Boundedness of sublinear operators and their commutators on generalized central Morrey spaces

Abstract

In this paper, we introduce the generalized central Morrey spaces ${\dot{B}}^{p,\phi }({\mathbb{R}}^n)$ and get the boundedness of a large class of rough operators on them. We also consider the CBMO estimates of their commutators on generalized central Morrey spaces. As applications, we obtain the boundedness characterizations of rough Hardy-Littlewood maximal function, rough Calderón-Zygmund singular integral, rough fractional integral, etc. on generalized central Morrey spaces.

MSC:42B20, 42B25.

1 Introduction

Let $\mathrm{\Omega }\in L^s(S^{n-1})$ be homogeneous of degree zero on ${\mathbb{R}}^n$, where $S^{n-1}$ denotes the unit sphere of ${\mathbb{R}}^n$ and $s>1$. We define $s^{\prime }=s/(s-1)$ for any $s>1$. Suppose that $T_{\mathrm{\Omega }}$ represents a sublinear operator, which satisfies that for any $f\in L^1({\mathbb{R}}^n)$ with compact support and $x\notin suppf$,

$$|T_{\mathrm{\Omega }}f(x)|\le C{\int }_{{\mathbb{R}}^n}\frac{|\mathrm{\Omega }(x-y)|}{{|x-y|}^n}|f(y)|dy,$$

(1.1)

where $C>0$ is an absolute constant. Similarly, for any $0<\alpha <n$, we assume that $T_{\mathrm{\Omega },\alpha }$ represents a sublinear operator, which satisfies that

$$|T_{\mathrm{\Omega },\alpha }f(x)|\le C{\int }_{{\mathbb{R}}^n}\frac{|\mathrm{\Omega }(x-y)|}{{|x-y|}^{n-\alpha }}|f(y)|dy$$

(1.2)

for any $f\in L^1({\mathbb{R}}^n)$ with compact support and $x\notin suppf$.

Let T be a linear operator. For a locally integrable function b on ${\mathbb{R}}^n$, we define the commutator $[T,b]$ by

$$[T,b]f(x)=b(x)Tf(x)-T(bf)(x)$$

(1.3)

for any suitable function f.

To study the local behavior of solutions to second-order elliptic partial differential equations, Morrey [1] introduced the classical Morrey spaces $L^{p,\lambda }({\mathbb{R}}^n)$. The readers can find more details in [2].

Let $1\le p<\mathrm{\infty }$ and $\lambda \ge 0$. $B(x_0,t)$ denotes a ball centered at $x_0$ of radius t. Morrey spaces $L^{p,\lambda }({\mathbb{R}}^n)$ are defined by

$$L^{p,\lambda }\left({\mathbb{R}}^n\right)=\{f\in L_{\mathrm{loc}}^p\left({\mathbb{R}}^n\right):{\parallel f\parallel }_{L^{p,\lambda }({\mathbb{R}}^n)}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{L^{p,\lambda }({\mathbb{R}}^n)}=\underset{x_0\in {\mathbb{R}}^n,t>0}{sup}{\left(\frac{1}{t^{\lambda }}{\int }_{B(x_0,t)}{|f(x)|}^{p}dx\right)}^{\frac{1}{p}}.$$

When $1\le p<\mathrm{\infty }$, $L^{p,0}({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)$ and $L^{p,n}({\mathbb{R}}^n)=L^{\mathrm{\infty }}({\mathbb{R}}^n)$; when $\lambda >n$, $L^{p,\lambda }({\mathbb{R}}^n)=\{0\}$.

Many authors have studied the mapping properties of many operators on Morrey spaces; see [3–5] and [6]. Alvarez et al. [7], in order to study the relationship between central BMO spaces and Morrey spaces, introduced λ-central bounded mean oscillation spaces and central Morrey spaces.

