Algebras of Calderón–Zygmund Operators Associated with Para-Accretive Functions on Spaces of Normal Homogeneous Type

Abstract

In this paper, by using several almost orthogonal estimates and a continuous Calderón reproducing formula associated with para-accretive functions, we obtain the algebras of generalized-product Calderón–Zygmund operators under the condition $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$.

1. Introduction

Early in 1952, Calderón and Zygmund [1] introduced singular integrals with convolution kernels, which commute with translations and generalize the Hilbert transform and the Riesz transforms. Next, they [2] proposed that a set of convolution singular integral operators form commutative algebras when these operators satisfy certain conditions. However, there are many non-convolution operators, such as the Cauchy integral on Lipschitz curves, the double layer potential on Lipschitz surfaces, the Calderón commutators, and the multilinear operators of Coifman and Meyer. We start by recalling the definition of a Calderón–Zygmund kernel. A continuous complex-valued function $K(x,y)$, defined on ${\mathbb{R}}^n\times {\mathbb{R}}^n\setminus \{x=y\}$, is called a Calderón–Zygmund kernel if there exist constants $C>0$ and a regularity exponent $\epsilon \in (0,1]$ such that

$$\begin{array}{cc}\hfill \left|K(x,y)\right|\le {C|x-y|}^{-n}& \end{array}$$

(1)

$$\begin{array}{cc}\hfill \left|K(x,y)-K\left(x^{\prime },y\right)\right|\le C{\left|x-x^{\prime }\right|}^{\epsilon }{|x-y|}^{-n-\epsilon }& if\left|x-x^{\prime }\right|\le |x-y|/2\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \left|K(x,y)-K\left(x,y^{\prime }\right)\right|\le C{\left|y-y^{\prime }\right|}^{\epsilon }{|x-y|}^{-n-\epsilon }& if\left|y-y^{\prime }\right|\le |x-y|/2.\hfill \end{array}$$

(3)

The smallest such constant C is denoted by ${\left|K\right|}_{CZ}$. A Calderón-–Zygmund singular integral operator T is a continuous linear operator from $C_0^{\infty }\left({\mathbb{R}}^n\right)$ to its dual space associated with the kernel $K(x,y)$ above, which can be represented by

$$\begin{array}{cc}\hfill \langle Tf,g\rangle & ={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}K(x,y)f\left(y\right)g\left(x\right)dydx,\hfill \end{array}$$

(4)

for test functions f and g with disjoint support. If the Calderón-–Zygmund singular integral operator T is bounded on $L^2$, then we call the operators the Calderón–Zygmund operator, and the norm of T is defined by ${\parallel T\parallel }_{\mathrm{CZ}}={\parallel T\parallel }_{L^2\mapsto L^2}+{\left|K\right|}_{\mathrm{CZ}}$.

Meyer demonstrated that this class of Calderón–Zygmund operators forms an algebra. If T is a Calderón–Zygmund operator, then its transpose $T^{*}$ is also a Calderón–Zygmund operator. Using a remarkable duality argument between the Hardy space $H^1$ and $BMO$, established by Fefferman [3], for any $f\in H^1$, $T\left(1\right)$ can be well defined by

$$\langle T\left(1\right),f\rangle =〈1,T^{*}f〉$$

given that T and its transpose $T^{*}$ are bounded operators mapping $H^1$ onto $L^1$, the condition $T\left(1\right)=0$ implies that $\int T^{*}f\left(x\right)dx=0$ holds for every $f\in C_{0,0}^{\eta }$. Likewise, $T^{*}\left(1\right)=0$ implies that $\int Tf\left(x\right)dx=0$ for all $f\in C_{0,0}^{\eta }$. Meyer and Coifman [4] proved that all generalized Calderón–Zygmund operators form non-commutative Banach algebras through orthonormal wavelet bases when $T\left(1\right)=T^{*}\left(1\right)=0$.

However, the $T1$ theorem cannot be applied to the following Cauchy integral:

$$\mathcal{C}\left(f\right)\left(x\right)={\displaystyle \frac{1}{\pi }}\mathrm{p}.\mathrm{v}.{\int }_{-\infty }^{\infty }{\displaystyle \frac{f\left(y\right)}{(x-y)+i\left(a\right(x)-a(y\left)\right)}}dy,$$

where the function $a\left(x\right)$ satisfies the Lipschitz condition. Replacing the function 1 in the $T1$ theorem with an accretive function b, defined as a bounded complex-valued function satisfying $0<\delta <\mathrm{Re}\left(b\right(x\left)\right)$ almost everywhere, leads to the conclusion that $C\left(f\right)$ is $L^2$-bounded on all Lipschitz curves.

Definition 1 

([5]). A bounded, complex-valued function b defined on X is said to be para-accretive if there exist constants $C,\gamma >0$ such that, for every cube $Q\subseteq {\mathbb{R}}^n$, there is a subcube $Q^{\prime }\subseteq Q$ with $\gamma \left|Q\right|\le \left|Q^{\prime }\right|$ satisfying

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\left|Q\right|}}\left|{\int }_{Q^{\prime }}b\left(x\right)dx\right|\ge C>0.\end{array}$$

(5)

Let $b_1$ and $b_2$ be para-accretive functions. A generalized singular integral operator T is a continuous linear operator that maps $b_1{C}_0^{\eta }$ onto the dual space ${\left(b_2{C}_0^{\eta }\right)}^{\prime }$ for every $\eta >0$. This operator is associated with a kernel $K(x,y)$ that satisfies conditions (1), (2), and (3). For any function $f,g\in C_0^{\eta }$ with disjoint supports, i.e., $supp\left(f\right)\cap supp\left(g\right)=⌀$, T is given by the following integral representation:

$$〈T{b}_{1}f,b_{2}g〉={\int \int }_{{\mathbb{R}}^n\times {\mathbb{R}}^n}g\left(x\right)b_2\left(x\right)K(x,y)b_1\left(y\right)f\left(y\right)dydx.$$

$T\left(b\right)$ can be well defined by

$$\langle T\left(b\right),f\rangle =〈b,T^{*}\left(f\right)〉\mathrm{for}\mathrm{all}f\in H^1$$

given that T and its adjoint $T^{*}$ are bounded operators mapping $H^1$ onto $L^1$, the condition $T\left(b\right)=0$ implies that $\int T^{*}f\left(x\right)b\left(x\right)dx=0$ for all $f\in H^1$. Similarly, $T^{*}\left(b\right)=0$ ensures that $\int Tf\left(x\right)b\left(x\right)dx=0$ holds for every $f\in H^1$. Meyer and Coifman [4] showed that all generalized Calderón–Zygmund operators form algebras through special wavelet bases associated with accretive or para-accretive functions when $T\left(b\right)=T^{*}\left(b\right)=0$ and b is an accretive or para-accretive function. In 2006, Han [6] verified that Calderón–Zygmund operators with $T\left(b\right)=T^{*}\left(b\right)=0$ form an algebra which uses the discrete Calderón reproducing formulas associated with para-accretive functions.

To generalize the theory of Calderón–Zygmund operators, in 1971, Coifman and Weiss [7] introduced spaces of homogeneous type, which have no translations, dilations, or Fourier transform. What is more, there are no orthonormal wavelet bases in spaces of homogeneous type. We now present the definition of spaces of homogeneous type. We say $(X,\mathrm{d})$ is a quasi-metric space if X is a non-empty set and $\mathrm{d}$ is a quasi-metric on X and $\mathrm{d}$ satisfying the following:

(a)

$\mathrm{d}(x,y)\ge 0$, and $\mathrm{d}(x,y)=0$ if and only if $x=y$;

(b)

$\mathrm{d}(x,y)=\mathrm{d}(y,x)$ for all $x,y\in X$;

(c)

there exists a constant $A\in [1,\infty )$ such that for all $x,y,z\in X$,

$$\begin{array}{c}\hfill \mathrm{d}(x,y)\le A\left(\mathrm{d}\right(x,z)+\mathrm{d}(z,y\left)\right).\end{array}$$

(6)

We define the quasi-metric ball as

$$B(x,r)=\{y\in X:\mathrm{d}(y,x)<r\},$$

for all $x\in X$ and $r>0$. Let $(X,\mathrm{d})$ be a quasi-metric space, and $\mu$ be a non-negative Borel measure on X. We say that $(X,\mathrm{d},\mu )$ is a space of homogeneous type if $\mu$ satisfies the doubling condition, i.e., there exists a positive constant $C<\infty$ such that for all $x\in X$ and $r>0$,

$$\begin{array}{c}\hfill \mu \left(B\right(x,2r\left)\right)\le C\mu \left(B\right(x,r\left)\right).\end{array}$$

(7)

Note that quasi-metric $\mathrm{d}$ may have no regularity. Although the quasi-metric ball B is a Borel set, it may not necessarily be an open set. To generalize the Fefferman–Stein maximal function characterization for Hardy spaces to spaces of homogeneous type, Macías and Segovia [8] showed that for any homogeneous space $(X,\mathrm{d},\mu )$, $\mathrm{d}$ can always be replaced by another equivalent quasi-metric $\rho$.

A space of homogeneous type $(X,\rho ,\mu )$ is said to be a space of normal homogeneous type if $\rho$ satisfies

$$\begin{array}{c}\hfill \left|\rho (x,y)-\rho \left(x^{\prime },y\right)\right|\le C\rho {\left(x,x^{\prime }\right)}^{\theta }{\left[\rho (x,y)+\rho \left(x^{\prime },y\right)\right]}^{1-\theta },\end{array}$$

(8)

for some constant $C>0$, some regularity exponent $0<\theta <1$ and for all $x,x^{\prime },y\in X$. Moreover, if the quasi-metric balls are defined by the quasi-metric $\rho$, that is, $B(x,r)=\{y\in X:\rho (y,x)<r\}$ for all $x\in X$ and $r>0$, then

$$\begin{array}{c}\hfill \mu \left(B\right(x,r\left)\right)\approx r.\end{array}$$

(9)

Obviously, property (9) is much stronger than the doubling condition (7).

Meyer has shown that many theories of Calderón–Zygmund singular integrals can still be applied to spaces of homogeneous type. Special spaces of homogeneous type include the Euclidean space, the n-torus in ${\mathbb{R}}^n$, the $C^{\infty }$-compact Riemann manifolds, the Lipschitz manifolds and the boundaries of bounded Lipschitz domains in ${\mathbb{R}}^n$, which are widely used in harmonic analysis and other analysis fields. To overcome the obstacle that wavelet bases are not available on spaces of homogeneous type, in 2003, Han and Lin [9] developed some useful tools such as discrete Calderón reproducing formulas, and proved the algebras of Calderón–Zygmund operators on spaces of homogeneous type.

In this paper, we consider the product space $(X_1\times X_2,{\rho }_1\times {\rho }_2,{\mu }_1\times {\mu }_2)$, where $(X_i,{\rho }_i,{\mu }_i)$, $i=1,2$ are the spaces of normal homogeneous type, along with a two-parameter family of dilations $(x,y)\mapsto \left({\delta }_{1}x,{\delta }_{2}y\right)$, $x\in X_1$, $y\in X_2$, ${\delta }_1>0$, ${\delta }_2>0$ instead of the classical one-parameter dilation. We may note that Fefferman and Stein [10] generalized the Calderón–Zygmund operators of convolution type to the two-parameter setting. Journé [11] introduced the product non-convolution singular integral operators. Pott and Vilarroya [12] verified a new type of the $T1$ theorem for product spaces. Han [13] proved a $Tb$ theorem on product spaces. On the product Carnot–Carathédory spaces, Liao [14] determined that all of the product singular integral operators in the Journé’s class, which satisfy $L^2$ boundedness, form an algebra under the conditions $T_1\left(1\right)=T_1^{*}\left(1\right)=T_2\left(1\right)=T_2^{*}\left(1\right)=0$. For the precise definition of T, refer to [14] [Definition 1.2].

The main purpose of this paper is to study the algebras of generalized-product Calderón–Zygmund operators associated with para-accretive functions in the two-parameter setting in spaces of the normal homogeneous type. To be more precise, all generalized-product Calderón–Zygmund operators form an algebra when $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$, where T is defined in Definition 4. This will extend the work of Han [6] to product spaces of normal homogeneous type, further generalize the results of Liao [14] to generalized-product Calderón–Zygmund operators associated with para-accretive functions, and establish some key, almost-orthogonal estimates. The new idea used here is to apply the continuous Calderón reproducing formulas associated with para-accretive functions. This new method is different from the orthonormal wavelet bases of Meyer and Coifman [4] and the discrete Calderón reproducing formulas of Han and Lin [6,9].

2. Preliminaries and the Main Theorem

In Section 2.1, we recall the definition of product singular integral operators in Journé’s class. In Section 2.2, we recall the generalized-product Calderón–Zygmund operators and the main theorem in this paper.

2.1. Product Singular Integral Operators in Journé’s Class

To formulate our results, we need to introduce the mixed-type conditions formulation for product singular integral operators, which is given by Martikainen [15] in Euclidean spaces. Nagel and Stein [16] introduced the representation in Journé’s class for spaces of homogeneous type.

Definition 2. 

