# Boundedness of Toeplitz type operator associated to singular integral operator satisfying a variant of Hörmander’s condition on ${L}^{p}$ spaces with variable exponent

## Abstract

In this paper, the boundedness for some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition on ${L}^{p}$ spaces with variable exponent is obtained by using a sharp estimate of the operator.

MSC:42B20, 42B25.

## 1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied (see [3, 4]). In [5, 6], some singular integral operators satisfying a variant of Hörmander’s condition and the boundedness for the operators and their commutators are obtained (see [6]). In [79], some Toeplitz type operators related to singular integral operators and strongly singular integral operators are introduced and the boundedness for the operators generated by $BMO$ and Lipschitz functions are obtained. In the last years, the theory of ${L}^{p}$ spaces with variable exponent has been developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [1013] and their references). Karlovich and Lerner study the boundedness of the commutators of singular integral operators on ${L}^{p}$ spaces with variable exponent (see [14]). Motivated by these papers, the main purpose of this paper is to introduce some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition and prove the boundedness for the operator on ${L}^{p}$ spaces with variable exponent by using a sharp estimate of the operator.

## 2 Preliminaries and results

First, let us introduce some notations. Throughout this paper, Q will denote a cube of ${R}^{n}$ with sides parallel to the axes. For any locally integrable function f and $\delta >0$, the sharp function of f is defined by

${f}_{\delta }^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}{\left(\frac{1}{|Q|}{\int }_{Q}{|f\left(y\right)-{f}_{Q}|}^{\delta }\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/\delta },$

where, and in what follows, ${f}_{Q}={|Q|}^{-1}{\int }_{Q}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$. It is well known that (see [1, 2])

${f}_{\delta }^{\mathrm{#}}\left(x\right)\approx \underset{Q\ni x}{sup}\underset{c\in C}{inf}{\left(\frac{1}{|Q|}{\int }_{Q}{|f\left(y\right)-c|}^{\delta }\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/\delta }.$

We write that ${f}_{\delta }^{\mathrm{#}}={f}^{\mathrm{#}}$ if $\delta =1$. We say that f belongs to $BMO\left({R}^{n}\right)$ if ${f}^{\mathrm{#}}$ belongs to ${L}^{\mathrm{\infty }}\left({R}^{n}\right)$ and define ${\parallel f\parallel }_{BMO}={\parallel {f}^{\mathrm{#}}\parallel }_{{L}^{\mathrm{\infty }}}$. Let M be the Hardy-Littlewood maximal operator defined by

$M\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}{|Q|}^{-1}{\int }_{Q}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy.$

For $k\in N$, we denote by ${M}^{k}$ the operator M iterated k times, i.e., ${M}^{1}\left(f\right)\left(x\right)=M\left(f\right)\left(x\right)$ and

Let Ψ be a Young function and $\stackrel{˜}{\mathrm{\Psi }}$ be the complementary associated to Ψ. We denote the Ψ-average by, for a function f,

${\parallel f\parallel }_{\mathrm{\Psi },Q}=inf\left\{\lambda >0:\frac{1}{|Q|}{\int }_{Q}\mathrm{\Psi }\left(\frac{|f\left(y\right)|}{\lambda }\right)\phantom{\rule{0.2em}{0ex}}dy\le 1\right\}$

and the maximal function associated to Ψ by

${M}_{\mathrm{\Psi }}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}{\parallel f\parallel }_{\mathrm{\Psi },Q}.$

The Young functions to be used in this paper are $\mathrm{\Psi }\left(t\right)=t{\left(1+logt\right)}^{r}$ and $\stackrel{˜}{\mathrm{\Psi }}\left(t\right)=exp\left({t}^{1/r}\right)$, the corresponding average and maximal functions are denoted by ${\parallel \cdot \parallel }_{L{\left(logL\right)}^{r},Q}$, ${M}_{L{\left(logL\right)}^{r}}$ and ${\parallel \cdot \parallel }_{exp{L}^{1/r},Q}$, ${M}_{exp{L}^{1/r}}$. Following [4], we know the generalized Hölder inequality,

