Journal of Function Spaces
Volume 2021 (2021), Article ID 6690963, 5 pages
https://doi.org/10.1155/2021/6690963

Research Article

On -Boundedness of -Pseudodifferential Operators

Jie Yang

College of Mathematics and System Science, Xinjiang University, Xinjiang 830046, China

Correspondence should be addressed to Jie Yang; yangjie1106@xju.edu.cn

Received 20 December 2020; Revised 6 February 2021; Accepted 16 February 2021; Published 20 February 2021

Academic Editor: Wenchang Sun

Copyright © 2021 Jie Yang. 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.

Abstract.

Let be the -pseudodifferential operators with symbol . When and , it is well known that is not always bounded in . In this paper, under the condition , we show that is bounded on .

1. Introduction and Main Results

Let and be the sets of all the symbol satisfying, for , , and ,

The classical pseudodifferential operator associated with the symbol is defined by

where is the Fourier transform of .

The generalized form of pseudodifferential operator is -pseudodifferential operator, which is introduced by Helffer in [1]. Helffer studied this operator to discuss the operator associated with the parameter , where is the Laplace operator and denotes some potential function for any . Moveover, the -pseudodifferential operator also provides a rigorous way to establish relationship each other for quantum physics and classical mechanics; see, for example, [2, 3]. Furthermore, by using -pseudodifferential operator, the Cauchy problem of semiclassical elliptic partial differential equations is studied in [3–5]. Because of these operators importance, many scholars have studied one; see, for example, [6–12] and the references given there.

Now, we consider the -pseudodifferential operator as follows:

where and

First of all, we review regularity theory for the classical pseudodifferential operators . In [13], Hörmander shown that is bounded in , when and . For , Ching [14] proved that is not bounded in . Moveover, for , Rodino [15] shown that is bounded in if and only if . However, the operator is not always -boundedness for ; see, for example, [13–15]. The necessary and sufficient conditions of -boundedness of are obtained by Higuchi [16] as . Recently, Aitemrar and Senoussaoui [6] studied boundedness and compactness on for -Fourier integral operators, where the symbol . Moveover, Aitemrar and Senoussaoui [9] also discussed regularity on Bessel potential spaces. Furthermore, Aitemrar and Senoussaoui [7] considered the symbol for -pseudodifferential operator, which belongs to the class , namely,

By using this condition and the compact set , they obtained the global -boundedness for -Fourier integral operators. The noncompact case for , they also proved -boundedness for . Motivated by this noncompact case, for , we consider the -boundedness of -pseudodifferential operators. In what follows, we mainly concentrate on the -boundedness for -pseudodifferential operator with the specific symbol . Our main result could be stated as follows:

Theorem 1. Let be the -pseudodifferential operators given by (3) with symbol , where is as in Definition 3. Then, for , there exists a constant such that

Remark 2. For the classical pseudodifferential operator, in [16, 17], they have constructed an example that the operator is not bounded in as Moveover, here, we remark that the symbol is different from the symbol in [6].

2. Definitions, Notations, and Preliminaries

Let be an -dimensional Euclidean space, , , , and . For any multi-index and , we let

and . Also, we also define and as follows:

Throughout this article, we denote by a positive constant which is independent of the main parameters, but it may vary from line to line. We sometimes write as shorthand for .

We give the class defined below, which will play significant role in our investigations.

Definition 3. Let be a real number. A function which is smooth in the frequency variable and bounded measurable in the spatial variable , belongs to the symbol class , if it satisfies, for all multi-indices ,

where is a nonnegative and decreasing function on and satisfies

Remark 4. If satisfies (10), then

To prove the main theorem, we need the following lemma.

Lemma 5. Let be the -pseudodifferential operators given by (3) with symbol and . Then, for , there exists a constant such that

The proof method of lemma is similar to Corollary 3.2 in [18], the details being omitted.

3. Proof of Main Result

In this section, we shall prove the main result Theorem 1.