Suppose that $f=f_1⨂f_2$ (i.e., $f(x_1,x_2)=f_1\left(x_1\right)f_2\left(x_2\right)$ for $x_1\in X_1$ and $x_2\in X_2)$ and $g=g_1⨂g_2$, $supp\left(f_i\right)\cap supp\left(g_i\right)=⌀$ with $i=1,2$, $\eta \in (0,\theta ]$. A linear operator T initially defined from $C_0^{\eta }\left(X_1\times X_2\right)=C_0^{\eta }\left(X_1\right)\otimes C_0^{\eta }\left(X_2\right)$ to its dual space is called a bi-parameter singular integral if for each $x_i,y_i\in X_i(i=1,2)$,

$$\begin{array}{cc}\hfill \langle Tf,g\rangle =& {\int }_{X_1}{\int }_{X_1}{\int }_{X_2}{\int }_{X_2}K\left(x_1,x_2,y_1,y_2\right)g_1\left(x_1\right)g_2\left(x_2\right)f_1\left(y_1\right)f_2\left(y_2\right)\mathrm{d}{\mu }_2\left(y_2\right)\mathrm{d}{\mu }_2\left(x_2\right)\hfill \\ \hfill & \times \mathrm{d}{\mu }_1\left(y_1\right)\mathrm{d}{\mu }_1\left(x_1\right).\hfill \end{array}$$

The kernel $K:\left(X_1\times X_2\times X_1\times X_2\right)\setminus \left\{\left(x_1,x_2,y_1,y_2\right):x_1=y_1\right. or \left.x_2=y_2\right\}\to C$ satisfies the following conditions:

(I) 

Size condition: $\left|K\left(x_1,x_2,y_1,y_2\right)\right|\le C{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}};$

(II) 

Mixed Hölder and size conditions:

$\left({\mathrm{II}}_1\right)\left|K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1^{\prime },x_2,y_1,y_2\right)\right|\le C{\left({\displaystyle \frac{{\rho }_1\left(x_1,x_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(x_1,x_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}$;

$\left({\mathrm{II}}_2\right)\left|K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2^{\prime },y_1,y_2\right)\right|\le C{\left({\displaystyle \frac{{\rho }_1\left(x_2,x_2^{\prime }\right)}{{\rho }_1\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_2\left(x_2,x_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}}$;

$\left({\mathrm{II}}_3\right)\left|K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2,y_1^{\prime },y_2\right)\right|\le C{\left({\displaystyle \frac{{\rho }_1\left(y_1,y_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(y_1,y_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}$;

$\left({\mathrm{II}}_4\right)\left|K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2,y_1,y_2^{\prime }\right)\right|\le C{\left({\displaystyle \frac{{\rho }_1\left(y_2,y_2^{\prime }\right)}{{\rho }_1\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_2\left(y_2,y_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}}$;

(III) 

Hölder conditions:

$\left({\mathrm{III}}_1\right)\left|\left[K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2,y_1,y_2^{\prime }\right)\right]-\left[K\left(x_1,x_2,y_1^{\prime },y_2\right)-K\left(x_1,x_2,y_1^{\prime },y_2^{\prime }\right)\right]\right|$

$\le C{\left({\displaystyle \frac{{\rho }_1\left(y_1,y_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\left({\displaystyle \frac{{\rho }_2\left(y_2,y_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(y_1,y_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}and{\rho }_2\left(y_2,y_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}};$

$\left({\mathrm{III}}_2\right)\left|\left[K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2^{\prime },y_1,y_2\right)\right]-\left[K\left(x_1^{\prime },x_2,y_1,y_2\right)-K\left(x_1^{\prime },x_2^{\prime },y_1,y_2\right)\right]\right|$

$\le C{\left({\displaystyle \frac{{\rho }_1\left(x_1,x_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\left({\displaystyle \frac{{\rho }_2\left(x_2,x_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(x_1,x_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}and{\rho }_2\left(x_2,x_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}},$

$\left({\mathrm{III}}_3\right)\left|\left[K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2^{\prime },y_1,y_2\right)\right]-\left[K\left(x_1,x_2,y_1^{\prime },y_2\right)-K\left(x_1,x_2^{\prime },y_1^{\prime },y_2\right)\right]\right|$

$\le C{\left({\displaystyle \frac{{\rho }_2\left(x_2,x_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\left({\displaystyle \frac{{\rho }_1\left(y_1,y_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(y_1,y_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}and{\rho }_2\left(y_2,y_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}};$

$\left({\mathrm{III}}_4\right)\left|\left[K\left(x_1,x_2,y_1,y_2\right)-K\left(x_1,x_2,y_1,y_2^{\prime }\right)\right]-\left[K\left(x_1^{\prime },x_2,y_1,y_2\right)-K\left(x_1^{\prime },x_2,y_1,y_2^{\prime }\right)\right]\right|$

$\le C{\left({\displaystyle \frac{{\rho }_1\left(x_1,x_1^{\prime }\right)}{{\rho }_1\left(x_1,y_1\right)}}\right)}^{\epsilon }{\left({\displaystyle \frac{{\rho }_2\left(y_2,y_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_1\left(x_1,y_1\right)}}{\displaystyle \frac{1}{{\rho }_2\left(x_2,y_2\right)}}$

$for{\rho }_1\left(x_1,x_1^{\prime }\right)\le {\displaystyle \frac{{\rho }_1\left(x_1,y_1\right)}{2A_1}}and{\rho }_2\left(y_2,y_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}},$

where $A_1,A_2$ are similar to A in (6) and correspond to ${\rho }_1,{\rho }_2$, respectively. Let $C_0^{\eta }\left(X_1\times X_2\right)$ denote the space of continuous functions f with compact support such that

$${\parallel f\parallel }_{C_0^{\eta }}:=\underset{\begin{array}{c}{x}_1\ne y_1\\ x_2\ne y_2\end{array}}{sup}{\displaystyle \frac{\left|f\left(x_1,x_2\right)-f\left(y_1,x_2\right)-f\left(x_1,y_2\right)+f\left(y_1,y_2\right)\right|}{\rho {(x_1,y_1)}^{\eta }\rho {(x_2,y_2)}^{\eta }}}<\infty .$$

We also need some Calderón–Zygmund structure in $X_1$ and $X_2$, respectively. We assume the kernel representation

$$\begin{array}{c}\hfill \langle T_{1}f,g\rangle ={\int }_{X_2}{\int }_{X_2}{K}^{f_1,g_1}\left(x_2,y_2\right)f_2\left(x_2\right)g_2\left(y_2\right)d{\mu }_2\left(x_2\right)d{\mu }_2\left(y_2\right).\end{array}$$

(10)

The kernel $K^{f_1,g_1}\left(x_2,y_2\right):(X_2\times X_2)\setminus \{(x_2,y_2)\in X_2\times X_2:x_2=y_2\}\to C$ is assumed to satisfy the following:

(IV) 

The size condition:

$\left|K^{f_1,g_1}\left(x_2,y_2\right)\right|\le C(f_1,g_1){\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}};$

(V) 

Hölder conditions:

$\left(V_1\right)\left|K^{f_1,g_1}\left(x_2,y_2\right)-K^{f_1,g_1}\left(x_2^{\prime },y_2\right)\right|\le C(f_1,g_1){\left({\displaystyle \frac{{\rho }_2\left(x_2,x_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}$

$for{\rho }_2\left(x_2,x_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}}$;

$\left(V_2\right)\left|K^{f_1,g_1}\left(x_2,y_2\right)-K^{f_1,g_1}\left(x_2,y_2^{\prime }\right)\right|\le C(f_1,g_1){\left({\displaystyle \frac{{\rho }_2\left(y_2,y_2^{\prime }\right)}{{\rho }_2\left(x_2,y_2\right)}}\right)}^{\epsilon }{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}$

$for{\rho }_2\left(x_2,x_2^{\prime }\right)\le {\displaystyle \frac{{\rho }_2\left(x_2,y_2\right)}{2A_2}}$;

(VI) 

We need the representations (10) and $C(f_1,g_1)\le C{\parallel f_1\parallel }_{L^2}{\parallel g_1\parallel }_{L^2}$.

We also assume the analogous representation and properties with a kernel $K^{f_2,g_2}$ when $supp\left(f_1\right)\cap supp\left(g_1\right)=⌀$.

If T is a linear operator associated with a kernel $K\left(x_1,x_2,y_1,y_2\right)$ which satisfies the conditions above, then T is said to be a product singular integral operator in Journé’s class when T is bounded on $L^2(X_1\times X_2)$.

2.2. Generalized-Product Calderón–Zygmund Operator

Next, we will introduce the generalized-product Calderón–Zygmund operators and the main results of this paper.

Definition 3 

([13]). Suppose that $b(x,y)=b_1\left(x\right)b_2\left(y\right)$, where $b_1$ and $b_2$ are para-accretive functions on $X_1$ and $X_2$, respectively. A generalized product singular integral operator is a continuous linear operator T from $b{C}_0^{\eta }(X_1\times X_2)$ to ${\left(b{C}_0^{\eta }(X_1\times X_2)\right)}^{*}$ for which there exists a kernel $K\left(x_1,x_2,y_1,y_2\right)$ that satisfies the size condition $\left(\mathrm{I}\right)$, the mixed Hölder and size conditions $\left(\mathrm{II}\right)$, and the Hölder conditions $\left(\mathrm{III}\right)$, such that, for all $f_1,g_1\in C_0^{\eta }\left(X_1\right)$ with $supp\left(f_1\right)\cap supp\left(g_1\right)=⌀$, and $f_2,g_2\in C_0^{\eta }\left(X_2\right)$ with $supp\left(f_2\right)\cap supp\left(g_2\right)=⌀$, where $M_b$ denotes the multiplication operator by b, that is, $M_{b}f=bf$.

$$\begin{array}{cc}\hfill & 〈M_{b}T{M}_b{f}_1\otimes f_2,g_1\otimes g_2〉\hfill \\ \hfill & ={\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{b}_2\left(x_2\right)b_1\left(x_1\right)g_1\left(x_1\right)g_2\left(x_2\right)K\left(x_1,x_2,y_1,y_2\right)b_2\left(y_2\right)b_1\left(y_1\right)f_1\left(y_1\right)f_2\left(y_2\right)\hfill \\ \hfill & \times \mathrm{d}{\mu }_2\left(y_2\right)\mathrm{d}{\mu }_2\left(x_2\right)\mathrm{d}{\mu }_1\left(y_1\right)\mathrm{d}{\mu }_1\left(x_1\right).\hfill \end{array}$$

Now, we need to explain the definitions of $T_1\left(b_1\right)=0$ and $T_1^{*}\left(b_1\right)=0$, where b is a para-accretive function. Denote that

$$C_{b,0}^{\eta }\left(X\right)=\left\{\psi \in C_0^{\eta }\left(X\right):{\int }_X\psi \left(y\right)b\left(y\right)d\mu \left(y\right)=0\right\}.$$

If $f_1$ is a bounded $C^{\eta }$ function, then for all $g_1\in C_{b_1,0}^{\eta }\left(X_1\right)$ and all $f_2,g_2\in C_0^{\eta }\left(X_2\right)$, $〈M_{b}T{M}_b{f}_1\otimes f_2,g_1\otimes g_2〉$ is well defined. Specifically, we define $T_1\left(b_1\right)=0$ if and only if

$$\begin{array}{c}\hfill 〈M_{b}T{M}_{b}1\otimes f_2,g_1\otimes g_2〉=0\end{array}$$

(11)

for all $g_1\in C_{b_1,0}^{\eta }\left(X_1\right)$ and $f_2,g_2\in C_0^{\eta }\left(X_2\right)$. Similarly, we can also define $T_1^{*}\left(b_1\right)=0$ if and only if

$$〈M_{b}T{M}_b{f}_1\otimes f_2,1\otimes g_2〉=0$$

for all $f_1\in C_{b_1,0}^{\eta }\left(X_1\right)$ and $f_2,g_2\in C_0^m\left(X_2\right)$. Exchanging the role of indices, we get the definition of $T_2\left(b_2\right)=0$ and $T_2^{*}\left(b_2\right)=0$.

Definition 4. 

Suppose that T is a generalized-product singular integral operator defined as in Definition 3. T is said to be a generalized-product Calderón–Zygmund operator if T extends to be a bounded operator on $L^2(X_1\times X_2)$.

Now, we formulate the main result of this paper, that is, that the set of generalized-product Calderón–Zygmund operators can form an algebra under certain conditions.

Theorem 1. 

Suppose that $b(x,y)=b_1\left(x\right)b_2\left(y\right)$, where $b_1$ and $b_2$ are para-accretive functions. T is a generalized-product Calderón–Zygmund operator defined as in Definition 4 and $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$. $\mathcal{A}$ is the collection of generalized-product Calderón–Zygmund operators in spaces of normal homogeneous type. $\mathcal{A}$ is an algebra if the multiplication is defined by $T_1\circ T_2=T_{1}b{T}_2$, where $b=b_1{b}_2$, for $T_1,T_2\in \mathcal{A}$.

Accordingly, as a special case of the product space, we also obtain that Calderón–Zygmund operators in spaces of normal homogeneous type can form algebras.

Corollary 1. 

Let $\mathcal{B}$ be the collection of Calderón–Zygmund operators T defined as in 4 satisfying $T\left(b\right)=T^{*}\left(b\right)=0$, where b is a para-accretive function. Then, $\mathcal{B}$ forms an algebra if the multiplication is defined by $T_1\circ T_2=T_{1}b{T}_2$ for $T_1,T_2\in \mathcal{B}$.

