Fault diagnosis of printing press bearing based on deformable convolution residual neural network

Qiumin Wu; Ziqi Zhu; Jiahui Tang; Yukang Xia; Qiumin Wu; Ziqi Zhu; Jiahui Tang; Yukang Xia

doi:10.3934/nhm.2023027

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 622-646. doi: 10.3934/nhm.2023027

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Research article Special Issues

Fault diagnosis of printing press bearing based on deformable convolution residual neural network

1.
Faculty of Printing, Packaging Engineering and Digital Media Technology, Xi'an University of Technology, Xi' an 710048, China
2.
School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, China

Received: 01 December 2022 Revised: 02 February 2023 Accepted: 06 February 2023 Published: 13 February 2023

Rolling bear is a major critical component of rotating machinery, as its working condition affects the performance of the equipment. As a result, the condition monitoring and fault diagnosis of bearings get more and more attention. However, the strong background noise makes it difficult to extract the bearing fault features exactly. Furthermore, regular gradient disappearance and overfit appear in traditional network model training. Therefore, taking the printing press bearings as the research object, an intelligent fault diagnosis method based on strong background noise is proposed. This method integrates frequency slice wavelet transform (FSWT), deformable convolution and residual neural network together, and realizes the high-precision fault diagnosis of the printing press bearings. First, FSWT is used to preprocess the original vibration signal to obtain bearing fault features in the time and frequency domain, reconstruct the signal in any frequency band and describe local features accurately. Second, the ResNet is selected as the base network, and the two-dimensional time-frequency diagrams (TFD) obtained by preprocessing are used as input. For the model that has a poor ability to extract subtle features under strong background noise, the deformable convolution layer is introduced to reconstruct the convolution layer of ResNet, called deformable convolution residual neural network (DC-ResNet). Finally, the effectiveness of this method is verified by using the data sets collected under experimental conditions and actual working conditions for fault diagnosis of the printing press. The results show that the DC-ResNet can classify different bearing faults under strong background noise, and the accuracy and stability are greatly improved, which the accuracy meets 93.90%. The intelligent fault diagnosis with high-precision of printing press bearings under complex working conditions is realized by the proposed method.

Keywords:

Citation: Qiumin Wu, Ziqi Zhu, Jiahui Tang, Yukang Xia. Fault diagnosis of printing press bearing based on deformable convolution residual neural network[J]. Networks and Heterogeneous Media, 2023, 18(2): 622-646. doi: 10.3934/nhm.2023027

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Abstract

1. Introduction and notations

Let $T$ be the Calderón-Zygmund singular integral operator and $b$ be a locally integrable function on $R^n$ . The commutator generated by $b$ and $T$ is defined by $[b, T]f = bT(f)-T(bf)$ . The investigation of the commutator begins with Coifman-Rochberg-Weiss pioneering study and classical result (see ^[6]). The classical result of Coifman, Rochberg and Weiss (see ^[6]) states that the commutator $[b, T]f = T(bf)-bTf$ is bounded on $L^p(R^n)$ for $1 < p < \infty$ if and only if $b\in BMO(R^n)$ . The major reason for considering the problem of commutators is that the boundedness of commutator can produces some characterizations of function spaces (see ^[1,6]). Chanillo (see ^[1]) proves a similar result when $T$ is replaced by the fractional integral operator. In ^[11], the boundedness properties of the commutators for the extreme values of $p$ are obtained. In recent years, the theory of Herz space and Herz type Hardy space, as a local version of Lebesgue space and Hardy space, have been developed (see ^[8,9,12,13]). The main purpose of this paper is to establish the endpoint continuity properties of some multilinear operators related to certain non-convolution type fractional singular integral operators on Herz and Herz type Hardy spaces.

First, let us introduce some notations (see ^{[8,9,10,12,13,15]}). Throughout this paper, $Q$ will denote a cube of $R^n$ with sides parallel to the axes. For a cube $Q$ and a locally integrable function $f$ , let $f_Q = |Q|^{-1}\int_Qf(x)dx$ and $f^{\#}(x) = \sup\limits_{Q\ni x}|Q|^{-1}\int_Q|f(y)-f_Q|dy$ . Moreover, $f$ is said to belong to $BMO(R^n)$ if $f^{\#} \in L^\infty$ and define $||f||_{BMO} = ||f^{\#}||_{L^\infty}$ ; We also define the central $BMO$ space by $CMO(R^n)$ , which is the space of those functions $f\in L_{loc}(R^n)$ such that

