## 1 Introduction and results

Let K be a kernel on $$\mathbb{R}\times\mathbb{R}\setminus\{ (x,x): x\in\mathbb{R}\}$$. Suppose that there exist two constants δ and C such that

\begin{aligned} &\bigl\vert K(x,y) \bigr\vert \leq\frac{C}{ \vert x-y \vert }\quad \mbox{for } x\neq y; \end{aligned}
(1.1)
\begin{aligned} &\bigl\vert K(x,y)-K\bigl(x',y\bigr) \bigr\vert \leq\frac{C \vert x-x' \vert ^{\delta}}{ \vert x-y \vert ^{1+\delta}}\quad \mbox{for } \vert x-y \vert \geq2 \bigl\vert x-x' \bigr\vert ; \end{aligned}
(1.2)
\begin{aligned} &\bigl\vert K(x,y)-K\bigl(x,y'\bigr) \bigr\vert \leq\frac{C \vert y-y' \vert ^{\delta}}{ \vert x-y \vert ^{1+\delta}} \quad \mbox{for } \vert x-y \vert \geq2 \bigl\vert y-y' \bigr\vert . \end{aligned}
(1.3)

We consider the family of operators $$T=\{T_{\epsilon}\}_{\epsilon>0}$$ given by

\begin{aligned} T_{\epsilon}f(x)= \int_{ \vert x-y \vert >\epsilon}K(x,y) f(y)\,dy. \end{aligned}
(1.4)

A common method of measuring the speed of convergence of the family $$T_{\epsilon}$$ is to consider the square functions

\begin{aligned} \Biggl(\sum_{i=1}^{\infty} \vert T_{\epsilon_{i}}f-T_{\epsilon _{i+1}}f \vert ^{2} \Biggr)^{1/2}, \end{aligned}

where $$\epsilon_{i}$$ is a monotonically decreasing sequence which approaches 0. For convenience, other expressions have also been considered. Let $$\{t_{i}\}$$ be a fixed sequence which decreases to zero. Following [1], the oscillation operator is defined as

\begin{aligned} \mathcal{O}(Tf) (x)= \Biggl(\sum_{i=1}^{\infty}\sup_{t_{i+1}\leq \epsilon_{i+1}< \epsilon_{i}\leq t_{i}} \bigl\vert T_{\epsilon _{i+1}}f(x)-T_{\epsilon_{i}}f(x) \bigr\vert ^{2} \Biggr)^{1/2} \end{aligned}

and the ρ-variation operator is defined as

\begin{aligned} \mathcal{V}_{\rho}(Tf) (x)=\sup_{\epsilon_{i}\searrow0 } \Biggl(\sum _{i=1}^{\infty}\bigl\vert T_{\epsilon_{i+1}}f(x)-T_{\epsilon_{i}}f(x) \bigr\vert ^{\rho}\Biggr)^{1/\rho}, \end{aligned}

where the sup is taken over all sequences of real number $$\{ \epsilon_{i}\}$$ decreasing to zero.

The oscillation and variation for some families of operators have been studied by many authors on probability, ergodic theory, and harmonic analysis; see [24]. Recently, some authors [58] researched the weighted estimates of the oscillation and variation operators for the commutators of singular integrals.

Let m be a positive integer, let b be a function on $$\mathbb{R}$$, and let $$R_{m+1}(b;x,y)$$ be the $$m+1$$th Taylor series remainder of b at x expander about y, i.e.

\begin{aligned} R_{m+1}(b;x,y)=b(x)-\sum_{\alpha\leq m} \frac{1}{\alpha!} b^{(\alpha )}(y) (x-y)^{\alpha}. \end{aligned}

We consider the family of operators $$T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon >0}$$, where $$T^{b}_{\epsilon}$$ are the multilinear singular integral operators of $$T_{\epsilon}$$,

\begin{aligned} T^{b}_{\epsilon}f(x)= \int_{ \vert x-y \vert >\epsilon}\frac{ R_{m+1}(b;x,y)}{ \vert x-y \vert ^{m}}K(x,y)f(y)\,dy. \end{aligned}
(1.5)

Note that when $$m=0$$, $$T^{b}_{\epsilon}$$ is just the commutator of $$T_{\epsilon}$$ and b, which is denoted by $$T_{\epsilon,b}$$, that is to say

\begin{aligned} T_{\epsilon,b} f(x)= \int_{ \vert x-y \vert >\epsilon}\bigl(b(x)-b(y)\bigr)K(x,y)f(y)\,dy. \end{aligned}
(1.6)

However, when $$m>0$$, $$T^{b}_{\epsilon}$$ is a non-trivial generation of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see [913]).

A locally integrable function b is said to be in Lipschitz space $$\mathrm{Lip}_{\beta}(\mathbb{R})$$ if

\begin{aligned} \Vert b \Vert _{\dot{\wedge}_{\beta}}=\sup_{I}\frac{1}{ \vert I \vert ^{1+\beta}} \int _{I} \bigl\vert b(x)-b_{I} \bigr\vert \,dx< \infty, \end{aligned}

where

\begin{aligned} b_{I}=\frac{1}{ \vert I \vert } \int_{I}b(x)\,dx. \end{aligned}

In this paper, we will study the boundedness of oscillation and variation operators for the family of the multilinear singular integral related to a Lipschitz function defined by (1.5) in weighted Lebesgue space. Our main results are as follows.

### Theorem 1.1

Suppose that $$K(x,y)$$ satisfies (1.1)-(1.3), $$b^{(m)}\in\dot{\wedge}_{\beta}$$, $$0<\beta\leq\delta<1$$, where δ is the same as in (1.2). Let $$\rho>2$$, $$T=\{T_{\epsilon}\} _{\epsilon>0}$$ and $$T^{b}=\{T^{b}_{\epsilon}\}_{\epsilon>0}$$ be given by (1.4) and (1.5), respectively. If $$\mathcal{O}(T)$$ and $$\mathcal{V}_{\rho}(T)$$ are bounded on $$L^{p_{0}}(\mathbb {R},dx)$$ for some $$1< p_{0}<\infty$$, then, for any $$1< p<1/\beta$$ with $$1/q=1/p-\beta$$, $$\omega\in A_{p,q}(\mathbb{R})$$, $$\mathcal{O}(T^{b})$$ and $$\mathcal{V}_{\rho}(T^{b})$$ are bounded from $$L^{p}(\mathbb{R},\omega^{p} \,dx)$$ into $$L^{q}(\mathbb{R},\omega^{q} \,dx)$$.