Remark 1. 

Han [6], through use of the discrete Calderón reproducing formulas associated with para-accretive functions, proved that Calderón–Zygmund operators form an algebra in Euclidean spaces ${\mathbb{R}}^n$ under the condition $T\left(b\right)=T^{*}\left(b\right)=0$. In this paper, we studied the algebras of generalized-product Calderón–Zygmund operators associated with para-accretive functions in a two-parameter setting in spaces of normal homogeneous type under the condition $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$. Our proof of Theorem 1 relies on continuous Calderón reproducing formulas associated with para-accretive functions, which distinguishes our approach from Han’s work. Since the spaces of normal homogeneous type $(X,\mathrm{d},\mu )$ contain the Euclidean spaces ${\mathbb{R}}^n$, Theorem 1 naturally contains Han’s results [6] for the one-parameter case. Moreover, Liao [14], using continuous Calderón reproducing formulas, demonstrated that all product singular integral operators in Journé’s class form an algebra in spaces of homogeneous type under the condition $T_1\left(1\right)=T_1^{*}\left(1\right)=T_2\left(1\right)=T_2^{*}\left(1\right)=0$, and the homogeneous spaces studied by Liao are product Carnot—Carathédory spaces. The new method in our paper extends Liao’s results to more general para-accretive functions. In order to obtain all of the generalized-product Calderón–Zygmund operators that form an algebra under the condition $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$, we also proved some key almost-orthogonal estimates.

3. The Proofs of Theorems

In Section 3.1, we recall the definition of an approximation of the identity and continuous Calderón reproducing formulas associated with a para-accretive function. This new method generalized Liao [14] to more general para-accretive functions. In Section 3.2, we prove some key almost-orthogonal estimates, which are important for the proof of Theorem 1. In Section 3.3, we give some useful estimates. Combining the above, we next prove the main results in Section 3.4.

3.1. Continuous Calderón Reproducing Formulas

To prove Theorem 1, we now recall the continuous Calderón reproducing formulas in product spaces of normal homogeneous type $(X_1\times X_2,{\rho }_1\times {\rho }_2,{\mu }_1\times {\mu }_2)$. Before starting the Calderón-type reproducing formula, we need an approximation of the identity.

Definition 5 

([17]). Let b be a para-accretive function. A sequence of operators ${\left\{S_k\right\}}_{k\in \mathbb{Z}}$ is considered to be an approximation of the identity associated with b if there exists a constant $C>0$ some $0<\theta \le 1$ such that for all $k\in \mathbb{Z}$ and all $x,x^{\prime },y$ and $y^{\prime }\in X$, the kernels $S_k(x,y)$ of $S_k$ satisfy the following conditions:

(i) 

$S_k(x,y)=0$ if $\rho (x,y)\ge C{2}^{-k}$ and $\left|S_k(x,y)\right|\le C{\displaystyle \frac{2^{-k\theta }}{{(2^{-k}+\rho (x,y))}^{1+\theta }}}$;

(ii) 

$\left|S_k(x,y)-S_k(x^{\prime },y)\right|\le C{\left({\displaystyle \frac{\rho (x,x^{\prime })}{2^{-k}+\rho (x,y)}}\right)}^{\theta }{\displaystyle \frac{2^{-k\theta }}{{\left(2^{-k}+\rho (x,y)\right)}^{1+\theta }}},for\rho (x,x^{\prime })\le {\displaystyle \frac{2^{-k}+\rho (x,y)}{2A}}$;

(iii) 

$\left|S_k(x,y)-S_k(x,y^{\prime })\right|\le C{\left({\displaystyle \frac{\rho (y,y^{\prime })}{2^{-k}+\rho (x,y)}}\right)}^{\theta }{\displaystyle \frac{2^{-k\theta }}{{\left(2^{-k}+\rho (x,y)\right)}^{1+\theta }}},for\rho (y,y^{\prime })\le {\displaystyle \frac{2^{-k}+\rho (x,y)}{2A}}$;

(iv) 

$\left|[S_k(x,y)-S_k(x,y^{\prime })]-[S_k(x^{\prime },y)-S_k(x^{\prime },y^{\prime })]\right|\le C{\left({\displaystyle \frac{\rho (x,x^{\prime })}{2^{-k}+\rho (x,y)}}\right)}^{\theta }{\left({\displaystyle \frac{\rho (y,y^{\prime })}{2^{-k}+\rho (x,y)}}\right)}^{\theta }$

${\displaystyle \frac{2^{-k\theta }}{{\left(2^{-k}+\rho (x,y)\right)}^{1+\theta }}}$, for $\rho (x,x^{\prime })\le {\displaystyle \frac{2^{-k}+\rho (x,y)}{2A}}$ and $\rho (y,y^{\prime })\le {\displaystyle \frac{2^{-k}+\rho (x,y)}{2A}}$;

(v) 

${\int }_X{S}_k(x,y)b\left(x\right)d\mu \left(y\right)=1$ for $k\in \mathbb{Z},x\in X$;

(vi) 

${\int }_X{S}_k(x,y)b\left(y\right)d\mu \left(x\right)=1$ for $k\in \mathbb{Z},y\in X$.

The existence of the above approximation of the identity has been established in [5].

Now, we state the continuous Calderón reproducing formulas in product spaces of normal homogeneous type as follows:

Lemma 1 

([13,18]). Suppose that ${\left\{S_{k_1}\right\}}_{k_1\in \mathbb{Z}}$ and ${\left\{S_{k_2}\right\}}_{k_2\in \mathbb{Z}}$ are approximations of the identity defined in Definition 5 and $D_{k_i}=S_{k_i}-S_{k_i-1}(i=1,2)$. Then, for all $f\in L^2(X_1\times X_2)$, we have

$$\begin{array}{cc}\hfill f(x,y)& =\sum _{k_1}\sum _{k_2}{\tilde{D}}_{k_1}{M}_{b_1}{\tilde{D}}_{k_2}{M}_{b_2}{D}_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}f(x,y)\hfill \\ \hfill & =\sum _{k_1}\sum _{k_2}{D}_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{\tilde{\tilde{D}}}_{k_1}{M}_{b_1}{\tilde{\tilde{D}}}_{k_2}{M}_{b_2}f(x,y),\hfill \end{array}$$

where the series converge in the norm of $L^2(X_1\times X_2)$. Moreover, ${\tilde{D}}_{k_i}(x,y)(i=1,2)$, the kernel of ${\tilde{D}}_{k_i}$, satisfies the following estimates: for $0<\epsilon <\theta$, there exists a constant $C>0$ depending on ε and θ such that

$$\left|{\tilde{D}}_{k_i}(x,y)\right|\le C{\displaystyle \frac{2^{-k_i\epsilon }}{{\left(2^{-k_i}+\rho (x,y)\right)}^{1+\epsilon }}},$$

$$\left|{\tilde{D}}_{k_i}(x,y)-{\tilde{D}}_{k_i}\left(x^{\prime },y\right)\right|\le C{\left({\displaystyle \frac{\rho \left(x,x^{\prime }\right)}{2^{-k_i}+\rho (x,y)}}\right)}^{\epsilon }{\displaystyle \frac{2^{-k_i\epsilon }}{{\left(2^{-k_i}+\rho (x,y)\right)}^{1+\epsilon }}},$$

$for\rho \left(x,x^{\prime }\right)\le {\displaystyle \frac{1}{2A}}\left(2^{-k_i}+\rho (x,y)\right),$

$${\int }_{X_i}{\tilde{D}}_{k_l}(x,y)b_i\left(y\right)d{\mu }_i\left(y\right)={\int }_{X_i}{\tilde{D}}_{k_i}(x,y)b_i\left(x\right)d{\mu }_i\left(x\right)=0.$$

${\tilde{\tilde{D}}}_{k_i}(x,y)(i=1,2)$, the kernel of ${\tilde{\tilde{D}}}_{k_i}$, satisfies the same conditions above but interchanging the positions of x and y.

Suppose that $\left\{D_{k_n}\right\}$, $\left\{{\tilde{D}}_{k_n}\right\}$, $\left\{D_{j_n}\right\}$, $\left\{{\tilde{\tilde{D}}}_{j_n}\right\}$, $\left\{D_{i_n}\right\}$, $\left\{{\tilde{D}}_{i_n}\right\}$, $\left\{D_{l_n}\right\}$, and $\left\{{\tilde{\tilde{D}}}_{l_n}\right\}$, $k,j,i,l\in \mathbb{Z}$, $n=1,2$, are families of operators given by Calderón reproducing formulas in Lemma 1. Suppose that $T_1,T_2$ are generalized-product Calderón–Zygmund operators; we can obtain the following representation of $T=T_2\circ T_1$ with respect to all these families: $\left\{D_{k_n}\right\}$, $\left\{{\tilde{D}}_{k_n}\right\}$, $\left\{D_{j_n}\right\}$, $\left\{{\tilde{\tilde{D}}}_{j_n}\right\}$, $\left\{D_{i_n}\right\}$, $\left\{{\tilde{D}}_{i_n}\right\}$, $\left\{D_{l_n}\right\}$ and $\left\{{\tilde{\tilde{D}}}_{l_n}\right\}$, $k,j,i,l\in \mathbb{Z}$, $n=1,2$,

$$\begin{array}{cc}\hfill Tf\left(x\right)=& T_2\circ T_1\left(f\right)\left(x_1,x_2\right)\hfill \\ \hfill =& \sum _{k_1,k_2}\sum _{j_1,j_2}\sum _{i_1,i_2}\sum _{l_1,l_2}{\tilde{D}}_{k_1}{M}_{b_1}{\tilde{D}}_{k_2}{M}_{b_2}\hfill \\ \hfill & \times D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}\hfill \\ \hfill & \times M_{b_1}{\tilde{\tilde{D}}}_{j_1}{M}_{b_2}{\tilde{\tilde{D}}}_{j_2}b{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}\hfill \\ \hfill & \times D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}\hfill \\ \hfill & \times {\tilde{\tilde{D}}}_{l_1}{M}_{b_1}{\tilde{\tilde{D}}}_{l_2}{M}_{b_2}\left(f\right)(x_1,x_2).\hfill \end{array}$$

Thus, we can show that $K(x_1,x_2,y_1,y_2)$, the kernel of T, can be written as

$$\begin{array}{cc}\hfill K(x_1,x_2,y_1,y_2)=& \sum _{k_1,k_2}\sum _{j_1,j_2}\sum _{i_1,i_2}\sum _{l_1,l_2}{\tilde{D}}_{k_1}{M}_{b_1}{\tilde{D}}_{k_2}{M}_{b_2}\hfill \\ \hfill & \times D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}\hfill \\ \hfill & \times M_{b_1}{\tilde{\tilde{D}}}_{j_1}{M}_{b_2}{\tilde{\tilde{D}}}_{j_2}b{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}\hfill \\ \hfill & \times D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}\hfill \\ \hfill & \times {\tilde{\tilde{D}}}_{l_1}{M}_{b_1}{\tilde{\tilde{D}}}_{l_2}{M}_{b_2}(x_1,x_2,y_1,y_2).\hfill \end{array}$$

(12)

3.2. Several Key Almost-Orthogonal Estimates

In order to verify Theorem 1, the following almost-orthogonal estimates play an important role:

Lemma 2. 

Suppose that ${\tilde{D}}_{j_i},{\tilde{\tilde{D}}}_{k_i}(j_i,k_i\in \mathbb{Z},i=1,2)$ are the same as in Lemma 1; then

$$\begin{array}{cc}\hfill & |{\tilde{\tilde{D}}}_{k_1}{M}_{b_1}{\tilde{\tilde{D}}}_{k_2}{M}_{b_2}{\tilde{D}}_{j_1}{M}_{b_1}{\tilde{D}}_{j_2}{M}_{b_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & \\ \hfill & \le C{2}^{-\left|k_1-j_1\right|\epsilon }{2}^{-\left|k_2-j_2\right|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{\left(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{\left(2^{-(k_2\wedge j_2)}+{\rho }_2(x_2,y_2)\right)}^{1+\epsilon }}}.\hfill \end{array}$$

(13)

Proof. 

