We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.
1 Introduction and main results
Let be a linear operator. Given a function , the commutator is defined by
It is very interesting that is a pseudo-differential operator because its theory plays an important role in many aspects of harmonic analysis and it has had quite a success in a linear setting. As one of the most meaningful branches, the study of bilinear pseudo-differential operators was motivated not only as generalizations of the theory of linear ones but also its natural appearance in Harmonic. This topic is continuous to attract many researchers.
Let be a Lipschitz function and . The estimates of the form
have been studied extensively. In particular, Calderón proved that (1.1) holds when is a pseudo-differential operator whose kernel is homogeneous of degree of in . Coifman and Meyer showed (1.1) when and is a symbol in the Hörmander class in [2,3], and this result was later extended by Auscher and Taylor in  to , where the class , which contains modulo symbols associated with smoothing operators, consists of symbols whose Fourier transforms in the first -dimensional variable are appropriately compactly supported. The method in the proofs of [2,3] mainly showed that, for each Lipschitz continuous function on , is a Calderón-Zygmund singular integral whose kernel constants are controlled by . For another thing, Auscher and Taylor proved (1.1) in two different ways: one method is based on the paraproducts while another is based on the Calderón-Zygmund singular integral operator approach that relies on the theorem. For a more systematic study of these (and even more general) spaces, we refer the readers to [5,6].
Given a bilinear operator and a function , the following two kinds of commutators are, respectively, defined by
In 2014, Bényi and Oh proved that (1.1) is also valid for this bilinear setting in . More precisely, given a bilinear pseudo-differential operator with in the bilinear Hörmander class and a Lipschitz function on , it was proved in [7, Theorem 1] that and are bilinear Calderón-Zygmund operators. The main aim of this article is to study (1.1) of on the products of Hardy spaces and Herz-type spaces with . Before stating our main results, we need to recall some definitions and notations.
We say that a function defined on is Lipschitz continuous if
Let , and . An infinitely differentiable function belongs to the bilinear Hörmander class if for all multi-indices there exists a positive constant such that
Given a , the bilinear pseudo-differential operator associated with is defined by
where denotes the Fourier transform of with respect to its first variable in , that is, for all . The class is defined as
Recently, many authors are interested in bilinear operators, which is a natural generalization of linear case. With further research, Bényi and Naibo proved the boundedness for the commutators of bilinear pseudo-differential operators and Lipschitz functions with on the Lebesgue spaces in . In 2018, Tao and Li proved that the boundedness of the commutators of bilinear pseudo-differential operators was also true on the classical and generalized Morrey spaces in . Motivated by the results mentioned above, a natural and interesting problem is to consider whether or not (1.1) is true on the products of Hardy spaces and Herz-type spaces with . The purpose of this article is to give a surely an answer. Our proofs are based on the pointwise estimates of the sharp maximal function proved in the next section.
Suppose that . Let and denote the kernel of and , respectively. We have
Then the following consequences are true.
[7, Lemma 3] If or , then we have
Let be a positive integer and be a locally integrable function defined away from the diagonal in and be a positive constant. We say that is a multilinear Calderón-Zygmund kernel if it satisfies the size condition that for all with for some ,
and satisfies the regularity condition that
whenever , and also that for each fixed with ,
The statements of our main theorems are presented as follows.
Let and b be a Lipschitz function on . Suppose that for all bounded functions supported on some cubes in with for ,
for every multi-index with , where and . Then extends boundedly from into for and with and , where for any denotes the integer not greater than .
Let and be a Lipschitz function on . Suppose , , , . If is bounded from into with controlled by , then is bounded from into
Throughout this article, for , is the conjugate index of , that is, . denotes the ball centered at with radius and . The boundedness of commutators on product of Hardy spaces is presented in Section 2. The boundedness of commutators on product of Herz-type spaces is given in Section 3.
2 Boundedness on product of Hardy spaces
 Let . The Hardy space is defined by
where with , and for any and , . Moreover, define
It is known that the definition of Hardy space does not depend on the choice of (see ).
 Let and with . For satisfying , a real-valued function is called a -atom centered at if
and is supported in a cube centered at ;
for every multi-index with .
Denote by the set of all continuous. Meda et al. proved the following result in , which ensures that a bounded linear operator on with or can be extended to be a bounded operator on .
 Let , with and satisfying . The quasi-norms and are equivalent on where and on when .
To prove Theorem 1.2, we need the boundedness of on products of Lebesgue spaces.
 If and is a Lipschitz function on , then the commutators are bilinear Calderón-Zygmund operators. In particular, are bounded from into for and and verify appropriate end-point boundedness properties. Moreover, the corresponding norms of the operators are controlled by .
Let and b be a Lipschitz function on . Suppose that is a -atom supported on and be a -atom supported on , with . Then
for any ,
while for any ,
for any ,
for any cube
Proof of Lemma 2.3
To obtain the conclusion of Lemma 2.3, we only prove (ii) and (iii). Together with (1.2), Theorem A and the vanishing moment of , it follows that for any ,
Similarly, applying the vanishing moment of , we obtain
and conclusion (ii) follows directly.
To prove conclusion (iii), we first observe that for any and ,
where is the Hardy-Littlewood maximal operator. By Hölder’s inequality, the fact is bounded on , Lemma 2.1 and the size condition of and , we have
This leads conclusion (iii) and completes the proof of Lemma 2.3.□
 Let . Then there exists a positive constant such that for all finite collections of cubes in and all nonnegative integrable functions with ,
where denotes the cube with the same center as and its side-length.
Proof of Theorem 1.1
By Lemma 2.1 and a density argument, it suffices to prove that is bounded from into , , we decompose as
where are -atoms in Definition 2.2. This means that are functions supported in cubes and satisfy the properties and . Without loss of generality, we may assume that for fixed and , . It is easy to see that there exists a cube such that
and , where is a constant independent of and with . Our purpose is to show
For this goal, we write
It follows from Lemma 2.4, (iii) of Lemma 2.3 and Hölder’s inequality that
We now turn to estimate . Since has vanishing moment up to order , we can subtract the Taylor polynomial of the function at the point with and . Set
and for all , write
Our assumption and with guarantee that we can choose such that and . On the other hand, we can verify that for all and , , and then for all ,
since we assume that . This, together with Lemma 2.2 and Hölder’s inequality, tells us that
The estimate for is cumbersome but straightforward. For , let