On the compactness of oscillation and variation of commutators

TL;DR

This paper investigates the compactness properties of oscillation and variation in commutators of singular integral operators, providing new characterizations and weighted compactness results in harmonic analysis.

Contribution

It introduces a novel characterization of CMO spaces through the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.

Findings

Weighted compactness results for oscillation and variation of commutators

New characterization of CMO(R^n) via compactness properties

Extension of compactness results to weighted Lebesgue spaces

Abstract

In this paper, we first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, we establish a new $CMO(\mathbb{R}^n)$ characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.

Equations

$$T_{\Omega }f(x):=\text{p.v.}{\int }_{{\mathbb{R}}^n}\frac{\Omega (x-y)}{|x-y{|}^n}f(y)dy,$$

$${\int }_{{\mathbb{S}}^{n-1}}\Omega (x^{\prime })d\sigma (x^{\prime })=0,$$

$$T^b(f)(x):=[b,T]f(x):=b(x)T(f)(x)-T(bf)(x)$$

$${\mathcal{V}}_{\rho }(\mathcal{W})(f)(x):=\underset{{ϵ}_i\downarrow 0}{\sup }{\left(\sum _{i=1}^{\infty }|W_{{ϵ}_{i+1}}f(x)-W_{{ϵ}_i}f(x){|}^{\rho }\right)}^{1/\rho },(\rho >2).$$

$$\mathcal{O}(\mathcal{W}f)(x):={\left(\sum _{i=1}^{\infty }\underset{t_{i+1}\le {ϵ}_{i+1}<{ϵ}_i\le t_i}{\sup }|W_{{ϵ}_{i+1}}f(x)-W_{{ϵ}_i}f(x){|}^2\right)}^{1/2},$$

$$T_{K}f(x)={\int }_{{\mathbb{R}}^n}K(x,y)f(y)dy\text{for all}x\notin \text{supp}f$$

$$|K(x,y)|\le \frac{C_K}{|x-y{|}^n}$$

$$|K(x,y)-K(x^{\prime },y)|+|K(y,x)-K(y,x^{\prime })|\le \frac{C_K}{|x-y{|}^n}{\left(\frac{|x-x^{\prime }|}{|x-y|}\right)}^{\gamma },$$

$$\Vert f{\Vert }_{\mathrm{BMO}({\mathbb{R}}^n)}:=\underset{Q\subset {\mathbb{R}}^n}{\sup }\mathcal{O}(f;Q)<\infty ,$$

$$f_Q:=\frac{1}{|Q|}{\int }_{Q}f(y)dy\mathrm{and}\mathcal{O}(f;Q):=\frac{1}{|Q|}{\int }_Q|f(x)-f_Q|dx.$$

$$[\omega {]}_{A_p}^{1/p}:=\underset{Q}{\sup }{\left(\frac{1}{|Q|}{\int }_Q\omega (x)dx\right)}^{1/p}{\left(\frac{1}{|Q|}{\int }_Q\omega (x{)}^{1-p^{\prime }}dx\right)}^{1/p^{\prime }}<\infty ,$$

$$\tilde{\mathcal{O}}(\mathcal{W}f)(x):={\left(\sum _{i=1}^{\infty }\underset{t_{i+1}\le {ϵ}_{i+1}<{ϵ}_i\le t_i}{\sup }|W_{{ϵ}_{i+1}}f(x)-W_{{ϵ}_i}f(x){|}^2\right)}^{1/2}+|W_{t_1}f(x)|.$$

$$\Vert {\mathcal{V}}_{\rho }({\mathcal{T}}_K^b){\Vert }_{L_{\omega }^p({\mathbb{R}}^n)\to L_{\omega }^p({\mathbb{R}}^n)}\lesssim \Vert b{\Vert }_{\mathrm{BMO}({\mathbb{R}}^n)}.$$

$$\Vert \mathcal{O}({\mathcal{T}}_K^b){\Vert }_{L_{\omega }^p({\mathbb{R}}^n)\to L_{\omega }^p({\mathbb{R}}^n)}\lesssim \Vert b{\Vert }_{\mathrm{BMO}({\mathbb{R}}^n)}.$$

$$\underset{N\to \infty }{\lim }{\int }_{|x|>N}|f(x){|}^p\omega (x)dx=0,\text{uniformly for all}f\in E.$$

$$\underset{\rho \to 0}{\lim }\underset{y\in B(0,\rho )}{\sup }{\int }_{{\mathbb{R}}^n}|f(x+y)-f(x){|}^p\omega (x)dx=0,\text{uniformly for all}f\in E.$$

$$\omega (\lambda Q)\le {\lambda }^{np}[\omega {]}_{A_p}\omega (Q).$$

$$\underset{N\to +\infty }{\lim }{\int }_{B(0,N{)}^c}\frac{\omega (x)}{|x{|}^{np}}dx=0,\underset{N\to +\infty }{\lim }{\int }_{B(0,N{)}^c}\frac{{\omega }^{1-p^{\prime }}(x)}{|x{|}^{n{p}^{\prime }}}dx=0.$$

$$\begin{array}{rl}{\int }_{B(0,N{)}^c}\frac{\omega (x)}{|x{|}^{np}}dx=& \sum _{j=0}^{\infty }{\int }_{2^{j}N\le |x|<2^{j+1}N}\frac{\omega (x)}{|x{|}^{np}}dx\\ \lesssim & \sum _{j=0}^{\infty }(2^{j}N{)}^{-np}\omega (B(0,2^{j+1}N))\\ \lesssim & \sum _{j=0}^{\infty }(2^{j}N{)}^{-np}(2^{j}N{)}^{n(p-ϵ)}=N^{-nϵ}\sum _{j=0}^{\infty }{2}^{-jnϵ}⟶0\end{array}$$

