Abstract
We study two sufficient conditions for the boundedness of a class of pseudodifferential operators with symbols in the Hölmander class on weighted Hardy spaces where belongs to Muckenhoupt class . The first one is an estimate from into . We get a better range of admissible and . The second one is a weighted version bounded for the operators on , and it is an addition to the literature.
1. Introduction
The purpose of this paper is to study some sufficient conditions for the boundedness of pseudodifferential operators T on weighted Hardy space , where the operators T have symbols in the Hölmander class . As in [1], for and , a symbol is a smooth function defined on such thatholds for all multi-indices , where is independent of x and ξ. We now assume that the symbol is smooth in both the spatial variable x and the frequency variable ξ.
Given , the pseudodifferential operator associated with the symbol is given bywhere denotes the Fourier transform of f. Moreover, we can express T by a kernel as (see, e.g., [2])
Pseudodifferential operators play an important role in the theory of partial differential equations. It is well known that the Hardy spaces coincide with the Lebesgue spaces when . The and weighted boundedness of the operator have been extensively studied. We refer to [1, 2, 3, 4] for the bounds and [5, 6, 7, 8] for the weighted bounds.
For , there is an estimate from into weak for the pseudodifferential operator (cf. [5]). As known, the Hardy space is an advantageous substitute for . The behavior of the pseudodifferential operator T on has attracted a lot of interest. For example, Alvarez and Hounie [5] have found that the pseudodifferential operator T with symbol in is bounded from into , where , and . Hounie and Kapp [9] have shown that the operator T with and is bounded from the local Hardy space into . Yabuta [10] has proved the operator T involving a modulus of continuity is bounded from into .
The bounds of the pseudodifferential operator T from the weighted Hardy space into the weighted Lebesgue space have also been studied. Yabuta [11] has found that the operator T is bounded from into , where and . In view of this, it is natural to look for a wide range of operator T in to study the bounds on the weighted Hardy space .
In this paper, we establish two estimates for the pseudodifferential operator T with symbols in . The first one is an estimate from into . We extend the result in Yabuta [11] to with and the operator T with , , and . Our first main result is stated as follows.
Theorem 1. Let , , and with , . If , then T is bounded from into , i.e., there exists a constant such that
The second one is an estimate on weighted Hardy spaces for the pseudodifferential operator T. It is well known that under certain conditions of , the operator T is bounded on (cf. [9, 12]). Alvarez and Hounie [5] have found that the pseudodifferential operator T is bounded on , where with , , and for some . It is natural to look for a weighted version estimate on . We now state our second main result.
Theorem 2. Let , , , and with , . Assume and . Then, T is bounded on ; i.e., there exists a constant such that
The remainder of this paper is organized as follows. In Section 2, we present some definitions and well-known results we use later. The aim of Section 3 is to set up the estimate from into for pseudodifferential operators T in . We develop a method to handle (see Proposition 1). The aim of Section 4 is to establish the estimate on weighted Hardy spaces for pseudodifferential operators T in .
Most of the notations we use are standard. C denotes a constant that may change from line to line and we write as shorthand for . If and , we mean . For a measurable set A, denotes the Lebesgue measure of A and the characteristic function. B will always denote a ball, and denotes the ball B dilated by t.
2. Notations and Auxiliary Lemma
In this section, we first present an auxiliary lemma about the pseudodifferential operator T associated with the kernel . Let be the class of Schwartz functions and be its dual space. The space of -function with compact support is denoted by . Pseudodifferential operators are bounded from to and so possess distribution kernels . Then, the following formula for the kernel is useful (cf. Proposition 3.1 in [9]; see also [5]).
Lemma 1. Let with , and associate with the pseudodifferential operator . Then, the distribution kernel of T is smooth away from the diagonal and is given bywhere satisfies for and the limit is taken in and independent of the choice of ψ. If and , satisfies the estimates
Moreover, for any multi-index and ,
The following useful bound for the pseudodifferential operator T is obtained by Michalowski et al. [7].
Lemma 2. Let , and . Then, for each and , there exists a constant such that
Remark 1. Obviously, the bounds of pseudodifferential operators T are established automatically.
Remark 2. satisfies the condition of m in Lemma 2.
The following useful bound of the pseudodifferential operator is obtained by Alvarez and Hounie [5].
Lemma 3. Let , , and . Then, the operator T is bounded on .
Remark 3. The range of here is and , respectively.
Remark 4. Obviously, satisfies the condition of m in Lemma 3.
Let . A nonnegative locally integrable function ω belongs to Muckenhoupt class , if there exists a constant , such that for all balls ,We denote .