$||f||_{CMO} = \sup\limits_{r > 1}|Q(0, r)|^{-1}\int_Q|f(y)-f_Q|dy < \infty.$

It is well-known that (see ^[9,10])

$||f||_{CMO}\approx\sup\limits_{r > 1}\inf\limits_{c\in C}|Q(0, r)|^{-1}\int_Q|f(x)-c|dx.$

For $k\in Z$ , define $B_k = \{x\in R^n: |x|\le 2^k\}$ and $C_k = B_k\setminus B_{k-1}$ . Denote by $\chi_k$ the characteristic function of $C_k$ and $\tilde\chi_k$ the characteristic function of $C_k$ for $k\ge1$ and $\tilde\chi_0$ the characteristic function of $B_0$ .

Definition 1. Let $0 < p < \infty$ and $\alpha\in R$ .

(1) The homogeneous Herz space $\dot K_p^\alpha(R^n)$ is defined by

$\dot K_p^\alpha(R^n) = \{f\in L_{loc}^p (R^n \setminus \{0\}): ||f||_{\dot K_p^\alpha} < \infty\},$

where

$||f||_{\dot K_p^\alpha} = \sum\limits_{k = -\infty}^\infty 2^{k\alpha}||f\chi_k||_{L^p};$

(2) The nonhomogeneous Herz space $K_p^\alpha(R^n)$ is defined by

$K_p^\alpha(R^n) = \{f\in L_{loc}^p (R^n): ||f||_{K_p^\alpha} < \infty\},$

where

$||f||_{K_p^\alpha} = \sum\limits_{k = 0}^\infty 2^{k\alpha}||f\tilde\chi_k||_{L^p}.$

If $\alpha = n(1-1/p)$ , we denote that $\dot K_p^\alpha(R^n) = \dot K_p(R^n)$ , $K_p^\alpha(R^n) = K_p(R^n)$ .

Definition 2. Let $0 < \delta < n$ and $1 < p < n/\delta$ . We shall call $B_p^{\delta}(R^n)$ the space of those functions $f$ on $R^n$ such that

$||f||_{B_p^{\delta}} = \sup\limits_{d > 1}d^{-n(1/p-\delta/n)}||f\chi_{Q(0, d)}||_{L^p} < \infty.$

Definition 3. Let $1 < p < \infty$ .

(1) The homogeneous Herz type Hardy space $H \dot K_p(R^n)$ is defined by

$H \dot K_p(R^n) = \{f\in S'(R^n): G(f)\in \dot K_p(R^n)\},$

where

$||f||_{H\dot K_p} = ||G(f)||_{\dot K_p}.$

(2) The nonhomogeneous Herz type Hardy space $HK_p(R^n)$ is defined by

$HK_p(R^n) = \{f\in S'(R^n): G(f)\in K_p(R^n)\},$

where

$||f||_{HK_p} = ||G(f)||_{K_p}.$

where $G(f)$ is the grand maximal function of $f$ .

The Herz type Hardy spaces have the atomic decomposition characterization.

Definition 4. Let $1 < p < \infty$ . A function $a(x)$ on $R^n$ is called a central $(n(1-1/p), p)$ -atom (or a central $(n(1-1/p), p)$ -atom of restrict type), if

1) Supp $a\subset B(0, d)$ for some $d > 0$ (or for some $d\ge 1$ ),

2) $||a||_{L^p}\le |B(0, d)|^{1/p-1}$ ,

3) $\int a(x)dx = 0$ .

Lemma 1. (see ^[9,13]) Let $1 < p < \infty$ . A temperate distribution $f$ belongs to $H\dot K_p(R^n)$ (or $HK_p(R^n)$ ) if and only if there exist central $(n(1-1/p), p)$ -atoms(or central $(n(1-1/p), p)$ -atoms of restrict type) $a_j$ supported on $B_j = B(0, 2^j)$ and constants $\lambda_j$ , $\sum_j|\lambda_j| < \infty$ such that $f = \sum_{j = -\infty}^\infty \lambda_ja_j$ (or $f = \sum_{j = 0}^\infty \lambda_j a_j$ )in the $S'(R^n)$ sense, and

$||f||_{H\dot K_p}(\ or \ ||f||_{HK_p})\approx\sum\limits_j|\lambda_j|.$

2. Theorems

In this paper, we will consider a class of multilinear operators related to some non-convolution type singular integral operators, whose definition are following.