$${\phi }_{\delta }(x):=\phi (\frac{x}{\delta }),K^{\delta }(x,y):=K(x,y)\cdot (1-{\phi }_{\delta }(x-y)),$$

$$T_{K^{\delta }}^b(f)(x):=b(x){\int }_{{\mathbb{R}}^n}{K}^{\delta }(x,y)f(y)dy-{\int }_{{\mathbb{R}}^n}{K}^{\delta }(x,y)b(y)f(y)dy.$$

$$\Vert \mathcal{O}({\mathcal{T}}_K^b)(f)-\mathcal{O}({\mathcal{T}}_{K^{\delta }}^b)(f){\Vert }_{L_{\omega }^p({\mathbb{R}}^n)}\lesssim \delta \Vert f{\Vert }_{L_{\omega }^p({\mathbb{R}}^n)},$$

$$|K^{\delta }(x,y)|\lesssim \frac{1}{|x-y{|}^n}(1-{\phi }_{\delta }(x-y))\le \frac{1}{|x-y{|}^n}{\chi }_{\{(x,y):|x-y|\ge \delta /2\}}\lesssim \frac{1}{{\delta }^n}.$$

$$|\mathcal{O}({\mathcal{T}}_K^b)(f)-\mathcal{O}({\mathcal{T}}_{K^{\delta }}^b)(f)|\le \mathcal{O}({\mathcal{T}}_K^b-{\mathcal{T}}_{K^{\delta }}^b)(f).$$

$$|b(x)-b(y)|\le \Vert \nabla b{\Vert }_{L^{\infty }({\mathbb{R}}^n)}|x-y|,\text{and}|{\phi }_{\delta }(x-y)|\le {\chi }_{\{(x,y):|x-y|\le \delta \}}(x,y),$$

$$|\mathcal{O}({\mathcal{T}}_K^b)(f)(x)-\mathcal{O}({\mathcal{T}}_{K^{\delta }}^b)(f)(x)|$$

$$\le {\left(\sum _{i=1}^{\infty }\underset{t_{i+1}\le {ϵ}_{i+1}<{ϵ}_i\le t_i}{\sup }{\left|{\int }_{{ϵ}_{i+1}<|x-y|\le {ϵ}_i}(b(x)-b(y)){\phi }_{\delta }(x-y)K(x,y)f(y)dy\right|}^2\right)}^{1/2}$$

$$\lesssim {\left(\sum _{i=1}^{\infty }\underset{t_{i+1}\le {ϵ}_{i+1}<{ϵ}_i\le t_i}{\sup }{\left|{\int }_{{ϵ}_{i+1}<|x-y|\le {ϵ}_i}{\phi }_{\delta }(x-y)\frac{|f(y)|}{|x-y{|}^{n-1}}dy\right|}^2\right)}^{1/2}$$

$$\le {\left(\sum _{i=1}^{\infty }{\left|{\int }_{t_{i+1}<|x-y|\le t_i}{\phi }_{\delta }(x-y)\frac{|f(y)|}{|x-y{|}^{n-1}}dy\right|}^2\right)}^{1/2}$$

$$\le \sum _{i=1}^{\infty }{\int }_{t_{i+1}<|x-y|\le t_i}{\phi }_{\delta }(x-y)\frac{|f(y)|}{|x-y{|}^{n-1}}dy\le {\int }_{B(x,\delta )}\frac{|f(y)|}{|x-y{|}^{n-1}}dy\lesssim \delta M(f)(x),$$

$$\Vert \mathcal{O}({\mathcal{T}}_K^b)(f)-\mathcal{O}({\mathcal{T}}_{K^{\delta }}^b)(f){\Vert }_{L_{\omega }^p({\mathbb{R}}^n)}\lesssim \delta \Vert M(f){\Vert }_{L_{\omega }^p({\mathbb{R}}^n)}\lesssim \delta \Vert f{\Vert }_{L_{\omega }^p({\mathbb{R}}^n)}$$

$$\{\mathcal{O}({\mathcal{T}}_K^b)(f):\Vert f{\Vert }_{L_{\omega }^p({\mathbb{R}}^n)}\le 1\}$$

Full text

Supported by the NSF of China (Nos.11771358, 11471041, 11701112, 11671414,11871254), and the NSF of Fujian Province of China (Nos.2015J01025, 2017J01011).

1. Introduction

The singular integral operator with homogeneous kernel is defined by

[TABLE]

where $\Omega$ is a homogeneous function of degree zero and satisfies the following mean value zero property:

[TABLE]

where $d\sigma$ is the spherical measure on the sphere $\mathbb{S}^{n-1}$. Given a locally integrable function $b$ and a linear operator $T$, the commutator $[b,T]$ is defined by

[TABLE]

for suitable functions $f$. The famous work of Coifman, Rochberg and Weiss [8] gave a characterization of $L^{p}$-boundedness of $[b,R_{j}]$, for every Riesz transform $R_{j}$. This result was improved by Uchiyama in his remarkable work [25], in which he showed that the commutator $[b,T_{\Omega}]$ with $\Omega\in Lip_{1}(\mathbb{S}^{n-1})$ is bounded (compact resp.) on $L^{p}({{\mathbb{R}}^{n}})$ if and only if the symbol $b$ is in ${{\rm BMO}(\mathbb{R}^{n})}$ (${{\rm CMO}(\mathbb{R}^{n})}$ resp.), where ${{\rm CMO}(\mathbb{R}^{n})}$ denotes the closure of $C_{c}^{\infty}(\mathbb{R}^{n})$ in the ${\rm BMO}(\mathbb{R}^{n})$ topology. Since then, the work on compactness of commutators of singular and fractional integral operators and its applications to PDE’s have been paid more and more attention; see, for example, [3, 4, 6, 7, 12, 15, 24] and the references therein. Recently, inspired by Lerner–Ombrosi–Rivera-Ríos [17], the first, third and fourth authors [11] give some new characterizations of the compact commutators of singular integrals via ${\rm CMO}(\mathbb{R}^{n})$.

