#### Abstract

Let a symbol belongs to the class with and . This paper mainly establish the boundedness of the class of all bi-parameter pseudo-differential operators associated with the symbol on the multi-parameter non-homogeneous Besov-Lipschitz and Triebel-Lizorkin spaces.

#### 1. Introduction

In this paper, we mainly consider the bi-parameter pseudo-differential operator for , where the symbol with , , , and , , for .

The pseudo-differential operator, which is formalised in Kohn and Nirenberg [1] and Hörmander [2] in 1960s, have been developed as a popular mathematical tool and widely applied in various fields of mathematics. Over the last few decades, the study of pseudo-differential operator have attracted many related mathematicians and many authors have been working on proving boundednesses of the one-parameter pseudo-differential operators on various classical spaces. It’s worth noting that the boundedness for the class was first treated by Calderón-Vaillancourt in [3], which means that the boundedness of all the derivatives of a symbol ensures the -boundedness of the corresponding operator. Further, Laptev [4] proved that any pseudo-differential operator with symbols in is a standard Calderón-Zygmund operator, hence the -boundedness is obvious. Hörmander [5] pointed out that the pseudo-differential operators with symbols in are bounded on if and only if , where , and (see also Fefferman [6] and Stein [7], Chapter VII, Section 5.12). Sugimoto [8] showed the pseudo-differential operators with symbols in are bounded on Besov spaces when , where , , and . Furthermore, Jun Park [9] studied the boundedness of pseudo-differential operators with symbols in on Triebel-Lizorkin spaces and Besov spaces when , where , and . Some weighted results about boundednesses of pseudo-differential operators also have been verified, such as in [10–12] and so on.

On the other hand, it’s well-known that there are two different directions being extended to developing the Calderón-Zygmund singular integral theory. One of the two extensions is non-convolution operators with a kernel, such as the Cauchy-operator on Lipschitz curves and the interested reader can see [13–16] for more reference. The other extension is due to R. Fefferman and E. M. Stein, they extended the classical singular integrals into the product situation in [17]. Later, Journé have unified these two different directions into a certain extent in [18], and studied a class of singular integral operators defined on a product of Euclidean spaces, the reader can see [16, 19–23] for further information on this topic. For the theory of multi-parameter singular integrals and function spaces plays an important role in many aspects of harmonic analysis, many authors have studied the pure product theory (which is also called multi-parameter harmonic analysis) over the last decades. For instance, the estimates for multi-parameter Coifman-Meyer Fourier multipliers in [24]; the estimates and Calderón-Vaillancourt type theorem for multi-parameter and multi-linear Fourier multipliers and pseudo-differential operators in [25, 26]. In particular, Sugimoto [27] showed that any pseudo-differential operators with symbols in Besov space implies the -boundedness. Furthermore, Tomita [28] generalized the results in [27] and studied the -boundedness of pseudo-differential operators with non-regular symbols, where . As some applications, Tomita [28] also obtained the -boundedness for symbols in Besov spaces and modulation spaces, respectively. Recently, the properties of multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces and the boundedness of multi-parameter singular integral operators on them have been established in [29–32].

Generally speaking, the pseudo-differential operators can be regarded as an extension for Fourier multipliers. On the other hand, the -modulation spaces , which were introduced by Gröbner’s PH.D. thesis [33], function as a kind of intermediate space between modulation spaces and the well-known Besov spaces. Here the parameter works as the “adjuster”, modulation spaces are the special -modulation spaces when and Besov spaces can be treated as the limit case of -modulation spaces when the parameter tends to . The difference between modulation spaces and Besov spaces can be referred to Tomita [28]. We also remark that the boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces have been obtained in [34]. The author in this paper and others have introduced the bi-parameter -modulation spaces with and bi-parameter modulation spaces in [35], and established the boundedness of bi-parameter pseudo-differential operators on these spaces. While not much is known about the boundedness of pseudo-differential operators on multi-parameter Besov-Lipschitz and Triebel-Lizorkin spaces:

The main purpose of this paper is to investigate the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces. Let us recall the definition of the bi-parameter pseudo-differential operators as the following.

For , be real pairs, where with . A function is said to belong to if there exists a positive constant such that the following inequality holds for all multi-indices , where , .

Denote the class of all bi-parameter pseudo-differential operators with by , for , where , .

Next we give the main results about the boundedness of bi-parameter pseudo-differential operators on the bi-parameter non-homogeneous Besov spaces and the bi-parameter non-homogeneous Triebel spaces.

Theorem 1. *Let , and with and , . Then the bi-parameter pseudo-differential operators defined in the form of (3) with symbols is bounded on the bi-parameter non-homogeneous Besov-Lipschitz . That is, there exists a positive constant such that
holds for all and .*

Theorem 2. *Let , and with and , . Then the bi-parameter pseudo-differential operators defined in the form of (3) with symbols is bounded on the bi-parameter non-homogeneous Triebel-Lizorkin spaces . That is, there exists a positive constant such that
holds for all and .*

The contents of this paper are organized as follows: In Section 2, we state some preliminary notation that will be used throughout this paper and introduce the bi-parameter non-homogeneous Besov-Lipschitz spaces and the bi-parameter non-homogeneous Triebel-Lizorkin spaces. Moveover, we give some necessary lemmas which will serve to prove the main theorems in Section 3. Since it is of importance for its own sake, we will give some self-contained proofs here. We give the detailed proof for Theorem 1 in Section 4 and the detailed proof for Theorem 2 in Section 5.

