Boundedness of Toeplitz-type operator associated to general integral operator on $L^p$ spaces with variable exponent

Abstract

In this paper, the boundedness for some Toeplitz-type operator related to some general integral operator on $L^p$ spaces with variable exponent is obtained by using a sharp estimate of the operator. The operator includes Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

MSC:42B20, 42B25.

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [3–5], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on $L^p(R^n)$ for $1<p<\mathrm{\infty }$. In [6–8], some Toeplitz-type operators associated to the singular integral operators and strongly singular integral operators were introduced, and the boundedness for the operators was obtained. In the last years, the theory of $L^p$ spaces with variable exponent was developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [9–13] and their references). Karlovich and Lerner studied the boundedness of the commutators of singular integral operators on $L^p$ spaces with variable exponent (see [12]). Motivated by these papers, in this paper, we have the purpose to introduce some Toeplitz-type operator related to some integral operator and BMO functions, and prove the boundedness for the operator on $L^p$ spaces with variable exponent by using a sharp estimate of the operator. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz 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

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

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

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

We write that $f^{\mathrm{\#}}=f_{\delta }^{\mathrm{\#}}$ if $\delta =1$. We say that f belongs to $\mathit{BMO}(R^n)$ if $f^{\mathrm{\#}}$ belongs to $L^{\mathrm{\infty }}(R^n)$ and define ${\parallel f\parallel }_{\mathit{BMO}}={\parallel f^{\mathrm{\#}}\parallel }_{L^{\mathrm{\infty }}}$. 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 Φ be a Young function and let $\tilde{\mathrm{\Phi }}$ be the complementary associated to Φ. For a function f, we denote the Φ-average by

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

and the maximal function associated to Φ by

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

The Young functions to be used in this paper are $\mathrm{\Phi }(t)=t{(1+logt)}^r$ and $\tilde{\mathrm{\Phi }}(t)=exp(t^{1/r})$, 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, 5], we know the generalized Hölder inequality

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

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

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

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

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

For $\lambda \in (0,1)$ and the 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 hold:

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

(1)

In this paper, we study some integral operators as follows (see [14, 15]).

Definition Let $F_t(x,y)$ be defined on $R^n\times R^n\times [0,+\mathrm{\infty })$, set

$$F_t(f)(x)={\int }_{R^n}{F}_t(x,y)f(y)dy$$

for every bounded and compactly supported function f. $F_t$ satisfies: for fixed $\delta >0$,

$$\parallel F_t(x-y)\parallel \le C|x-y{|}^{-n}$$

and

$$\parallel F_t(y-x)-F_t(z-x)\parallel \le C|y-z{|}^{\delta }|x-z{|}^{-n-\delta }$$

if $2|y-z|\le |x-z|$. We define that $T(f)(x)=\parallel F_t(f)(x)\parallel$.

Let H be the Banach space $H=\{h:\parallel h\parallel <\mathrm{\infty }\}$. For each fixed $x\in R^n$, we view $F_t(f)(x)$ and $F_t^b(f)(x)$ as the mappings from $[0,+\mathrm{\infty })$ to H. Set

$$T(f)(x)=\parallel F_t(f)(x)\parallel .$$

Moreover, let b be a locally integrable function on $R^n$. The Toeplitz-type operator related to T is defined by

$$T^b(f)=\parallel F_t^b(f)\parallel ,$$

where

$$F_t^b(f)=\sum _{k=1}^m{F}_t^{k,1}{M}_b{F}_t^{k,2}(f),$$

$F_t^{k,1}(f)$ are $F_t(f)$ or ±I (the identity operator), $T^{k,2}(f)=\parallel F_t^{k,2}(f)\parallel$ are the operators for $k=1,\dots ,m$ and $M_b(f)=bf$.

Note that the commutator is a particular operator of the Toeplitz-type operator $T^b$. The Toeplitz-type operators $T^b$ are the non-trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis, and they have been widely studied by many authors (see [4, 5]). In recent years, the boundedness of classical operators on spaces $L^{p(\cdot )}(R^n)$ have attracted great attention (see [9–13] and their references). The main purpose of this paper is twofold. First, we establish a sharp estimate for the operator $T^b$; and second, we prove the boundedness for the operator on $L^p$ spaces with variable exponent by using the sharp estimate. In Section 4, we give some applications of the theorems in this paper.

We shall prove the following theorems.