$\frac{1}{|Q|}{\int }_{Q}|f\left(y\right)g\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\le {\parallel f\parallel }_{\mathrm{\Psi },Q}{\parallel g\parallel }_{\stackrel{˜}{\mathrm{\Psi }},Q},$

and the following inequality: for $r,{r}_{j}\ge 1$, $j=1,\dots ,l$ with $1/r=1/{r}_{1}+\cdots +1/{r}_{l}$ and any $x\in {R}^{n}$, $b\in BMO\left({R}^{n}\right)$,

$\begin{array}{c}{\parallel f\parallel }_{L{\left(logL\right)}^{1/r},Q}\le {M}_{L{\left(logL\right)}^{1/r}}\left(f\right)\le C{M}_{L{\left(logL\right)}^{l}}\left(f\right)\le C{M}^{l+1}\left(f\right),\hfill \\ {\parallel f-{f}_{Q}\parallel }_{exp{L}^{r},Q}\le C{\parallel f\parallel }_{BMO},\hfill \\ |{f}_{{2}^{k+1}Q}-{f}_{2Q}|\le Ck{\parallel f\parallel }_{BMO}.\hfill \end{array}$

The non-increasing rearrangement of a measurable function f on ${R}^{n}$ is defined by

${f}^{\ast }\left(t\right)=inf\left\{\lambda >0:|\left\{x\in {R}^{n}:|f\left(x\right)|>\lambda \right\}|\le t\right\}\phantom{\rule{1em}{0ex}}\left(0

For $\lambda \in \left(0,1\right)$ and a measurable function f on ${R}^{n}$, the local sharp maximal function of f is defined by

${M}_{\lambda }^{\mathrm{#}}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}\underset{c\in C}{inf}{\left(\left(f-c\right){\chi }_{Q}\right)}^{\ast }\left(\lambda |Q|\right).$

Let $p:{R}^{n}\to \left[1,\mathrm{\infty }\right)$ be a measurable function. Denote by ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$ the sets of all Lebesgue measurable functions f on ${R}^{n}$ such that $m\left(\lambda f,p\right)<\mathrm{\infty }$ for some $\lambda =\lambda \left(f\right)>0$, where

$m\left(f,p\right)={\int }_{{R}^{n}}{|f\left(x\right)|}^{p\left(x\right)}\phantom{\rule{0.2em}{0ex}}dx.$

The sets become Banach spaces with respect to the following norm:

${\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}=inf\left\{\lambda >0:m\left(f/\lambda ,p\right)\le 1\right\}.$

Denote by $M\left({R}^{n}\right)$ the sets of all measurable functions $p:{R}^{n}\to \left[1,\mathrm{\infty }\right)$ such that the Hardy-Littlewood maximal operator M is bounded on ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$ and the following holds:

$1<{p}_{-}=ess\underset{x\in {R}^{n}}{inf}p\left(x\right),\phantom{\rule{2em}{0ex}}ess\underset{x\in {R}^{n}}{sup}p\left(x\right)={p}_{+}<\mathrm{\infty }.$
(1)

In recent years, the boundedness of classical operators on spaces ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$ has attracted a great deal of attention (see [1014] and their references).

Definition 1 Let $\mathrm{\Phi }=\left\{{\varphi }_{1},\dots ,{\varphi }_{l}\right\}$ be a finite family of bounded functions in ${R}^{n}$. For any locally integrable function f, the Φ sharp maximal function of f is defined by

${M}_{\mathrm{\Phi }}^{\mathrm{#}}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}\underset{\left\{{c}_{1},\dots ,{c}_{l}\right\}}{inf}\frac{1}{|Q|}{\int }_{Q}|f\left(y\right)-\sum _{i=1}^{l}{c}_{i}{\varphi }_{i}\left({x}_{Q}-y\right)|\phantom{\rule{0.2em}{0ex}}dy,$

where the infimum is taken over all m-tuples $\left\{{c}_{1},\dots ,{c}_{l}\right\}$ of complex numbers and ${x}_{Q}$ is the center of Q. For $\eta >0$, let

${M}_{\mathrm{\Phi },\eta }^{\mathrm{#}}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}\underset{\left\{{c}_{1},\dots ,{c}_{l}\right\}}{inf}{\left(\frac{1}{|Q|}{\int }_{Q}{|f\left(y\right)-\sum _{i=1}^{l}{c}_{j}{\varphi }_{i}\left({x}_{Q}-y\right)|}^{\eta }\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/\eta }.$

Remark We note that ${M}_{\mathrm{\Phi }}^{\mathrm{#}}\approx {M}_{\mathrm{\Phi }}^{\mathrm{#}}\left(f\right)$ if $l=1$ and ${\varphi }_{1}=1$.