This paper is devoted to a first contribution to the weighted $L^{p}$-compactness of the oscillation and variation of the commutator of singular integral operator. To state our main results, we first recall some notations.

For a one-parameter family of operators $\mathcal{W}:=\{W_{t}\}_{t>0}$, the variation of $\mathcal{W}$ is defined by

[TABLE]

In general, the boundedness estimate of variation operators can fail when $\rho\leq 2$, see the case of martingales in [22].

Next, we recall the definition of the oscillation operator of $\mathcal{W}$:

[TABLE]

where $\{t_{j}\}$ is a decreasing sequence of positive numbers converging to [math].

The variation inequality was first proved by Lépingle [16] for martingales. Then, Bourgain [1] proved the variation inequality for the ergodic averages of a dynamic system. Since then, the oscillation and variation have been the active subject of recent research in the field of probability, ergodic theory and harmonic analysis. In 2000, Campbell et al. [2] established the $L^{p}({{\mathbb{R}}^{n}})$-boundedness of variation for truncated Hilbert transform and then extended to higher dimensional case in [13]. For the weighted boundedness result one can see [9], [18] and [19].

We say that $T_{K}$ is a Calderón-Zygmund operator on $\mathbb{R}^{n}$ if $T_{K}$ is bounded on $L^{2}(\mathbb{R}^{n})$ and it admits the following representation

[TABLE]

with kernel $K$ satisfying the size condition

[TABLE]

and a smoothness condition

[TABLE]

for all $|x-y|>2|x-x^{\prime}|$, where $C_{K}>0$, $\gamma>0$.

Definition 1.1**.**

The space of functions with bounded mean oscillation, denoted by ${\rm BMO}(\mathbb{R}^{n})$, consists of all $f\in L_{loc}^{1}(\mathbb{R}^{n})$ such that

[TABLE]

where

[TABLE]

The following class of ${A_{p}}$ was introduced by Muckenhoupt [20] to study the weighted norm inequalities of Hardy-Littlewood maximal operators.

Definition 1.2**.**

For $1<p<\infty$, the Muckenhoupt class ${A_{p}}$ is the set of non-negative locally integrable functions $\omega$ such that

[TABLE]

where $1/p+1/p^{\prime}=1$.

Our main results can be formulated as follows.

Theorem 1.3**.**

Let $1<p<\infty$, $\omega\in{A_{p}}$ and $b\in{{\rm CMO}(\mathbb{R}^{n})}$. We have the following two statements.

1. (1)

The $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{O}(\mathcal{T}_{K})$ implies the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-compactness of $\mathcal{O}(\mathcal{T}_{K}^{b})$; 2. (2)

The $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K})$ implies the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-compactness of $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$.

In order to establish the necessity and equivalent characterization of compact oscillation operator, we define the modified oscillation by

[TABLE]

This variant of oscillation is necessary for the following Theorem 1.4 and Corollary 1.5, since one can choose a function $b\notin{{\rm BMO}(\mathbb{R}^{n})}$ such that $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ in Theorem 1.4 and Corollary 1.5 is a compact operator on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. We put the details in Appendix A.

Theorem 1.4**.**

Let $1<p<\infty$, $b\in L_{loc}^{1}(\mathbb{R}^{n})$ and $\omega\in{A_{p}}$. Let $\Omega$ be a bounded measurable function on $\mathbb{S}^{n-1}$, which does not change sign and is not equivalent to zero on some open subset of $\mathbb{S}^{n-1}$. Then we have the following two statements.

1. (1)

Let $\{t_{j}\}_{j=1}^{\infty}$ be a sequence with $\sup_{i\in\mathbb{Z}}|\{j:2^{i}\leq|t_{j}|<2^{i+1}\}|<\infty$. Then the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-compactness of $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ implies $b\in{{\rm CMO}(\mathbb{R}^{n})}$; 2. (2)

The $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-compactness of $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$ implies $b\in{{\rm CMO}(\mathbb{R}^{n})}$.

Corollary 1.5**.**

Let $1<p<\infty$, $b\in L_{loc}^{1}(\mathbb{R}^{n})$, $\omega\in{A_{p}}$ and $\Omega\in Lip_{1}(\mathbb{S}^{n-1})$ with $\Omega\not\equiv 0$. Then

1. (1)

$b\in{{\rm CMO}(\mathbb{R}^{n})}\Longleftrightarrow\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$* is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$;* 2. (2)

$b\in{{\rm CMO}(\mathbb{R}^{n})}\Longleftrightarrow\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$* is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$.*

This paper is organized as follows. Section 2 is devoted to the proof of the sufficiency of compactness, i.e., Theorem 1.3. It is well known that the Fréchet-Kolmogorov theorem is a powerful tool in the study of compactness of commutators of singular integral operators, see, for example, [25]. In the proof of Theorem 1.3, we also use the weighted Fréchet-Kolmogorov theorem (see Lemma 2.2) to prove the compactness of $\mathcal{O}(\mathcal{T}_{K}^{b})$ and $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$. However, due to the special structures of oscillation and variation, the argument here is more complicated. Moreover, compared to the known case of singular integral operators, the regularity of commutator of oscillation or variation of a singular integral operator comes from not only the regularity of symbol $b$ and the kernel $K$, but also the smallness of corresponding measurable sets degenerated by the annuluses in the definition of oscillation or variation.