Theorem 1 Let T be the integral operator as defined in Definition, $0<\delta <1$ and $b\in \mathit{BMO}(R^n)$. If $F_t^1(g)=0$ 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 $\tilde{x}\in R^n$,

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

Theorem 2 Let T be the integral operator as defined in Definition, $p(\cdot )\in M(R^n)$ and $b\in \mathit{BMO}(R^n)$. If $F_t^1(g)=0$ for any $g\in L^u(R^n)$ ($1<u<\mathrm{\infty }$) and $T^{k,2}$ are the bounded 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,

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

Corollary Let $[b,T](f)=bT(f)-T(bf)$ be the commutator generated by the integral operator 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 [5, p.485]

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

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

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

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

Lemma 3 [14]

Let T be the integral operator as defined in Definition. Then T is bounded from $L^1(R^n)$ to $W{L}^1(R^n)$.

Lemma 4 [12]

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 [12]

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 [12, 16]

Let $\delta >0$, $0<\lambda <1$ and $f\in L_{\mathrm{loc}}^{\delta }(R^n)$. Then

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

Lemma 7 [17]

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 [12]

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

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

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 $f\in C_0^{\mathrm{\infty }}(R^n)$ and some constant $C_0$, that the following inequality holds:

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

Without loss of generality, we may assume that $T^{k,1}$ are T ($k=1,\dots ,m$). Fix a cube $Q=Q(x_0,d)$ and $\tilde{x}\in Q$. We write, by $F_t^1(g)=0$,

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

Then

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q\right|T^b(f)(x)-\parallel f_2(x_0)\parallel {|}^{\delta }dx)}^{1/\delta }\hfill \\ ={\left(\frac{1}{|Q|}{\int }_Q\right|\parallel F_t^b(f)(x)\parallel -\parallel f_2(x_0)\parallel {|}^{\delta }dx)}^{1/\delta }\le {\left(\frac{1}{|Q|}{\int }_Q{\parallel F_t^b(f)(x)-f_2(x_0)\parallel }^{\delta }dx\right)}^{1/\delta }\hfill \\ \le C{\left(\frac{1}{|Q|}{\int }_Q{\parallel f_1(x)\parallel }^{\delta }dx\right)}^{1/\delta }+C{\left(\frac{1}{|Q|}{\int }_Q{\parallel f_2(x)-f_2(x_0)\parallel }^{\delta }dx\right)}^{1/\delta }=I_1+I_2.\hfill \end{array}$$

For $I_1$, by the weak $(L^1,L^1)$ boundedness of T (see Lemma 3) and Kolmogorov’s inequality (see Lemma 1), we obtain

$$\begin{array}{c}{\left(\frac{1}{|Q|}{\int }_Q{\parallel F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{F}_t^{k,2}(f)(x)\parallel }^{\delta }dx\right)}^{1/\delta }\hfill \\ ={\left(\frac{1}{|Q|}{\int }_Q\right|T^{k,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{k,2}(f)(x){|}^{\delta }dx)}^{1/\delta }\hfill \\ \le \frac{|Q{|}^{1/\delta -1}}{|Q{|}^{1/\delta }}\frac{{\parallel T^{k,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{k,2}(f){\chi }_Q\parallel }_{L^{\delta }}}{{\parallel {\chi }_Q\parallel }_{L^{\delta /(1-\delta )}}}\hfill \\ \le \frac{C}{|Q|}{\parallel T^{k,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{T}^{k,2}(f)\parallel }_{W{L}^1}\hfill \\ \le \frac{C}{|Q|}{\int }_{R^n}|M_{(b-b_{2Q}){\chi }_{2Q}}{T}^{k,2}(f)(x)|dx\hfill \\ \le C|Q{|}^{-1}{\parallel M_{(b-b_{2Q}){\chi }_{2Q}}{T}^{k,2}(f)\parallel }_{L^1}\hfill \\ \le C|Q{|}^{-1}{\int }_{2Q}|b(x)-b_{2Q}\left|\right|T^{k,2}(f)(x)|dx\hfill \\ \le C{\parallel b-b_Q\parallel }_{expL,2Q}{\parallel T^{k,2}(f)\parallel }_{L(logL),2Q}\hfill \\ \le C{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{k,2}(f))(\tilde{x}).\hfill \end{array}$$

Thus

$$\begin{array}{rcl}{I}_1& \le & C\sum _{k=1}^m{\left(\frac{1}{|Q|}{\int }_Q{\parallel F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{2Q}}{F}_t^{k,2}(f)(x)\parallel }^{\delta }dx\right)}^{1/\delta }\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}\sum _{k=1}^m{M}^2(T^{k,2}(f))(\tilde{x}).\end{array}$$