Let $m$ be a positive integer and $A$ be a function on $R^n$ . We denote that

$R_{m+1}(A;x, y) = A(x)-\sum\limits_{|\beta|\le m}\frac{1}{\beta!}D^\beta A(y)(x-y)^\beta$

and

$Q_{m+1}(A;x, y) = R_m(A;x, y)-\sum\limits_{|\beta| = m}\frac{1}{\beta!}D^\beta A(x)(x-y)^\beta.$

Definition 5. Fixed $\varepsilon > 0$ and $0 < \delta < n$ . Let $T_\delta : S\to S'$ be a linear operator. $T_\delta$ is called a fractional singular integral operator if there exists a locally integrable function $K(x, y)$ on $R^n \times R^n$ such that

$T_\delta(f)(x) = \int_{R^n} K(x, y)f(y)dy$

for every bounded and compactly supported function $f$ , where $K$ satisfies:

$|K(x, y)|\le C|x-y|^{-n+\delta}$

and

$|K(y, x)-K(z, x)|+|K(x, y)-K(x, z)|\le C|y-z|^\varepsilon|x-z|^{-n-\varepsilon+\delta}$

if $2|y-z|\le |x-z|$ . The multilinear operator related to the fractional singular integral operator $T_\delta$ is defined by

$T_\delta^A(f)(x) = \int_{R^n}\frac{R_{m+1}(A;x, y)}{|x-y|^m}K(x, y)f(y)dy;$

We also consider the variant of $T_\delta^A$ , which is defined by

$\tilde T_\delta^A(f)(x) = \int_{R^n}\frac{Q_{m+1}(A;x, y)}{|x-y|^m}K(x, y)f(y)dy.$

Note that when $m = 0$ , $T_\delta^A$ is just the commutators of $T_\delta$ and $A$ (see ^[1,6,11,14]). It is well known that multilinear operator, as a non-trivial extension of commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see ^[3,4,5]). In ^[7], the weighted $L^p$ ( $p > 1$ )-boundedness of the multilinear operator related to some singular integral operator are obtained. In ^[2], the weak ( $H^1$ , $L^1$ )-boundedness of the multilinear operator related to some singular integral operator are obtained. In this paper, we will study the endpoint continuity properties of the multilinear operators $T_\delta^A$ and $\tilde T_\delta^A$ on Herz and Herz type Hardy spaces.

Now we state our results as following.

Theorem 1. Let $0 < \delta < n$ , $1 < p < n/\delta$ and $D^\beta A\in BMO(R^n)$ for all $\beta$ with $|\beta| = m$ . Suppose that $T_\delta^A$ is the same as in Definition 5 such that $T_\delta$ is bounded from $L^p(R^n)$ to $L^q(R^n)$ for any $p, q\in (1, +\infty]$ with $1/q = 1/p-\delta/n$ . Then $T_\delta^A$ is bounded from $B_p^{\delta}(R^n)$ to $CMO(R^n)$ .

Theorem 2. Let $0 < \delta < n$ , $1 < p < n/\delta$ , $1/q = 1/p-\delta/n$ and $D^\beta A\in BMO(R^n)$ for all $\beta$ with $|\beta| = m$ . Suppose that $\tilde T_\delta^A$ is the same as in Definition 5 such that $\tilde T_\delta^A$ is bounded from $L^p(R^n)$ to $L^q(R^n)$ for any $p, q\in (1, +\infty)$ with $1/q = 1/p-\delta/n$ . Then $\tilde T_\delta^A$ is bounded from $H\dot K_p(R^n)$ to $\dot K_q^\alpha(R^n)$ with $\alpha = n(1-1/p)$ .

Theorem 3. Let $0 < \delta < n$ , $1 < p < n/\delta$ and $D^\beta A\in BMO(R^n)$ for all $\beta$ with $|\beta| = m$ . Suppose that $\tilde T_\delta^A$ is the same as in Definition 5 such that $\tilde T_\delta^A$ is bounded from $L^p(R^n)$ to $L^q(R^n)$ for any $p, q\in (1, +\infty)$ with $1/q = 1/p-\delta/n$ . Then the following two statements are equivalent:

(ⅰ) $\tilde T_\delta^A$ is bounded from $B_p^{\delta}(R^n)$ to $CMO(R^n)$ ;

(ⅱ) for any cube $Q$ and $z\in 3Q\setminus 2Q$ , there is

$\frac{1}{|Q|}\int_Q\left|\sum\limits_{|\beta| = m}\frac{1}{\beta!}|D^\beta A(x)-(D^\beta A)_Q| \int_{(4Q)^c}K_{\beta}(z, y)f(y)dy\right|dx\le C||f||_{B_p^{\delta}},$

where $K_{\beta}(z, y) = \frac{(z-y)^\beta}{|z-y|^m}K(z, y)$ for $|\beta| = m$ .