The necessity conditions of compactness will be dealt with in Section 3. By establishing two claims A and B, we reduce our cases to the known cases in [11]. Then, Theorem 1.4 can be proved. Appendix A is used to clarify the reasonableness of the modified oscillation in our results for the necessity.

We remark that all conclusions of this article can de extended to the high order commutator case (oscillation and variation of high order commutators) as in [11]. We omit such more complicated expression form just for concise, and leave the details for the interested readers.

Throughout this paper, we will adopt the following notations. Let $C$ be a positive constant which is independent of the main parameters. The notation $X\lesssim Y$ denotes the statement that $X\leq CY$, $X\sim Y$ means $X\lesssim Y\lesssim X$. For a given cube $Q$, we use $c_{Q}$, $l_{Q}$ and $\chi_{Q}$ to denote the center, side length and characteristic function of $Q$. We also denote $(A\setminus B)\cup(B\setminus A)$ by $A\triangle B$. For any point $x_{0}\in{{\mathbb{R}}^{n}}$ and sets $E,F\subset{{\mathbb{R}}^{n}}$, $E+x_{0}:=\{y+x_{0}:y\in E\}$ and $E-F:=\{x-y:x\in E,y\in F\}$.

2. Compactness of the oscillation operators

In this part, we study the compactness property of oscillation and variation. Thanks to the so called conjugation method (see, for example, [21]) and John-Nirenberg inequality, the boundedness of $\mathcal{O}(\mathcal{T}_{K}^{b})$ and $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$ can be deduced by the weighted boundedness of $\mathcal{O}(\mathcal{T}_{K})$ and $\mathcal{V}_{\rho}(\mathcal{T}_{K})$ respectively. Precisely, we have the following lemma.

Lemma 2.1**.**

Let $1<p<\infty$, $\omega\in{A_{p}}$ and $b\in{{\rm BMO}(\mathbb{R}^{n})}$. We have the following two statements.

1. (1)

The $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K})$ implies the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$ with

[TABLE] 2. (2)

The $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{O}(\mathcal{T}_{K})$ implies the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{O}(\mathcal{T}_{K}^{b})$ with

[TABLE]

The conclusion (1) of Lemma 2.1 was proved in [5, Theorem 1.1]. Since the proof of (2) in Lemma 2.1 is similar, we omit the details here.

Now, we turn to the proof of Theorem 1.3. We recall the weighted Fréchet-Kolmogorov theorem [7] as follows.

Lemma 2.2**.**

Let $p\in(1,\infty)$ and $\omega\in A_{p}$. A subset $E$ of $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ is precompact (or totally bounded) if the following statements hold:

1. (a)

$E$* is bounded, i.e., $sup_{f\in E}\|f\|_{L_{\omega}^{p}(\mathbb{R}^{n})}\lesssim 1$;* 2. (b)

$E$* uniformly vanishes at infinity, that is,*

[TABLE] 3. (c)

$E$* is uniformly equicontinuous, that is,*

[TABLE]

Then, we collect some basic properties of the Muckenhoupt class ${A_{p}}$. One can see [10] for the proofs of (i)-(iii) of Lemma 2.3.

Lemma 2.3**.**

Let $1<p<\infty$.

• (i)

$\omega\in A_{p}\Longleftrightarrow\omega^{1-p^{\prime}}=\omega^{-p^{\prime}/p}\in A_{p^{\prime}}$.

• (ii)

If $\omega\in A_{p}$, there exists a small constant $\epsilon$ depending only on $n$, $p$ and $[\omega]_{A_{p}}$ such that $\omega\in A_{p-\epsilon}$.

• (iii)

For all $\lambda>1$, and all cubes $Q$,

[TABLE]

• (iv)

If $\omega\in A_{p}$, we have

[TABLE]

Proof.

We only give the proof of (iv). Since $\omega\in A_{p}$, there exists $\epsilon>0$ such that $\omega\in A_{p-\epsilon}$. Write

[TABLE]

as $N\rightarrow+\infty$, where we use property (iii) in the second inequality. Similarly, by property (i), we get $\omega^{1-p^{\prime}}\in A_{p^{\prime}}$, then the second equality of (iv) follows. ∎

We now present the proof of Theorem 1.3. We point out that since there is some essential difference between the arguments for oscillation and variation, we give the proofs of (1) and (2), respectively.

Proof of (1) in Theorem 1.3.

Assume that $\mathcal{O}(\mathcal{T}_{K})$ is bounded on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ and $b\in{{\rm CMO}(\mathbb{R}^{n})}$. Using Lemma 2.1 (1), we see that $\mathcal{O}(\mathcal{T}_{K}^{b})$ is also bounded on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. Moreover, by the definition of ${{\rm CMO}(\mathbb{R}^{n})}$, it suffices to show $\mathcal{O}(\mathcal{T}_{K}^{b})$ is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ for $b\in C_{c}^{\infty}(\mathbb{R}^{n})$. To this end, we follow the idea in [15] and consider smooth truncated singular integral operators. Take $\varphi\in C_{c}^{\infty}(\mathbb{R}^{n})$ supported on $B(0,1)$ such that $\varphi=1$ on $B(0,1/2)$, $0\leq\varphi\leq 1$. Let $\delta>0$ be a small constant,

[TABLE]

and

[TABLE]

We first claim that for any $f\in L^{p}_{\omega}({{\mathbb{R}}^{n}})$,

[TABLE]

where the implicit constant is independent of $f$. In fact, A simple calculation yields that the kernel $K^{\delta}$ also satisfies (1.4) and (1.5) with $C_{K}$ replaced by certain constant $C$ therein. Moreover, $K^{\delta}$ is a bounded function since for any $x,y$ with $x\not=y$,

[TABLE]

By the sub-linearity of oscillation, we have

[TABLE]

From this and the fact that for any $x$ and $y$,

[TABLE]

we further deduce that for any $x$,

[TABLE]

where $M(f)$ is the Hardy-littlewood maximal function of $f$, and the implicit constant is independent of $x,\delta$ and $f$. This via the boundedness of $M(f)$ on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ implies that

[TABLE]

and shows the claim (2.1).