### Corollary 1.1

Suppose that $$K(x,y)$$ satisfies (1.1)-(1.3), $$b\in\dot{\wedge}_{\beta}$$, $$0<\beta\leq\delta<1$$, where δ is the same as in (1.2). Let $$\rho>2$$, $$T=\{T_{\epsilon}\} _{\epsilon>0}$$ and $$T_{b}=\{T_{b,\epsilon}\}_{\epsilon>0}$$ be given by (1.4) and (1.6), respectively. If $$\mathcal{O}(T)$$ and $$\mathcal{V}_{\rho}(T)$$ are bounded on $$L^{p_{0}}(\mathbb {R},dx)$$ for some $$1< p_{0}<\infty$$, then, for any $$1< p<1/\beta$$ with $$1/q=1/p-\beta$$, $$\omega\in A_{p,q}(\mathbb{R})$$, $$\mathcal{O}(T_{b})$$ and $$\mathcal{V}_{\rho}(T_{b})$$ are bounded from $$L^{p}(\mathbb{R},\omega^{p} \,dx)$$ into $$L^{q}(\mathbb{R},\omega^{q} \,dx)$$.

In this paper, we shall use the symbol $$A\lesssim B$$ to indicate that there exists a universal positive constant C, independent of all important parameters, such that $$A\leq CB$$. $$A\thickapprox B$$ means that $$A\lesssim B$$ and $$B\lesssim A$$.

## 2 Some preliminaries

### 2.1 Weight

A weight ω is a nonnegative, locally integrable function on $$\mathbb{R}$$. The classical weight theories were introduced by Muckenhoupt and Wheeden in [14] and [15].

A weight ω is said to belong to the Muckenhoup class $$A_{p}(\mathbb{R})$$ for $$1< p<\infty$$, if there exists a constant C such that

\begin{aligned} \biggl(\frac{1}{ \vert I \vert } \int_{I}\omega(x)\,dx \biggr) \biggl(\frac {1}{ \vert I \vert } \int_{I}\omega(x)^{-\frac{1}{p-1}}\,dx \biggr)^{p-1}\leq C \end{aligned}

for every interval I. The class $$A_{1}(\mathbb{R})$$ is defined by replacing the above inequality with

\begin{aligned} \frac{1}{ \vert I \vert } \int_{I}\omega(x)\,dx\lesssim \mathop{\operatorname{ess}\operatorname{inf}}_{x\in I} w(x)\quad\mbox{for every ball } I\subset\mathbb{R}. \end{aligned}

When $$p=\infty$$, we define $$A_{\infty}(\mathbb{R})=\bigcup_{1\leq p<\infty}A_{p}(\mathbb{R})$$.

A weight $$\omega(x)$$ is said to belong to the class $$A_{p,q}(\mathbb{R})$$, $$1< p\leq q<\infty$$, if

\begin{aligned} \biggl(\frac{1}{ \vert I \vert } \int_{I}\omega(x)^{q}\,dx \biggr)^{1/q} \biggl(\frac {1}{ \vert I \vert } \int_{I}\omega(x)^{-p'}\,dx \biggr)^{1/p'}\leq C. \end{aligned}

It is well known that if $$\omega\in A_{p.q}(\mathbb{R})$$, then $$\omega^{q}\in A_{\infty}(\mathbb{R})$$.

### 2.2 Function of $$\mathrm{Lip}_{\beta}(\mathbb{R})$$

The function of $$\mathrm{Lip}_{\beta}(\mathbb{R})$$ has the following important properties.

### Lemma 2.1

Let $$b\in \mathrm{Lip}_{\beta}(\mathbb{R})$$. Then

1. (1)

$$1\leq p<\infty$$

\begin{aligned} \sup_{I}\frac{1}{ \vert I \vert ^{\beta}} \biggl(\frac{1}{ \vert I \vert } \int _{I} \bigl\vert b(x)-b_{I} \bigr\vert ^{p}\,dx \biggr)^{1/p}\leq C \Vert b \Vert _{\dot{\wedge}_{\beta}}; \end{aligned}
2. (2)

for any $$I_{1}\subset I_{2}$$,

\begin{aligned} \frac{1}{ \vert I_{2} \vert } \int_{I_{2}} \bigl\vert b(y)-b_{I_{1}} \bigr\vert \,dy \lesssim\frac { \vert I_{2} \vert }{ \vert I_{1} \vert } \vert I_{2} \vert ^{\beta} \Vert b \Vert _{\dot{\wedge}_{\beta}}. \end{aligned}

### 2.3 Maximal function

We recall the definition of Hardy-Littlewood maximal operator and fractional maximal operator. The Hardy-Littlewood maximal operator is defined by

\begin{aligned} M(f) (x)=\sup_{I\ni x}\frac{1}{ \vert I \vert } \int_{I} \bigl\vert f(y) \bigr\vert \,dy. \end{aligned}

The fractional maximal function is defined as

\begin{aligned} M_{\beta,r}(f) (x)=\sup_{I\ni x} \biggl(\frac{1}{ \vert I \vert ^{1-r\beta}} \int _{I} \bigl\vert f(y) \bigr\vert ^{r}\,dy \biggr)^{1/r} \end{aligned}

for $$1\leq r<\infty$$. In order to simplify the notation, we set $$M_{\beta}(f)(x)=M_{\beta,1}(f)(x)$$.

### Lemma 2.2

Let $$1< p<\infty$$ and $$\omega\in A_{\infty}(\mathbb{R})$$. Then

\begin{aligned} \Vert M f \Vert _{L^{p}(\omega)}\lesssim \bigl\Vert M^{\sharp}f \bigr\Vert _{L^{p}(\omega)} \end{aligned}

for all f such that the left hand side is finite.

### Lemma 2.3

Suppose $$0<\beta<1$$, $$1\leq r< p<1/\beta$$, $$1/q=1/p-\beta$$. If $$\omega\in A_{p,q}(\mathbb{R})$$, then

\begin{aligned} \Vert M_{\beta,r} f \Vert _{L^{q}(\omega^{q})}\lesssim \Vert f \Vert _{L^{p}(\omega^{p})}. \end{aligned}

### 2.4 Taylor series remainder

The following lemma gives an estimate on Taylor series remainder.

### Lemma 2.4

[10] Let b be a function on $$\mathbb{R}$$ and $$b^{(m)}\in L^{s}(\mathbb {R})$$ for any $$s>1$$. Then

\begin{aligned} \bigl\vert R_{m}(b;x,y) \bigr\vert \lesssim \vert x-y \vert ^{m} \biggl(\frac{1}{ \vert I_{x}^{y} \vert } \int _{I_{x}^{y}} \bigl\vert b^{(m)}(z) \bigr\vert ^{s}\,dz \biggr)^{1/s}, \end{aligned}

where $$I_{x}^{y}$$ is the interval $$(x-5 \vert x-y \vert , x+5 \vert x-y \vert )$$.