Remark. Theorem 2 is also hold for nonhomogeneous Herz and Herz type Hardy space.

3. Proofs of theorems

To prove the theorem, we need the following lemma.

Lemma 2. (see ^[5]) Let $A$ be a function on $R^n$ and $D^\beta A\in L^q(R^n)$ for $|\beta| = m$ and some $q > n$ . Then

$|R_m(A;x, y)|\le C|x-y|^m\sum\limits_{|\beta| = m}\left(\frac{1}{|\tilde Q(x, y)|}\int_{\tilde Q(x, y)}|D^\beta A(z)|^q dz\right)^{1/q},$

where $\tilde Q(x, y)$ is the cube centered at $x$ and having side length $5\sqrt{n}|x-y|$ .

Proof of Theorem 1. It suffices to prove that there exists a constant $C_Q$ such that

$\frac{1}{|Q|}\int_Q|T_\delta^A(f)(x)-C_Q|dx \le C||f||_{B_p^{\delta}}$

holds for any cube $Q = Q(0, d)$ with $d > 1$ . Fix a cube $Q = Q(0, d)$ with $d > 1$ . Let $\tilde Q = 5\sqrt{n}Q$ and $\tilde A(x) = A(x)-\sum\limits_{|\beta| = m}\frac{1}{\beta!}(D^\beta A)_{\tilde Q}x^\beta$ , then $R_{m+1}(A; x, y) = R_{m+1}(\tilde A; x, y)$ and $D^\beta\tilde A = D^\beta A-(D^\beta A)_{\tilde Q}$ for all $\beta$ with $|\beta| = m$ . We write, for $f_1 = f\chi_{\tilde Q}$ and $f_2 = f\chi_{R^n\setminus\tilde Q}$ ,

$\begin{eqnarray*} T_\delta^A(f)(x)& = &\int_{R^n}\frac{R_{m+1}(\tilde A; x, y)}{|x-y|^m}K(x, y)f(y)dy = \int_{R^n}\frac{R_m(\tilde A; x, y)}{|x-y|^m}K(x, y)f_1(y)dy \\ &\;& -\sum\limits_{|\beta| = m}\frac{1}{\beta!}\int_{R^n}\frac{K(x, y)(x-y)^\beta}{|x-y|^m}D^\beta\tilde A(y)f_1(y)dy +\int_{R^n}\frac{R_{m+1}(\tilde A; x, y)}{|x-y|^m}K(x, y)f_2(y)dy, \end{eqnarray*}$

then

$\begin{eqnarray*} &\;& \frac{1}{|Q|}\int_Q\left|T_\delta^A(f)(x)-T_\delta^{\tilde A}(f_2)(0)\right|dx \le\frac{1}{|Q|}\int_Q\left|T_\delta\left(\frac{R_m(\tilde A;x, \cdot)}{|x-\cdot|^m}f_1\right)(x)\right|dx \\ &\;& +\sum\limits_{|\beta| = m}\frac{1}{\beta!}\frac{1}{|Q|}\int_Q\left|T_\delta\left(\frac{(x-\cdot)^\beta} {|x-\cdot|^m}D^\beta\tilde A f_1\right)(x)\right|dx+\left|T_\delta^{\tilde A}(f_2)(x)-T_\delta^{\tilde A}(f_2)(0)\right|dx \\ &: = & I+II+III. \end{eqnarray*}$

For $I$ , note that for $x\in Q$ and $y\in \tilde Q$ , using Lemma 2, we get

$R_m(\tilde A; x, y)\le C|x-y|^m\sum\limits_{|\beta| = m}||D^\beta A||_{BMO},$

thus, by the $L^p(R^n)$ to $L^q(R^n)$ -boundedness of $T_\delta^A$ for $1 < p, q < \infty$ with $1/q = 1/p-\delta/n$ , we get

$\begin{eqnarray*} I &\le& \frac{C}{|Q|}\int_Q \left|T_\delta\left(\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}f_1\right)(x)\right|dx \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\left(\frac{1}{|Q|}\int_Q|T_\delta(f_1)(x)|^qdx\right)^{1/q} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}|Q|^{-1/q}||f_1||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}r^{-n(1/p-\delta/n)}||f\chi_{\tilde Q}||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}}. \end{eqnarray*}$