For $I_2$, we get, for $x\in Q$,

$$\begin{array}{c}\parallel F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{F}_t^{k,2}(f)(x)-F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{F}_t^{k,2}(f)(x_0)\parallel \hfill \\ \le {\int }_{{(2Q)}^c}|b(y)-b_{2Q}|\parallel F_t(x,y)-F_t(x_0,y)\parallel |T^{k,2}(f)(y)|dy\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}{\int }_{2^{j}d\le |y-x_0|<2^{j+1}d}|b(y)-b_{2Q}|\frac{|x-x_0{|}^{\delta }}{|x_0-y{|}^{n+\delta }}|T^{k,2}(f)(y)|dy\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}{2}^{-j\delta }\frac{1}{|2^{j+1}Q|}{\int }_{2^{j+1}Q}|b(y)-b_{2Q}||T^{k,2}(f)(y)|dy\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}{2}^{-j\delta }{\parallel b-b_{2Q}\parallel }_{expL,2^{j+1}Q}{\parallel T^{k,2}(f)\parallel }_{L(logL),2^{j+1}Q}\hfill \\ \le C\sum _{j=1}^{\mathrm{\infty }}j{2}^{-j\delta }{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{k,2}(f))(\tilde{x})\hfill \\ \le C{\parallel b\parallel }_{\mathit{BMO}}{M}^2(T^{k,2}(f))(\tilde{x}).\hfill \end{array}$$

Thus

$$\begin{array}{rcl}{I}_2& \le & \frac{C}{|Q|}{\int }_Q\sum _{k=1}^m\parallel F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{F}_t^{k,2}(f)(x)-F_t^{k,1}{M}_{(b-b_{2Q}){\chi }_{{(2Q)}^c}}{F}_t^{k,2}(f)(x_0)\parallel dx\\ \le & C{\parallel b\parallel }_{\mathit{BMO}}\sum _{k=1}^m{M}^2(T^{k,2}(f))(\tilde{x}).\end{array}$$

These complete 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 $g\in L^{p^{\prime }(\cdot )}(R^n)$,

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

Thus, by Lemma 8,

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

This completes the proof of Theorem 2. □

4 Applications

In this section we shall apply Theorems 1 and 2 of the paper to some particular operators such as Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Application 1 Littlewood-Paley operator.

Fixed $\epsilon >0$. Let ψ be a fixed function which satisfies:

• (1) ${\int }_{R^n}\psi (x)dx=0$,

• (2) $|\psi (x)|\le C{(1+|x|)}^{-(n+1)}$,

• (3) $|\psi (x+y)-\psi (x)|\le C|y{|}^{\epsilon }{(1+|x|)}^{-(n+1+\epsilon )}$ when $2|y|<|x|$.

Let ${\psi }_t(x)=t^{-n}\psi (x/t)$ for $t>0$ and $F_t(f)(x)={\int }_{R^n}f(y){\psi }_t(x-y)dy$. The Littlewood-Paley operator is defined by (see [18])

$$g_{\psi }(f)(x)={\left({\int }_0^{\mathrm{\infty }}\right|F_t(f)(x){|}^2\frac{dt}{t})}^{1/2}.$$

Set H to be the space

$$H=\{h:\parallel h\parallel ={\left({\int }_0^{\mathrm{\infty }}\right|h(t){|}^{2}dt/t)}^{1/2}<\mathrm{\infty }\}.$$

Let b be a locally integrable function on $R^n$. The Toeplitz-type operator related to the Littlewood-Paley operator is defined by

$$g_{\psi }^b(f)(x)={\left({\int }_0^{\mathrm{\infty }}\right|F_t^b(f)(x){|}^2\frac{dt}{t})}^{1/2},$$

where

$$F_t^b=\sum _{k=1}^m{F}_t^{k,1}{M}_b{F}_t^{k,2},$$

$F_t^{k,1}$ are $F_t$ or ±I (the identity operator), $T^{k,2}=\parallel F_t^{k,2}\parallel$ are the bounded linear operators on $L^p(R^n)$ for $1<p<\mathrm{\infty }$ and $k=1,\dots ,m$, $M_b(f)=bf$. Then, for each fixed $x\in R^n$, $F_t^b(f)(x)$ may be viewed as the mapping from $[0,+\mathrm{\infty })$ to H, and it is clear that

$$g_{\psi }^b(f)(x)=\parallel F_t^b(f)(x)\parallel ,g_{\psi }(f)(x)=\parallel F_t(f)(x)\parallel .$$

It is easy to see that $g_{\psi }^b$ satisfies the conditions of Theorems 1 and 2 (see [14, 15, 19]), thus Theorems 1 and 2 hold for $g_{\psi }^b$.