### 2.5 Oscillation and variation operators

We consider the operator

\begin{aligned} \mathcal{O}'(Tf) (x)= \Biggl(\sum_{i=1}^{\infty}\sup_{t_{i+1}< \delta _{i}< t_{i}} \bigl\vert T_{t_{i+1}}f(x)-T_{\delta_{i}}f(x) \bigr\vert ^{2} \Biggr)^{1/2}. \end{aligned}

It is easy to check that

\begin{aligned} \mathcal{O}'(Tf)\thickapprox\mathcal{O}(Tf). \end{aligned}

Following [4], we denote by E the mixed norm Banach space of two variable function h defined on $$\mathbb{R}\times\mathbb{N}$$ such that

\begin{aligned} \Vert h \Vert _{E}\equiv \biggl(\sum _{i} \Bigl(\sup_{s} \bigl\vert h(s,i) \bigr\vert \Bigr)^{2} \biggr)^{1/2}< \infty. \end{aligned}

Given $$T=\{T_{\epsilon}\}_{\epsilon>0}$$, where $$T_{\epsilon}$$ defined as (1.4), for a fixed decreasing sequence $$\{t_{i}\}$$ with $$t_{i}\searrow0$$, let $$J_{i}=(t_{i+1},t_{i}]$$ and define the E-valued operator $$\mathcal{U}(T): f\rightarrow\mathcal{U}(T)f$$ by

\begin{aligned} \mathcal{U}(T)f(x)=\bigl\{ T_{t_{i+1}}f(x)-T_{s}f(x)\bigr\} _{s\in J_{i},i\in\mathbb{N}}= \biggl\{ \int_{\{t_{i+1}< \vert x-y \vert < s\} }K(x,y)f(y)\,dy \biggr\} _{s\in J_{i},i\in\mathbb{N}}. \end{aligned}

Then

\begin{aligned} \mathcal{O}'(Tf) (x)={}& \bigl\Vert \mathcal{U}(T)f(x) \bigr\Vert _{E}= \bigl\Vert \bigl\{ T_{t_{i+1}}f(x)-T_{s}f(x) \bigr\} _{s\in J_{i},i\in\mathbb{N}} \bigr\Vert _{E} \\ ={}& \biggl\Vert \biggl\{ \int_{\{t_{i+1}< \vert x-y \vert < s\}}K(x,y)f(y)\,dy \biggr\} _{s\in J_{i},i\in\mathbb{N}} \biggr\Vert _{E}. \end{aligned}

On the other hand, let $$\Theta=\{\beta: \beta=\{\epsilon_{i}\} ,\epsilon_{i}\in\mathbb{R},\epsilon_{i}\searrow0\}$$. We denote by $$F_{\rho}$$ the mixed norm space of two variable functions $$g(i,\beta)$$ such that

\begin{aligned} \Vert g \Vert _{F_{\rho}}\equiv\sup_{\beta}\biggl(\sum _{i} \bigl\vert g(i,\beta) \bigr\vert ^{\rho}\biggr)^{1/\rho}. \end{aligned}

We also consider the $$F_{\rho}$$-valued operator $$\mathcal {V}(T):f\rightarrow\mathcal{V}(T)f$$ given by

\begin{aligned} \mathcal{V}(T)f(x)=\bigl\{ T_{t_{i+1}}f(x)-T_{t_{i}}f(x)\bigr\} _{\beta=\{ \epsilon_{i}\}\in\Theta}. \end{aligned}

Then

\begin{aligned} \mathcal{V}_{\rho}(T)f(x)= \bigl\Vert \mathcal{V}(T)f(x) \bigr\Vert _{F_{\rho}}. \end{aligned}

Next, let B be a Banach space and φ be a B-valued function, we define the sharp maximal operator as follows:

\begin{aligned} \varphi^{\sharp}(x)=\sup_{x\in I}\frac{1}{ \vert I \vert } \int_{I} \biggl\Vert \varphi (y)-\frac{1}{ \vert I \vert } \int_{I}\varphi(z)\,dz \biggr\Vert _{B}\,dy \thickapprox\sup_{ x\in I}\inf_{c} \frac{1}{ \vert I \vert } \int_{I} \bigl\Vert \varphi(y)-c \bigr\Vert _{B}\,dy. \end{aligned}

Then

\begin{aligned} M^{\sharp}\bigl(\mathcal{O}'(Tf)\bigr)\leq2\bigl( \mathcal{U}(T)f\bigr)^{\sharp}(x) \end{aligned}

and

\begin{aligned} M^{\sharp}\bigl(\mathcal{\mathcal{V}}_{\rho}(Tf)\bigr)\leq2\bigl( \mathcal {V}(T)f\bigr)^{\sharp}(x). \end{aligned}

Finally, let us recall some results about oscillation and variation operators.

### Lemma 2.5

([5])

Suppose that $$K(x,y)$$ satisfies (1.1)-(1.3), $$\rho >2$$. Let $$T=\{T_{\epsilon}\}_{\epsilon>0}$$ be given by (1.4). If $$O(T)$$ and $$V_{\rho}(T)$$ are bounded on $$L^{p_{0}}(R)$$ for some $$1< p_{0}<\infty$$, then, for any $$1< p<\infty$$, $$\omega\in A_{p}(\mathbb{R})$$,

\begin{aligned} \bigl\Vert \mathcal{O}'(Tf) \bigr\Vert _{L^{p}(\omega)}\leq \bigl\Vert \mathcal{O}(Tf) \bigr\Vert _{L^{p}(\omega)}\lesssim \Vert f \Vert _{L^{p}(\omega)} \end{aligned}

and

\begin{aligned} \bigl\Vert \mathcal{V}_{\rho}(Tf) \bigr\Vert _{L^{p}(\omega)}\lesssim \Vert f \Vert _{L^{p}(\omega)}. \end{aligned}

## 3 The proof of main results

Note that if $$\omega\in A_{p,q}(\mathbb{R})$$, then $$\omega^{q}\in A_{\infty}(\mathbb{R})$$. By Lemma 2.2 and Lemma 2.3, we only need to prove

\begin{aligned} M^{\sharp}\bigl(\mathcal{O}'\bigl(T^{b} \bigr)f\bigl)(x) \lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge }_{\beta}} \bigl( M_{\beta,r}(f) (x)+ M_{\beta}(f) (x) \bigr) \end{aligned}
(3.1)

and

\begin{aligned} M^{\sharp}\bigl(\mathcal{V}_{\rho}\bigl(T^{b} \bigr)f\bigl)(x) \lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot {\wedge}_{\beta}} \bigl( M_{\beta,r}(f) (x)+ M_{\beta}(f) (x) \bigr) \end{aligned}
(3.2)

hold for any $$1< r<\infty$$.