For the sake of simplicity, we only consider the case $k_1>j_1$, $k_2>j_2$. The other three cases are similar. Through the cancellation condition

$${\int }_{X_1}{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)b_1\left(z_1\right)d{\mu }_1\left(z_1\right)=0,{\int }_{X_2}{\tilde{\tilde{D}}}_{k_2}(x_2,z_2)b_2\left(z_2\right)d{\mu }_2\left(z_2\right)=0,$$

we have

$$\begin{array}{cc}\hfill & |{\tilde{\tilde{D}}}_{k_1}{M}_{b_1}{\tilde{\tilde{D}}}_{k_2}{M}_{b_2}{\tilde{D}}_{j_1}{M}_{b_1}{\tilde{D}}_{j_2}{M}_{b_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & =|{\int }_{X_1}{\int }_{X_2}{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)b_1\left(z_1\right){\tilde{\tilde{D}}}_{k_2}(x_2,z_2)b_2\left(z_2\right)\left[{\tilde{D}}_{j_1}(z_1,y_1)-{\tilde{D}}_{j_1}(x_1,y_1)\right]\hfill \\ \hfill & \times b_1\left(y_1\right)\left[{\tilde{D}}_{j_2}(z_2,y_2)-{\tilde{D}}_{j_2}(x_2,y_2)\right]b_2\left(y_2\right)d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)|\hfill \\ \hfill & \le C{\int }_{X_1}{\int }_{X_2}|{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)\left|\right|{\tilde{\tilde{D}}}_{k_2}(x_2,z_2)\left|\right|{\tilde{D}}_{j_1}(z_1,y_1)-{\tilde{D}}_{j_1}(x_1,y_1)|\hfill \\ \hfill & \times |{\tilde{D}}_{j_2}(z_2,y_2)-{\tilde{D}}_{j_2}(x_2,y_2)|d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right).\hfill \end{array}$$

We only give the estimate for the term ${\int }_{X_1}|{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)\left|\right|{\tilde{D}}_{j_1}(z_1,y_1)-{\tilde{D}}_{j_1}(x_1,y_1)|d{\mu }_1\left(z_1\right)$. Note that the support conditions for ${\tilde{\tilde{D}}}_{k_1}$ and ${\tilde{D}}_{j_1}$, namely

$${\rho }_1(x_1,z_1)<C{2}^{-k_1},{\rho }_1(z_1,y_1)<C{2}^{-j_1},$$

for all $x_1,y_1,z_1\in X_1$, we can derive

$${\rho }_1(x_1,y_1)\le A\left({\rho }_1(x_1,z_1)+{\rho }_1(z_1,y_1)\right)\le AC{2}^{-k_1}+AC{2}^{-j_1}\le AC{2}^{-j_1}.$$

If ${\rho }_1(x_1,z_1)>{\displaystyle \frac{2^{-k_1}+{\rho }_1(x_1,y_1)}{2A}}$, then ${\displaystyle \frac{2A{\rho }_1(x_1,z_1)}{2^{-k_1}}}>1$. Using the size condition in ${\tilde{\tilde{D}}}_{k_1}$ and ${\tilde{D}}_{j_1}$, we have

$$\begin{array}{cc}\hfill & {\int }_{X_1}|{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)\left|\right|{\tilde{D}}_{j_1}(z_1,y_1)-{\tilde{D}}_{j_1}(x_1,y_1)|d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\int }_{X_1}{\displaystyle \frac{2^{-k_1\epsilon }}{{\left(2^{-k_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}{\left({\displaystyle \frac{2A{\rho }_1(x_1,z_1)}{2^{-k_1}}}\right)}^{\epsilon }\hfill \\ \hfill & \times \left({\displaystyle \frac{2^{-j_1\epsilon }}{{\left(2^{-j_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}+{\displaystyle \frac{2^{-j_1\epsilon }}{{\left(2^{-j_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}\right)d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\int }_{{\chi }_{\{{\rho }_1(x_1,z_1)<C{2}^{-k_1}\}}}{2}^{k_1}{2}^{k_1\epsilon }{2}^{j_1}{\rho }_1{(x_1,z_1)}^{\epsilon }d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{2}^{-(k_1-j_1)\epsilon }{2}^{j_1\epsilon }{\chi }_{\{{\rho }_1(x_1,y_1)<C{2}^{-j_1}\}}\hfill \\ \hfill & \approx C{2}^{-(k_1-j_1)\epsilon }{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

(14)

If ${\rho }_1(x_1,z_1)\le {\displaystyle \frac{2^{-k_1}+{\rho }_1(x_1,y_1)}{2A}}$, taking advantage of the size condition in ${\tilde{\tilde{D}}}_{k_1}$ and the smooth condition in ${\tilde{D}}_{j_1}$, we get

$$\begin{array}{cc}\hfill & {\int }_{X_1}|{\tilde{\tilde{D}}}_{k_1}(x_1,z_1)\left|\right|{\tilde{D}}_{j_1}(z_1,y_1)-{\tilde{D}}_{j_1}(x_1,y_1)|d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\int }_{X_1}{\displaystyle \frac{2^{-k_1\epsilon }}{{\left(2^{-k_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{{\rho }_1\left(z_1,x_1\right)}{2^{-j_1}+{\rho }_1(x_1,y_1)}}\right)}^{\epsilon }{\displaystyle \frac{2^{-j_1\epsilon }}{{\left(2^{-j_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\int }_{{\chi }_{\{{\rho }_1(x_1,z_1)<C{2}^{-k_1}\}}}{2}^{k_1}{\rho }_1{(z_1,x_1)}^{\epsilon }{2}^{j_1(1+\epsilon )}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{2}^{-(k_1-j_1)\epsilon }{2}^{j_1}{\chi }_{\{{\rho }_1(x_1,z_1)<C{2}^{-k_1}\}}\hfill \\ \hfill & \approx C{2}^{-(k_1-j_1)\epsilon }{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

(15)

By applying the same method,

$$\begin{array}{c}\hfill {\int }_{X_2}|{\tilde{\tilde{D}}}_{k_2}(x_2,z_2)\left|\right|{\tilde{D}}_{j_2}(z_2,y_2)-{\tilde{D}}_{j_2}(x_2,y_2)|d{\mu }_2\left(z_2\right)\le C{2}^{-(k_2-j_2)\epsilon }{\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}.\end{array}$$

Therefore, we get this estimate (13). □

Suppose T is defined as in Definition 4, $D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)$. The kernel of $D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}$ satisfies the following almost-orthogonal estimate, which generalizes the corollary of Zheng [19] in Euclidean space:

Lemma 3. 

Let T be the same as in Definition 4. If $T_1\left(b_1\right)=T_2\left(b_2\right)=T_1^{*}\left(b_1\right)=T_2^{*}\left(b_2\right)=0$, then

$$\begin{array}{cc}\hfill & |D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & \le C{2}^{-|k_1-j_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

Proof. 

For the different $k_1$ and $k_2$, $j_1$ and $j_2$, we consider the following four cases: $\left(\mathrm{I}\right)$ $k_1\le j_1$, $k_2\le j_2$; $\left(\mathrm{II}\right)$ $k_1\le j_1$, $k_2>j_2$; $\left(\mathrm{III}\right)$ $k_1>j_1$, $k_2\le j_2$; $\left(\mathrm{IV}\right)$ $k_1>j_1$, $k_2>j_2$. Here, we only give the details of the first case $\left(\mathrm{I}\right)$. The other cases are similar. For $x_1,y_1\in X_1$, $x_2,y_2\in X_2$, we split $\left(\mathrm{I}\right)$ into four cases:

(I₁)

${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)\ge 4A^{2}C{2}^{-k_2}$,

(I₂)

${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$,

(I₃)

${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)\ge 4A^{2}C{2}^{-k_2}$,

(I₄)

${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$.

First, we consider the case $\left({\mathrm{I}}_1\right)$: $k_1\le j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)\ge 4A^{2}C{2}^{-k_2}$. Noticing the Definition 5, we have ${\rho }_1(x_1,u_1)\le C{2}^{-k_1}$, ${\rho }_2(x_2,u_2)\le C{2}^{-k_2}$, ${\rho }_1(v_1,y_1)\le C{2}^{-j_1}$, ${\rho }_2(v_2,y_2)\le C{2}^{-j_2}$. We easily get

$$\begin{array}{cc}\hfill {\rho }_1(u_1,y_1)& \ge A{\rho }_1(x_1,y_1)-A{\rho }_1(u_1,x_1)\ge A{\rho }_1(x_1,y_1)-AC{2}^{-k_1}\hfill \\ \hfill & \ge A{\rho }_1(x_1,y_1)-{\displaystyle \frac{{\rho }_1(x_1,y_1)}{4A}}={\displaystyle \frac{4A^2-1}{4A}}{\rho }_1(x_1,y_1)\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1),\hfill \\ \hfill {\rho }_1(v_1,y_1)& \le C{2}^{-j_1}\le C{2}^{-k_1}\le {\displaystyle \frac{{\rho }_1(x_1,y_1)}{4A^2}}.\hfill \end{array}$$

Combining the above, for all $u_1,v_1,y_1\in X_1$, $u_2,v_2,y_2\in X_2$, we have ${\rho }_1(v_1,y_1)\le {\displaystyle \frac{{\rho }_1(u_1,y_1)}{2A}}$. Similarly, ${\rho }_2(v_2,y_2)\le {\displaystyle \frac{{\rho }_2(u_2,y_2)}{2A}}$, which together with the cancellation condition

$${\int }_{X_1}{D}_{j_1}(v_1,y_1)b_1\left(v_1\right)d{\mu }_1\left(v_1\right)=0,{\int }_{X_2}{D}_{j_2}(v_2,y_2)b_2\left(v_2\right)d{\mu }_2\left(v_2\right)=0$$

and the Hölder condition of the kernel K in (${\mathrm{III}}_1$) of Definition 2, we obtain,

$$\begin{array}{cc}\hfill & \left|D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\right|\hfill \\ \hfill & =|{\int \int }_{X_1\times X_2}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)D_{k_2}(x_2,u_2)b_2\left(u_2\right){\int \int }_{X_1\times X_2}[K(u_1,u_2,v_1,v_2)-K(u_1,u_2,y_1,v_2)\hfill \\ \hfill & - K(u_1,u_2,v_1,y_2)+K(u_1,u_2,y_1,y_2)]b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)\hfill \\ \hfill & \times d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)|\hfill \\ \hfill & \le C{\int \int }_{X_1\times X_2}{\int \int }_{X_1\times X_2}|D_{k_1}(x_1,u_1)||b_1\left(u_1\right)||D_{k_2}(x_2,u_2)||b_2\left(u_2\right)|{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}{\displaystyle \frac{{\rho }_2{(v_2,y_2)}^{\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}\hfill \\ \hfill & \times d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\left|b_1\left(j_1\right)\right|\left|D_{j_1}(v_1,y_1)\right|\left|b_2\left(v_2\right)\right|\left|D_{j_2}(v_2,y_2)\right|d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right).\hfill \end{array}$$

Combining ${\rho }_1(u_1,y_1)\ge {\displaystyle \frac{4A^2-1}{4A}}{\rho }_1(x_1,y_1)$ and ${\epsilon }^{\prime }<\epsilon$, which yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}& \le C{\displaystyle \frac{2^{-j_1\epsilon }}{{\rho }_1{(x_1,y_1)}^{1+\epsilon }}}\le C{\displaystyle \frac{2^{-j_1\epsilon }}{{\left(2^{-k_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}\hfill \\ \hfill & =C{2}^{(k_1-j_1)\epsilon }{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_1(\epsilon -{\epsilon }^{\prime })}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{\epsilon -{\epsilon }^{\prime }}}}\hfill \\ \hfill & \le C{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

(16)

Similarly, we have ${\displaystyle \frac{{\rho }_2{(v_2,y_2)}^{\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}\le C{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}$. By the above estimates, we get

$$\begin{array}{cc}\hfill & \left|D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\right|\hfill \\ \hfill & \le C{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}C{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}\hfill \\ \hfill & \times {\int \int }_{X_1\times X_2}|D_{k_1}(x_1,u_1)||b_1\left(u_1\right)||D_{k_2}(x_2,u_2)||b_2\left(u_2\right)|d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill & \times \left|b_1\left(v_1\right)\right|\left|D_{j_1}(v_1,y_1)\right|\left|b_2\left(v_2\right)\right|\left|D_{j_2}(v_2,y_2)\right|d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)\hfill \\ \hfill & \le C{2}^{(k_1-j_1){\epsilon }^{\prime }}{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

Second, for the case $\left({\mathrm{I}}_2\right)$: $k_1\le j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$. Since $T_2^{*}\left(b_2\right)=0$, we have

$$\begin{array}{cc}\hfill & D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\hfill \\ \hfill & ={\int \int }_{X_1\times X_2}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)D_{k_2}(x_2,u_2)b_2\left(u_2\right){\int \int }_{X_1\times X_2}K(u_1,u_2,v_1,v_2)\hfill \\ \hfill & \times b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill & ={\int \int }_{X_1\times X_2}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right){\int \int }_{X_1\times X_2}K(u_1,u_2,v_1,v_2)\hfill \\ \hfill & \times b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right).\hfill \end{array}$$

Let a smooth cut-off function ${\xi }_0\in C^{\infty }\left(\mathbb{R}\right)$ be 1 in the unit ball and 0 outside its double. Set ${\xi }_1=1-{\xi }_0$. Then, we write

$$\begin{array}{cc}\hfill & D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\hfill \\ \hfill & ={\int \int }_{X_1\times X_2}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right)({\xi }_0\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\hfill \\ \hfill & + {\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)){\int \int }_{X_1\times X_2}K(u_1,u_2,v_1,v_2)b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)\hfill \\ \hfill & \times d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill & :={\mathrm{I}}_{21}+{\mathrm{I}}_{22}.\hfill \end{array}$$

