Shaoyong HE(何少勇)+

Department of Mathematics，Huzhou University，Huzhou 313000，China E-mail : hsyongmath@zjnu.edu.cn

Jiecheng CHEN(陳杰誠(chéng))

Department of Mathematics,Zhejiang Normal University，Jinhua 321004，China E-mail : jcchen @zjnu.edu.cn

It is well-known that classical Lipschitz spaces play a significant role in harmonic analysis and partial differential equations. More precisely, Han, Han, Li and Tan [19] constructed flag Lipschitz spaces on Heisenberg groups and proved that Marcinkiewicz multipliers are bounded on them. In addition, multi-parameter Lipschitz spaces associated with mixed homogeneities were studied in Han and Han [18], and Tan and Han [40]. Very recently, the first author and his collaborators, Zheng, Chen and others, established in [41] a necessary and sufficient condition for the boundedness of Journ′e’s product singular integral operators on the product Lipschitz spaces;namely,supposing that T is a singular integral operator in Journ′e’s class with a regularity exponent ε ∈(0,1], then T is bounded on Lip(α1,α2) if and only if T11=T21=0 for 0 ＜max{α1,α2}＜ε, where Lip(α1,α2)denotes the product homogeneous Lipschitz spaces on Rn1× Rn2introduced in [41]. T1(1) = T2(1) = 0 will be explained below. In [28], we introduced multi-parameter inhomogeneous Lipschitz spaces and established the boundedness of multi-parameter pseudodifferential operators on multi-parameter inhomogeneous Lipschitz spaces.

Motivated by those works,in this note,we investigate bi-parameter mixed Lipschitz spaces.To do this,we first introduce the following notations: for x1,u ∈Rn1and x2,v ∈Rn2,we denote that

Now the bi-parameter mixed Lipschitz space is defined as follows:

Definition 1.1 Let α1,α2> 0. The mixed Lipschitz space is defined as the space of all continuous functions f defined on Rn1×Rn2such that

1. when 0 ＜α1,α2＜1,

As is well-known, the theory of one-parameter singular integral operators has been generalized in two ways. First, the convolution singular integral operators have been replaced by non-convolution singular integrals associated with a kernel in the following sense:

Definition 1.3 A locally integrable function defined from the diagonal x=y in Rn×Rnis called a one-parameter Calder′on-Zygmund kernel if there exist constants C >0 and a regularity exponent ε ∈(0,1] such that

for all f,g,h,k ∈C∞0(R) with supp (f)∩supp (g)=?, and a pair (K1,K2) of δCZ-δ-standard kernels (see [30], p. 63).

Now we give the definition of singular integral operators in the mixed Journ′e class on Rn1×Rn2that are suitable for our bi-parameter mixed Lipschitz spaces.

and similarly,

Then T can be said to be in a mixed Journ′e class with regularity exponents ε ∈(0,1]and δ >0 if T satisfies the following conditions:

Our last main result is the following boundedness on bi-parameter mixed Lipschitz spaces of singular integral operators in a mixed Journ′e class:

Theorem 1.5 Let T be a singular integral operator in a mixed Journ′e class with regularity exponents ε ∈(0,1] and δ > 0. If T1(1) = T2(1) = 0, then T is bounded on Lipmix(α1,α2)for 0 ＜α1＜min{ε,δ} and 0 ＜α2＜ε. Conversely, if T is bounded on Lipmix(α1,α2), then T2(1)=0.

The rest of this paper is organized as follows: in Section 2,we will give the proof of Theorem 1.2. Theorem 1.5 will be proved in Section 3.

Throughout this paper, the letter C stands for a positive constant which is independent of the essential variables, but whose value may vary from line to line. We use the notation A ≈B to denote that there exists a positive constant C such that C-1B ≤A ≤CB. Let a ∧b=min{a,b}.

2 Proof of Theorem 1.2

To establish the Littlewood-Paley characterization of bi-parameter mixed Lipschitz spaces,we recall the following well-known one-parameter almost orthogonality estimates (see, for example, [14, 15]):

Case 2 j ≥1,k ∈Z.

We only consider |u| ＜1. If |u| ≥1, the proof below is even more simple. Let m1, m2be the unique integer such that 2-m1-1≤|u| ＜2-m1and 2-m2-1≤|u| ＜2-m2. Now we split the above series by

3 Proof of Theorem 1.5

for any f,g ∈Cη0(Rn) with supports in B(0,r). It is easy to check that the L2boundedness of T implies that T satisfies the weak boundedness property. This property can be generalized to multi-parameter operators.

This completes the proof of Lemma 3.7. □

Repeating the same argument as that of Lemma 3.7, we can obtain the following result:

This completes the proof of Theorem 1.5. □