We will prove only inequality (3.1), since (3.2) can be obtained by a similar argument. Fix f and $$x_{0}$$ with an interval $$I=(x_{0}-l,x_{0}+l)$$. Write $$f=f_{1}+f_{2}=f\chi_{5I}+f\chi_{\mathbb{R}\setminus5I}$$, and let

\begin{aligned} C_{I}= \biggl\{ \int_{\{t_{i+1}< \vert x_{0}-y \vert < s\}}\frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y)f_{2}(y)\,dy \biggr\} _{s\in J_{i},i\in \mathbb{N}}=\mathcal{U}\bigl(T^{b}\bigr)f_{2}(x_{0}). \end{aligned}

Then

\begin{aligned} \mathcal{U}\bigl(T^{b}\bigr)f (x) ={}& \biggl\{ \int_{\{t_{i+1}< \vert x-y \vert < s\}}\frac {R_{m+1}({b};x,y)}{ \vert x-y \vert ^{m}}K(x,y)f(y)\,dy \biggr\} _{s\in J_{i},i\in\mathbb {N}} \\ ={}& \mathcal{U}(T) \biggl(\frac{R_{m+1}({b};x,\cdot)}{ \vert x-\cdot \vert ^{m}}f_{1} \biggr)+\mathcal{U} \bigl(T^{b}\bigr)f_{2}(x). \end{aligned}

Therefore

\begin{aligned} &\frac{1}{ \vert I \vert } \int_{I} \bigl\Vert \mathcal{U}\bigl(T^{b}\bigr)f (x)-C_{I} \bigr\Vert _{E}\,dx \\ &\quad \leq \frac{1}{ \vert I \vert } \int_{I} \biggl\Vert \mathcal{U}(T) \biggl( \frac {R_{m+1}({b};x,\cdot)}{ \vert x-\cdot \vert ^{m}}f_{1} \biggr) \biggr\Vert _{E}\,dx + \frac{1}{ \vert I \vert } \int_{I} \bigl\Vert \mathcal{U}\bigl(T^{b} \bigr)f_{2}(x)-\mathcal {U}\bigl(T^{b}\bigr)f_{2}(x_{0}) \bigr\Vert _{E}\,dx \\ &\quad = M_{1}+M_{2}. \end{aligned}

For $$x\in I$$, $$k=0,-1,-2,\ldots$$ , let $$E_{k}=\{y:2^{k-1}\cdot6l\leq \vert y-x \vert <2^{k}\cdot6l\}$$, let $$I_{k}=\{y: \vert y-x \vert <2^{k}\cdot6l\}$$, and let $${b}_{k}(z)=b(z)-\frac{1}{m!}(b^{(m)})_{I_{k}}z^{m}$$. By [10] we have $$R_{m+1}({b};x,y)=R_{m+1}(b_{k};x,y)$$ for any $$y\in E_{k}$$.

By Lemma 2.5, we know $$\mathcal{O}'(T)$$ is bounded on $$L^{u}(\mathbb{R})$$ for $$u>1$$. Then, using Hölder’s inequality, we deduce

\begin{aligned} M_{1} \lesssim{}& \biggl(\frac{1}{ \vert I \vert } \int_{I} \biggl\Vert \mathcal{U}(T) \biggl( \frac {R_{m+1}({b};x,\cdot)}{ \vert x-\cdot \vert ^{m}}f_{1} \biggr) \biggr\Vert ^{u}_{E}\,dx \biggr)^{1/u} \\ \lesssim& \biggl(\frac{1}{ \vert I \vert } \int_{\{y: \vert y-x \vert < 6l\}} \biggl\vert \frac {R_{m+1}({b};\cdot,y)}{ \vert y-\cdot \vert ^{m}}f(y) \biggr\vert ^{u}\,dy \biggr)^{1/u} \\ =& \Biggl(\frac{1}{ \vert I \vert }\sum_{k=-\infty}^{0} \int_{E_{k}} \biggl\vert \biggl(\frac{R_{m+1}({b}_{k};\cdot,y)}{ \vert y-\cdot \vert ^{m}}f(y) \biggr) \biggr\vert ^{r}\,dy \Biggr)^{1/r} \\ \lesssim& \Biggl(\frac{1}{ \vert I \vert }\sum_{k=-\infty}^{0} \int_{E_{k}} \biggl\vert \biggl( \biggl(\frac{R_{m}({b_{k}};\cdot,y)}{ \vert y-\cdot \vert ^{m}}- \frac{1}{m!}\frac {(y-\cdot)^{m}{b}_{k}^{(m)}(y) }{ \vert y-\cdot \vert ^{m}} \biggr)f(y) \biggr) \biggr\vert ^{u}\,dy \Biggr)^{1/u} \\ \lesssim& \Biggl(\frac{1}{ \vert I \vert }\sum_{k=-\infty}^{0} \int_{E_{k}} \biggl\vert \frac{R_{m}({b_{k}};\cdot,y)}{ \vert y-\cdot \vert ^{m}}f(y) \biggr\vert ^{u}\,dy \Biggr)^{1/u} \\ &{} + \Biggl(\frac{1}{ \vert I \vert }\sum_{k=-\infty}^{0} \int_{E_{k}} \biggl\vert \frac {1}{m!} \frac{(y-\cdot)^{m}{b}_{k}^{(m)}(y) }{ \vert y-\cdot \vert ^{m}}f(y) \biggr\vert ^{u}\,dy \Biggr)^{1/u} \\ =& M_{11}+M_{12}. \end{aligned}

By Lemma 2.4 and Lemma 2.1,

\begin{aligned} \bigl\vert R_{m}({b}_{k};x,y) \bigr\vert \lesssim& \vert x-y \vert ^{m} \biggl(\frac{1}{ \vert I_{x}^{y} \vert } \int _{I_{x}^{y}} \bigl\vert {b}_{k}^{(m)}(z) \bigr\vert ^{s}\,dz \biggr)^{1/s} \\ \lesssim& \vert x-y \vert ^{m} \biggl(\frac{1}{2^{k}\cdot30l} \int _{ \vert y-x \vert < 2^{k}\cdot30l} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{I_{k}} \bigr\vert ^{s}\,dz \biggr)^{1/s} \\ \lesssim& \vert x-y \vert ^{m}\bigl(2^{k}l \bigr)^{\beta}\bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}. \end{aligned}