Definition 2 Given a positive and locally integrable function f in ${R}^{n}$, we say that f satisfies the reverse Hölder condition (write this as $f\in R{H}_{\mathrm{\infty }}\left({R}^{n}\right)$) if, for any cube Q centered at the origin, we have

$0<\underset{x\in Q}{sup}f\left(x\right)\le C\frac{1}{|Q|}{\int }_{Q}f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$

In this paper, we study some singular integral operator as follows (see [5]).

Definition 3 Let $K\in {L}^{2}\left({R}^{n}\right)$ and satisfy

$\begin{array}{c}{\parallel K\parallel }_{{L}^{\mathrm{\infty }}}\le C,\hfill \\ |K\left(x\right)|\le C{|x|}^{-n}.\hfill \end{array}$

There exist functions ${B}_{1},\dots ,{B}_{l}\in {L}_{\mathrm{loc}}^{1}\left({R}^{n}-\left\{0\right\}\right)$ and $\mathrm{\Phi }=\left\{{\varphi }_{1},\dots ,{\varphi }_{l}\right\}\subset {L}^{\mathrm{\infty }}\left({R}^{n}\right)$ such that ${|det\left[{\varphi }_{j}\left({y}_{i}\right)\right]|}^{2}\in R{H}_{\mathrm{\infty }}\left({R}^{nl}\right)$, and for a fixed $\delta >0$ and any $|x|>2|y|>0$,

$|K\left(x-y\right)-\sum _{i=1}^{l}{B}_{i}\left(x\right){\varphi }_{i}\left(y\right)|\le C\frac{{|y|}^{\delta }}{{|x-y|}^{n+\delta }}.$

For $f\in {C}_{0}^{\mathrm{\infty }}$, we define a singular integral operator related to the kernel K by

$T\left(f\right)\left(x\right)={\int }_{{R}^{n}}K\left(x-y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.$

Let b be a locally integrable function on ${R}^{n}$ and T be a singular integral operator with variable Calderón-Zygmund kernels. The Toeplitz type operator associated to T is defined by

${T}_{b}=\sum _{j=1}^{m}{T}^{j,1}{M}_{b}{T}^{j,2},$

where ${T}^{j,1}$ are the singular integral operators T with variable Calderón-Zygmund kernels or ±I (the identity operator), ${T}^{j,2}$ are the linear operators for $j=1,\dots ,m$ and ${M}_{b}\left(f\right)=bf$.

Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 3 (see [2, 4]).

We shall prove the following theorems.

Theorem 1 Let T be a singular integral operator as in Definition  3, $0<\delta <1$ and $b\in BMO\left({R}^{n}\right)$. If ${T}_{1}\left(g\right)=0$ for any $g\in {L}^{u}\left({R}^{n}\right)$ ($1), then there exists a constant $C>0$ such that for any $f\in {L}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ and $\stackrel{˜}{x}\in {R}^{n}$,

${M}_{\mathrm{\Phi },\delta }^{\mathrm{#}}\left({T}_{b}\left(f\right)\right)\left(\stackrel{˜}{x}\right)\le C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right).$

Theorem 2 Let T be a singular integral operator as in Definition  3, $p\left(\cdot \right)\in M\left({R}^{n}\right)$ and $b\in BMO\left({R}^{n}\right)$. If ${T}_{1}\left(g\right)=0$ for any $g\in {L}^{u}\left({R}^{n}\right)$ ($1) and ${T}^{j,2}$ are the bounded linear operators on ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$ for $k=1,\dots ,m$, then ${T}_{b}$ is bounded on ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$, that is,

${\parallel {T}_{b}\left(f\right)\parallel }_{{L}^{p\left(\cdot \right)}}\le C{\parallel b\parallel }_{BMO}{\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}.$

Corollary Let $\left[b,T\right]\left(f\right)=bT\left(f\right)-T\left(bf\right)$ be a commutator generated by the singular integral operators T and b. Then Theorems  1 and 2 hold for $\left[b,T\right]$.