Let $\lambda <\frac{1}{n}$ and $1<p<\mathrm{\infty }$. A function $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$ belongs to the λ-central bounded mean oscillation spaces ${\mathit{CBMO}}^{p,\lambda }({\mathbb{R}}^n)$ if

$${\parallel f\parallel }_{{\mathit{CBMO}}^{p,\lambda }}=\underset{r>0}{sup}{\left(\frac{1}{{|B(0,r)|}^{1+\lambda p}}{\int }_{B(0,r)}{|f(x)-f_{B(0,r)}|}^p\right)}^{\frac{1}{p}}<\mathrm{\infty },$$

where $f_{B(0,r)}=\frac{1}{|B(0,r)|}{\int }_{B(0,r)}f(y)dy$. If two functions which differ by a constant are regarded as functions in the spaces ${\mathit{CBMO}}^{p,\lambda }({\mathbb{R}}^n)$, then ${\mathit{CBMO}}^{p,\lambda }({\mathbb{R}}^n)$ spaces become Banach spaces. ${\mathit{CBMO}}^{p,\lambda }({\mathbb{R}}^n)$ spaces become the spaces of constants when $\lambda <-\frac{1}{p}$ and they coincide with $L^p({\mathbb{R}}^n)$ modulo constants when $\lambda =-\frac{1}{p}$.

Let $\lambda \in \mathbb{R}$ and $1<p<\mathrm{\infty }$. The central Morrey spaces ${\dot{B}}^{p,\lambda }({\mathbb{R}}^n)$ are defined by

$${\parallel f\parallel }_{{\dot{B}}^{p,\lambda }}=\underset{r>0}{sup}{\left(\frac{1}{{|B(0,r)|}^{1+\lambda p}}{\int }_{B(0,r)}{|f(x)|}^p\right)}^{\frac{1}{p}}<\mathrm{\infty }.$$

It follows that ${\dot{B}}^{p,\lambda }({\mathbb{R}}^n)$ spaces are Banach spaces continuously included in ${\mathit{CBMO}}^{p,\lambda }({\mathbb{R}}^n)$ spaces. ${\dot{B}}^{p,\lambda }({\mathbb{R}}^n)$ spaces reduce to $\{0\}$ when $\lambda <-\frac{1}{p}$, and it is true that ${\dot{B}}^{p,-\frac{1}{p}}({\mathbb{R}}^n)=L^p({\mathbb{R}}^n)$, ${\dot{B}}^{p,0}({\mathbb{R}}^n)={\dot{B}}^p({\mathbb{R}}^n)$.

Recently, Guliyev [8] introduced the generalized Morrey spaces $L^{p,\phi }({\mathbb{R}}^n)$, where $\phi (x,r)$ is a positive measurable function on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$ and $1\le p<\mathrm{\infty }$. For all functions $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$, the generalized Morrey spaces $L^{p,\phi }({\mathbb{R}}^n)$ are defined by

$${\parallel f\parallel }_{L^{p,\phi }({\mathbb{R}}^n)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\phi {(x,r)}^{-1}{|B(x,r)|}^{-\frac{1}{p}}{\parallel f\parallel }_{L^p(B(x,r))}<\mathrm{\infty }.$$

Obviously, if $\phi (x,r)=r^{\frac{\lambda -n}{p}}$, $L^{p,\lambda }({\mathbb{R}}^n)=L^{p,\phi }({\mathbb{R}}^n)$.

When $\mathrm{\Omega }\equiv 1$, Guliyev obtained the sufficient condition on ${\phi }_1$ and ${\phi }_2$

$${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}+1}}dt\le C{\phi }_2(r)$$

for the boundedness of $T_{\mathrm{\Omega }}$ satisfying (1.1) from $L^{p,{\phi }_1}({\mathbb{R}}^n)$ to $L^{p,{\phi }_2}({\mathbb{R}}^n)$ in [9] and gave the condition on the pair of $({\phi }_1,{\phi }_2)$

$${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r)$$

for the boundedness of $T_{\mathrm{\Omega },\alpha }$ satisfying (1.2) from $L^{p,{\phi }_1}({\mathbb{R}}^n)$ to $L^{q,{\phi }_2}({\mathbb{R}}^n)$ in [10], where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$.

Inspired by the above, we consider the boundedness of sublinear operators on the following generalized central Morrey spaces and give the λ-central bounded mean oscillation estimates for linear operator commutators.