Then

\begin{aligned} M_{11} \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} l^{\beta}\Biggl(\frac{1}{ \vert I \vert }\sum _{k=-\infty}^{0}2^{k\beta u} \int _{E_{k}} \bigl\vert f(y) \bigr\vert ^{u}\,dy \Biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} l^{\beta}\Biggl(\frac {1}{ \vert I \vert }\sum_{k=-\infty}^{0} \int_{E_{k}} \bigl\vert f(y) \bigr\vert ^{u}\,dy \Biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} l^{\beta}\biggl(\frac {1}{ \vert I \vert } \int_{7I} \bigl\vert f(y) \bigr\vert ^{u}\,dy \biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} l^{\beta}\biggl(\frac {1}{ \vert I \vert } \int_{7I} \bigl\vert f(y) \bigr\vert ^{r}\,dy \biggr)^{1/r} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta,r}(f) (x_{0}). \end{aligned}

Since $${b}_{k}^{(m)}(y)=b^{(m)}(y)-(b^{(m)})_{I_{k}}$$, then, applying Hölder’s inequality and Lemma 2.1, we get

\begin{aligned} M_{12} \lesssim & \Biggl(\frac{1}{ \vert I \vert }\sum _{k=-\infty}^{0} \int _{E_{k}} \bigl\vert \bigl(b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{I_{k}}\bigr)f(y) \bigr\vert ^{u}\,dy \Biggr)^{1/u} \\ \lesssim& \biggl(\frac{1}{ \vert I \vert }\sum_{k=-\infty}^{0} \biggl( \int _{I_{k}} \bigl\vert f(y) \bigr\vert ^{r}\,dy \biggr)^{u/r} \biggl( \int_{I_{k}} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{I_{k}} \bigr\vert ^{\frac {ur}{r-u}} \biggr)^{1-u/r} \biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \Biggl( \frac {1}{ \vert I \vert }\sum_{k=-\infty}^{0} \biggl( \int_{I_{k}} \bigl\vert f(y) \bigr\vert ^{r}\,dy \biggr)^{u/r} \vert I_{k} \vert ^{\beta u+1-u/r} \Biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta ,r}(f) (x_{0}) \Biggl(\frac{1}{ \vert I \vert }\sum _{k=-\infty}^{0} \vert I_{k} \vert \Biggr)^{1/u} \\ \lesssim& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta,r}(f) (x_{0}). \end{aligned}

We now estimate $$M_{2}$$. For $$x\in I$$, we have

\begin{aligned} & \bigl\Vert \mathcal{U}\bigl(T^{b}\bigr)f_{2}(x)- \mathcal{U}\bigl(T^{b}\bigr)f_{2}(x_{0}) \bigr\Vert _{E} \\ &\quad = \biggl\Vert \biggl\{ \int_{\{t_{i+1}< \vert x-y \vert < s\}}\frac {R_{m+1}({b};x,y)}{ \vert x-y \vert ^{m}}K(x,y)f_{2}(y)\,dy \\ &\qquad{}- \int_{\{t_{i+1}< \vert x_{0}-y \vert < s\}}\frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y)f_{2}(y)\,dy \biggr\} _{s\in J_{i}, i\in\mathbb{N}} \biggr\Vert _{E} \\ &\quad \leq \biggl\Vert \biggl\{ \int_{\{t_{i+1}< \vert x-y \vert < s\}} \biggl(\frac {R_{m+1}({b};x,y)}{ \vert x-y \vert ^{m}}K(x,y)- \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y) \biggr)f_{2}(y)\,dy\biggr\} _{s\in J_{i}, i\in\mathbb{N}} \biggr\Vert _{E} \\ &\qquad{} + \biggl\Vert \biggl\{ \int_{R} \bigl(\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y)-\chi _{\{t_{i+1}< \vert x_{0}-y \vert < s\}}(y) \bigr)\frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y)f_{2}(y)\,dy \biggr\} _{s\in J_{i}, i\in \mathbb{N}} \biggr\Vert _{E} \\ &\quad = N_{1}+N_{2}. \end{aligned}

For $$k=0,1,2,\ldots$$ , let $$F_{k}=\{y:2^{k}\cdot4l\leq \vert y-x_{0} \vert <2^{k+1}\cdot4l\}$$, let $$\widetilde{I}_{k}=\{y: \vert y-x_{0} \vert <2^{k}\cdot4l\}$$, and let $$\widetilde{b}_{k}(z)=b(z)-\frac {1}{m!}(b^{(m)})_{\widetilde{I}_{k}}z^{m}$$. Note that

\begin{aligned} & \frac{R_{m+1}({b};x,y)}{ \vert x-y \vert ^{m}}K(x,y) -\frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y) \\ &\quad =\frac{R_{m+1}(\widetilde{b}_{k};x,y)}{ \vert x-y \vert ^{m}}K(x,y) -\frac{R_{m+1}(\widetilde{b}_{k};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y) \\ &\quad = \frac{1}{ \vert x-y \vert ^{m}} \bigl(R_{m}(\widetilde {b}_{k};x,y)-R_{m}( \widetilde{b}_{k};x_{0},y) \bigr)K(x,y) \\ &\qquad{} +R_{m}(\widetilde{b}_{k};x_{0},y) \biggl( \frac{1}{ \vert x-y \vert ^{m}}-\frac {1}{ \vert x_{0}-y \vert ^{m}} \biggr)K(x,y) \\ &\qquad{} - \frac{1}{m!}\widetilde{b}_{k}^{(m)}(y) \biggl( \frac {(x-y)^{m}}{ \vert x-y \vert ^{m}}-\frac{(x_{0}-y)^{m}}{ \vert x_{0}-y \vert ^{m}} \biggr)K(x,y) \\ &\qquad{} + \frac{R_{m+1}(\widetilde{b}_{k};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \bigl(K(x,y)-K(x_{0},y) \bigr). \end{aligned}