For $II$ , taking $1 < s < p$ such that $1/r = 1/s-\delta/n$ , by the $(L^s, L^r)$ -boundedness of $T_\delta$ and Holder's inequality, we gain

$\begin{eqnarray*} II &\le& \frac{C}{|Q|}\int_Q \left|T_\delta\left(\sum\limits_{|\beta| = m}(D^\beta A-(D^\beta A)_{\tilde Q})f_1\right)(x)\right|dx \\ &\le& C\sum\limits_{|\beta| = m}\left(\frac{1}{|Q|}\int_Q\left|T_\delta\left((D^\beta A-(D^\beta A)_{\tilde Q}\right)f_1)(x)\right|^rdx\right)^{1/r} \\ &\le& C|Q|^{-1/r}\sum\limits_{|\beta| = m}||(D^\beta A-(D^\beta A)_{\tilde Q})f_1||_{L^s} \\ &\le& C|Q|^{-1/r}||f_1||_{L^p}\sum\limits_{|\beta| = m}\left(\frac{1}{|Q|}\int_{\tilde Q} |D^\beta A(y)-(D^\beta A)_{\tilde Q}|^{ps/(p-s)}dy\right)^{(p-s)/(ps)}|Q|^{(p-s)/(ps)} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}r^{-n/q}||f\chi_{\tilde Q}||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}}. \end{eqnarray*}$

To estimate $III$ , we write

$\begin{eqnarray*} T_\delta^{\tilde A}(f_2)(x)-T_\delta^{\tilde A}(f_2)(0)& = &\int_{R^n}\left[\frac{K(x, y)}{|x-y|^m}-\frac{K(0, y)}{|y|^m}\right]R_m(\tilde A;x, y)f_2(y)dy \\ &\;& +\int_{R^n}\frac{K(0, y)f_2(y)}{|y|^m}[R_m(\tilde A; x, y)-R_m(\tilde A; 0, y)]dy \\ &\;& -\sum\limits_{|\beta| = m}\frac{1}{\beta!}\int_{R^n}\left(\frac{K(x, y)(x-y)^\beta}{|x-y|^m}-\frac{K(0, y)(-y)^\beta}{|y|^m}\right)D^\beta\tilde A(y)f_2(y)dy \\ &: = & III_1+III_2+III_3. \end{eqnarray*}$

By Lemma 2 and the following inequality (see ^[15])

$|b_{Q_1}-b_{Q_2}|\le C\log(|Q_2|/|Q_1|)||b||_{BMO} \ for \ Q_1 \subset Q_2,$

we know that, for $x\in Q$ and $y\in 2^{k+1}\tilde Q\setminus 2^k\tilde Q$ ,

$\begin{eqnarray*} |R_m(\tilde A;x, y)|&\le& C|x-y|^m\sum\limits_{|\beta| = m}(||D^\beta A||_{BMO}+|(D^\beta A)_{\tilde Q(x, y)}-(D^\beta A)_{\tilde Q}|) \\ &\le& C k|x-y|^m\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}. \end{eqnarray*}$

Note that $|x-y|\sim |y|$ for $x\in Q$ and $y\in R^n\setminus\tilde Q$ , we obtain, by the condition of $K$ ,

$\begin{eqnarray*} |III_1|&\le& C\int_{R^n}\left(\frac{|x|}{|y|^{m+n+1-\delta}}+\frac{|x|^\varepsilon} {|y|^{m+n+\varepsilon-\delta}}\right)|R_m(\tilde A; x, y)||f_2(y)|dy \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = 0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q}k\left(\frac{|x|} {|y|^{n+1-\delta}}+\frac{|x|^\varepsilon}{|y|^{n+\varepsilon-\delta}}\right)|f(y)|dy \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = 1}^\infty k(2^{-k}+2^{-\varepsilon k})(2^k r)^{-n(1/p-\delta/n)}||f\chi_{2^k\tilde Q}||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = 1}^\infty k(2^{-k}+2^{-\varepsilon k})||f||_{B_p^{\delta}} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}}. \end{eqnarray*}$

For $III_2$ , by the formula (see ^[5]):

$R_m(\tilde A; x, y)-R_m(\tilde A; x_0, y) = \sum\limits_{|\gamma| < m}\frac{1}{\gamma!}R_{m-|\gamma|}(D^\gamma\tilde A; x, x_0)(x-y)^\gamma$

and Lemma 2, we have

$|R_m(\tilde A; x, y)-R_m(\tilde A; x_0, y)|\le C\sum\limits_{|\gamma| < m}\sum\limits_{|\beta| = m}|x-x_0|^{m-|\gamma|}|x-y|^{|\gamma|}||D^\beta A||_{BMO},$

thus, similar to the estimates of $III_1$ , we get

$|III_2| \le C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = 0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q} \frac{|x|}{|y|^{n+1-\delta}}|f(y)|dy \le C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}}.$

