Anti-plane interfacial waves in a square lattice

Victor A. Eremeyev; Victor A. Eremeyev

doi:10.3934/nhm.2025004

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 52-64. doi: 10.3934/nhm.2025004

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Research article Special Issues

Anti-plane interfacial waves in a square lattice

Victor A. Eremeyev ^,

Department of Civil and Environmental Engineering and Architecture (DICAAR), University of Cagliari, Via Marengo 2, Cagliari 09-123, CA, Italy

Received: 31 October 2024 Revised: 26 December 2024 Accepted: 06 January 2025 Published: 10 January 2025

Using the lattice dynamics approach, we discussed the propagation of interfacial waves localized near the interface in an infinite square lattice. The interface has been modeled as a single-particle layer of material particles with masses and elastic bonds different from those in the bulk. In this lattice structure there were anti-plane interface waves, i.e., waves that decayed exponentially with distance from the interface. Such waves could be useful for determining material properties in the vicinity of the interface. We obtained equations of motion and analyzed the corresponding dispersion relations for steady-state solutions. Here, the dispersion equation related the circular frequency to the wave number. In addition, we provided a comparison of the dispersion relations with those derived within the Gurtin-Murdoch surface elasticity. To do this, we have used the scaling law that links the continuum and discrete models. Unlike the continuum model, in the discrete model the wave number was limited by the first Brillouin zone, whereas in the continuum model it took a range from zero to infinity. The detailed parametric analysis was given for the discrete model. Finally, other models of interfaces in the case of a square lattice were discussed.

Keywords:

Citation: Victor A. Eremeyev. Anti-plane interfacial waves in a square lattice[J]. Networks and Heterogeneous Media, 2025, 20(1): 52-64. doi: 10.3934/nhm.2025004

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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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