Now observe that to show $\mathcal{O}(\mathcal{T}_{K}^{b})$ is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$, we only need to show that

[TABLE]

is precompact. Then by (2.1), it suffices to show that for $\delta$ small enough, the set

[TABLE]

is precompact.

We now use Lemma 2.2 to show that $A(\mathcal{O}(\mathcal{T}_{K^{\delta}}^{b}))$ is precompact. Firstly, note that (2.1) also yields the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{O}(\mathcal{T}_{K^{\delta}}^{b})$. Then we see that $A(\mathcal{O}(\mathcal{T}_{K^{\delta}}^{b}))$ is a bounded set in $L^{p}_{\omega}({{\mathbb{R}}^{n}})$, and (a) of Lemma 2.2 is true.

To show (b), without loss of generality, we assume that $b$ is supported in a cube $Q$ centered at the origin. For $x\in(2Q)^{c}$, by (1.4) for $K^{\delta}$, the Hölder inequality and $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$, we have

[TABLE]

Taking $N>2$, we then have

[TABLE]

which tends to zero as $N$ tends to infinity, where we use (iv) in Lemma 2.3. Thus, Lemma 2.2 (b) holds.

It remains to prove that $A(\mathcal{O}(\mathcal{T}_{K^{\delta}}^{b}))$ satisfies Lemma (c). Taking $z\in\mathbb{R}^{n}$ with $|z|\leq\frac{\delta}{8}$, then

[TABLE]

Moreover, for each $i,$ we write

[TABLE]

We first estimate $I_{2}(i)$. Since $b\in C_{c}^{\infty}(\mathbb{R}^{n})$, we have

[TABLE]

which yields that

[TABLE]

Furthermore, assume that there exists $i_{0}\in\mathbb{N}:=\{1,2,\,\cdots\}$ such that $t_{i_{0}+1}<\delta\leq t_{i_{0}}$. Then we see that for a. e. $x$,

[TABLE]

Observe that for a. e. $x$,

[TABLE]

and

[TABLE]

Moreover, take $\tilde{\epsilon}_{i_{0}+1},\tilde{\epsilon}_{i_{0}}\in[t_{i_{0}+1},t_{i_{0}}]$ such that

[TABLE]

We then have

[TABLE]

Therefore, we conclude that for a. e. $x$,

[TABLE]

where the implicit constant is independent of $i_{0}$, $\delta$, $f$ and $x$. From the above two estimates, the boundedness of $\mathcal{O}(\mathcal{T}_{K})(f)$ and $M(f)$ on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ and $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$, we get

[TABLE]

Next, we turn to the estimate of $I_{3}(i)$. Observe that $K^{\delta}(x+z,y)$ and $K^{\delta}(x,y)$ vanish when $|x-y|\leq\frac{\delta}{4}$. Then by (1.5) for $K^{\delta}$

[TABLE]

where $\gamma$ is as in (1.5).

From this, we further have

[TABLE]

where the implicit constant is independent of $f$, $x$, $\delta$ and $z$. Thus,

[TABLE]

Finally, we proceed to the estimate of $I_{1}(i)$. Note that

[TABLE]

if and only if at least one of the following four statements holds:

• (i)

$\epsilon_{i+1}<|x+z-y|\leq\epsilon_{i}$ and $|x-y|\leq\epsilon_{i+1}$;

• (ii)

$\epsilon_{i+1}<|x+z-y|\leq\epsilon_{i}$ and $|x-y|>\epsilon_{i}$;

• (iii)

$\epsilon_{i+1}<|x-y|\leq\epsilon_{i}$ and $|x+z-y|\leq\epsilon_{i+1}$;

• (iv)

$\epsilon_{i+1}<|x-y|\leq\epsilon_{i}$ and $|x+z-y|>\epsilon_{i}$.

This further implies the following four cases:

• (i’)

$\epsilon_{i+1}<|x+z-y|\leq\epsilon_{i+1}+|z|$;

• (ii’)

$\epsilon_{i}<|x-y|\leq\epsilon_{i}+|z|$;

• (iii’)

$\epsilon_{i+1}<|x-y|\leq\epsilon_{i+1}+|z|$;

• (iv’)

$\epsilon_{i}<|x+z-y|\leq\epsilon_{i}+|z|$.

We then have that

[TABLE]

By similarity, we only estimate

[TABLE]

Observe that if $\epsilon_{i+1}<2|z|$, then by the fact that $|z|\leq|x-y|/2$, we have that ${\rm I}_{1,\,1,\,i}=0$. Thus, assume that $i_{1}\in\mathbb{N}$ such that $t_{i_{1}+1}<2|z|\leq t_{i_{1}}$. We see that for any $i\geq i_{1}+1$, ${\rm I}_{1,\,1,\,i}=0.$ Therefore,

[TABLE]

Moreover, for $i=i_{1}$, we may assume that $\epsilon_{i_{1}+1}\geq 2|z|$. Then for $i\in\{1,2,\cdots,i_{1}\}$, since $\epsilon_{i+1}\geq 2|z|$, for $r\in(1,2)$ such that $w\in A_{p/r}$, by (i’) and the Hölder inequality, we have

[TABLE]

Since $r<2$, we then conclude that

[TABLE]

Now since $w\in A_{p/r}$, by the boundedness of $M$, we see that

[TABLE]

The equicontinuity of $A(\mathcal{O}(\mathcal{T}_{K^{\delta}}^{b}))$ follows from the combination of (2)-(2.6). We have now completed this proof. ∎

Proof of (2) in Theorem 1.3.