By Minkowski’s inequalities and $$\Vert \{\chi_{\{ t_{i+1}< \vert x-y \vert <s\}}\}_{s\in J_{i}, i\in\mathbb{N}} \Vert _{E}\leq1$$, we obtain

\begin{aligned} N_{1}\leq{}& \int_{\mathbb{R}} \bigl\Vert \{\chi_{\{t_{i+1}< \vert x-y \vert < s\} } \}_{s\in J_{i}, i\in\mathbb{N}} \bigr\Vert _{E} \\ &{}\times \biggl\vert \frac{R_{m+1}(\widetilde{b}_{k};x,y)}{ \vert x-y \vert ^{m}}K(x,y) -\frac{R_{m+1}(\widetilde{b}_{k};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y) \biggr\vert \bigl\vert f_{2}(y) \bigr\vert \,dy \\ \leq{}& \sum _{k=0}^{\infty}\int_{F_{k}}\frac{1}{ \vert x-y \vert ^{m}} \bigl\vert R_{m}( \widetilde{b}_{k};x,y)-R_{m}(\widetilde{b}_{k};x_{0},y) \bigr\vert \bigl\vert K(x,y) \bigr\vert \bigl\vert f_{2}(y) \bigr\vert \,dy \\ &{}+ \sum_{k=0}^{\infty}\int_{F_{k}} \bigl\vert R_{m}(\widetilde{b}_{k};x_{0},y) \bigr\vert \biggl\vert \frac{1}{ \vert x-y \vert ^{m}}-\frac{1}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \bigl\vert K(x,y) \bigr\vert \bigl\vert f_{2}(y) \bigr\vert \,dy \\ &{}+ \sum_{k=0}^{\infty}\int_{F_{k}}\frac{1}{m!} \bigl\vert \widetilde {b}_{k}^{(m)}(y) \bigr\vert \biggl\vert \frac{(x-y)^{m}}{ \vert x-y \vert ^{m}}-\frac {(x_{0}-y)^{m}}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \bigl\vert K(x,y) \bigr\vert \bigl\vert f_{2}(y) \bigr\vert \,dy \\ &{}+ \sum_{k=0}^{\infty}\int_{F_{k}} \biggl\vert \frac{R_{m+1}(\widetilde {b}_{k};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \bigl\vert K(x,y)-K(x_{0},y) \bigr\vert \bigl\vert f_{2}(y) \bigr\vert \,dy \\ ={}& N_{11}+N_{12}+N_{13}+N_{14}. \end{aligned}

From the mean value theorem, there exists $$\eta\in I$$ such that

\begin{aligned} R_{m}(\widetilde{b}_{k};x,y)-R_{m}(\widetilde {b}_{k};x_{0},y)=(x-x_{0})R_{m-1}\bigl( \widetilde{b}_{k}';\eta,y\bigr). \end{aligned}

For $$\eta, x\in I$$, $$y\in F_{k}$$, we have $$\vert y-x_{0} \vert \thickapprox \vert y-x \vert \thickapprox \vert y-\eta \vert$$ and $$5 \vert y-\eta \vert \approx5 \vert y-x_{0} \vert \leq 2^{k+1}\cdot20l$$. By Lemma 2.4 and Lemma 2.1 we get

\begin{aligned} \bigl\vert R_{m-1}\bigl(\widetilde{b}'_{k}; \eta,y\bigr) \bigr\vert \lesssim{}& \vert \eta-y \vert ^{m-1} \biggl( \frac{1}{ \vert I_{\eta}^{y} \vert } \int_{I_{\eta}^{y}} \bigl\vert \widetilde {b}_{k}^{(m)}(z) \bigr\vert ^{s}\,dz \biggr)^{1/s} \\ \lesssim{}& \vert x-y \vert ^{m-1} \biggl(\frac{1}{2^{k+1}\cdot20l} \int _{ \vert z-x_{0} \vert < 2^{k+1}\cdot20l} \bigl\vert {b}^{(m)}(z)- \bigl(b^{(m)}\bigr)_{\widetilde {I}_{k}} \bigr\vert ^{s}\,dz \biggr)^{1/s} \\ \lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \bigl(2^{k}l\bigr)^{\beta} \vert x-y \vert ^{m-1}. \end{aligned}

Then

\begin{aligned} \bigl\vert R_{m}(\widetilde{b}_{k};x,y)-R_{m}( \widetilde{b}_{k};x_{0},y) \bigr\vert \lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}\bigl(2^{k}l \bigr)^{\beta} \vert x-x_{0} \vert \vert x-y \vert ^{m-1}. \end{aligned}

Since $$\vert K(x,y) \vert \leq C \vert x_{0}-y \vert ^{-1}$$,

\begin{aligned} N_{11}\lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}\sum_{k=0}^{\infty}\bigl(2^{k}l\bigr)^{\beta}\int_{2^{k}\cdot4l\leq \vert x_{0}-y \vert < 2^{k+1}\cdot4l}\frac{l}{(2^{k}\cdot4l)^{2}} \bigl\vert f(y) \bigr\vert \,dy \\ \lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}\sum _{k=0}^{\infty}\frac{1}{2^{k}} \frac{(2^{k}l)^{\beta}}{2^{k}l} \int_{ \vert x_{0}-y \vert < 2^{k+1}\cdot4l} \bigl\vert f(y) \bigr\vert \,dy \\ \lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}M_{\beta}(f) (x_{0}). \end{aligned}

For $$N_{12}$$, since $$x\in I$$, $$y\in F_{k}$$,

\begin{aligned} \bigl\vert R_{m}(\widetilde{b}_{k};x,y) \bigr\vert \lesssim \vert x-y \vert ^{m} \biggl(\frac {1}{ \vert I_{x}^{y} \vert } \int_{I_{x}^{y}} \bigl\vert \widetilde{b}_{k}^{(m)}(z) \bigr\vert ^{s}\,dz \biggr)^{1/s}\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}\bigl(2^{k}l \bigr)^{\beta} \vert x-y \vert ^{m} \end{aligned}

and

\begin{aligned} \biggl\vert \frac{1}{ \vert x-y \vert ^{m}}-\frac{1}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \lesssim \frac { \vert x-x_{0} \vert }{ \vert x-y \vert ^{m+1}}. \end{aligned}

Thus

\begin{aligned} N_{12}\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}\sum_{k=0}^{\infty}\bigl(2^{k}l\bigr)^{\beta}\int_{2^{k}\cdot4l\leq \vert x_{0}-y \vert < 2^{k+1}\cdot4l}\frac {l}{(2^{k}\cdot4l)^{2}} \bigl\vert f(y) \bigr\vert \,dy \lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge }_{\beta}}M_{\beta}(f) (x_{0}). \end{aligned}