Definition 1.1 Let $\phi (r)$ be a positive measurable function on ${\mathbb{R}}_{+}$ and $1<p<\mathrm{\infty }$. We denote by ${\dot{B}}^{p,\phi }({\mathbb{R}}^n)$ the generalized central Morrey spaces, the spaces of all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$ with finite quasinorm

$${\parallel f\parallel }_{{\dot{B}}^{p,\phi }({\mathbb{R}}^n)}=\underset{r>0}{sup}\phi {(r)}^{-1}{|B(0,r)|}^{-\frac{1}{p}}{\parallel f\parallel }_{L^p(B(0,r))}.$$

We can recover the spaces ${\dot{B}}^{p,\lambda }({\mathbb{R}}^n)$ under the choice $\phi (r)=r^{n\lambda }$.

Recall that in 1994 the doctoral thesis [11] by Guliyev (see also [12–15]) introduced the local Morrey-type space $L{M}_{p\theta ,\omega }$ given by

$${\parallel f\parallel }_{L{M}_{p\theta ,\omega }}={\parallel \omega (r){\parallel f\parallel }_{L^p(B(0,r))}\parallel }_{L^{\theta }(0,\mathrm{\infty })}<\mathrm{\infty },$$

where ω is a positive measurable function defined on $(0,\mathrm{\infty })$. The main purpose of [11] (also of [12–15]) is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type space $L{M}_{p\theta ,\omega }$. In a series of papers by Burenkov, H Guliyev and V Guliyev, etc. (see [16–21]), some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators and singular integral operators in local Morrey-type spaces $L{M}_{p\theta ,\omega }$ were given.

Particularly, if $\theta =\mathrm{\infty }$, $L{M}_{p\theta ,\omega }=L{M}_{p,\omega }$, then the generalized central Morrey spaces ${\dot{B}}^{p,\phi }({\mathbb{R}}^n)$ are the same spaces as the local Morrey spaces $L{M}_{p,\omega }$ with $\omega (r)=\phi {(r)}^{-1}{r}^{-\frac{n}{p}}$.

The following statements were proved in [11] (see also [14]).

Theorem A Let $1<p<\mathrm{\infty }$ and $({\phi }_1,{\phi }_2)$ satisfy the condition

$${\int }_r^{\mathrm{\infty }}{\phi }_1(t)\frac{dt}{t}\le C{\phi }_2(r),$$

where C does not depend on r. Then the Calderón-Zygmund operator T is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Theorem B Let $1<p<\mathrm{\infty }$, $0<\alpha <\frac{n}{p}$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ and $({\phi }_1,{\phi }_2)$ satisfy the condition

$${\int }_r^{\mathrm{\infty }}{t}^{\alpha }{\phi }_1(t)\frac{dt}{t}\le C{\phi }_2(r),$$

where C does not depend on r. Then the Riesz potential $I_{\alpha }$ is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{q,{\phi }_2}$.

From Lemmas 4.4 and 5.3 in [9] we get the following for the generalized central (local) Morrey spaces ${\dot{B}}^{p,\phi }$.

Theorem C Let $1<p<\mathrm{\infty }$, T be a sublinear operator satisfying that for any $f\in L^1({\mathbb{R}}^n)$ with compact support and $x\notin suppf$,

$$|Tf(x)|\le C{\int }_{{\mathbb{R}}^n}\frac{|f(y)|}{{|x-y|}^n}dy,$$

and bounded on $f\in L^p({\mathbb{R}}^n)$. Let also the pair $({\phi }_1,{\phi }_2)$ satisfy the condition

$${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}+1}}dt\le C{\phi }_2(r),$$

where C does not depend on r. Then the operator T is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Theorem D Let $1<p<\mathrm{\infty }$, $0<\alpha <\frac{n}{p}$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, $T_{\alpha }$ be a sublinear operator satisfying that for any $f\in L^1({\mathbb{R}}^n)$ with compact support and $x\notin suppf$,

$$|T_{\alpha }f(x)|\le C{\int }_{{\mathbb{R}}^n}\frac{|f(y)|}{{|x-y|}^{n-\alpha }}dy,$$

and bounded from $f\in L^p({\mathbb{R}}^n)$ to $f\in L^q({\mathbb{R}}^n)$. Let also the pair $({\phi }_1,{\phi }_2)$ satisfy the condition

$${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r),$$

where C does not depend on r. Then the operator $T_{\alpha }$ is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{q,{\phi }_2}$.

