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).