First we need a dyadic partition of unity, let be the annulus

where and . When with , we decompose the operator as

The first term in (13) is bounded on from Lemma 5. After a change of variables, we have

The kernel of the operator is given by

Let

Then

and satisfying

Now, we deal with the high frequency component of . Let . Then

The kernel of the operator reads

We define

By this, we have

Now, we claim that

In fact, by using (18), we have

Next, we consider the following differential operators, for

So

From this and (25), it follows that

Moreover, by (27) and (24), we further obtain

Integration by parts yields

Thus,

which implies that

and hence

By Young’s inequality, we obtain

Therefore, we have

Namely

Next, we need the Littlewood-Paley decomposition. Let be a smooth radial function which equals to one on the unit ball centric at the origin and supported on its concentric double. Set and . Then, for any ,

and for . Thus

Furthermore, we have

where ; in order to simplify, let

And hence

We use Remark 4 here. This finishes the proof of Theorem 1.

Data Availability

The data is available at https://journals.tubitak.gov.tr/math/issues/mat-18-42-4/mat-42-4-14-1610-104.pdf.

Conflicts of Interest

The author declares that he/she has no conflicts of interest.

Acknowledgments

The author would like to express his deep thanks to the referees for their very careful reading and useful comments which do improve the presentation of this article. The research of first author is supported by the National Natural Science Foundation of China (11561065) and the Doctoral Foundation of Xingjiang University (Grants No. 62008031).

References

B. Helffer and D. Robert, “Calcul fonctionnel par la transformation de Mellin et operateurs admissibles,” Journal of Functional Analysis, vol. 53, no. 3, pp. 246–268, 1983.

D. Robert and B. Helffer, “Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques,” Annales de l'institut Fourier, vol. 31, no. 3, pp. 169–223, 1981.

D. Robert, “On semiclassical approximation,” in Progress in Mathematics, vol. 68, Birkhäuser Boston, Inc., Boston, MA, 1987.

B. Helffer, “Théorie spectrale pour des opérateurs globalement elliptiques,” Astérisque, vol. 112, 1984.

L. Hörmander, “The Weyl calculus of pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 32, no. 3, pp. 359–443, 1979.

C. Aitemrar and A. Senoussaoui, “h-admissible Fourier integral operators,” Turkish Journal of Mathematics, vol. 40, no. 3, pp. 553–568, 2016.

C. Aitemrar and A. Senoussaoui, “On the global Lp boundedness of a general class of h-Fourier integral operators,” Turkish Journal of Mathematics, vol. 42, no. 4, pp. 1726–1737, 2018.

J. Butler, “Global h-Fourier integral operators with complex-valued phase functions,” The Bulletin of the London Mathematical Society, vol. 34, no. 4, pp. 479–489, 2002.

A. O. Farouk and A. Senoussaoui, “The boundedness of -admissible Fourier integral operators on Bessel potential spaces,” Turkish Journal of Mathematics, vol. 43, no. 5, pp. 2125–2141, 2019.

C. Harrat and A. Senoussaoui, “On a class of h-Fourier integral operators,” Demonstratio Mathematica, vol. 47, pp. 595–607, 2014.

V. S. Rabinovich, “Local exponential estimates for h-pseudo-differential operators with operator-valued symbols. Pseudo-differential operators: analysis, applications and computations,” in Pseudo-Differential Operators: Analysis, Applications and Computations, pp. 173–190, Birkhäuser/Springer Basel, AG, Basel, 2011.

A. Senoussaoui, “Opérateurs h-admissibles matriciels ásymbole opérateur,” African Diaspora Journal of Mathematics, vol. 4, no. 1, pp. 7–26, 2007.

L. Hörmander, “L² continuity of pseudo-diferential operators,” Communications on Pure and Applied Mathematics, vol. 24, pp. 529–535, 1971.

C. H. Ching, “Pseudo-differential operators with nonregular symbols,” Journal of Differential Equations, vol. 11, pp. 436–447, 1972.

L. Rodino, “On the boundedness of pseudo-differential operators in the class L_(ρ,1)^m,” Proceedings of the American Mathematical Society, vol. 58, pp. 211–215, 1976.

Y. Higuchi and M. Nagase, “On the L²-boundedness of pseudo-differential operators,” Kyoto Journal of Mathematics, vol. 28, pp. 133–139, 1988.

A. Senoussaoui, “On the unboundedness of a class of Fourier integral operators on L^2 (R^n),” Journal of Mathematical Analysis and Applications, vol. 405, no. 2, pp. 700–705, 2013.

M. Nagase, “On some classes of L^p-bounded pseudodifferential operators,” Osaka Journal of Mathematics, vol. 23, no. 2, pp. 425–440, 1986.