For $III_3$ , by Holder's inequality, similar to the estimates of $III_1$ , we get

$\begin{eqnarray*} |III_3|&\le&C\sum\limits_{|\beta| = m}\sum\limits_{k = 0}^\infty\int_{2^{k+1}\tilde Q\setminus2^k\tilde Q}\left(\frac{|x|} {|y|^{n+1-\delta}}+\frac{|x|^\varepsilon}{|y|^{n+\varepsilon-\delta}}\right)|D^\beta \tilde A(y)||f(y)|dy \\ &\le& C\sum\limits_{|\beta| = m}\sum\limits_{k = 1}^\infty (2^{-k}+2^{-\varepsilon k})(2^k r)^{-n(1/p-\delta/n)} \left(|2^k\tilde Q|^{-1}\int_{2^k\tilde Q}|D^\beta A(y)-(D^\beta A)_{\tilde Q}|^{p'}dy\right)^{1/p'}||f\chi_{2^k\tilde Q}||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = 1}^\infty (2^{-k}+2^{-\varepsilon k})(2^k r)^{-n(1/p-\delta/n)} ||f\chi_{2^k\tilde Q}||_{L^p} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}}. \end{eqnarray*}$

Thus

$III\le C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{B_p^{\delta}},$

which together with the estimates for $I$ and $II$ yields the desired result. This finishes the proof of Theorem 1.

Proof of Theorem 2. Let $f\in H\dot K_p(R^n)$ , by Lemma 1, $f = \sum_{j = -\infty}^\infty\lambda_ja_j$ , where $a_j's$ are the central $(n(1-1/p), p)$ -atom with supp $a_j \subset B_j = B(0, 2^j)$ and $||f||_{H\dot K_p}\approx\sum_j|\lambda_j|$ . We write

$\begin{eqnarray*} ||\tilde T_\delta^A(f)||_{\dot K_q^\alpha}& = &\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}||\chi_k\tilde T_\delta^A(f)||_{L^q} \\ &\le&\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = -\infty}^{k-1}|\lambda_j|||\chi_k\tilde T_\delta^A(a_j)||_{L^q} +\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = k}^\infty|\lambda_j|||\chi_k\tilde T_\delta^A(a_j)||_{L^q} \\ & = & J+JJ. \end{eqnarray*}$

For $JJ$ , by the $(L^p, L^q)$ -boundedness of $\tilde T_\delta^A$ for $1/q = 1/p-\delta/n$ , we get

$\begin{eqnarray*} JJ &\le& C\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = k}^\infty|\lambda_j|||a_j||_{L^p}\le C\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = k}^\infty|\lambda_j|2^{jn(1/p-1)} \\ &\le& C\sum\limits_{j = -\infty}^\infty|\lambda_j|\sum\limits_{k = -\infty}^j2^{(k-j)n(1-1/p)}\le C\sum\limits_{j = -\infty}^\infty|\lambda_j|\le C||f||_{H\dot K_p}. \end{eqnarray*}$

To obtain the estimate of $J$ , we denote that $\tilde A(x) = A(x)-\sum_{|\beta| = m}\frac{1}{\beta!}(D^\beta A)_{2B_j}x^\beta$ . Then $Q_m(A; x, y) = Q_m(\tilde A; x, y)$ and $Q_{m+1}(A; x, y) = R_m(A; x, y)-\sum_{|\beta| = m}\frac{1}{\beta!} (x-y)^\beta D^\beta A(x)$ . We write, by the vanishing moment of $a$ and for $x \in C_k$ with $k\ge j+1$ ,

$\begin{eqnarray*} \tilde T_\delta^A(a_j)(x)& = & \int_{R^n}\frac{K(x, y)R_m(A;x, y)}{|x-y|^m}a_j(y)dy-\sum\limits_{|\beta| = m}\frac{1}{\beta!} \int_{R^n}\frac{K(x, y)D^\beta\tilde A(x)(x-y)^\beta}{|x-y|^m}a_j(y)dy \\ & = & \int_{R^n}\left[\frac{K(x, y)}{|x-y|^m}-\frac{K(x, 0)}{|x|^m}\right]R_m(\tilde A;x, y)a_j(y)dy \\ &\;& +\int_{R^n}\frac{K(x, 0)}{|x|^m}[R_m(\tilde A;x, y)-R_m(\tilde A; x, 0)]a_j(y)dy \\ &\;& -\sum\limits_{|\beta| = m}\frac{1}{\beta!}\int_{R^n}\left[\frac{K(x, y)(x-y)^\beta}{|x-y|^m}-\frac{K(x, 0)x^\beta}{|x|^m}\right] D^\beta\tilde A(x)a_j(y)dy. \end{eqnarray*}$