2 Sublinear operator with rough kernel

Theorem E Let ω be a positive weight function on $(0,\mathrm{\infty })$. The inequality

$$ess\underset{t>0}{sup}{\nu }_2(t)H_{\omega }g(t)\le cess\underset{t>0}{sup}{\nu }_1(t)g(t)$$

holds for all non-negative and non-increasing g on $(0,\mathrm{\infty })$ if and only if

$$A:=\underset{t>0}{sup}\frac{{\nu }_2(t)}{t}{\int }_0^t\frac{\omega (r)dr}{ess{sup}_{0<\tau <r}{\nu }_1(\tau )}<\mathrm{\infty },$$

and $c\approx A$, where the $H_{\omega }$ is the weighted Hardy operator

$$H_{\omega }g(t):=\frac{1}{t}{\int }_0^{t}g(r)\omega (r)dr,0<t<\mathrm{\infty }.$$

Note that Theorem E can be proved analogously to Theorem 1 in [22]; particularly, when $\omega \equiv 1$, it was proved in [23].

In this section we are going to discuss the boundedness of $T_{\mathrm{\Omega }}$ and $T_{\mathrm{\Omega },\alpha }$ on generalized central Morrey spaces.

Lemma 2.1 Let $1<p<\mathrm{\infty }$, $T_{\mathrm{\Omega }}$ be a sublinear operator and satisfy (1.1) with $\mathrm{\Omega }\in L^s(S^{n-1})$.

When $s^{\prime }\le p$ and $T_{\mathrm{\Omega }}$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$, then the inequality

$${\parallel T_{\mathrm{\Omega }}f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{-\frac{n}{p}-1}{\parallel f\parallel }_{L^p(B(0,t))}dt$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$; or $p<s$ and $T_{\mathrm{\Omega }}$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$, then the inequality

$${\parallel T_{\mathrm{\Omega }}f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{-\frac{n}{p}+\frac{n}{s}-1}{\parallel f\parallel }_{L^p(B(0,t))}dt$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$.

Proof Let $1<p<\mathrm{\infty }$. For any $r>0$, set $B=B(0,r)$ and $2B=B(0,2r)$. We write

$$f(x)=f(x){\chi }_{2B}(x)+f(x){\chi }_{{(2B)}^c}(x):=f_1(x)+f_2(x)$$

and have

$${\parallel T_{\mathrm{\Omega }}f\parallel }_{L^p(B)}\le {\parallel T_{\mathrm{\Omega }}{f}_1\parallel }_{L^p(B)}+{\parallel T_{\mathrm{\Omega }}{f}_2\parallel }_{L^p(B)}.$$

Since $T_{\mathrm{\Omega }}{f}_1$ is bounded on $L^p({\mathbb{R}}^n)$, it follows that

$${\parallel T_{\mathrm{\Omega }}{f}_1\parallel }_{L^p(B)}\le {\parallel T_{\mathrm{\Omega }}{f}_1\parallel }_{L^p({\mathbb{R}}^n)}\le C{\parallel f_1\parallel }_{L^p({\mathbb{R}}^n)}=C{\parallel f\parallel }_{L^p(2B)},$$

where the constant $C>0$ is independent of f.

It is known that $x\in B$, $y\in {(2B)}^c$, which implies $\frac{1}{2}|y|\le |x-y|\le \frac{3}{2}|y|$. Thus

$$|T_{\mathrm{\Omega }}{f}_2(x)|\le C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^n}.$$

• (i) When $s^{\prime }\le p$ and by Fubini’s theorem, we have

  $$\begin{array}{r}{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^n}\\ =C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|{\int }_{|y|}^{\mathrm{\infty }}\frac{dt}{t^{n+1}}dy\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |y|<t}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p}-\frac{1}{s}}\frac{dt}{t^{n+1}}\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{p}+1}}.\end{array}$$

Hence, for all $p\in (0,\mathrm{\infty })$, the inequality

$${\parallel T_{\mathrm{\Omega }}{f}_2\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{p}+1}}$$

holds.