Assume that $\mathcal{V}_{\rho}(\mathcal{T}_{K})$ is bounded on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ and $b\in{{\rm CMO}(\mathbb{R}^{n})}$. Take $K^{\delta}(x,y)$ with $\delta>0$ as in the proof of (1). Arguing as in (2.3), we also have that for any $f\in L^{p}_{\omega}({{\mathbb{R}}^{n}})$,

[TABLE]

and obtain the boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K^{\delta}}^{b})$ via this inequality and the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$, which follows from Lemma 2.1 and the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K})$. Moreover, by a similar argument, to show $\mathcal{V}_{\rho}(\mathcal{T}_{K}^{b})$ is compact, we only need to verify the set

[TABLE]

is precompact, where $b\in C_{c}^{\infty}(\mathbb{R}^{n})$.

Without loss of generality, we assume that $b$ is supported in a cube $Q$ centered at the origin. By the boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K^{\delta}}^{b})$, $A(\mathcal{V}_{\rho}(\mathcal{T}_{K^{\delta}}^{b}))$ is a bounded set on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. Therefore condition (a) of Lemma 2.2 holds.

Again, by the same argument as in the proof of (1) in Theorem 1.3, we have that for any $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$ and $x\notin 2^{N}Q$ with $N>2$,

[TABLE]

And hence,

[TABLE]

tends to zero as $N$ tends to infinity. This proves condition (b) of Lemma 2.2.

It remains to prove that $A(\mathcal{V}_{\rho}(\mathcal{T}_{K^{\delta}}^{b}))$ is uniformly equicontinuous in $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. Take $z\in\mathbb{R}^{n}$ with $|z|\leq\frac{\delta}{8}\leq l_{Q}/2$. By the sub-linearity of $\mathcal{V}_{\rho}$, we have

[TABLE]

Write

[TABLE]

Observe that $J_{2}(i)$ is dominated by

[TABLE]

which yields that

[TABLE]

Furthermore,

[TABLE]

Here the implicit constant is independent of the choice of $\delta$, $f$ and $x$. Thus, by the boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{K})(f)$ and $M(f)$, for any $f$ with $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$,

[TABLE]

Observe that $K^{\delta}(x+z,y)$ and $K^{\delta}(x,y)$ vanish when $|x-y|\leq\frac{\delta}{4}$. Then by (1.5), $J_{3}(i)$ is dominated by

[TABLE]

By the same argument as the estimate of $I_{3}$ in the proof of (1) of Theorem 1.3, we get that for any $f$ such that $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$,

[TABLE]

Finally, we turn to the estimate of $J_{1}(i)$. Denote

[TABLE]

Recalling that $b$ is supported in $Q$ and $|z|\leq\delta/8\leq l_{Q}/2$, we have that for any $x\in\mathbb{R}^{n}$,

[TABLE]

This and the sub-linearity of $\mathcal{V}_{\rho}$ imply that

[TABLE]

We start with the estimate of $L_{1}$. For some large positive constant $N$, denote $f_{1}:=f\chi_{(2^{N}Q)^{c}}$, $f_{2}:=f\chi_{2^{N}Q}$,

[TABLE]

and

[TABLE]

Then we write

[TABLE]

Recall $|K^{\delta}(x+z,y)|\lesssim\frac{1}{|x+z-y|^{n}}\sim\frac{1}{|y|^{n}}$ for $x\in 2Q$ and $y\in(2^{N}Q)^{c}$. For $x\in 2Q$, we have

[TABLE]

where $\beta^{(1)}_{N}\rightarrow 0$ as $N\rightarrow\infty$ by (iv) in Lemma 2.3.

Next, recall $|K_{\delta}(x+z,y)|\lesssim\frac{1}{\delta^{n}}$. For fixed $N>0$, $x\in 2Q$, we have

[TABLE]

where in the last-to-second inequality, we use the fact that for any $f$ with $\|f\|_{L^{p}_{\omega}({{\mathbb{R}}^{n}})}\leq 1$,

[TABLE]

and the constant depends on $p$, $Q$ and $\omega$.

For the last term, we claim that,

[TABLE]

uniformly for all $x\in 2Q$, $\{\epsilon_{i}\}$ and $i\in\mathbb{N}$, where $\beta^{(2)}_{N,|z|}\rightarrow 0$ as $|z|\rightarrow 0$ for any fixed $N$.

In fact, for any $\{\epsilon_{i}\}$, assume $i_{0}\in\mathbb{N}$ be such that

[TABLE]

Then we see that for any $i$ such that $i\leq i_{0}-1$,

[TABLE]

Moreover, we may further assume that $\epsilon_{i_{0}}\leq\sqrt{n}(2+2^{N})l_{Q}+\delta$. Then for all $i\geq i_{0}$, by the fact that

[TABLE]

we get

[TABLE]

Therefore the claim follows.

This claim and the fact $\omega^{1-p^{\prime}}\chi_{2^{N}Q}\in L^{1}({{\mathbb{R}}^{n}})$ yield that

[TABLE]

Hence,

[TABLE]

Combination of the above estimates for $L_{1}^{1}$ and $L_{1}^{2}$ yields that

[TABLE]

Taking sufficient large $N$ and sufficient small $|z|$, we can make $L_{1}$ arbitrary small.

Now, we turn to the estimate of $L_{2}$.