Similar to the proof of Theorem 1, we obtain

$\begin{eqnarray*} |\tilde T_\delta^A(a_j)(x)|&\le& C\int_{R^n}\left[\frac{|y|}{|x|^{m+n+1-\delta}}+\frac{|y|^\varepsilon}{|x|^{m+n+\varepsilon-\delta}}\right] |R_m(\tilde A; x, y)||a_j(y)|dy \\ &\;& +C\sum\limits_{|\beta| = m}\int_{R^n}\left[\frac{|y|}{|x|^{n+1-\delta}}+\frac{|y|^\varepsilon}{|x|^{n+\varepsilon-\delta}}\right]|D^\beta\tilde A(x)||a_j(y)|dy \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\left[\frac{2^j}{2^{k(n+1-\delta)}}+\frac{2^{j\varepsilon}} {2^{k(n+\varepsilon-\delta)}}\right]+C\sum\limits_{|\beta| = m}\left[\frac{2^j}{2^{k(n+1-\delta)}}+\frac{2^{j\varepsilon}} {2^{k(n+\varepsilon-\delta)}}\right]|D^\beta\tilde A(x)|, \end{eqnarray*}$

thus

$\begin{eqnarray*} J&\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = -\infty}^{k-1}|\lambda_j| \left[\frac{2^j}{2^{k(n+1-\delta)}}+\frac{2^{j\varepsilon}}{2^{k(n+\varepsilon-\delta)}}\right]2^{kn/q} \\ &\;&+C\sum\limits_{|\beta| = m}\sum\limits_{k = -\infty}^\infty 2^{kn(1-1/p)}\sum\limits_{j = -\infty}^{k-1}|\lambda_j|\left[\frac{2^j}{2^{k(n+1-\delta)}} +\frac{2^{j\varepsilon}}{2^{k(n+\varepsilon-\delta)}}\right]\left(\int_{B_k}|D^\beta\tilde A(x)|^q dx \right)^{1/q} \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{k = -\infty}^\infty 2^{kn(1-\delta/n)}\sum\limits_{j = -\infty}^{k-1}|\lambda_j| \left[\frac{2^j}{2^{k(n+1-\delta)}}+\frac{2^{j\varepsilon}}{2^{k(n+\varepsilon-\delta)}}\right] \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{j = -\infty}^\infty|\lambda_j|\sum\limits_{k = j+1}^\infty[2^{j-k}+ 2^{(j-k)\varepsilon}] \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}\sum\limits_{j = -\infty}^\infty|\lambda_j| \\ &\le& C\sum\limits_{|\beta| = m}||D^\beta A||_{BMO}||f||_{H\dot K_p}. \end{eqnarray*}$

This completes the proof of Theorem 2.

Proof of Theorem 3. For any cube $Q = Q(0, r)$ with $r > 1$ , let $f \in B_p^{\delta}$ and $\tilde A(x) = A(x) -\sum\limits_{|\beta| = m}\frac{1}{\beta!}(D^\beta A)_{\tilde Q}x^\beta$ . We write, for $f = f\chi_{4Q}+f\chi_{(4Q)^c} = f_1+f_2$ and $z\in 3Q\setminus2Q$ ,

$\begin{eqnarray*} \tilde T_\delta^A(f)(x)& = &\tilde T_\delta^A(f_1)(x)+\int_{R^n}\frac{R_m(\tilde A; x, y)}{|x-y|^m}K(x, y)f_2(y)dy \\ &\;& -\sum\limits_{|\beta| = m}\frac{1}{\beta!}(D^\beta A(x)-(D^\beta A)_Q)(T_{\delta, \beta}(f_2)(x)-T_{\delta, \beta}(f_2)(z)) \\ &\;& -\sum\limits_{|\beta| = m}\frac{1}{\beta!}(D^\beta A(x)-(D^\beta A)_Q)T_{\delta, \beta}(f_2)(z) \\ & = & I_1(x)+I_2(x)+I_3(x, z)+I_4(x, z), \end{eqnarray*}$

where $T_{\delta, \beta}$ is the singular integral operator with the kernel $\frac{(x-y)^\beta}{|x-y|^m}K(x, y)$ for $|\beta| = m$ . Note that $(I_4(\cdot, z))_Q = 0$ , we have

