 1982.
View at: Publisher Site | Google Scholar
L. Tang, “Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators,” Journal of Functional Analysis, vol. 262, no. 4, pp. 1603–1629, 2012.
View at: Publisher Site | Google Scholar
A. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Mathematica, vol. 88, pp. 85–139, 1952.
View at: Publisher Site | Google Scholar
R. Coifman and Y. Meyer, “Au delà des opérateurs pseudo-differentiels,” Astérisque, no. 57, 1979.
View at: Google Scholar
R. R. Coifman, A. McIntosch, and Y. Meyer, “L'integrale de Cauchy Definit un Operateur Borne sur Pour Les Courbes Lipschitziennes,” Annals of Mathematics, vol. 116, no. 2, pp. 361–387, 1982.
View at: Publisher Site | Google Scholar
J. L. Journé, Calderón-Zygmund operators, pseudo-differential operators andthe Cauchy integral of Calderón, Springer-Verlag, 1994.
R. Fefferman and E. M. Stein, “Singular integrals on product spaces,” Advances in Mathematics, vol. 45, no. 2, pp. 117–143, 1982.
View at: Publisher Site | Google Scholar
J. L. Journé, “Calderón-Zygmund operators on product spaces,” Revista Matemática Iberoamericana, vol. 1, pp. 55–91, 1985.
View at: Publisher Site | Google Scholar
R. Fefferman, “Calderón-Zygmund theory for product domains: spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 83, no. 4, pp. 840–843, 1986.
View at: Publisher Site | Google Scholar
R. Fefferman, “Harmonic analysis on product spaces,” Annals of Mathematics, vol. 126, no. 1, pp. 109–130, 1987.
View at: Publisher Site | Google Scholar
J. L. Journé, “A covering lemma for product spaces,” Proceedings of American Mathematical Society, vol. 96, no. 4, pp. 593–598, 1986.
View at: Publisher Site | Google Scholar
J. L. Journé, “Two problems of Calderón-Zygmund theory on product spaces,” Annales de l'institut Fourier, vol. 38, no. 1, pp. 111–132, 1988.
View at: Publisher Site | Google Scholar
J. Pipher, “Journé’s covering lemma and its extension to higher dimensions,” Duke Mathematical Journal, vol. 53, no. 3, pp. 683–690, 1986.
View at: Publisher Site | Google Scholar
C. Muscalu, J. Pipher, T. Tao, and C. Thiele, “Bi-parameter paraproducts,” Acta Mathematica, vol. 193, no. 2, pp. 269–296, 2004.
View at: Publisher Site | Google Scholar
W. Dai and G. Lu, “ estimates for multi-linear and multi-parameter pseudo-differential operators,” Bulletin de la Société Mathématique de France, vol. 143, no. 3, pp. 567–597, 2015.
View at: Publisher Site | Google Scholar
L. Huang and J. Chen, “The boundedness of multi-linear and multi-parameter pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 20, no. 2, pp. 801–815, 2021.
View at: Publisher Site | Google Scholar
M. Sugimoto, “-boundedness of pseudo-differential operators satisfying Besov estimates I,” Journal of the Mathematical Society of Japan, vol. 40, pp. 105–122, 1988.
View at: Google Scholar
N. Tomita, “On the -boundedness of pseudo-differential operators with non-regular symbols,” Arkiv för Matematik, vol. 49, no. 1, pp. 175–197, 2011.
View at: Publisher Site | Google Scholar
W. Ding and G. Lu, “Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators,” Transactions of the American Mathematical Society, vol. 368, no. 10, pp. 7119–7152, 2016.
View at: Publisher Site | Google Scholar
W. Ding, G. Lu, and Y. Zhu, “Multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition,” Forum Mathematicum, vol. 28, no. 1, pp. 25–42, 2016.
View at: Publisher Site | Google Scholar
W. Ding and Y. Zhu, “Duality of weighted multiparameter Triebel-Lizorkin spaces,” Acta Mathematica Sinica, English Series, vol. 37, no. 4, pp. 1083–1104, 2017.
View at: Publisher Site | Google Scholar
G. Lu and Y. Zhu, “Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters,” Acta Mathematica Sinica, English Series, vol. 29, no. 1, pp. 39–52, 2013.
View at: Publisher Site | Google Scholar
P. Gröbner, Banachraume glatter Funktionen und Zerlegungsmethoden [Ph.D. thesis], University of Vienna, 1992.
L. Chen, G. Lu, and X. Luo, “Boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces,” Nonlinear Analysis, vol. 134, pp. 55–69, 2016.
View at: Publisher Site | Google Scholar
C. Xu and L. Huang, “Boundedness of bi-parameter pseudo-differential operators on bi-parameter -modulation spaces,” Nonlinear Analysis, vol. 180, pp. 20–40, 2019.
View at: Publisher Site | Google Scholar
L. Hörmander, “Estimates for translation invariant operators in spaces,” Acta Mathematica, vol. 104, no. 1-2, pp. 93–140, 1960.
View at: Publisher Site | Google Scholar
H. Triebel, “Theory of Function Spaces,” Monographs in Mathematics, Birkhäuser Verlag, Basel, 1983.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Chengdan Xu. 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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