• (ii) When $p<s$, by Fubini’s theorem and the Minkowski inequality, we get

  $$\begin{array}{rcl}{\parallel T_{\mathrm{\Omega }}{f}_2\parallel }_{L^p(B)}& \le & {\left({\int }_B{\left|{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\right|}^{p}dx\right)}^{\frac{1}{p}}\\ \le & {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dx\right)}^{\frac{1}{s}}{|B|}^{\frac{1}{p}-\frac{1}{s}}dy\frac{dt}{t^{n+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{p}-\frac{n}{s}+1}}.\end{array}$$

On the other hand, for any $q>0$, we have

$${\parallel f\parallel }_{L^p(2B)}=C{r}^{\frac{n}{q}}{\parallel f\parallel }_{L^p(2B)}{\int }_{2r}^{\mathrm{\infty }}\frac{dt}{t^{\frac{n}{q}+1}}\le C{r}^{\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.$$

Combining the above estimates, we complete the proof of Lemma 2.1. □

Theorem 2.2 Let $1<p<\mathrm{\infty }$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. Let $T_{\mathrm{\Omega }}$ be a sublinear operator satisfying (1.1) and bounded on $L^p({\mathbb{R}}^n)$ for $p>1$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $p<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $T_{\mathrm{\Omega }}$ is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Proof When $s^{\prime }\le p$, by Lemma 2.1 and Theorem E, for $\omega \equiv 1$, $p>1$, we have

$$\begin{array}{rcl}{\parallel T_{\mathrm{\Omega }}f\parallel }_{{\dot{B}}^{p,{\phi }_2}}& \le & C\underset{r>0}{sup}{\phi }_2{(r)}^{-1}{\int }_r^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{p}+1}}\\ =& C\underset{r>0}{sup}{\phi }_2{(r)}^{-1}{\int }_0^{r^{-\frac{n}{p}}}{\parallel f\parallel }_{L^p(B(0,t^{-\frac{p}{n}}))}dt\\ =& C\underset{r>0}{sup}{\phi }_2{\left(r^{-\frac{p}{n}}\right)}^{-1}{\int }_0^r{\parallel f\parallel }_{L^p(B(0,t^{-\frac{p}{n}}))}dt\\ \le & C\underset{r>0}{sup}{\phi }_1{\left(r^{-\frac{p}{n}}\right)}^{-1}r{\parallel f\parallel }_{L^p(B(0,r^{-\frac{p}{n}}))}=C{\parallel f\parallel }_{{\dot{B}}^{p,{\phi }_1}}.\end{array}$$

For the case of $p<s$, we can use the same method to prove the desirable conclusion. □

The Calderón-Zygmund operator with rough kernel ${\tilde{T}}_{\mathrm{\Omega }}$ has the following integral expression:

$${\tilde{T}}_{\mathrm{\Omega }}f(x)={\int }_{{\mathbb{R}}^n}K(x,y)f(y)\mathrm{\Omega }(y)dy$$

for any test function f and $x\notin suppf$. The kernel is a locally integral function defined away from the diagonal satisfying the size condition

$$K(x,y)\le C\frac{1}{{|x-y|}^n}$$

for all $x,y\in {\mathbb{R}}^n$ and $x\ne y$.

$f\in L_{\mathrm{loc}}^1$, the rough Hardy-Littlewood maximal function $M_{\mathrm{\Omega }}$ is defined by

$$M_{\mathrm{\Omega }}f(x)=\underset{t>0}{sup}\frac{1}{|B(x,t)|}{\int }_{B(x,t)}|\mathrm{\Omega }(y)||f(y)|dy.$$

Then we can get the following corollary.

Corollary 2.3 Let $1<p<\mathrm{\infty }$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}+1}}\le C{\phi }_2(r),$$

• (ii) when $p<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{p}-\frac{n}{s}+1}}\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then $M_{\mathrm{\Omega }}$ and ${\tilde{T}}_{\mathrm{\Omega }}$ are both bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

In the following statements, the boundedness of $T_{\mathrm{\Omega },\alpha }$ satisfying (1.2) in generalized central Morrey spaces is proved.

Lemma 2.4 Let $0<\alpha <n$, $1<p<\frac{n}{\alpha }$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, $T_{\mathrm{\Omega },\alpha }$ be a sublinear operator and satisfy (1.2) with $\mathrm{\Omega }\in L^s(S^{n-1})$.