For ${\mathrm{I}}_{21}$, by the cancellation condition ${\int }_{X_1}{D}_{j_1}(v_1,y_1)b_1\left(v_1\right)d{\mu }_1\left(v_1\right)=0$, we have

$$\begin{array}{cc}\hfill {\mathrm{I}}_{21}=& {\displaystyle {\int \int }_{X_1\times X_2}}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right){\xi }_0\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\hfill \\ \hfill & \times {\displaystyle {\int \int }_{X_1\times X_2}}K(u_1,u_2,v_1,v_2)b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)\hfill \\ \hfill & d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill =& {\displaystyle {\int \int }_{X_1\times X_2}}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right){\xi }_0\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\hfill \\ \hfill & \times {\displaystyle {\int \int }_{X_1\times X_2}}\left[K(u_1,u_2,v_1,v_2)-K(u_1,u_2,y_1,v_2)\right]b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)\hfill \\ \hfill & \times d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right).\hfill \end{array}$$

Now, set $\psi \left(v_2\right)=D_{j_2}(v_2,y_2)$, $\phi \left(v_2\right)=\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]{\xi }_0\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)$ and for each $u_1,v_1\in X_1$, $u_2,v_2\in X_2$, define

$$\begin{array}{c}\hfill 〈\psi b_2,K_1(u_1,v_1)b_2\phi 〉={\int }_{X_2}{\int }_{X_2}\psi \left(v_2\right)b_2\left(v_2\right)K(u_1,u_2,v_1,v_2)b_2\left(u_2\right)\phi \left(u_2\right)d{\mu }_2\left(u_2\right)d{\mu }_2\left(v_2\right),\end{array}$$

where $K_1(u_1,v_1)$ is also a Calderón–Zygmund operator acting on $X_2$ with kernel

$K_1(u_1,v_1)(u_2,v_2)=K(u_1,u_2,v_1,v_2)$. Then,

$$\begin{array}{cc}\hfill |{\mathrm{I}}_{21}| =& |{\int }_{X_1}{\int }_{X_1}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)〈\psi b_2,\left[K_1(u_1,v_1)-K_1(u_1,y_1)\right]b_2\phi 〉b_1\left(v_1\right)D_{j_1}(v_1,y_1)d{\mu }_1\left(v_1\right)\hfill \\ \hfill & d{\mu }_2\left(u_2\right)|\hfill \\ \hfill \le & {\int }_{X_1}{\int }_{X_1}|D_{k_1}(x_1,u_1)||〈\psi b_2,\left[K_1(u_1,v_1)-K_1(u_1,y_1)\right]b_2\phi 〉||D_{j_1}(v_1,y_1)|d{\mu }_1\left(v_1\right)d{\mu }_1\left(u_1\right).\hfill \end{array}$$

The $L^2(X_1\times X_2)$ boundedness of T implies the following weak boundedness property [17]:

$$\begin{array}{cc}\hfill & \left|〈\psi b_2,\left[K_1(u_1,v_1)-K_1(u_1,y_1)\right]b_2\phi 〉\right|\hfill \\ \hfill & \le C{2}^{-j_2(1+2\eta )}{\parallel \phi \parallel }_{C_0^{\eta }}{\parallel \psi \parallel }_{C_0^{\eta }}{\parallel K_1(u_1,v_1)-K_1(u_1,y_1)\parallel }_{CZ}\hfill \\ \hfill & \le C{2}^{-j_2(1+2\eta )}\left(2^{-(j_2-k_2)\epsilon }{2}^{k_2}{2}^{j_2\eta }\right)\left(2^{j_2}{2}^{j_2\eta }\right){\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{(k_2-j_2)\epsilon }{2}^{k_2}{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}},\hfill \end{array}$$

where ${\rho }_1(v_1,y_1)<{\displaystyle \frac{1}{2A}}{\rho }_1(u_1,y_1)$. Since $j_2\ge k_2$, $\epsilon >{\epsilon }^{\prime }$, through (16), it is easy to obtain that

$$\begin{array}{cc}\hfill |{\mathrm{I}}_{21}| \le & C{2}^{(k_2-j_2){\epsilon }^{\prime }}{2}^{k_2}{\int }_{X_1}{\int }_{X_1}\left|D_{k_1}(x_1,u_1)\right|{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}\left|D_{j_1}(v_1,y_1)\right|d{\mu }_1\left(v_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{(k_2-j_2){\epsilon }^{\prime }}{2}^{k_2}{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

Next, we deal with the term ${\mathrm{I}}_{22}$. Using the cancellation condition

$${\int }_{X_1}{D}_{j_1}(v_1,y_1)b_1\left(v_1\right)d{\mu }_1\left(v_1\right)=0,{\int }_{X_2}{D}_{j_2}(v_2,y_2)b_2\left(v_2\right)d{\mu }_2\left(v_2\right)=0,$$

we write

$$\begin{array}{cc}\hfill {\mathrm{I}}_{22}=& {\int \int }_{X_1\times X_2}{D}_{k_1}(x_1,u_1)b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right){\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\hfill \\ \hfill & \times {\int \int }_{X_1\times X_2}\left[K(u_1,u_2,v_1,v_2)-K(u_1,u_2,y_1,v_2)-K(u_1,u_2,v_1,y_2)+K(u_1,u_2,y_1,y_2)\right]\hfill \\ \hfill & \times b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right).\hfill \end{array}$$

Combining the estimate of Lemma 1, the Hölder condition of the kernel K in (${\mathrm{III}}_1$) of Definition 2, and the estimate of (16), the similar methods in (14) and (15) could be used to estimate $\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]$. We now consider the case where ${\rho }_2(u_2,y_2)\le {\displaystyle \frac{2^{-k_2}+{\rho }_2(x_2,u_2)}{2A_2}}$.

$$\begin{array}{cc}\hfill {\mathrm{I}}_{22}\le & {\int \int }_{X_1\times X_2}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,u_1))}^{1+\epsilon }}}{\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{2^{-k_2}+{\rho }_2(x_2,u_2)}}\right)}^{{\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,u_2))}^{1+{\epsilon }^{\prime }}}}\hfill \\ \hfill & \times {\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right){\int \int }_{X_1\times X_2}{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}{\displaystyle \frac{{\rho }_2{(v_2,y_2)}^{\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{\int \int }_{X_1\times X_1}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,u_1))}^{1+\epsilon }}}{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(v_1\right)d{\mu }_1\left(u_1\right)\hfill \\ \hfill & \times {\int \int }_{X_2\times X_2}{\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{2^{-k_2}+{\rho }_2(x_2,u_2)}}\right)}^{{\epsilon }^{\prime }}{2}^{k_2}{\displaystyle \frac{{\rho }_2{(v_2,y_2)}^{\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}{\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}{\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\hfill \\ \hfill & \times d{\mu }_2\left(v_2\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\int \int }_{X_2\times X_2}{\displaystyle \frac{1}{{\rho }_2{(u_2,y_2)}^{1+\epsilon -{\epsilon }^{\prime }}}}{\left({\displaystyle \frac{1}{2^{-k_2}+{\rho }_2(x_2,u_2)}}\right)}^{{\epsilon }^{\prime }}\hfill \\ \hfill & \times 2^{k_2}{\rho }_2{(v_2,y_2)}^{\epsilon }{\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)d{\mu }_2\left(v_2\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{2}^{k_2}{2}^{k_2{\epsilon }^{\prime }}{2}^{-j_2\epsilon }{\int }_{X_2}{\displaystyle \frac{1}{{\rho }_2{(u_2,y_2)}^{1+\epsilon -{\epsilon }^{\prime }}}}{\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{2}^{k_2}{2}^{k_2{\epsilon }^{\prime }}{2}^{-j_2\epsilon }{2}^{-j_2({\epsilon }^{\prime }-\epsilon )}\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

For the term $\left({\mathrm{I}}_3\right)$: $k_1\le j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)\ge 4A^{2}C{2}^{-k_2}$, using the similar method with $\left({\mathrm{I}}_2\right)$, we can get that

$$\left|{\mathrm{I}}_3\right|\le C{2}^{(k_1-j_1){\epsilon }^{\prime }}{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.$$

For the case $\left({\mathrm{I}}_4\right)$: $k_1\le j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$ and ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$. Given the fact that $T_1^{*}\left(b_1\right)=T_2^{*}\left(b_2\right)=0$, we write

$$\begin{array}{cc}\hfill & D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\hfill \\ \hfill & ={\int \int }_{X_1\times X_2}\left[D_{k_1}(x_1,u_1)-D_{k_1}(x_1,y_1)\right]b_1\left(u_1\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right)\hfill \\ \hfill & \times {\int \int }_{X_1\times X_2}K(u_1,u_2,v_1,v_2)b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)\hfill \\ \hfill & \times d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right).\hfill \end{array}$$

Let a smooth cut-off function ${\varphi }_0\in C^{\infty }\left(\mathbb{R}\right)$ be 1 in the unit ball and 0 outside its double. Set ${\varphi }_1=1-{\varphi }_0$. We can write the following four parts:

$$\begin{array}{cc}\hfill & D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)\hfill \\ \hfill & ={\int \int }_{X_1\times X_2}\left[D_{k_1}(x_1,u_1)-D_{k_1}(x_1,y_1)\right]b_1\left(u_1\right)\left[{\varphi }_0\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right)+{\varphi }_1\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right)\right]\hfill \\ \hfill & \times \left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]b_2\left(u_2\right)\left[{\xi }_0\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)+{\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)\right]\hfill \\ \hfill & \times {\int \int }_{X_1\times X_2}K(u_1,u_2,v_1,v_2)b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)\hfill \\ \hfill & \times d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill & :={\mathrm{I}}_{41}+{\mathrm{I}}_{42}+{\mathrm{I}}_{43}+{\mathrm{I}}_{44}.\hfill \end{array}$$

We only provide the details of ${\mathrm{I}}_{44}$, since ${\mathrm{I}}_{41}$, ${\mathrm{I}}_{42}$, and ${\mathrm{I}}_{43}$ are similar to ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$. Using the cancellation conditions ${\int }_{X_1}{D}_{j_1}(v_1,y_1)b_1\left(v_1\right)d{\mu }_1\left(v_1\right)=0$ and ${\int }_{X_2}{D}_{j_2}(v_2,y_2)b_2\left(v_2\right)d{\mu }_2\left(v_2\right)=0$, we can write

$$\begin{array}{cc}\hfill {\mathrm{I}}_{44}=& {\int \int }_{X_1\times X_2}\left[D_{k_1}(x_1,u_1)-D_{k_1}(x_1,y_1)\right]b_1\left(u_1\right){\varphi }_1\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right)\left[D_{k_2}(x_2,u_2)-D_{k_2}(x_2,y_2)\right]\hfill \\ \hfill & \times b_2\left(u_2\right){\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right){\int \int }_{X_1\times X_2}[K(u_1,u_2,v_1,v_2)-K(u_1,u_2,y_1,v_2)-K(u_1,u_2,v_1,y_2)\hfill \\ \hfill & + K(u_1,u_2,y_1,y_2)]b_1\left(v_1\right)D_{j_1}(v_1,y_1)b_2\left(v_2\right)D_{j_2}(v_2,y_2)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right).\hfill \end{array}$$

We also need to use the estimate of Lemma 1 and the Hölder condition of the kernel K in $\left({\mathrm{III}}_1\right)$ in Definition 2. Since ${\rho }_1(v_1,y_1)\le C{2}^{-j_1}$, ${\varphi }_1:{\rho }_1(u_1,y_1)>4A^{2}C{2}^{-j_1}$, we obtain ${\rho }_1(v_1,y_1)\le {\displaystyle \frac{1}{2A_1}}{\rho }_1(u_1,y_1)$. Similarly, ${\rho }_2(v_2,y_2)\le {\displaystyle \frac{1}{2A_2}}{\rho }_2(u_2,y_2)$. Combining the above facts, we obtain

