Commutators of Pseudodifferential Operators on Weighted Hardy Spaces
In this paper, we establish an endpoint estimate for the commutator, , of a class of pseudodifferential operators with symbols in Hörmander class . In particular, there exists a nontrivial subspace of such that, when belongs to this subspace, the commutators is bounded from into , which we extend the well-known result of Calderón-Zygmund operators.
The purpose of this paper is to find out a proper subspace of such that, the commutators of pseudodifferential operators is bounded on weighted Hardy space , where the operators associated with the symbols in the Hölmander class and . As in , we firstly recall some notations and lemmas. For and , a symbol is a smooth function defined on such thatholds for all multi-indices , where is independent of and (see, e.g., ).
Given an infinitely differentiable function with compact supports and symbol , the pseudodifferential operator is defined bywhere is the Fourier transform of and we write . Moreover, the operator can be expressed by a distribution kernel as (see, e.g., )
Let and be a Calderón-Zygmund operator. A classical result in  stated that the commutator operators , defined byis bounded on for . However, it fails to be of weak type and of type when (see, [5, 6]). Instead, some endpoint theories are provided.
Remark that if the symbol satisfies some particular assumptions, pseudodifferential operator in is a Calderón-Zygmund operator (see, ). Correspondingly, when and , the boundness of on Lebesgue space for was considered (see, e.g., [8–10]).
It is widely known that is an advantageous substitute for . The behavior of commutator on has also attracted a lot of interest. For example, when (see, ), Yang et al.  obtained that is bounded from into , where with ; Hung and Ky  established an estimate for on local Hardy space . Very recently, Deng and Long  got an estimate for from into weak , where .
For , there are numerous papers dealing with the weighted boundedness of the commutators for and we refer to [15–20] for more details, where and . A nature question is that can one establish an estimate for on weighted Hardy spaces ?
In general, the commutators is not bounded from the weighted Hardy spaces into the weighted Lebesgue spaces if is not a constant function, even is a Calderón-Zygmund operator. It is worthy to pointing out that in , Liang et al. found a proper subspaces of , such that, the commutators of Calderón-Zygmund operator is bounded on weighted Hardy spaces. Motivated by this result, we wonder whether there exists a nontrivial subspace of such that when belongs to this subspace, the commutators of pseudodifferential operator is bounded on .
The main concern of this paper is to give an answer to the above question. For this purpose, we recall the definition of the Muckenhoupt weights . A nonnegative measurable function is said to be in the Muckenhoupt class for , ifand for , ifwhere the supremum is taken over all balls and .
As known, if , then for some . We thus write to denote the critical index of . For a measurable set , we denote . The following lemma provides a way to compare and of a set (see ).
Lemma 1. Let and . Then, there exists a constant such thatfor all balls and measurable subsets .
Definition 1. Let and . A locally integrable function is said to belong to ifHere, and the supremum is taken over all balls with center and radius .
We point out that the space has been studied in [21, 23, 24]. A locally integrable function is said to be in ifwhere the supremum is taken over all balls .
In , the space is proved to be a subspace of , and not be a trivial space since it contains the Lipschitz function with compact support and also that,
Lemma 2. Let and . Then, there exists a constant such that, for any and any ball , The first main result is stated as follow.
Theorem 1. Let , satisfies and . Assume that the pseudodifferential operator with , andThen, the commutator is bounded from into ; i.e., there exists a constant such that, for all ,Finally, we make some conventions on notations. denotes a positive constant may change from line to line and we write as shorthand for . If and , we mean . For a measurable set , denotes the Lebesgue measure of . will always denote a ball and denotes the ball dilated by .
2. Notations and Technical Lemmas
In this section, we begin our story by presenting an estimate about the pseudodifferential operator 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 1 in , see also ).
Lemma 3. Let with and associate with the pseudodifferential operator . Then, the distribution kernel of is smooth away from the diagonal and is given bywhere satisfies for and the limit is taken in and independent of the choice of . If and , satisfies the estimatesMoreover, for any multi-index and ,In , the following is derived from Lemma 3.
Lemma 4. Let and and the pseudodifferential operator associated with the distribution kernel . Then, for any and every , we havewhere .
Let and . We denote by the weighted Lebesgue space of all measurable functions satisfyingWhen , is defined to be the same as , the following useful bounds for pseudodifferential operator are due to Michalowski et al. .
Lemma 5. Let with and . Then, for each and , there exists a constant such thatLet with . Then, for any , the maximal function of a distribution is defined bywhere for any . Let . Then, the maximal function is bounded on if and only if . Analogous to the classical Hardy space, the weighted Hardy space can be defined in terms of maximal functions.
Definition 2. Let . The weighted Hardy space is defined bywhich is independent of the choice of . Moreover, we define .
Definition 3. Let be a weight with critical index . An-atom is a function satisfyingand for every multi-index with . Conventionally, means for all -atoms .
The Hardy space is spanned by all of -atoms (see ). Namely,in the sense of , where each is an -atom and satisfiesMoreover, .
Deng et al.  got some sufficient conditions for the boundedness of pseudodifferential operators on weighted Hardy space .
Lemma 6. Let , , , and with , . If , then is bounded from into , i.e., there exists a constant such that
3. The Proof of Theorem 1
In this section, we establish the sufficient condition for the boundedness of from into . As in , we need the following proposition.
Proposition 1. Write . Let and . Assume that the pseudodifferential operator with , and . Then, there exists a constant such that, for any and -atom ,where .
Proof. It suffices to show thatandFor , it is easy to see thatThen, by Lemma 6, the boundedness of the operator from to , we conclude thatAlso, by Hölder inequality, Lemma 2 and the boundedness of on (Lemma 5), we haveThus, (26) holds.
For , by the moment condition of -atoms , we haveThen, we apply Lemma 4 to getFinally, by using Lemma 2 again and combing the inequalitywith Lemma 1 and the condition , we deduceand which suggests that (27) holds. Thus, we finish the proof of Proposition 1.
Now, we are ready to give the proofs of Theorem 1.
Proof. Write . Let and . According to the atomic characterization of and Proposition 1, it is reduced to showing thatholds for each -atom related to some ball .
Then, by the boundedness of from to as in Lemma 6, we just need to proveFinally, (36) is equivalent to establishingfor with , since as in Definition 2.
In order to get (37), we considerFor , combining Hölder’s inequality with the weighted boundedness of the maximal function and Lemma 2, we haveFor , noting that for every and any , we getHence, and it completes the proof of Theorem 1.
The author confirms that no data were used to support this study. All references used were listed.
Conflicts of Interest
The author declares that he has no conflicts of interest.
The author would like to express his thanks to Xiangtan University for part of the work completed here during his study period. This work was supported by Changsha Normal University, and the article processing charge is sponsored by her. This study was also supported by the Key Scientic Research Projects of Hunan Education Department (21A0617).
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