## 3 Proofs of theorems

To prove the theorems, we need the following lemmas.

Lemma 1 ([[1], p.485])

Let $0. We define that for any function $f\ge 0$ and $1/r=1/p-1/q$,

${\parallel f\parallel }_{W{L}^{q}}=\underset{\lambda >0}{sup}\lambda {|\left\{x\in {R}^{n}:f\left(x\right)>\lambda \right\}|}^{1/q},\phantom{\rule{2em}{0ex}}{N}_{p,q}\left(f\right)=\underset{E}{sup}{\parallel f{\chi }_{E}\parallel }_{{L}^{p}}/{\parallel {\chi }_{E}\parallel }_{{L}^{r}},$

where the sup is taken for all measurable sets E with $0<|E|<\mathrm{\infty }$. Then

${\parallel f\parallel }_{W{L}^{q}}\le {N}_{p,q}\left(f\right)\le {\left(q/\left(q-p\right)\right)}^{1/p}{\parallel f\parallel }_{W{L}^{q}}.$

Lemma 2 ([4])

Let ${r}_{j}\ge 1$ for $j=1,\dots ,l$, we denote that $1/r=1/{r}_{1}+\cdots +1/{r}_{l}$. Then

$\frac{1}{|Q|}{\int }_{Q}|{f}_{1}\left(x\right)\cdots {f}_{l}\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le {\parallel f\parallel }_{exp{L}^{{r}_{1}},Q}\cdots {\parallel f\parallel }_{exp{L}^{{r}_{l}},Q}{\parallel g\parallel }_{L{\left(logL\right)}^{1/r},Q}.$

Lemma 3 (see [5])

Let T be a singular integral operator as in Definition  3. Then T is weak bounded of $\left({L}^{1},{L}^{1}\right)$.

Lemma 4 ([13])

Let $p:{R}^{n}\to \left[1,\mathrm{\infty }\right)$ be a measurable function satisfying (1). Then ${L}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ is dense in ${L}^{p\left(\cdot \right)}\left({R}^{n}\right)$.

Lemma 5 ([14])

Let $f\in {L}_{\mathrm{loc}}^{1}\left({R}^{n}\right)$ and g be a measurable function satisfying

Then

${\int }_{{R}^{n}}|f\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le {C}_{n}{\int }_{{R}^{n}}{M}_{{\lambda }_{n}}^{\mathrm{#}}\left(f\right)\left(x\right)M\left(g\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$

Lemma 6 ([5, 14])

Let $\delta >0$, $0<\lambda <1$, $f\in {L}_{\mathrm{loc}}^{\delta }\left({R}^{n}\right)$ and $\mathrm{\Phi }=\left\{{\varphi }_{1},\dots ,{\varphi }_{m}\right\}\subset {L}^{\mathrm{\infty }}\left({R}^{n}\right)$ such that ${|det\left[{\varphi }_{j}\left({y}_{i}\right)\right]|}^{2}\in R{H}_{\mathrm{\infty }}\left({R}^{nm}\right)$. Then

${M}_{\lambda }^{\mathrm{#}}\left(f\right)\left(x\right)\le {\left(1/\lambda \right)}^{1/\delta }{M}_{\mathrm{\Phi },\delta }^{\mathrm{#}}\left(f\right)\left(x\right).$

Lemma 7 ([13, 14])

Let $p:{R}^{n}\to \left[1,\mathrm{\infty }\right)$ be a measurable function satisfying (1). If $f\in {L}^{p\left(\cdot \right)}\left({R}^{n}\right)$ and $g\in {L}^{{p}^{\prime }\left(\cdot \right)}\left({R}^{n}\right)$ with ${p}^{\prime }\left(x\right)=p\left(x\right)/\left(p\left(x\right)-1\right)$, then fg is integrable on ${R}^{n}$ and

${\int }_{{R}^{n}}|f\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}{\parallel g\parallel }_{{L}^{{p}^{\prime }\left(\cdot \right)}}.$

Lemma 8 ([14])

Let $p:{R}^{n}\to \left[1,\mathrm{\infty }\right)$ be a measurable function satisfying (1). Set

${\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}^{\prime }=sup\left\{{\int }_{{R}^{n}}|f\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx:f\in {L}^{p\left(\cdot \right)}\left({R}^{n}\right),g\in {L}^{{p}^{\prime }\left(\cdot \right)}\left({R}^{n}\right)\right\}.$

Then ${\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}\le {\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}^{\prime }\le C{\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}$.