As it was pointed out by one of the referees our work seems to have overlapping with our previous work [35]. The mainly thought to prove the main theorems maybe seem similar. However, we also give some new results, such as some norm estimates related to the bi-parameter pseudo-differential operators with different symbols (see lemma 16, lemma 17 and lemma 18). In particular, the method to prove Theorem 2 is slightly different from the idea to deal with Theorem 1, where we have applied the bi-parameter -multiplier estimates (see lemma 19). These new lemmas are key tools in our proofs.

#### 2. Preliminary Notation and Definition

In this section, we fix notations and provide some preliminary definitions that will be used in this paper.

General notation. We will denote universal positive constants by which can be different line to line. If and are two quantities (typically nonnegative), we use to denote the statement for some absolute constant . Also, we use to denote the statement and .

Let and denote the Schwartz space of rapidly decreasing smooth functions and its dual space of tempered distributions, respectively, with their usual topologies. Write , for any multi-index , denote and .

For , the bi-parameter Lebesgue space is equipped with the (quasi)-norm

When ,

For , we denote by the set of all complex number sequences such that with the usual modification for .

Through out the following paper, we denote and , and . We use denote the Fourier transform and denote the inverse Fourier transform. That is, for any .

We also use denote the partial Fourier transform of functions with respect to the -variable and the -variable, and denote the inverse partial Fourier transform of functions with respect to the -variable and the -variable. That is, for any .

A class of Fourier multiplier operators can be stated as follow:

In order to introduce the definition of bi-parameter non-homogeneous Besov-Lipschitz spaces and non-homogeneous Triebel-Lizorkin spaces, we first recall the definition of the test function which is defined on .

Let denote the space of Schwartz function and be its dual. The test function defined on can be given by where , . Set

*Definition 3. *Let be the collection of all the systems that satisfy the following conditions:
(i)supp , supp for , ;(ii) for all , ;(iii)There exists a positive number such that the inequality holds for all non-negative integer and every multi-index and , .

*Definition 4. *Let , . The bi-parameter non-homogeneous Besov-Lipschitz space is defined by
where
with a slight modification in the case of and . Here are non-negative integers.

*Definition 5. *Let , . The bi-parameter non-homogeneous Triebel-Lizorkin space is defined by
where
with a slight modification in the case of and . Here are non-negative integers.

It should pointed out that the definition of bi-parameter non-homogeneous Besov-Lipschitz space and the definition of bi-parameter non-homogeneous Triebel-Lizorkin space are independent with the choosing of and . Definition 4 and Definition 5 are identical to definitions of weighted Besov-Lipschitz and Triebel-Lizorkin spaces defined in Sugimoto [7, 27] with the weight . We also refer the reader to [29, 32] for the bi-parameter homogeneous cases.

For simplicity and convenience, we will choose and denote the inverse Fourier transform of by , that is, . Denote and for all , , . Hence we can deduce , for , and where denotes the identity operator. When the integer , we denote by for convenience. Hence we have

For short, the bi-parameter non-homogeneous Besov-Lipschitz space will be called as the bi-parameter Besov space and the bi-parameter non-homogeneous Triebel-Lizorkin space called as the bi-parameter Triebel space in the next. According to the above, the bi-parameter Besov space and the bi-parameter Triebel space can also be defined as follows:

*Definition 6. *Let , . The bi-parameter Besov space is defined by
where

*Definition 7. *Let , . The bi-parameter Triebel space is defined by
where

According to (18) and (20), the symbol can be decomposed as where the sequence and are the decomposition constituting bi-parameter Besov spaces and Triebel space, and by the support condition of in Definition 3 for . By using Definition 6 and Definition 7, we have where

#### 3. Necessary Lemmas

In this section, we show some necessary lemmas that will be used to prove the main theorem in this paper. Let be a compact subset of and be a sequence of compact subsets of . Write where for .

The following lemmas (Lemma 8 to Lemma 10) are requirements to prove the main theorem and they have been considered in our previous work [35], we will omit the detailed proof here.

Lemma 8. *Let , and . Then we have the estimates
holds for all , where the implicit constant is independent of and .*

Lemma 9. (see [36]). *Let , then there exists a constant such that
for all sequences of locally integrable functions on , where denotes the strong maximal operator. The strong maximal operator is defined by
for any , where , and is a locally integrable function on .*

Lemma 10. *Let , for any and any function whose Fourier transform is supported in , assume that is an integrable function. Then there exists two constants and (which depend only on ) such that
where denotes the strong maximal operator.*

By Lemma 9 and Lemma 10, we can deduce the following Lemma 11.

Lemma 11. *Let , and . If, then we have
holds for all , where the implicit constant is independent of and .*

Lemma 12. *Let . If , then we have the estimates
holds for all .*

*Proof. *We only need to prove by Lemma 11. Since
By the definition of the sequence , we get
Let and for in Lemma 11, hence we have .

By Lemma 12, we can obtain the following lemma due to Triebel [37].

Lemma 13. *Let and . If , then we have the estimates
holds for all .*

For any and with , the bi-parameter Bessel potential can be denoted as . By adopting the main idea for Besov spaces and Triebel spaces in Triebel [37], we give the following lemma for the bi-parameter case.

Lemma 14. *Let , , with , . Then
*(i)*The mapping is isomorphic*(ii)*The mapping is isomorphic*

To prove Theorem 1 and Theorem 2, we will change the formulation of slightly as follows. Choosing a function and satisfy on , and besides with .

Set

Then, we have on the support of . Thus, we can deduce where