It is well known that implies for all . Also, if , then for some . We thus write to denote the critical index of ω. For a measurable set E, we denote . The following lemma provides a way to compare and of a set E (see [13]).
Lemma 4. Let and . Then, there exists a constant such thatfor all balls B and measurable subsets .
Given a weight function ω on , we denote by the weighted Lebesgue space of all functions f satisfying
When , is . Analogous to the classical Hardy space, the weighted Hardy space can be defined in terms of maximal functions.
Definition 1. Let . The weighted Hardy space is defined bywhere is a fixed function with and for any . Moreover, we define .
Remark 5. Definition 1 is independent of the choice of (see [14]).
Definition 2. Let ω be a weight with the critical index . An atom with respect to ω is a function a satisfyingand for every multi-index α with .
The Hardy space is a linear space spanned by all of atoms with respect to ω. Namely, if and only if f can be written as (see [13])in the sense of , where each is an -atom with respect to ω and satisfiesMoreover, .
Definition 3. Let T be a pseudodifferential operator in . We say if for all with compact support and .
3. The Proof of Theorem 1
In this section, we prove that the pseudodifferential operators T in are bounded from into .
Proposition 1. Let , and . Assume pseudodifferential operator with , and . Then, there exists a constant such thatholds for all -atoms a with respect to ω, where .
Proof. Inspired by the proof of Lemma 3.2 in [15], we consider two cases about the radius r.
Case 1. When . For every and , we haveHence, by (8) and properties of -atoms with respect to ω, we havefor all . Thus,
Case 2. When . For every and , by moments condition, we haveBy the mean value theorem, , and (7), we havewhere we take and use the fact that if . Let us now consider two subcases.
Subcase 1. If , then, for any and ,Similar to the case , we getSince and , we haveNoting , it easy to see . This implies (17).
Subcase 2. If . Since , (22) yieldsIn view of (20) and (26), we finish the proof of Proposition 1.
Proof. The proof of Theorem 1 is motivated by the atomic decomposition for . Let . We obtain an atomic decomposition of f satisfying (15) and (16). So, to prove that the pseudodifferential operators T are bounded from into , it suffices to show that for each -atom a with respect to ω, we have . Recall that an -atom a with respect to ω is a function satisfyingfor some ball .
Now, let a be such an atom and writeIt is easy to estimate the term . Using Hölder inequality and -boundedness for the pseudodifferential operator T (see Remark 1), we getwhere C is independent of a.
For the term , we writeBy Proposition 1, we getsince . Combing (29) and (31), we finish the proof of Theorem 1.
4. The Proof of Theorem 2
In this section, we establish the weighted norm inequality on weighted Hardy spaces for pseudodifferential operators T in .
Proof. Without loss of generality, we assume , where . Fix and . By (15), it is sufficient to show that for each -atom a with respect to ω, with C independent of a. In order to do this, one can suppose and writeFor the term , by Hölder inequality, -boundedness of the maximal function , -boundedness of the pseudodifferential operator T, and (14), we getwhere C is independent of a.
To estimate , we first estimate for . For any , since (see Definition 3), we haveFor the term , by the mean value theorem and Hölder’s inequality, we havewhere depends on , and , and . Here, we use the inequalitiesand , and -boundedness of the pseudodifferential operator T (see Lemma 3).
To estimate and , we first estimate when and consider two cases about r.
Case 3. If , then for every and , we have . Hence, by (8), we have
Case 4. In the case of , we have . Thus, for every , from , (7) yieldswhere we use the fact that if and .
Let us now continue to estimate . When , using the mean value theorem and (37), we haveHere, we use the fact that under the condition of .
Similarly, in the case of , by the moments condition for a, the mean value condition, and (38), we getFor the term , we haveSince , we have . Thus,Meanwhile,So, in the case of , by (37), (42), and (43), we haveIn the case of , by (38), (42), and (43), we havesince .
Let . In view of (35), (39), (40), (44), and (45), we shall unify these formulas. Firstly, implies . Secondly, implies . Therefore,holds for any . Finally, since holds for and , we haveUsing these three facts, we haveNote that . Then, in all cases, we have andThis concludes the proof of Theorem 2.
Data Availability
The author confirm that no data were used to support this study. All References used were listed.
Disclosure
This study is a part of research work done by Yu-long Deng, a PhD student, under the supervision of the second author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was partially supported by the School of Mathematics and Computational Science, Xiangtan University, China, and Institute of Computational Mathematics, School of Science, Hunan University of Science and Engineering, China. This study was also supported by the Scientific Research Projects of Hunan Eduction Department (18C1073). The article processing charge is shared by them.