Application 2 Marcinkiewicz operator.

Fixed $0<\gamma \le 1$. Let Ω be homogeneous of degree zero on $R^n$ with ${\int }_{S^{n-1}}\mathrm{\Omega }(x^{\prime })d\sigma (x^{\prime })=0$. Assume that $\mathrm{\Omega }\in {Lip}_{\gamma }(S^{n-1})$. Set $F_t(f)(x)={\int }_{|x-y|\le t}\frac{\mathrm{\Omega }(x-y)}{|x-y{|}^{n-1}}f(y)dy$. The Marcinkiewicz operator is defined by (see [20])

$${\mu }_{\mathrm{\Omega }}(f)(x)={\left({\int }_0^{\mathrm{\infty }}\right|F_t(f)(x){|}^2\frac{dt}{t^3})}^{1/2}.$$

Set H to be the space

$$H=\{h:\parallel h\parallel ={\left({\int }_0^{\mathrm{\infty }}\right|h(t){|}^{2}dt/t^3)}^{1/2}<\mathrm{\infty }\}.$$

Let b be a locally integrable function on $R^n$. The Toeplitz-type operator related to the Marcinkiewicz operator is defined by

$${\mu }_{\mathrm{\Omega }}^b(f)(x)={\left({\int }_0^{\mathrm{\infty }}\right|F_t^b(f)(x){|}^2\frac{dt}{t^3})}^{1/2},$$

where

$$F_t^b=\sum _{k=1}^m{F}_t^{k,1}{M}_b{F}_t^{k,2},$$

$F_t^{k,1}$ are $F_t$ or ±I (the identity operator), $T^{k,2}=\parallel F_t^{k,2}\parallel$ are the bounded linear operators on $L^p(R^n)$ for $1<p<\mathrm{\infty }$ and $k=1,\dots ,m$, $M_b(f)=bf$. Then it is clear that

$${\mu }_{\mathrm{\Omega }}^b(f)(x)=\parallel F_t^b(f)(x)\parallel ,{\mu }_{\mathrm{\Omega }}(f)(x)=\parallel F_t(f)(x)\parallel .$$

It is easy to see that ${\mu }_{\mathrm{\Omega }}^b$ satisfies the conditions of Theorems 1 and 2 (see [14, 15, 20]), thus Theorems 1 and 2 hold for ${\mu }_{\mathrm{\Omega }}^b$.

Application 3 Bochner-Riesz operator.

Let $\delta >(n-1)/2$, $F_t^{\delta }(f\widehat{)(}\xi )={(1-t^2|\xi {|}^2)}_{+}^{\delta }\hat{f}(\xi )$ and $B_t^{\delta }(z)=t^{-n}{B}^{\delta }(z/t)$ for $t>0$. The maximal Bochner-Riesz operator is defined by (see [17])

$$B_{\delta ,\ast }(f)(x)=\underset{t>0}{sup}|F_t^{\delta }(f)(x)|.$$

Set H to be the space $H=\{h:\parallel h\parallel ={sup}_{t>0}|h(t)|<\mathrm{\infty }\}$. Let b be a locally integrable function on $R^n$. The Toeplitz-type operator related to the maximal Bochner-Riesz operator is defined by

$$B_{\delta ,\ast }^b(f)(x)=\underset{t>0}{sup}|B_{\delta ,t}^b(f)(x)|,$$

where

$$B_{\delta ,t}^b=\sum _{k=1}^m{F}_t^{k,1}{M}_b{F}_t^{k,2},$$

$F_t^{k,1}$ are $F_t$ or ±I (the identity operator), $T^{k,2}=\parallel F_t^{k,2}\parallel$ are the bounded linear operators on $L^p(R^n)$ for $1<p<\mathrm{\infty }$ and $k=1,\dots ,m$, $M_b(f)=bf$. Then

$$B_{\delta ,\ast }^b(f)(x)=\parallel B_{\delta ,t}^b(f)(x)\parallel ,B_{\ast }^{\delta }(f)(x)=\parallel B_t^{\delta }(f)(x)\parallel .$$

It is easy to see that $B_{\delta ,\ast }^b$ satisfies the conditions of Theorems 1 and 2 (see [14, 15]), thus Theorems 1 and 2 hold for $B_{\delta ,\ast }^b$.

Authors’ information

All authors read and approved the final manuscript.