[TABLE]

First, we deal with $L_{2}^{1}$. Recall $|K^{\delta}(x+z,y)|\lesssim\frac{1}{|x+z-y|^{n}}\sim\frac{1}{|x^{n}}$ for $y\in Q$, $x\in(2^{\widetilde{N}}Q)^{c}$. We have

[TABLE]

where $\beta^{(3)}_{\widetilde{N}}\rightarrow 0$ as $\widetilde{N}\rightarrow\infty$ by (iv) in Lemma 2.3. Then, for fixed $\widetilde{N}$, by the same technique as in the estimate of $L_{1}^{1}$,

[TABLE]

where $\beta^{(4)}_{\widetilde{N},|z|}\rightarrow 0$ as $|z|\rightarrow 0$ for any fixed $\widetilde{N}$. Hence,

[TABLE]

Therefore, we conclude that the set $A(\mathcal{V}_{\rho}(\mathcal{T}_{K^{\delta}}^{b}))$ is uniformly equicontinuous in $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ and hence Lemma 2.2 (c) holds, which finishes the proof of Theorem 1.3 (2). ∎

3. Necessity of compact oscillation and variation of commutators

This section is devoted to the proof of Theorem 1.4. We follow the approach of [11]. For any measurable function $f$, let $f^{*}$ be the non-increasing rearrangement of $f$, namely, for any $t\in(0,\infty)$,

[TABLE]

Recall the John-Strömberg equivalence (see [14] and [23]) of a function $f\in\rm BMO(\mathbb{R}^{n})$

[TABLE]

where for $0<\lambda<1$, the local mean oscillation of $f$ over a cube $Q$ is defined by

[TABLE]

In [11], the following equivalent characterization of ${{\rm CMO}(\mathbb{R}^{n})}$ in terms of the local mean oscillation was established.

Lemma 3.1**.**

([11]) Let $f\in{{\rm BMO}(\mathbb{R}^{n})}$. Then $f\in{{\rm CMO}(\mathbb{R}^{n})}$ if and only if the following three conditions hold:

1. (1)

$\lim_{r\rightarrow 0}\sup\limits_{|Q|=r}a_{\lambda}(f;Q)=0$, 2. (2)

$\lim_{r\rightarrow\infty}\sup\limits_{|Q|=r}a_{\lambda}(f;Q)=0$, 3. (3)

$\lim_{d\rightarrow\infty}\sup\limits_{Q\cap[-d,d]^{n}=\emptyset}a_{\lambda}(f;Q)=0$.

In order to deal with the necessity conditions for the compact oscillation and variation of commutators, we recall following two lemmas from [11].

Lemma 3.2** (lower estimates).**

Let $\omega\in{A_{p}}$, $\lambda\in(0,1)$ and $b$ be a real-valued measurable function. For a given cube $Q$, there exists a cube $P$ with the same side length of $Q$ satisfying $|c_{Q}-c_{P}|=k_{0}l_{Q}\ (k_{0}>10\sqrt{n})$, and measurable sets $E\subset Q$ with $|E|=\frac{\lambda}{2}|Q|$, and $F\subset P$ with $|F|=\frac{1}{2}|Q|$, such that for $f:=(\int_{F}\omega(x)dx)^{-1/p}\chi_{F}$, and any measurable set $B$ with $|B|\leq\frac{\lambda}{8}|Q|$,

[TABLE]

Lemma 3.3** (upper estimates).**

Let $b\in{{\rm BMO}(\mathbb{R}^{n})}$, $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ and $\omega\in{A_{p}}$. For a given cube $Q$, denote by $F$ the set associated with $Q$ mentioned in Lemma 3.2. Let $f:=(\int_{F}\omega(x)dx)^{-1/p}\chi_{F}$. Then, there exists a positive constant $\delta$ such that

[TABLE]

for sufficient large $d$, where the implicit constant is independent of $d$ and $Q$.

Claim A: Under the assumptions of Theorem 1.4, Lemma 3.2 is also valid if we replace $T_{\Omega}^{b}$ by $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ or by $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$.

Proof of Claim A.

Arguing as in the proof of [11, Proposition 4.2], we see that for any given cube $Q$, the sets $P$, $E$ and $F$ exist. Moreover, for $f:=(\int_{F}\omega(x)dx)^{-1/p}\chi_{F}$, the following function

[TABLE]

does not change sign on $E\times F$. Hence, for $x\in E$,

[TABLE]

Observe that for any $x\in E$, $y\in F$,

[TABLE]

where $Q_{0}$ is the cube centered at origin with side length 1. By this fact and the assumption

[TABLE]

there are only finite terms which are non-zero in the series in (3). Thus,

[TABLE]

and the implicit constant depends on $\{t_{i}\}_{i}$ but not on $x$, $Q$, $P$, $E$ and $F$. Therefore, by this fact and Lemma 3.2, (3.2) with $T_{\Omega}^{b}$ replaced by $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ holds.

On the other hand, for any $x\in E$,

[TABLE]

It follows that $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})(f)(x)=T_{\Omega}^{b}(f)(x)$ for $x\in E$, which implies that (3.2) with $T_{\Omega}^{b}$ replaced by $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$ holds. ∎

We further have the following corollary directly follows from Lemma 3.2 and Claim A.

Corollary 3.4**.**

Let $1<p<\infty$, $b\in L_{loc}^{1}(\mathbb{R}^{n})$ and $\omega\in{A_{p}}$. Let $\Omega$ be a measurable function on $\mathbb{S}^{n-1}$, which does not change sign and is not equivalent to zero on some open subset of $\mathbb{S}^{n-1}$. Then,

1. (1)

let $\{t_{j}\}_{j=1}^{\infty}$ be a sequence with $\sup_{i\in\mathbb{Z}}|\{j:2^{i}\leq|t_{j}|\leq 2^{i+1}\}|<\infty$, then the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ implies $b\in{{\rm BMO}(\mathbb{R}^{n})}$; 2. (2)

the $L^{p}_{\omega}({{\mathbb{R}}^{n}})$-boundedness of $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$ implies $b\in{{\rm BMO}(\mathbb{R}^{n})}$.

