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 [7–9], 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 [10–13] 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

$$f_{\delta }^{\mathrm{\#}}(x)=\underset{Q\ni x}{sup}{\left(\frac{1}{|Q|}{\int }_Q{|f(y)-f_Q|}^{\delta }dy\right)}^{1/\delta },$$

where, and in what follows, $f_Q={|Q|}^{-1}{\int }_{Q}f(x)dx$. It is well known that (see [1, 2])

$$f_{\delta }^{\mathrm{\#}}(x)\approx \underset{Q\ni x}{sup}\underset{c\in C}{inf}{\left(\frac{1}{|Q|}{\int }_Q{|f(y)-c|}^{\delta }dy\right)}^{1/\delta }.$$

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

$$M(f)(x)=\underset{Q\ni x}{sup}{|Q|}^{-1}{\int }_Q|f(y)|dy.$$

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

$$M^k(f)(x)=M(M^{k-1}(f))(x)\text{when }k\ge 2.$$

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

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

and the maximal function associated to Ψ by

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

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

$$\frac{1}{|Q|}{\int }_Q|f(y)g(y)|dy\le {\parallel f\parallel }_{\mathrm{\Psi },Q}{\parallel g\parallel }_{\tilde{\mathrm{\Psi }},Q},$$

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 $x\in R^n$, $b\in BMO(R^n)$,

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

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

$$f^{\ast }(t)=inf\{\lambda >0:\left|\{x\in R^n:|f(x)|>\lambda \}\right|\le t\}(0<t<\mathrm{\infty }).$$

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

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

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

$$m(f,p)={\int }_{R^n}{|f(x)|}^{p(x)}dx.$$

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

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

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

$$1<p_{-}=ess\underset{x\in R^n}{inf}p(x),ess\underset{x\in R^n}{sup}p(x)=p_{+}<\mathrm{\infty }.$$

(1)

In recent years, the boundedness of classical operators on spaces $L^{p(\cdot )}(R^n)$ has attracted a great deal of attention (see [10–14] and their references).

Definition 1 Let $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_l\}$ 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

$$M_{\mathrm{\Phi }}^{\mathrm{\#}}(f)(x)=\underset{Q\ni x}{sup}\underset{\{c_1,\dots ,c_l\}}{inf}\frac{1}{|Q|}{\int }_Q|f(y)-\sum _{i=1}^l{c}_i{\varphi }_i(x_Q-y)|dy,$$

where the infimum is taken over all m-tuples $\{c_1,\dots ,c_l\}$ of complex numbers and $x_Q$ is the center of Q. For $\eta >0$, let

$$M_{\mathrm{\Phi },\eta }^{\mathrm{\#}}(f)(x)=\underset{Q\ni x}{sup}\underset{\{c_1,\dots ,c_l\}}{inf}{\left(\frac{1}{|Q|}{\int }_Q{|f(y)-\sum _{i=1}^l{c}_j{\varphi }_i(x_Q-y)|}^{\eta }dy\right)}^{1/\eta }.$$

Remark We note that $M_{\mathrm{\Phi }}^{\mathrm{\#}}\approx M_{\mathrm{\Phi }}^{\mathrm{\#}}(f)$ 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 $f\in R{H}_{\mathrm{\infty }}(R^n)$) if, for any cube Q centered at the origin, we have

$$0<\underset{x\in Q}{sup}f(x)\le C\frac{1}{|Q|}{\int }_{Q}f(y)dy.$$

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

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

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

There exist functions $B_1,\dots ,B_l\in L_{\mathrm{loc}}^1(R^n-\{0\})$ and $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_l\}\subset L^{\mathrm{\infty }}(R^n)$ such that ${|det[{\varphi }_j(y_i)]|}^2\in R{H}_{\mathrm{\infty }}(R^{nl})$, and for a fixed $\delta >0$ and any $|x|>2|y|>0$,

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

For $f\in C_0^{\mathrm{\infty }}$, we define a singular integral operator related to the kernel K by

$$T(f)(x)={\int }_{R^n}K(x-y)f(y)dy.$$

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 $M_b(f)=bf$.