Proof of Theorem 1 It suffices to prove for $f\in {L}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ and some constant ${C}_{0}$ that the following inequality holds:

${\left(\frac{1}{|Q|}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{C}_{0}{|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }\le C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right),$

where Q is any cube centered at ${x}_{0}$, ${C}_{0}={\sum }_{j=1}^{m}{\sum }_{i=1}^{l}{g}_{j}^{i}{\varphi }_{i}\left({x}_{0}-x\right)$ and ${g}_{j}^{i}={\int }_{{R}^{n}}{B}_{i}\left({x}_{0}-y\right){M}_{\left(b-{b}_{2Q}\right){\chi }_{{\left(2Q\right)}^{c}}}{T}^{j,2}\left(f\right)\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$. Without loss of generality, we may assume that ${T}^{j,1}$ are T ($j=1,\dots ,m$). Let $\stackrel{˜}{x}\in Q$. Fix a cube $Q=Q\left({x}_{0},d\right)$ and $\stackrel{˜}{x}\in Q$. Write

${T}_{b}\left(f\right)\left(x\right)={T}_{b-{b}_{2Q}}\left(f\right)\left(x\right)={T}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}\left(f\right)\left(x\right)+{T}_{\left(b-{b}_{2Q}\right){\chi }_{{\left(2Q\right)}^{c}}}\left(f\right)\left(x\right)={f}_{1}\left(x\right)+{f}_{2}\left(x\right).$

Then

$\begin{array}{rl}{\left(\frac{1}{|Q|}{\int }_{Q}|{T}_{b}\left(f\right)\left(x\right)-{C}_{0}{|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }\le & C{\left(\frac{1}{|Q|}{\int }_{Q}{|{f}_{1}\left(x\right)|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }\\ +C{\left(\frac{1}{|Q|}{\int }_{Q}{|{f}_{2}\left(x\right)-{C}_{0}|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }=I+II.\end{array}$

For I, by Lemmas 1, 2 and 3, we obtain

$\begin{array}{r}{\left(\frac{1}{|Q|}{\int }_{Q}{|{T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}{T}^{j,2}\left(f\right)\left(x\right)|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }\\ \phantom{\rule{1em}{0ex}}\le {|Q|}^{-1}\frac{{\parallel {T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}{T}^{j,2}\left(f\right){\chi }_{Q}\parallel }_{{L}^{\delta }}}{{|Q|}^{1/\delta -1}}\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{-1}{\parallel {T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}{T}^{j,2}\left(f\right)\parallel }_{W{L}^{1}}\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{-1}{\parallel {M}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}{T}^{j,2}\left(f\right)\parallel }_{{L}^{1}}\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{-1}{\int }_{2Q}|b\left(x\right)-{b}_{2Q}||{T}^{j,2}\left(f\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\parallel b-{b}_{2Q}\parallel }_{expL,2Q}{\parallel {T}^{j,2}\left(f\right)\parallel }_{L\left(logL\right),2Q}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel b\parallel }_{BMO}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right),\end{array}$

thus

$I\le C\sum _{j=1}^{m}{\left(\frac{1}{|Q|}{\int }_{Q}{|{T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{2Q}}{T}^{j,2}\left(f\right)\left(x\right)|}^{\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/\delta }\le C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right).$