$$\begin{array}{cc}\hfill \left|{\mathrm{I}}_{44}\right|\le & C{\int \int }_{X_1\times X_2}\left({\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,u_1))}^{1+\epsilon }}}+{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\right){\varphi }_1\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right)\hfill \\ \hfill & \times \left({\displaystyle \frac{2^{-k_2\epsilon }}{{(2^{-k_2}+{\rho }_2(x_2,u_2))}^{1+\epsilon }}}+{\displaystyle \frac{2^{-k_2\epsilon }}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\right){\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right){\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{2^{-k_1}}}\right)}^{{\epsilon }^{\prime }}\hfill \\ \hfill & \times {\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{2^{-k_2}}}\right)}^{{\epsilon }^{\prime }}{\int \int }_{X_1\times X_2}{\displaystyle \frac{{\rho }_1{(v_1,y_1)}^{\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}{\displaystyle \frac{{\rho }_2{(v_2,y_2)}^{\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{\int \int }_{X_1\times X_1}(2^{k_1}+2^{k_1}){\varphi }_1\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right){\displaystyle \frac{2^{-j_1\epsilon }}{{\rho }_1{(u_1,y_1)}^{1+\epsilon }}}{\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{2^{-k_1}}}\right)}^{{\epsilon }^{\prime }}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times d{\mu }_1\left(v_1\right)d{\mu }_1\left(u_1\right){\int \int }_{X_2\times X_2}(2^{k_2}+2^{k_2}){\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right){\displaystyle \frac{2^{-j_2\epsilon }}{{\rho }_2{(u_2,y_2)}^{1+\epsilon }}}{\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{2^{-k_2}}}\right)}^{{\epsilon }^{\prime }}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}d{\mu }_2\left(v_2\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{-j_1\epsilon }{2}^{k_1{\epsilon }^{\prime }}{2}^{k_1}{\int }_{X_1}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(v_1\right){\int }_{X_1}{\displaystyle \frac{1}{{\rho }_1{(u_1,y_1)}^{1+\epsilon -{\epsilon }^{\prime }}}}{\varphi }_1\left({\displaystyle \frac{{\rho }_1(u_1,y_1)}{4A^{2}C{2}^{-j_1}}}\right)\hfill \\ \hfill & \times d{\mu }_1\left(u_1\right)2^{-j_2\epsilon }{2}^{k_2{\epsilon }^{\prime }}{2}^{k_2}{\int }_{X_2}{\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}d{\mu }_1\left(v_2\right){\int }_{X_2}{\displaystyle \frac{1}{{\rho }_2{(u_2,y_2)}^{1+\epsilon -{\epsilon }^{\prime }}}}\hfill \\ \hfill & \times {\xi }_1\left({\displaystyle \frac{{\rho }_2(u_2,y_2)}{4A^{2}C{2}^{-j_2}}}\right)d{\mu }_2\left(u_2\right)\hfill \\ \hfill \le & C{2}^{-j_1\epsilon }{2}^{k_1{\epsilon }^{\prime }}{2}^{k_1}{2}^{-j_1({\epsilon }^{\prime }-\epsilon )}{\int }_{X_1}{\displaystyle \frac{2^{-j_1\epsilon }}{{(2^{-j_1}+{\rho }_1(v_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(v_1\right)\hfill \\ \hfill & \times 2^{-j_2\epsilon }{2}^{k_2{\epsilon }^{\prime }}{2}^{k_2}{2}^{-j_2({\epsilon }^{\prime }-\epsilon )}{\int }_{X_2}{\displaystyle \frac{2^{-j_2\epsilon }}{{(2^{-j_2}+{\rho }_2(v_2,y_2))}^{1+\epsilon }}}d{\mu }_1\left(v_2\right)\hfill \\ \hfill \le & C{2}^{(k_1-j_1){\epsilon }^{\prime }}{2}^{(k_2-j_2){\epsilon }^{\prime }}{\displaystyle \frac{2^{-k_1{\epsilon }^{\prime }}}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-k_2{\epsilon }^{\prime }}}{{(2^{-k_2}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

In conclusion, for the case $\left({\mathrm{IV}}_2\right)$ $k_1>j_1$, $k_2>j_2$, ${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-j_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-j_2}$, combining $T_2\left(b_2\right)=0$; for the case $\left({\mathrm{IV}}_4\right)$ $k_1>j_1$, $k_2>j_2$, ${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$, using $T_1\left(b_1\right)=T_2\left(b_2\right)=0$; for the case $\left({\mathrm{II}}_2\right)$ $k_1\le j_1$, $k_2>j_2$, ${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-k_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-j_2}$, combining $T_2\left(b_2\right)=0$; for the case $\left({\mathrm{II}}_4\right)$ $k_1\le j_1$, $k_2>j_2$, ${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-k_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-j_2}$, using $T_1^{*}\left(b_1\right)=T_2^{*}\left(b_2\right)=0$; for the case $\left({\mathrm{III}}_2\right)$ $k_1>j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)\ge 4A^{2}C{2}^{-j_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$, combining $T_2^{*}\left(b_2\right)=0$; for the case $\left({\mathrm{III}}_4\right)$ $k_1>j_1$, $k_2\le j_2$, ${\rho }_1(x_1,y_1)<4A^{2}C{2}^{-j_1}$, ${\rho }_2(x_2,y_2)<4A^{2}C{2}^{-k_2}$, using $T_1^{*}\left(b_1\right)=T_2^{*}\left(b_2\right)=0$. Hence, the proof of Lemma 3 is complete. □

Using the same method as in Lemma 3, we obtain the following results:

Lemma 4. 

Let T be the same as in Definition 4. If $T_1\left(b_1\right)=T_2\left(b_2\right)=0$, then

$$\begin{array}{cc}\hfill & |D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & \le C(1+|k_1-j_1\left|\right)(1+|k_2-j_2\left|\right)(2^{-(k_1-j_1)\epsilon }\wedge 1)(2^{-(k_2-j_2)\epsilon }\wedge 1)\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+\rho (x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+\rho (x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

Lemma 5. 

Let T be the same as in Definition 4. If $T_1^{*}\left(b_1\right)=T_2^{*}\left(b_2\right)=0$, then

$$\begin{array}{cc}\hfill & |D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}T{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & \le C(1+|k_1-j_1\left|\right)(1+|k_2-j_2\left|\right)(2^{-(j_1-k_1)\epsilon }\wedge 1)(2^{-(j_2-k_2)\epsilon }\wedge 1)\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+\rho (x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+\rho (x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

3.3. Some Useful Estimates

We also need the following results:

Proposition 1. 

Let $l_1,l_2,i_1,i_2,j_1,j_2,k_1,k_2\in \mathbb{Z}$ and $0<{\epsilon }^{\prime }<\epsilon$, then

$$\begin{array}{cc}\hfill & \sum _{i_1}\sum _{i_2}{2}^{-|l_1-i_1|\epsilon }{2}^{-|l_2-i_2|\epsilon }{2}^{-|i_1-j_1|\epsilon }{2}^{-|i_2-j_2|\epsilon }{\displaystyle \frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(l_2\wedge i_2\wedge j_2)\epsilon }}{{(2^{-(l_2\wedge i_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{2}^{-|l_2-j_2|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(l_2\wedge j_2)\epsilon }}{{(2^{-(l_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

Proof. 

For clarity, we only consider ${\displaystyle \sum _{i_1}{2}^{-|l_1-i_1|\epsilon }{2}^{-|i_1-j_1|\epsilon }\frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}$. By symmetry, we may consider only three cases where $l_1\le j_1\le i_1$, $l_1\le i_1\le j_1$, and $i_1\le l_1\le j_1$. We first consider the case $l_1\le j_1\le i_1$, then

$$\begin{array}{cc}\hfill & \sum _{l_1\le j_1\le i_1}{2}^{-|l_1-i_1|\epsilon }{2}^{-|i_1-j_1|\epsilon }{\displaystyle \frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\sum _{l_1\le j_1\le i_1}{2}^{(l_1-i_1)\epsilon }{2}^{(j_1-i_1)\epsilon }{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{(l_1-j_1)\epsilon }{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

For the case $l_1\le i_1\le j_1$,

$$\begin{array}{cc}\hfill & \sum _{l_1\le i_1\le j_1}{2}^{-|l_1-i_1|\epsilon }{2}^{-|i_1-j_1|\epsilon }{\displaystyle \frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\sum _{l_1\le i_1\le j_1}{2}^{(l_1-i_1)\epsilon }{2}^{(i_1-j_1)\epsilon }{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\sum _{l_1\le i_1\le j_1}{2}^{(l_1-j_1)\epsilon }{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le (j_1-l_1)2^{(l_1-j_1)\epsilon }{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(l_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Finally, if $i_1\le l_1\le j_1$, then

$$\begin{array}{cc}\hfill & \sum _{i_1\le l_1\le j_1}{2}^{-|l_1-i_1|\epsilon }{2}^{-|i_1-j_1|\epsilon }{\displaystyle \frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\sum _{i_1\le l_1\le j_1}{2}^{(i_1-l_1)\epsilon }{2}^{(i_1-j_1)\epsilon }{\displaystyle \frac{2^{-i_1\epsilon }}{{(2^{-i_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-(l_1+j_1)\epsilon }(\sum _{\begin{array}{c}{i}_1\le l_1\\ 2^{-l_1}\le 2^{-i_1}\le {\rho }_1(x_1,y_1)\end{array}}{\displaystyle \frac{2^{i_1\epsilon }}{{\rho }_1{(x_1,y_1)}^{1+\epsilon }}}\hfill \\ \hfill & +\sum _{\begin{array}{c}{i}_1\le l_1\\ 2^{-l_1}\le {\rho }_1(x_1,y_1)\le 2^{-i_1}\end{array}}{\displaystyle \frac{2^{i_1\epsilon }}{2^{-i_1(1+\epsilon )}}}+\sum _{\begin{array}{c}{i}_1\le l_1\\ {\rho }_1(x_1,y_1)\le 2^{-l_1}\le 2^{-i_1}\end{array}}{\displaystyle \frac{2^{i_1\epsilon }}{2^{-i_1(1+\epsilon )}}})\hfill \\ \hfill & :={\mathrm{I}}_1+{\mathrm{I}}_2+{\mathrm{I}}_3.\hfill \end{array}$$

Note that $2^{-l_1}\le {\rho }_1(x_1,y_1)$ implies that $2^{-l_1}+{\rho }_1(x_1,y_1)\le 2{\rho }_1(x_1,y_1)$. Thus, we get

$$\begin{array}{cc}\hfill {\mathrm{I}}_1\le & C{2}^{-(l_1+j_1)\epsilon }\sum _{\begin{array}{c}{i}_1\le l_1\\ 2^{-l_1}\le 2^{-i_1}\le {\rho }_1(x_1,y_1)\end{array}}{\displaystyle \frac{2^{i_1\epsilon }{2}^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{2}^{l_1\epsilon }\hfill \\ \hfill \le & C{2}^{-(j_1-l_1)\epsilon }{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill \le & C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

For the term ${\mathrm{I}}_2$, when $2^{-l_1}\le {\rho }_1(x_1,y_1)\le 2^{-i_1}$, then ${\rho }_1(x_1,y_1)+2^{-l_1}\le 2·2^{-i_1}$. Therefore, we have

$$\begin{array}{cc}\hfill {\mathrm{I}}_2& \le C{2}^{-(l_1+j_1)\epsilon }\sum _{\begin{array}{c}{i}_1\le l_1\\ 2^{-l_1}\le {\rho }_1(x_1,y_1)\le 2^{-i_1}\end{array}}{\displaystyle \frac{2^{i_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-(l_1+j_1)\epsilon }\sum _{\begin{array}{c}{i}_1\le l_1\\ 2^{-l_1}\le {\rho }_1(x_1,y_1)\le 2^{-i_1}\end{array}}{\displaystyle \frac{2^{i_1\epsilon }{2}^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{2}^{l_1}\epsilon \hfill \\ \hfill & \le C{2}^{-(j_1-l_1)\epsilon }{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Using a similar method for ${\mathrm{I}}_3$, we obtain

$$\begin{array}{c}\hfill I_3\le C{2}^{-|l_1-j_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\end{array}$$

In the same way, we can deal with ${\displaystyle \sum _{i_2}{2}^{-|l_2-i_2|\epsilon }{2}^{-|i_2-j_2|\epsilon }\frac{2^{-(l_2\wedge i_2\wedge j_2)\epsilon }}{{(2^{-(l_2\wedge i_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}$. □

Next, using the same method as in Proposition 1, we can obtain the following result:

Remark 2. 

Let $l_1,l_2,i_1,i_2,j_1,j_2,k_1,k_2\in \mathbb{Z}$ and $0<{\epsilon }^{″}<{\epsilon }^{\prime }<\epsilon$, then

$$\begin{array}{cc}\hfill & \sum _{i_1}\sum _{i_2}{2}^{-|l_1-i_1|{\epsilon }^{\prime }}{2}^{-|l_2-i_2|{\epsilon }^{\prime }}{2}^{-|i_1-j_1|{\epsilon }^{\prime }}{2}^{-|i_2-j_2|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(l_1\wedge i_1\wedge j_1)\epsilon }}{{(2^{-(l_1\wedge i_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(l_2\wedge i_2\wedge j_2)\epsilon }}{{(2^{-(l_2\wedge i_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{2}^{-|l_1-j_1|{\epsilon }^{\prime \prime }}{2}^{-|l_2-j_2|{\epsilon }^{\prime \prime }}{\displaystyle \frac{2^{-(l_1\wedge j_1){\epsilon }^{\prime }}}{{(2^{-(l_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-(l_2\wedge j_2){\epsilon }^{\prime }}}{{(2^{-(l_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}.\hfill \end{array}$$

Proposition 2. 

Let $l_1,l_2,k_1,k_2\in \mathbb{Z}$ and $\epsilon >0$, then

$$\begin{array}{cc}\hfill & \sum _{l_1}\sum _{l_2}\sum _{k_1}\sum _{k_2}{2}^{-|k_1-l_1|\epsilon }{2}^{-|k_2-l_2|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}.\hfill \end{array}$$

Proof. 

For simplicity, we only estimate ${\displaystyle \sum _{l_1}\sum _{k_1}{2}^{-|k_1-l_1|\epsilon }\frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}.}$

$$\begin{array}{cc}\hfill & \sum _{l_1}\sum _{k_1}{2}^{-|k_1-l_1|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ & \le \sum _{l_1}\sum _{k_1\ge l_1}{2}^{(l_1-k_1)\epsilon }{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}+\sum _{l_1}\sum _{k_1<l_1}{2}^{(k_1-l_1)\epsilon }{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le \sum _{l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}+\sum _{k_1}\sum _{l_1>k_1}{\displaystyle \frac{2^{(k_1-l_1)\epsilon }{2}^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \le \sum _{l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}+\sum _{k_1}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Next, we provide an estimate for ${\displaystyle \sum _{l_1}\frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}$.