As for $$N_{13}$$, due to

\begin{aligned} \biggl\vert \frac{(x-y)^{m}}{ \vert x-y \vert ^{m}}-\frac{(x_{0}-y)^{m}}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \lesssim \frac{ \vert x-x_{0} \vert }{ \vert x-y \vert }, \end{aligned}

and noting $$\widetilde{b}_{k}^{(m)}(y)=b^{(m)}(y)-(b^{(m)})_{\widetilde {I}_{k}}$$, we have

\begin{aligned} N_{13}\lesssim{}& \sum_{k=0}^{\infty}\int_{F_{k}} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert \frac { \vert x-x_{0} \vert }{ \vert x_{0}-y \vert ^{2}} \bigl\vert f(y) \bigr\vert \,dy \\ \lesssim{}& \sum_{k=0}^{\infty}\frac{1}{2^{k}}\frac{1}{2^{k}\cdot 4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot4l} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde {I}_{k}} \bigr\vert \bigl\vert f(y) \bigr\vert \,dy \\ \lesssim{}& \sum_{k=0}^{\infty}\frac{1}{2^{k}} \biggl(\frac {1}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot4l} \bigl\vert f(y) \bigr\vert ^{r}\,dy \biggr)^{1/r} \\ &{} \times \biggl(\frac{1}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot 4l} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert ^{r'}\,dy \biggr)^{1/r'} \\ \lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}M_{r,\beta }(f) (x_{0})\sum_{k=0}^{\infty}\frac{1}{2^{k}}\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot {\wedge}_{\beta}}M_{\beta,r}(f) (x_{0}). \end{aligned}

Notice

\begin{aligned} \bigl\vert R_{m+1}(\widetilde{b}_{k}; x_{0},y) \bigr\vert &\leq \bigl\vert R_{m}(\widetilde {b}_{k};x_{0},y) \bigr\vert +\frac{1}{m!} \bigl\vert \widetilde{b}_{k}^{(m)}(y) (x_{0}-y)^{m} \bigr\vert \\ & \lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \bigl(2^{k}l\bigr)^{\beta} \vert x_{0}-y \vert ^{m}+ \bigl\vert b^{(m)}(y)-\bigl(b^{(m)} \bigr)_{\widetilde{I}_{k}} \bigr\vert \vert x_{0}-y \vert ^{m} \end{aligned}

and by (1.2),

\begin{aligned} \bigl\vert K(x,y)-K(x_{0},y) \bigr\vert \lesssim \frac{ \vert x-x_{0} \vert ^{\delta}}{ \vert x_{0}-y \vert ^{1+\delta}}. \end{aligned}

Similar to the estimates for $$N_{11}$$, we have

\begin{aligned} \sum_{k=0}^{\infty}\int_{F_{k}}\frac{ \vert R_{m}(\widetilde {b}_{k};x_{0},y) \vert }{ \vert x-y \vert ^{m}}\frac{ \vert x-x_{0} \vert ^{\delta}}{ \vert x_{0}-y \vert ^{1+\delta }} \bigl\vert f(y) \bigr\vert \,dy\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}M_{\beta}(f) (x_{0}). \end{aligned}

Similar to the estimates for $$N_{13}$$, we have

\begin{aligned} \sum_{k=0}^{\infty}\int_{F_{k}}\frac{ \vert \widetilde {b}_{k}^{(m)}(y)(x_{0}-y)^{m} \vert }{ \vert x-y \vert ^{m}}\frac{ \vert x-x_{0} \vert ^{\delta}}{ \vert x_{0}-y \vert ^{1+\delta}} \bigl\vert f(y) \bigr\vert \,dy\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge }_{\beta}}M_{\beta,r}(f) (x_{0}). \end{aligned}

Then

\begin{aligned} N_{14}\lesssim \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \bigl(M_{\beta}(f) (x_{0})+M_{\beta,r}(f) (x_{0}) \bigr). \end{aligned}

Finally, let us estimate $$N_{2}$$. Notice that the integral

\begin{aligned} \int_{R} \bigl(\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y)-\chi_{\{ t_{i+1}< \vert x_{0}-y \vert < s\}}(y) \bigr) \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y)f_{2}(y)\,dy \end{aligned}

will be non-zero in the following cases:

1. (i)

$$t_{i+1}< \vert x-y \vert <s$$ and $$\vert x_{0}-y \vert \leq t_{i+1}$$;

2. (ii)

$$t_{i+1}< \vert x-y \vert <s$$ and $$\vert x_{0}-y \vert \geq s$$;

3. (iii)

$$t_{i+1}< \vert x_{0}-y \vert <s$$ and $$\vert x-y \vert \leq t_{i+1}$$;

4. (iv)

$$t_{i+1}< \vert x_{0}-y \vert <s$$ and $$\vert x-y \vert \geq s$$.

In case (i) we have $$t_{i+1}< \vert x-y \vert \leq \vert x_{0}-x \vert + \vert x_{0}-y \vert <l+t_{i+1}$$ as $$\vert x-x_{0} \vert < l$$. Similarly, in case (iii) we have $$t_{i+1}< \vert x_{0}-y \vert <l+t_{i+1}$$ as $$\vert x-x_{0} \vert < l$$. In case (ii) we have $$s< \vert x_{0}-y \vert <l+s$$ and in case (iv) we have $$s< \vert x-y \vert <l+s$$. By (1.1) and taking $$1< t< r$$, we have

\begin{aligned} &\int_{\mathbb{R}} \bigl(\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y)-\chi_{\{ t_{i+1}< \vert x_{0}-y \vert < s\}}(y) \bigr) \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}}K(x_{0},y)f_{2}(y)\,dy \\ &\quad \lesssim \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y)\chi_{\{ t_{i+1}< \vert x-y \vert < l+t_{i+1}\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \frac { \vert f_{2}(y) \vert }{ \vert x_{0}-y \vert }\,dy \\ &\qquad{} + \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y)\chi_{\{ s< \vert x_{0}-y \vert < l+s\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \frac { \vert f_{2}(y) \vert }{ \vert x_{0}-y \vert }\,dy \\ &\qquad{} + \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x_{0}-y \vert < s\}}(y)\chi_{\{ t_{i+1}< \vert x_{0}-y \vert < l+t_{i+1}\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \frac { \vert f_{2}(y) \vert }{ \vert x_{0}-y \vert }\,dy \\ &\qquad{} + \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x_{0}-y \vert < s\}}(y)\chi_{\{ s< \vert x-y \vert < l+s\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert \frac { \vert f_{2}(y) \vert }{ \vert x_{0}-y \vert }\,dy \\ &\quad \lesssim l^{1/t'} \biggl( \int_{\mathbb{R}}\chi_{\{ t_{i+1}< \vert x-y \vert < s\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy \biggr)^{1/t} \\ &\qquad{} + l^{1/t'} \biggl( \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x_{0}-y \vert < s\} }(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy \biggr)^{1/t}. \end{aligned}

Then

\begin{aligned} N_{2}\lesssim{}& l^{1/t'} \biggl\Vert \biggl\{ \biggl( \int_{\mathbb{R}}\chi _{\{t_{i+1}< \vert x-y \vert < s\}}(y) \biggl\vert \frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy \biggr)^{1/t}\biggr\} _{s\in J_{i}, i\in \mathbb{N}} \biggr\Vert _{E} \\ &{} +l^{1/t'} \biggl\Vert \biggl\{ \biggl( \int_{\mathbb{R}}\chi_{\{ t_{i+1}< \vert x_{0}-y \vert < s\}}(y) \biggl\vert \frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy \biggr)^{1/t}\biggr\} _{s\in J_{i}, i\in \mathbb{N}} \biggr\Vert _{E} \\ ={}& N_{21}+N_{22}. \end{aligned}