$\tilde T_\delta^A(f)(x)-(\tilde T_\delta^A(f))_Q = I_1(x)-(I_1(\cdot))_Q+I_2(x)-I_2(z)-[I_2(\cdot)-I_2(z)]_Q-I_3(x, z)+(I_3(x, z))_Q-I_4(x, z).$

By the $(L^p, L^q)$ -bounded of $\tilde T_\delta^A$ , we get

$\frac{1}{|Q|}\int_Q |I_1(x)|dx \le \left(\frac{1}{|Q|}\int_Q|\tilde T_\delta^A(f_1)(x)|^qdx\right)^{1/q} \le C|Q|^{-1/q}||f_1||_{L^p}\le C||f||_{B_p^{\delta}}.$

Similar to the proof of Theorem 1, we obtain

$|I_2(x)-I_2(z)|\le C||f||_{B_p^{\delta}}$

and

$\frac{1}{|Q|}\int_Q |I_3(x, z)|dx \le C||f||_{B_p^{\delta}}.$

Then integrating in $x$ on $Q$ and using the above estimates, we obtain the equivalence of the estimate

$\frac{1}{|Q|}\int_Q |\tilde T_\delta^A(f)(x)-(\tilde T_\delta^A(f))_Q|dx \le C||f||_{B_p^{\delta}}$

and the estimate

$\frac{1}{|Q|}\int_Q |I_4(x, z)|dx \le C||f||_{B_p^{\delta}}.$

This completes the proof of Theorem 3.

4. Applications

In this section we shall apply the theorems of the paper to some particular operators such as the Calderón-Zygmund singular integral operator and fractional integral operator.

Application 1. Calderón-Zygmund singular integral operator.

Let $T$ be the Calderón-Zygmund operator defined by (see ^[10,11,15])

$T(f)(x) = \int_{R^n} K(x, y)f(y)dy,$

the multilinear operator related to $T$ is defined by

$T^A(f)(x) = \int_{R^n}\frac{R_{m+1}(A; x, y)}{|x-y|^m}K(x, y)f(y)dy.$

Then it is easily to see that $T$ satisfies the conditions in Theorems 1–3, thus the conclusions of Theorems 1–3 hold for $T^A$ .

Application 2. Fractional integral operator with rough kernel.

For $0 < \delta < n$ , let $T_\delta$ be the fractional integral operator with rough kernel defined by (see ^[2,7])

$T_\delta f(x) = \int_{R^n}\frac{\Omega(x-y)}{|x-y|^{n-\delta}}f(y)dy,$

the multilinear operator related to $T_\delta$ is defined by

$T_\delta^A f(x) = \int_{R^n}\frac{R_{m+1}(A;x, y)}{|x-y|^{m+n-\delta}}\Omega(x-y)f(y)dy,$

where $\Omega$ is homogeneous of degree zero on $R^n$ , $\int_{S^{n-1}}\Omega(x')d\sigma(x') = 0$ and $\Omega\in Lip_\varepsilon(S^{n-1})$ for some $0 < \varepsilon\le 1$ , that is there exists a constant $M > 0$ such that for any $x, y\in S^{n-1}$ , $|\Omega(x)-\Omega(y)|\le M|x-y|^\varepsilon$ . Then $T_\delta$ satisfies the conditions in Theorem 1. In fact, for $supp f \subset (2Q)^c$ and $x\in Q = Q(x_0, d)$ , by the condition of $\Omega$ , we have (see ^[16])

$\left|\frac{\Omega(x-y)}{|x-y|^{n-\delta}}-\frac{\Omega(x_0-y)}{|x_0-y|^{n-\delta}}\right| \le C\left(\frac{|x-x_0|^\varepsilon}{|x_0-y|^{n+\varepsilon-\delta}}+\frac{|x-x_0|}{|x_0-y|^{n+1-\delta}}\right),$

thus, the conclusions of Theorems 1–3 hold for $T_\delta^A$ .

Acknowledgements

The author would like to express his deep gratitude to the referee for his/her valuable comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant No. 11901126), the Scientific Research Funds of Hunan Provincial Education Department. (Grant No. 19B509).

Conflict of interest

The authors declare that they have no competing interests.

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