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 $b\in BMO(R^n)$. If $T_1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$), then there exists a constant $C>0$ such that for any $f\in L_0^{\mathrm{\infty }}(R^n)$ and $\tilde{x}\in R^n$,

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

Theorem 2 Let T be a singular integral operator as in Definition  3, $p(\cdot )\in M(R^n)$ and $b\in BMO(R^n)$. If $T_1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$) and $T^{j,2}$ are the bounded linear operators on $L^{p(\cdot )}(R^n)$ for $k=1,\dots ,m$, then $T_b$ is bounded on $L^{p(\cdot )}(R^n)$, that is,

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

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

3 Proofs of theorems

To prove the theorems, we need the following lemmas.

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

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

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

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

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

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

$$\frac{1}{|Q|}{\int }_Q|f_1(x)\cdots f_l(x)g(x)|dx\le {\parallel f\parallel }_{exp{L}^{r_1},Q}\cdots {\parallel f\parallel }_{exp{L}^{r_l},Q}{\parallel g\parallel }_{L{(logL)}^{1/r},Q}.$$

Lemma 3 (see [5])

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

Lemma 4 ([13])

Let $p:R^n\to [1,\mathrm{\infty })$ be a measurable function satisfying (1). Then $L_0^{\mathrm{\infty }}(R^n)$ is dense in $L^{p(\cdot )}(R^n)$.

Lemma 5 ([14])

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

$$\left|\{x\in R^n:|g(x)|>\alpha \}\right|<\mathrm{\infty }\mathit{\text{for all }}\alpha >0.$$

Then

$${\int }_{R^n}|f(x)g(x)|dx\le C_n{\int }_{R^n}{M}_{{\lambda }_n}^{\mathrm{\#}}(f)(x)M(g)(x)dx.$$

Lemma 6 ([5, 14])

Let $\delta >0$, $0<\lambda <1$, $f\in L_{\mathrm{loc}}^{\delta }(R^n)$ and $\mathrm{\Phi }=\{{\varphi }_1,\dots ,{\varphi }_m\}\subset L^{\mathrm{\infty }}(R^n)$ such that ${|det[{\varphi }_j(y_i)]|}^2\in R{H}_{\mathrm{\infty }}(R^{nm})$. Then

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

Lemma 7 ([13, 14])

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

$${\int }_{R^n}|f(x)g(x)|dx\le C{\parallel f\parallel }_{L^{p(\cdot )}}{\parallel g\parallel }_{L^{p^{\prime }(\cdot )}}.$$

Lemma 8 ([14])

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

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

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

Proof of Theorem 1 It suffices to prove for $f\in L_0^{\mathrm{\infty }}(R^n)$ and some constant $C_0$ that the following inequality holds:

$${\left(\frac{1}{|Q|}{\int }_Q\right|T_b(f)(x)-C_0{|}^{\delta }dx)}^{1/\delta }\le C{\parallel b\parallel }_{BMO}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}),$$

where Q is any cube centered at $x_0$, $C_0={\sum }_{j=1}^m{\sum }_{i=1}^l{g}_j^i{\varphi }_i(x_0-x)$ and $g_j^i={\int }_{R^n}{B}_i(x_0-y)M_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(y)dy$. Without loss of generality, we may assume that $T^{j,1}$ are T ($j=1,\dots ,m$). Let $\tilde{x}\in Q$. Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. Write

$$T_b(f)(x)=T_{b-b_{2Q}}(f)(x)=T_{(b-b_{2Q}){\chi }_{2Q}}(f)(x)+T_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}(f)(x)=f_1(x)+f_2(x).$$