For II, we get, for $x\in Q$,

$\begin{array}{r}|{T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{{\left(2Q\right)}^{c}}}{T}^{j,2}\left(f\right)\left(x\right)-\sum _{i=1}^{l}{g}_{j}^{i}{\varphi }_{i}\left({x}_{0}-x\right)|\\ \phantom{\rule{1em}{0ex}}\le |{\int }_{{R}^{n}}\left(K\left(x-y\right)-\sum _{i=1}^{l}{B}_{i}\left({x}_{0}-y\right){\varphi }_{i}\left({x}_{0}-x\right)\right)\left(b\left(y\right)-{b}_{2Q}\right){\chi }_{{\left(2Q\right)}^{c}}\left(y\right){T}^{j,2}\left(f\right)\left(y\right)\phantom{\rule{0.2em}{0ex}}dy|\\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{\mathrm{\infty }}{\int }_{{2}^{k}\phantom{\rule{0.2em}{0ex}}d\le |y-{x}_{0}|<{2}^{k+1}d}|K\left(x-y\right)-\sum _{i=1}^{l}{B}_{i}\left({x}_{0}-y\right){\varphi }_{i}\left({x}_{0}-x\right)||b\left(y\right)-{b}_{2Q}||{T}^{j,2}\left(f\right)\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\sum _{k=1}^{\mathrm{\infty }}{\int }_{{2}^{k}\phantom{\rule{0.2em}{0ex}}d\le |y-{x}_{0}|<{2}^{k+1}d}\frac{{|x-{x}_{0}|}^{\delta }}{{|y-{x}_{0}|}^{n+\delta }}|b\left(y\right)-{b}_{2Q}||{T}^{j,2}\left(f\right)\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\sum _{k=1}^{\mathrm{\infty }}\frac{{d}^{\delta }}{{\left({2}^{k}d\right)}^{n+\delta }}{\left({2}^{k}d\right)}^{n}{\parallel b-{b}_{2Q}\parallel }_{expL,{2}^{k+1}Q}{\parallel {T}^{j,2}\left(f\right)\parallel }_{L\left(logL\right),{2}^{k+1}Q}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel b\parallel }_{BMO}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right)\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }\\ \phantom{\rule{1em}{0ex}}\le C{\parallel b\parallel }_{BMO}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right),\end{array}$

thus

$\begin{array}{rcl}II& \le & \frac{C}{|Q|}{\int }_{Q}\sum _{j=1}^{m}|{T}^{j,1}{M}_{\left(b-{b}_{2Q}\right){\chi }_{{\left(2Q\right)}^{c}}}{T}^{j,2}\left(f\right)\left(x\right)-\sum _{i=1}^{l}{g}_{j}^{i}{\varphi }_{i}\left({x}_{0}-x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(\stackrel{˜}{x}\right).\end{array}$

This completes the proof of Theorem 1. □

Proof of Theorem 2 By Lemmas 4-7, we get, for $f\in {L}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ and $g\in {L}^{{p}^{\prime }\left(\cdot \right)}\left({R}^{n}\right)$,

$\begin{array}{rcl}{\int }_{{R}^{n}}|{T}_{b}\left(f\right)\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx& \le & C{\int }_{{R}^{n}}{M}_{{\lambda }_{n}}^{\mathrm{#}}\left({T}_{b}\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C{\int }_{{R}^{n}}{M}_{\mathrm{\Phi },\delta }^{\mathrm{#}}\left({T}_{b}\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{\int }_{{R}^{n}}{M}^{2}\left({T}^{j,2}\left(f\right)\right)\left(x\right)M\left(g\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{\parallel {M}^{2}\left({T}^{j,2}\left(f\right)\right)\parallel }_{{L}^{p\left(\cdot \right)}}{\parallel M\left(g\right)\parallel }_{{L}^{{p}^{\prime }\left(\cdot \right)}}\\ \le & C{\parallel b\parallel }_{BMO}\sum _{j=1}^{m}{\parallel {T}^{j,2}\left(f\right)\parallel }_{{L}^{p\left(\cdot \right)}}{\parallel M\left(g\right)\parallel }_{{L}^{{p}^{\prime }\left(\cdot \right)}}\\ \le & C{\parallel b\parallel }_{BMO}{\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}{\parallel g\parallel }_{{L}^{{p}^{\prime }\left(\cdot \right)}},\end{array}$

thus, by Lemma 8,

${\parallel {T}_{b}\left(f\right)\parallel }_{{L}^{p\left(\cdot \right)}}\le {\parallel b\parallel }_{BMO}{\parallel f\parallel }_{{L}^{p\left(\cdot \right)}}.$

This completes the proof of Theorem 2. □

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Correspondence to Qiufen Feng.

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Feng, Q. Boundedness of Toeplitz type operator associated to singular integral operator satisfying a variant of Hörmander’s condition on ${L}^{p}$ spaces with variable exponent. J Inequal Appl 2014, 369 (2014). https://doi.org/10.1186/1029-242X-2014-369

• variable ${L}^{p}$ space