Suppose that there exists a $j\in \mathbb{Z}$ such that $2^{-j}<{\rho }_1(x_1,y_1)<2^{-j+1}$.

$$\begin{array}{cc}\hfill & \sum _{l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\left(\sum _{j\ge l_1}+\sum _{j<l_1}\right){\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & =\mathrm{I}+\mathrm{II}.\hfill \end{array}$$

For $\mathrm{I}$, since $j\ge l_1$, we have $2^{-j}\le 2^{-l_1}$. Using $2^{-j}<{\rho }_1(x_1,y_1)<2^{-j+1}$, we obtain ${\rho }_1(x_1,y_1)\approx 2^{-j}\le 2^{-l_1}$. Then,

$$\begin{array}{cc}\hfill \mathrm{I}=& \sum _{j\ge l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\le \sum _{j\ge l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{2^{-l_1(1+\epsilon )}}}\hfill \\ \hfill =& \sum _{j\ge l_1}{2}^{l_1}=\sum _{j\ge l_1}{2}^{l_1-j}·2^j\approx \sum _{j\ge l_1}{2}^{l_1-j}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}\hfill \\ \hfill \le & {\displaystyle \frac{C}{{\rho }_1(x_1,y_1)}}.\hfill \end{array}$$

For $\mathrm{II}$, since $j<l_1$, we have $2^{-j}\ge 2^{-l_1}$. Using assumption condition ${\rho }_1(x_1,y_1)>2^{-j}$, it follows that $2^{-l_1}+{\rho }_1(x_1,y_1)>2^{-j}$. What is more, if ${\rho }_1(x_1,y_1)\approx 2^{-j}$, then

$$\begin{array}{cc}\hfill \mathrm{II}=& \sum _{j<l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{(2^{-l_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\le \sum _{j<l_1}{\displaystyle \frac{2^{-l_1\epsilon }}{2^{-j(1+\epsilon )}}}\hfill \\ \hfill =& \sum _{j<l_1}{2}^{-(l_1-j)\epsilon }{2}^j\approx {\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}\sum _{j<l_1}{2}^{-(l_1-j)\epsilon }\hfill \\ \hfill \le & {\displaystyle \frac{C}{{\rho }_1(x_1,y_1)}}.\hfill \end{array}$$

Therefore,

$${\displaystyle \sum _{l_1}\sum _{k_1}{2}^{-|k_1-l_1|\epsilon }\frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}\le C\frac{1}{{\rho }_1(x_1,y_1)}.}$$

Using the same method, we have

$${\displaystyle \sum _{l_2}\sum _{k_2}{2}^{-|k_2-l_2|\epsilon }\frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}\le C\frac{1}{{\rho }_2(x_2,y_2)}.}$$

Finally, we obtain

$$\begin{array}{cc}\hfill & \sum _{l_1}\sum _{l_2}\sum _{k_1}\sum _{k_2}{2}^{-|k_1-l_1|\epsilon }{2}^{-|k_2-l_2|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}.\hfill \end{array}$$

□

Proposition 3. 

Let $l_1,l_2,i_1,i_2\in \mathbb{Z}$ and $\epsilon >0$. Then,

$$\begin{array}{cc}\hfill & {\int }_{X_2}{\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-l_2\epsilon }}{{\left(2^{-l_2}+{\rho }_2(x_2,z_2)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_2\epsilon }}{{\left(2^{-i_2}+{\rho }_2(z_2,y_2)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-(l_1\wedge i_1)\epsilon }}{{(2^{-(l_1\wedge i_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(l_2\wedge i_2)\epsilon }}{{(2^{-(l_2\wedge i_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

Proof. 

For the sake of simplicity, we only need to consider the estimate for

$${\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right).$$

In the first case, $l_1\le i_1$,

$$\begin{array}{cc}\hfill & {\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & =\left({\int }_{{\rho }_1(x_1,z_1)\ge {\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)}+{\int }_{{\rho }_1(x_1,z_1)<{\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)}\right){\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & :={\mathrm{I}}_1+{\mathrm{I}}_2.\hfill \end{array}$$

To estimate ${\mathrm{I}}_1$, because of ${\rho }_1(x_1,z_1)\ge {\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)$, we write

$$\begin{array}{cc}\hfill {\mathrm{I}}_1& \le {\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}{\int }_{X_1}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}.\hfill \end{array}$$

To estimate ${\mathrm{I}}_2$, consider the first case $2^{-l_1}\ge \rho (x_1,y_1)$; then,

$$\begin{array}{cc}\hfill {\mathrm{I}}_2& \le C{2}^l{\int }_{X_1}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{2}^l\le C{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}.\hfill \end{array}$$

If $2^{-l_1}<\rho (x_1,y_1)$, using $l\le i$, we have $2^{-i}\le 2^{-l_1}<\rho (x_1,y_1)$; since ${\rho }_1(x_1,z_1)<{\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)$ implies ${\rho }_1(z_1,y_1)\ge {\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)$, then

$$\begin{array}{cc}\hfill {\mathrm{I}}_2& \le {\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}{\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le {\displaystyle \frac{2^{-i_1\epsilon }}{{\rho }_1{(x_1,y_1)}^{1+\epsilon }}}{\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-i_1\epsilon }}{{\rho }_1{(x_1,y_1)}^{1+\epsilon }}}\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}.\hfill \end{array}$$

Combining the two cases, we have ${\mathrm{I}}_2\le C{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}.$ Together with the estimate for ${\mathrm{I}}_1$, we obtain

$$\begin{array}{cc}\hfill & {\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}\hfill \\ \hfill & =C{\displaystyle \frac{2^{-(l_1\wedge i_1)\epsilon }}{{(2^{-(l_1\wedge i_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Similarly, for $l_1>i_1$,

$$\begin{array}{cc}\hfill & {\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(x_1,y_1)\right)}^{1+\epsilon }}}.\hfill \end{array}$$

Therefore, we can obtain

$$\begin{array}{cc}\hfill & {\int }_{X_1}{\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_1\epsilon }}{{\left(2^{-i_1}+{\rho }_1(z_1,y_1)\right)}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-(l_1\wedge i_1)\epsilon }}{{(2^{-(l_1\wedge i_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Using the same method, we have

$$\begin{array}{cc}\hfill & {\int }_{X_2}{\displaystyle \frac{2^{-l_2\epsilon }}{{\left(2^{-l_2}+{\rho }_1(x_2,z_2)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-i_2\epsilon }}{{\left(2^{-i_2}+{\rho }_1(z_2,y_2)\right)}^{1+\epsilon }}}d{\mu }_2\left(z_2\right)\hfill \\ \hfill & \le C{\displaystyle \frac{2^{-(l_2\wedge i_2)\epsilon }}{{(2^{-(l_2\wedge i_2)}+{\rho }_1(x_2,y_2))}^{1+\epsilon }}}.\hfill \end{array}$$

Therefore, we finish the proof of Proposition 3. □

3.4. Proof of Theorem 1

We now prove Theorem 1.

Proof of Theorem 1. 

For $0<{\epsilon }^{″}<{\epsilon }^{\prime }<\epsilon$, applying Lemmas 1–3, Propositions 1–3, and Remark 2, we have

$$\begin{array}{cc}\hfill & \left|K\right(x_1,x_2,y_1,y_2\left)\right|\hfill \\ \hfill & =|\sum _{k_1,k_2}\sum _{j_1,j_2}\sum _{i_1,i_2}\sum _{l_1,l_2}{\tilde{D}}_{k_1}{M}_{b_1}{\tilde{D}}_{k_2}{M}_{b_2}{D}_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}\hfill \\ \hfill & \times M_{b_1}{\tilde{\tilde{D}}}_{j_1}{M}_{b_2}{\tilde{\tilde{D}}}_{j_2}b{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}{D}_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}\hfill \\ \hfill & \times {\tilde{\tilde{D}}}_{l_1}{M}_{b_1}{\tilde{\tilde{D}}}_{l_2}{M}_{b_2}(x_1,x_2,y_1,y_2)|\hfill \\ \hfill & \le \sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}|{\tilde{D}}_{k_1}(x_1,z_1)||b_1\left(z_1\right)|\hfill \\ \hfill & \times |{\tilde{D}}_{k_2}(x_2,z_2)||b_2\left(z_2\right)||D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(z_1,z_2,v_1,v_2)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{j_1}{M}_{b_1}{\tilde{\tilde{D}}}_{j_2}{M}_{b_2}{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}(v_1,v_2,u_1,u_2)|\hfill \\ \hfill & \times |D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}(u_1,u_2,w_1,w_2)||{\tilde{\tilde{D}}}_{l_1}(w_1,y_1)||b_1\left(y_1\right)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{l_2}(w_2,y_2)||b_2\left(y_2\right)|d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & \le C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-k_1\epsilon }}{{\left(2^{-k_1}+{\rho }_1(x_1,z_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-k_2\epsilon }}{{\left(2^{-k_2}+{\rho }_2(x_2,z_2)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times 2^{-|k_1-j_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{\displaystyle \frac{2^{-(k_1\wedge j_1)\epsilon }}{{(2^{-(k_1\wedge j_1)}+{\rho }_1(z_1,v_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2)\epsilon }}{{(2^{-(k_2\wedge j_2)}+{\rho }_2(z_2,v_2))}^{1+\epsilon }}}\hfill \\ \hfill & \times 2^{-\left|j_1-i_1\right|\epsilon }{2}^{-\left|j_2-i_2\right|\epsilon }{\displaystyle \frac{2^{-(j_1\wedge i_1)\epsilon }}{{\left(2^{-(j_1\wedge i_1)}+{\rho }_1(v_1,u_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-(j_2\wedge i_2)\epsilon }}{{\left(2^{-(j_2\wedge i_2)}+{\rho }_2(v_2,u_2)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times 2^{-\left|i_1-l_1\right|\epsilon }{2}^{-\left|i_2-l_2\right|\epsilon }{\displaystyle \frac{2^{-(i_1\wedge l_1)\epsilon }}{{\left(2^{-(i_1\wedge l_1)}+{\rho }_1(u_1,w_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-(i_2\wedge l_2)\epsilon }}{{\left(2^{-(i_2\wedge l_2)}+{\rho }_2(u_2,w_2)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-l_1\epsilon }}{{\left(2^{-l_1}+{\rho }_1(w_1,y_1)\right)}^{1+\epsilon }}}{\displaystyle \frac{2^{-l_2\epsilon }}{{\left(2^{-l_2}+{\rho }_2(w_2,y_2)\right)}^{1+\epsilon }}}\hfill \\ \hfill & \times d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & \le C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C\sum _{k_1}\sum _{k_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-i_1|{\epsilon }^{\prime }}{2}^{-|i_1-l_1|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(k_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge i_1\wedge l_1)}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & \times 2^{-|k_2-i_2|{\epsilon }^{\prime }}{2}^{-|i_2-l_2|{\epsilon }^{\prime }}{\displaystyle \frac{2^{-(k_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge i_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & \le C\sum _{k_1}\sum _{k_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-l_1|{\epsilon }^{″}}{2}^{-|k_2-l_2|{\epsilon }^{″}}{\displaystyle \frac{2^{-(k_1\wedge l_1){\epsilon }^{\prime }}}{{(2^{-(k_1\wedge l_1)}+{\rho }_1(x_1,y_1))}^{1+{\epsilon }^{\prime }}}}{\displaystyle \frac{2^{-(k_2\wedge l_2){\epsilon }^{\prime }}}{{(2^{-(k_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+{\epsilon }^{\prime }}}}\hfill \\ \hfill & \le C{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}},\hfill \end{array}$$

Therefore, we obtain that $K(x_1,x_2,y_1,y_2)$ satisfies condition $\left(\mathrm{I}\right)$ in Definition 2.