Notice

\begin{aligned} \bigl\vert R_{m+1}(\widetilde{b}_{k};x_{0},y) \bigr\vert \lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \bigl(2^{k}l\bigr)^{\beta} \vert x_{0}-y \vert ^{m}+ \bigl\vert b^{(m)}(y)-\bigl(b^{(m)} \bigr)_{\widetilde{I}_{k}} \bigr\vert \vert x_{0}-y \vert ^{m}. \end{aligned}

Choosing $$1< r< p$$ with $$t=\sqrt{r}$$, we have

\begin{aligned} N_{21}\lesssim{}& l^{1/t'}\biggl\{ \sum _{i\in\mathbb{N}}\sup_{s\in J_{i}} \biggl( \int_{\mathbb{R}}\chi_{\{t_{i+1}< \vert x-y \vert < s\}}(y) \biggl\vert \frac{R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy \biggr)^{2/t}\biggr\} ^{1/2} \\ \lesssim{}& l^{1/t'}\biggl\{ \sum_{i\in\mathbb{N}} \int_{\mathbb {R}}\chi_{\{t_{i+1}< \vert x-y \vert < t_{i}\}}(y) \biggl\vert \frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\biggr\} ^{1/t} \\ \lesssim{}& l^{1/t'}\biggl\{ \int_{\mathbb{R}} \biggl\vert \frac {R_{m+1}({b};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\biggr\} ^{1/t} \\ \lesssim{}& l^{1/t'}\Biggl\{ \sum_{k=0}^{\infty}\int_{F_{k}} \biggl\vert \frac {R_{m+1}(\widetilde{b}_{k};x_{0},y)}{ \vert x_{0}-y \vert ^{m}} \biggr\vert ^{t} \frac{ \vert f_{2}(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\Biggr\} ^{1/t} \\ \lesssim{}& \bigl\Vert {b}^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}}l^{1/t'} \Biggl\{ \sum_{k=0}^{\infty}\bigl(2^{k}l \bigr)^{\beta t} \int_{F_{k}}\frac { \vert f(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\Biggr\} ^{1/t} \\ &{} +l^{1/t'}\Biggl\{ \sum_{k=0}^{\infty}\int_{F_{k}} \bigl( \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert \bigr)^{t} \frac{ \vert f(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\Biggr\} ^{1/t}. \end{aligned}

But

\begin{aligned} & l^{1/t'}\Biggl\{ \sum_{k=0}^{\infty}\bigl(2^{k}l\bigr)^{\beta t} \int_{F_{k}}\frac { \vert f(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\Biggr\} ^{1/t} \\ &\quad \lesssim l^{1/t'} \Biggl(\sum_{k=1}^{\infty}\frac{(2^{k}l)^{\beta t}}{(2^{k}\cdot4l)^{t}} \int_{ \vert x_{0}-y \vert < 2^{k+1}\cdot4l} \bigl\vert f(y) \bigr\vert ^{t}\,dy \Biggr)^{1/t} \\ &\quad \lesssim \Biggl(\sum_{k=1}^{\infty}\frac{1}{2^{k(t-1)}}\frac {(2^{k}l)^{\beta t}}{2^{k}\cdot5l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot 5l} \bigl\vert f(y) \bigr\vert ^{t}\,dy \Biggr)^{1/t} \\ &\quad \lesssim \Biggl(\sum_{k=1}^{\infty}\frac{1}{2^{k(t-1)}} \biggl(\frac {(2^{k}l)^{\beta t^{2}}}{2^{k}\cdot5l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot 5l} \bigl\vert f(y) \bigr\vert ^{t^{2}}\,dy \biggr)^{1/t} \Biggr)^{1/t} \\ &\quad \lesssim \Biggl(\sum_{k=1}^{\infty}\frac{1}{2^{k(t-1)}} \Biggr)^{1/t}M_{\beta,r}(f) (x_{0}) \lesssim M_{\beta,r}(f) (x_{0}) \end{aligned}

and

\begin{aligned} & l^{1/t'}\Biggl\{ \sum_{k=0}^{\infty}\int_{F_{k}} \bigl( \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert \bigr)^{t} \frac{ \vert f(y) \vert ^{t}}{ \vert x_{0}-y \vert ^{t}}\,dy\Biggr\} ^{1/t} \\ &\quad \lesssim \Biggl(\sum_{k=0}^{\infty}\frac{1}{2^{k(t-1)}}\frac {1}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot 4l} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert ^{t} \bigl\vert f(y) \bigr\vert ^{t}\,dy \Biggr)^{1/t} \\ &\quad \lesssim \Biggl(\sum_{k=0}^{\infty}\frac{1}{2^{k(t-1)}} \biggl(\frac {1}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot4l} \bigl\vert f(y) \bigr\vert ^{t^{2}}\,dy \biggr)^{1/t} \\ &\qquad{}\times \biggl(\frac{1}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot 4l} \bigl\vert b^{(m)}(y)- \bigl(b^{(m)}\bigr)_{\widetilde{I}_{k}} \bigr\vert ^{tt'} \biggr)^{1/t'} \Biggr)^{1/t} \\ &\quad \lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} \Biggl(\sum _{k=0}^{\infty}\frac{1}{2^{k(t-1)}} \biggl( \frac{(2^{k}\cdot4l)^{r\beta }}{2^{k}\cdot4l} \int_{ \vert x_{0}-y \vert < 2^{k}\cdot4l} \bigl\vert f(y) \bigr\vert ^{t^{2}}\,dy \biggr)^{1/t} \Biggr)^{1/t} \\ &\quad \lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta ,r}(f) (x_{0}) \Biggl(\sum_{k=0}^{\infty}\frac{1}{2^{k(t-1)}} \Biggr)^{1/t} \\ &\quad \lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta,r}(f) (x_{0}). \end{aligned}

Therefore

\begin{aligned} N_{21}\lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta,r}(f) (x_{0}). \end{aligned}

Similarly,

\begin{aligned} N_{22}\lesssim \bigl\Vert b^{(m)} \bigr\Vert _{\dot{\wedge}_{\beta}} M_{\beta,r}(f) (x_{0}). \end{aligned}

This completes the proof of (3.1). Hence, Theorem 1.1 is proved.