Moreover, by combining Corollary 3.4, [19, Corollary 1.4] and the boundedness of $T_{\Omega,t}^{b}$, we have another corollary on the characterization of bounded $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ and $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$.

Corollary 3.5**.**

Let $1<p<\infty$, $b\in L_{loc}^{1}(\mathbb{R}^{n})$ and $\omega\in{A_{p}}$, $\Omega\in Lip_{1}(\mathbb{S}^{n-1})$ and $\Omega\not\equiv 0$. Let $\{t_{j}\}_{j=1}^{\infty}$ be a sequence with $\sup_{i\in\mathbb{Z}}|\{j:2^{i}\leq|t_{j}|<2^{i+1}\}|<\infty$ in the definition of $\widetilde{\mathcal{O}}$. Then

1. (1)

$b\in{{\rm BMO}(\mathbb{R}^{n})}\Longleftrightarrow\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$* is bounded on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$;* 2. (2)

$b\in{{\rm BMO}(\mathbb{R}^{n})}\Longleftrightarrow\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$* is bounded on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$.*

Claim B: Lemma 3.3 is also valid if we replace $T_{\Omega}^{b}$ by $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ or by $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$.

Proof of Claim B.

It follows from the definitions of $\widetilde{\mathcal{O}}(\mathcal{T}_{\Omega}^{b})$ and $\mathcal{V}_{\rho}(\mathcal{T}_{\Omega}^{b})$ that for any cube $Q$, $d\in\mathbb{N}$ large enough, $x\in 2^{d+1}Q\backslash 2^{d}Q$ and the function $f:=(\int_{F}\omega(x)dx)^{-1/p}\chi_{F}$ satisfies

[TABLE]

Then arguing as in the proof of [11, Proposition 4.4], we have that

[TABLE]

where the implicit constant is independent of $d$ and $Q$. Then the desired conclusion follows. ∎

Proof of Theorem 1.4.

Using Lemma 3.1, Claim A and B, the proof of Theorem 1.4 is just a repetition of the proof of [11, Theorem 1.4]. We omit the details here. ∎

Proof of Corollary 1.5.

By a similar argument as in the proof of (2) of Theorem 1.3, one can verify that $T_{\Omega,t_{1}}^{b}$ is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. Then, the sufficiency follows from [19, Corollary 1.4], Theorem 1.3 and the compactness of $T_{\Omega,t_{1}}^{b}$. The necessity follows from Theorem 1.4. ∎

Appendix A

In this section, we give an example of oscillation of $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ such that $b\notin{{\rm CMO}(\mathbb{R}^{n})}$ and $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ is compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$.

To begin with, take $b$ to be a smooth function on ${{\mathbb{R}}^{n}}$ such that

[TABLE]

One can check that $b\notin{{\rm BMO}(\mathbb{R}^{n})}$ by

[TABLE]

However, assume $\Omega\in Lip({{\mathbb{R}}^{n}})$. We find that $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ is a compact operator on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$.

In fact, let $\varphi$ be a smooth bump function with $0\leq\varphi\leq 1$, supported in the ball $\{\xi:|\xi|<2\}$ and be equal to 1 on the ball $\{\xi:|\xi|\leq 1\}$. For any positive number $N\in\mathbb{N}$ we define $\varphi_{N}(x):=\varphi(x/N)$. Define

[TABLE]

We claim that $\{\mathcal{O}_{N}(\mathcal{T}_{\Omega}^{b})\}_{N\in\mathbb{N}}$ is a sequence of compact operators on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$. In fact, for any fixed $N$, we have

[TABLE]

for every $x,y$ such that $\varphi_{N}(x)\neq 0$ and $|x-y|\leq t_{1}$. From this and the definition of $\mathcal{O}_{N}(\mathcal{T}_{\Omega}^{b})$, we have

[TABLE]

Observe that $b_{2N+t_{1}}\in C_{c}^{\infty}({{\mathbb{R}}^{n}})\subset{{\rm CMO}(\mathbb{R}^{n})}$. Then the compactness of $\mathcal{O}(\mathcal{T}_{\Omega}^{b_{2N+t_{1}}})$ follows from Theorem 1.3. Since $\varphi_{N}$ is a bounded operator on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$ as a pointwise multiplier, the operator $\mathcal{O}_{N}(\mathcal{T}_{\Omega}^{b})=\varphi_{N}(x)\mathcal{O}(\mathcal{T}_{\Omega}^{b_{2N+t_{1}}})$, as the product of a bounded operator and a compact operator, is also compact on $L^{p}_{\omega}({{\mathbb{R}}^{n}})$.

Finally, we claim that $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ is the limit of $\mathcal{O}_{N}(\mathcal{T}_{\Omega}^{b})$ in the sense of operator norm, as $N\rightarrow\infty$. Then the compactness of $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$ follows.

Write

[TABLE]

Denote by $t_{1}$ the first term of the sequence $\{t_{j}\}_{j=1}^{\infty}$ in the definition of $\mathcal{O}(\mathcal{T}_{\Omega}^{b})$. By the definition of $b$ and the mean value theorem, we have

[TABLE]

for all $|x-y|\leq t_{1}$.

Hence,

[TABLE]

It follows that

[TABLE]

as $N\rightarrow\infty$. We have now completed this proof.