Then

$$\begin{array}{rl}{\left(\frac{1}{|Q|}{\int }_Q\right|T_b(f)(x)-C_0{|}^{\delta }dx)}^{1/\delta }\le & C{\left(\frac{1}{|Q|}{\int }_Q{|f_1(x)|}^{\delta }dx\right)}^{1/\delta }\\ +C{\left(\frac{1}{|Q|}{\int }_Q{|f_2(x)-C_0|}^{\delta }dx\right)}^{1/\delta }=I+II.\end{array}$$

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

$$\begin{array}{r}{\left(\frac{1}{|Q|}{\int }_Q{|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)(x)|}^{\delta }dx\right)}^{1/\delta }\\ \le {|Q|}^{-1}\frac{{\parallel T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f){\chi }_Q\parallel }_{L^{\delta }}}{{|Q|}^{1/\delta -1}}\\ \le C{|Q|}^{-1}{\parallel T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)\parallel }_{W{L}^1}\\ \le C{|Q|}^{-1}{\parallel M_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)\parallel }_{L^1}\\ \le C{|Q|}^{-1}{\int }_{2Q}|b(x)-b_{2Q}||T^{j,2}(f)(x)|dx\\ \le C{\parallel b-b_{2Q}\parallel }_{expL,2Q}{\parallel T^{j,2}(f)\parallel }_{L(logL),2Q}\\ \le C{\parallel b\parallel }_{BMO}{M}^2(T^{j,2}(f))(\tilde{x}),\end{array}$$

thus

$$I\le C\sum _{j=1}^m{\left(\frac{1}{|Q|}{\int }_Q{|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{j,2}(f)(x)|}^{\delta }dx\right)}^{1/\delta }\le C{\parallel b\parallel }_{BMO}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}).$$

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

$$\begin{array}{r}|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(x)-\sum _{i=1}^l{g}_j^i{\varphi }_i(x_0-x)|\\ \le |{\int }_{R^n}(K(x-y)-\sum _{i=1}^l{B}_i(x_0-y){\varphi }_i(x_0-x))(b(y)-b_{2Q}){\chi }_{{(2Q)}^c}(y)T^{j,2}(f)(y)dy|\\ \le \sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}d\le |y-x_0|<2^{k+1}d}|K(x-y)-\sum _{i=1}^l{B}_i(x_0-y){\varphi }_i(x_0-x)||b(y)-b_{2Q}||T^{j,2}(f)(y)|dy\\ \le C\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}d\le |y-x_0|<2^{k+1}d}\frac{{|x-x_0|}^{\delta }}{{|y-x_0|}^{n+\delta }}|b(y)-b_{2Q}||T^{j,2}(f)(y)|dy\\ \le C\sum _{k=1}^{\mathrm{\infty }}\frac{d^{\delta }}{{(2^{k}d)}^{n+\delta }}{\left(2^{k}d\right)}^n{\parallel b-b_{2Q}\parallel }_{expL,2^{k+1}Q}{\parallel T^{j,2}(f)\parallel }_{L(logL),2^{k+1}Q}\\ \le C{\parallel b\parallel }_{BMO}{M}^2(T^{j,2}(f))(\tilde{x})\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k\delta }\\ \le C{\parallel b\parallel }_{BMO}{M}^2(T^{j,2}(f))(\tilde{x}),\end{array}$$

thus

$$\begin{array}{rcl}II& \le & \frac{C}{|Q|}{\int }_Q\sum _{j=1}^m|T^{j,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{T}^{j,2}(f)(x)-\sum _{i=1}^l{g}_j^i{\varphi }_i(x_0-x)|dx\\ \le & C{\parallel b\parallel }_{BMO}\sum _{j=1}^m{M}^2(T^{j,2}(f))(\tilde{x}).\end{array}$$

This completes the proof of Theorem 1. □

Proof of Theorem 2 By Lemmas 4-7, we get, for $f\in L_0^{\mathrm{\infty }}(R^n)$ and $g\in L^{p^{\prime }(\cdot )}(R^n)$,

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

thus, by Lemma 8,

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

This completes the proof of Theorem 2. □