Next, we verify that $K(x_1,x_2,y_1,y_2)$ satisfies condition $\left(\mathrm{II}\right)$ in Definition 2. Let

$${\mathrm{\Gamma }}_1=\left\{z_1:{\rho }_1\left(x_1,x_1^{\prime }\right)\le {\displaystyle \frac{2^{-k_1}+{\rho }_1\left(x_1,z_1\right)}{2A_1}}\right\}\mathrm{and}{\mathrm{\Gamma }}_2=\left\{z_1:{\rho }_1\left(x_1,x_1^{\prime }\right)>{\displaystyle \frac{2^{-k_1}+{\rho }_1\left(x_1,z_1\right)}{2A_1}}\right\},$$

we write

$$\begin{array}{cc}\hfill & \left|K(x_1,x_2,y_1,y_2)-K(x_1^{\prime },x_2,y_1,y_2)\right|\hfill \\ \hfill & \le \sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}|{\tilde{D}}_{k_1}(x_1,z_1)-{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|\hfill \\ \hfill & \times |b_1\left(z_1\right)||{\tilde{D}}_{k_2}(x_2,z_2)||b_2\left(z_2\right)||D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(z_1,z_2,v_1,v_2)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{j_1}{M}_{b_1}{\tilde{\tilde{D}}}_{j_2}{M}_{b_2}{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}(v_1,v_2,u_1,u_2)|\hfill \\ \hfill & \times |D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}(u_1,u_2,w_1,w_2)||{\tilde{\tilde{D}}}_{l_1}(w_1,y_1)||b_1\left(y_1\right)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{l_2}(w_2,y_2)||b_2\left(y_2\right)|d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & \le C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{{\mathrm{\Gamma }}_1}\hfill \\ \hfill & \times |{\tilde{D}}_{k_1}(x_1,z_1)-{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|\hfill \\ \hfill & \times |{\tilde{D}}_{k_2}(x_2,z_2)||D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(z_1,z_2,v_1,v_2)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{k_1}{M}_{b_1}{\tilde{\tilde{D}}}_{k_2}{M}_{b_2}{\tilde{D}}_{j_1}{M}_{b_1}{\tilde{D}}_{j_2}{M}_{b_2}(v_1,v_2,u_1,u_2)|\hfill \\ \hfill & \times |D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}(u_1,u_2,w_1,w_2)||{\tilde{\tilde{D}}}_{l_1}(w_1,y_1)||{\tilde{\tilde{D}}}_{l_2}(w_2,y_2)|\hfill \\ \hfill & \times d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & + C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{{\mathrm{\Gamma }}_2}\hfill \\ \hfill & \times \left(|{\tilde{D}}_{k_1}(x_1,z_1)|+|{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|\right)\hfill \\ \hfill & \times |{\tilde{D}}_{k_2}(x_2,z_2)||D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(z_1,z_2,v_1,v_2)|\hfill \\ \hfill & \times |{\tilde{\tilde{D}}}_{k_1}{M}_{b_1}{\tilde{\tilde{D}}}_{k_2}{M}_{b_2}{\tilde{D}}_{j_1}{M}_{b_1}{\tilde{D}}_{j_2}{M}_{b_2}(v_1,v_2,u_1,u_2)|\hfill \\ \hfill & \times |D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}(u_1,u_2,w_1,w_2)||{\tilde{\tilde{D}}}_{l_1}(w_1,y_1)||{\tilde{\tilde{D}}}_{l_2}(w_2,y_2)|\hfill \\ \hfill & \times d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & :=J_1+J_2.\hfill \end{array}$$

For $0<{\epsilon }^{‴}<{\epsilon }^{″}<{\epsilon }^{\prime }<\epsilon$, then

$$\begin{array}{cc}\hfill J_1\le & C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }\hfill \\ \hfill & \times {\int }_{{\mathrm{\Gamma }}_1}|{\tilde{D}}_{k_1}(x_1,z_1)-{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}},\hfill \end{array}$$

Using Lemma 1,

$$\begin{array}{cc}\hfill & {\int }_{{\mathrm{\Gamma }}_1}|{\tilde{D}}_{k_1}(x_1,z_1)-{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\int }_{X_1}{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,z_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,z_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C\left({\int }_{{\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)\le {\rho }_1(x_1,z_1)}+{\int }_{{\rho }_1(x_1,z_1)<{\displaystyle \frac{1}{2A}}{\rho }_1(x_1,y_1)}\right){\left({\displaystyle \frac{{\rho }_1(x,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,z_1)}}\right)}^{{\epsilon }^{‴}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \le C{\left({\displaystyle \frac{{\rho }_1(x,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}\hfill \\ \hfill & + C{2}^{k_1{\epsilon }^{‴}}{\rho }_1{(x_1,x_1^{\prime })}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}.\hfill \end{array}$$

Applying Proposition 1, 2, and Remark 2, we get

$$\begin{array}{cc}\hfill J_1\le & C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }\hfill \\ \hfill & \times {\left({\displaystyle \frac{{\rho }_1(x,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill & + C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }\hfill \\ \hfill & \times 2^{k_1{\epsilon }^{‴}}{\rho }_1{(x_1,x_1^{\prime })}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }+{\rho }_1(x_1,y_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill \le & C{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}.\hfill \end{array}$$

Note that if $z_1\in {\mathrm{\Gamma }}_2$, then ${\rho }_1(x_1,x_1^{\prime })\ge C(2^{-k_1}+{\rho }_1(x_1,z_1))$ and ${\rho }_1(x_1,x_1^{\prime })\ge C(2^{-k_1}+{\rho }_1(x_1^{\prime },z_1))$. Then,

$$\begin{array}{cc}\hfill J_2\le & C\sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }\hfill \\ \hfill & \times {\int }_{{\mathrm{\Gamma }}_2}\left(\left|{\tilde{D}}_{k_1}(x_1,z_1)\right|+\left|{\tilde{D}}_{k_1}(x_1^{\prime },z_1)\right|\right){\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}+{\rho }_2(x_2,y_2))}^{1+\epsilon }}}\hfill \\ \hfill \le & C{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}\sum _{k_1}\sum _{j_1}\sum _{i_1}\sum _{l_1}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }\hfill \\ \hfill & \times {\int }_{X_1}({\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,z_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+\rho (x_1,z_1))}^{1+\epsilon }}}\hfill \\ \hfill & + {\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1^{\prime },z_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1^{\prime },z_1))}^{1+\epsilon }}})\hfill \\ \hfill & \times {\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill \le & C{\left({\displaystyle \frac{\rho (x_1,x_1^{\prime })}{{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}+{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{{\rho }_1(x_1^{\prime },y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1^{\prime },y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}\hfill \\ \hfill \le & C{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}},\hfill \end{array}$$

where in the last inequality we use the fact that if $\rho (x_1,x_1^{\prime })\le {\displaystyle \frac{1}{2A_1}}\rho (x_1,y_1)$, then $\rho (x_1^{\prime },y_1)\ge {\displaystyle \frac{1}{2A_1}}\rho (x_1,y_1)$. Therefore, $K(x_1,x_2,y_1,y_2)$ satisfies condition $\left({\mathrm{II}}_1\right)$ in Definition 2, replacing $\epsilon$ with ${\epsilon }^{‴}$.

Similarly, $K(x_1,x_2,y_1,y_2)$ satisfies conditions $\left({\mathrm{II}}_2\right),\left({\mathrm{II}}_3\right)$ and $\left({\mathrm{II}}_4\right)$ in Definition 2.

Now, we check that the kernel $K(x_1,x_2,y_1,y_2)$ satisfies the condition $\left(\mathrm{III}\right)$ in Definition 2, and we only consider $\left({\mathrm{III}}_2\right)$.

Let

$${\mathrm{\Gamma }}_3=\left\{z_2:{\rho }_2\left(x_2,x_2^{\prime }\right)\le {\displaystyle \frac{2^{-k_2}+{\rho }_2\left(x_2,z_2\right)}{2A_2}}\right\}$$

and

$${\mathrm{\Gamma }}_4=\left\{z_2:{\rho }_2\left(x_2,x_2^{\prime }\right)>{\displaystyle \frac{2^{-k_2}+{\rho }_2\left(x_2,z_2\right)}{2A_2}}\right\}.$$

We now write

$$\begin{array}{cc}\hfill & \left|K(x_1,x_2,y_1,y_2)-K(x_1^{\prime },x_2,y_1,y_2)-K(x_1,x_2^{\prime },y_1,y_2)+K(x_1^{\prime },x_2^{\prime },y_1,y_2)\right|\hfill \\ \hfill & \le \sum _{k_1}\sum _{k_2}\sum _{j_1}\sum _{j_2}\sum _{i_1}\sum _{i_2}\sum _{l_1}\sum _{l_2}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}{\int }_{X_2}{\int }_{X_1}\hfill \\ \hfill & \times \left({\int }_{{\mathrm{\Gamma }}_3}{\int }_{{\mathrm{\Gamma }}_1}+{\int }_{{\mathrm{\Gamma }}_4}{\int }_{{\mathrm{\Gamma }}_1}+{\int }_{{\mathrm{\Gamma }}_3}{\int }_{{\mathrm{\Gamma }}_2}+{\int }_{{\mathrm{\Gamma }}_4}{\int }_{{\mathrm{\Gamma }}_2}\right)|{\tilde{D}}_{k_1}(x_1,z_1)-{\tilde{D}}_{k_1}(x_1^{\prime },z_1)|\left|b_1\left(z_1\right)\right|\hfill \\ \hfill & \times |{\tilde{D}}_{k_2}(x_2,z_2)-{\tilde{D}}_{k_2}(x_2^{\prime },z_2)|\left|b_2\left(z_2\right)\right|\hfill \\ \hfill & \times \left|D_{k_1}{M}_{b_1}{D}_{k_2}{M}_{b_2}{T}_2{M}_{b_1}{D}_{j_1}{M}_{b_2}{D}_{j_2}(z_1,z_2,v_1,v_2)\right|\hfill \\ \hfill & \times \left|{\tilde{\tilde{D}}}_{j_1}{M}_{b_1}{\tilde{\tilde{D}}}_{j_2}{M}_{b_2}{\tilde{D}}_{i_1}{M}_{b_1}{\tilde{D}}_{i_2}{M}_{b_2}(v_1,v_2,u_1,u_2)\right|\hfill \\ \hfill & \times \left|D_{i_1}{M}_{b_1}{D}_{i_2}{M}_{b_2}{T}_1{M}_{b_1}{D}_{l_1}{M}_{b_2}{D}_{l_2}(u_1,u_2,w_1,w_2)\right|\left|{\tilde{\tilde{D}}}_{l_1}(w_1,y_1)\right|\left|b_1\left(y_1\right)\right|\left|{\tilde{\tilde{D}}}_{l_2}(w_2,y_2)\right|\left|b_2\left(y_2\right)\right|\hfill \\ \hfill & \times d{\mu }_1\left(z_1\right)d{\mu }_2\left(z_2\right)d{\mu }_1\left(v_1\right)d{\mu }_2\left(v_2\right)d{\mu }_1\left(u_1\right)d{\mu }_1\left(u_2\right)d{\mu }_1\left(w_1\right)d{\mu }_1\left(w_2\right)\hfill \\ \hfill & :=J_3+J_4+J_5+J_6.\hfill \end{array}$$

Let $0<{\epsilon }^{‴}<{\epsilon }^{″}<{\epsilon }^{\prime }<\epsilon$. Using the same estimates for $J_1,J_2$, we obtain

$$\begin{array}{cc}\hfill J_3\le & C\sum _{k_1}\sum _{j_1}\sum _{i_1}\sum _{l_1}{2}^{-|k_1-j_1|\epsilon }{2}^{-|j_1-i_1|\epsilon }{2}^{-|i_1-l_1|\epsilon }{\int }_{X_1}{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{2^{-k_1}+{\rho }_1(x_1,z_1)}}\right)}^{{\epsilon }^{‴}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-k_1\epsilon }}{{(2^{-k_1}+{\rho }_1(x_1,z_1))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)\epsilon }}{{(2^{-(k_1\wedge j_1\wedge i_1\wedge l_1)}+{\rho }_1(z_1,y_1))}^{1+\epsilon }}}d{\mu }_1\left(z_1\right)\hfill \\ \hfill & \times \sum _{k_2}\sum _{j_2}\sum _{i_2}\sum _{l_2}{2}^{-|k_2-j_2|\epsilon }{2}^{-|j_2-i_2|\epsilon }{2}^{-|i_2-l_2|\epsilon }{\int }_{X_2}{\left({\displaystyle \frac{{\rho }_2(x_2,x_2^{\prime })}{2^{-k_2}+{\rho }_2(x_2,z_2)}}\right)}^{{\epsilon }^{‴}}\hfill \\ \hfill & \times {\displaystyle \frac{2^{-k_2\epsilon }}{{(2^{-k_2}+{\rho }_2(x_2,z_2))}^{1+\epsilon }}}{\displaystyle \frac{2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)\epsilon }}{{(2^{-(k_2\wedge j_2\wedge i_2\wedge l_2)}+{\rho }_2(z_2,y_2))}^{1+\epsilon }}}d{\mu }_2\left(z_2\right)\hfill \\ \hfill \le & C{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\left({\displaystyle \frac{{\rho }_2(x_2,x_2^{\prime })}{{\rho }_2(x_2,y_2)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}.\hfill \end{array}$$

Similarly, for $n=4,5,6$, we have

$$J_n\le C{\left({\displaystyle \frac{{\rho }_1(x_1,x_1^{\prime })}{{\rho }_1(x_1,y_1)}}\right)}^{{\epsilon }^{‴}}{\left({\displaystyle \frac{{\rho }_2(x_2,x_2^{\prime })}{{\rho }_2(x_2,y_2)}}\right)}^{{\epsilon }^{‴}}{\displaystyle \frac{1}{{\rho }_1(x_1,y_1)}}{\displaystyle \frac{1}{{\rho }_2(x_2,y_2)}}.$$

Thus, we obtain that $K(x_1,x_2,y_1,y_2)$ satisfies the condition $\left({\mathrm{III}}_2\right)$ in Definition 2, replacing $\epsilon$ with ${\epsilon }^{‴}$.

Since $T_1,T_2$ are $L^2$ bounded operators, then $T=T_2\circ T_1=T_{2}b{T}_1$ is bounded on $L^2$. When $T=T_2\circ T_1$ and $T_1\left(b_1\right)=T_1^{*}\left(b_1\right)=T_2\left(b_2\right)=T_2^{*}\left(b_2\right)=0$, it is easy to verify that $T\left(b\right)=T^{*}\left(b\right)=0$ when $b=b_1{b}_2$. Therefore, we finish the proof of Theorem 1. □