When $s^{\prime }\le p$ and $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$, then the inequality

$${\parallel T_{\mathrm{\Omega },\alpha }f\parallel }_{L^q(B(0,r))}\le C{r}^{\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{t}^{-\frac{n}{q}-1}{\parallel f\parallel }_{L^p(B(0,t))}dt$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$; or $q<s$ and $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$, then the inequality

$${\parallel T_{\mathrm{\Omega },\alpha }f\parallel }_{L^q(B(0,r))}\le C{r}^{\frac{n}{q}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{-\frac{n}{q}+\frac{n}{s}-1}{\parallel f\parallel }_{L^p(B(0,t))}dt$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^p({\mathbb{R}}^n)$.

Proof Let $0<\alpha <n$, $1<p<\frac{n}{\alpha }$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$. For any $r>0$, set $B=B(0,r)$ and $2B=B(0,2r)$. We write

$$f(x)=f(x){\chi }_{2B}(x)+f(x){\chi }_{{(2B)}^c}(x):=f_1(x)+f_2(x)$$

and have

$${\parallel T_{\mathrm{\Omega },\alpha }f\parallel }_{L^q(B)}\le {\parallel T_{\mathrm{\Omega },\alpha }{f}_1\parallel }_{L^q(B)}+{\parallel T_{\mathrm{\Omega },\alpha }{f}_2\parallel }_{L^q(B)}.$$

Since $T_{\mathrm{\Omega },\alpha }{f}_1$ is bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$, it follows that

$${\parallel T_{\mathrm{\Omega },\alpha }{f}_1\parallel }_{L^q(B)}\le {\parallel T_{\mathrm{\Omega },\alpha }{f}_1\parallel }_{L^q({\mathbb{R}}^n)}\le C{\parallel f_1\parallel }_{L^p({\mathbb{R}}^n)}=C{\parallel f\parallel }_{L^p(2B)},$$

where the constant $C>0$ is independent of f.

Since $x\in B$, $y\in {(2B)}^c$, thus

$$|T_{\mathrm{\Omega },\alpha }{f}_2(x)|\le C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^{n-\alpha }}.$$

• (i) When $s^{\prime }\le p$ and by Fubini’s theorem, we have

  $$\begin{array}{r}{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^{n-\alpha }}\\ =C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|{\int }_{|y|}^{\mathrm{\infty }}\frac{dt}{t^{n+1-\alpha }}dy\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |y|<t}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1-\alpha }}\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p}-\frac{1}{s}}\frac{dt}{t^{n+1-\alpha }}\\ \le C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.\end{array}$$

Hence, for all $p\in (0,\mathrm{\infty })$, the inequality

$${\parallel T_{\mathrm{\Omega },\alpha }{f}_2\parallel }_{L^q(B)}\le C{r}^{\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}$$

holds.

• (ii) When $q<s$, by Fubini’s theorem and the Minkowski inequality, we get

  $$\begin{array}{rcl}{\parallel T_{\mathrm{\Omega },\alpha }{f}_2\parallel }_{L^p(B)}& \le & {\left({\int }_B{\left|{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1-\alpha }}\right|}^{q}dx\right)}^{\frac{1}{q}}\\ \le & {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{q}dx\right)}^{\frac{1}{q}}dy\frac{dt}{t^{n+1-\alpha }}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dx\right)}^{\frac{1}{s}}{|B|}^{\frac{1}{q}-\frac{1}{s}}dy\frac{dt}{t^{n+1-\alpha }}\\ \le & C{r}^{\frac{n}{q}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}+1-\alpha }}\\ \le & C{r}^{\frac{n}{q}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}.\end{array}$$

Similarly, combining the above estimates, we finish this proof. □

Theorem 2.5 Let $0<\alpha <n$, $1<p<\frac{n}{\alpha }$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. Let $T_{\mathrm{\Omega },\alpha }$ be a sublinear operator $T_{\mathrm{\Omega }}$ satisfying (1.2) and bounded from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $q<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $T_{\mathrm{\Omega },\alpha }$ is bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{q,{\phi }_2}$.

Proof When $s^{\prime }\le p$, by Lemma 2.1 and Theorem E, for $1<p<\frac{n}{\alpha }$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$, we have

$$\begin{array}{rcl}{\parallel T_{\mathrm{\Omega },\alpha }f\parallel }_{{\dot{B}}^{q,{\phi }_2}}& \le & C\underset{r>0}{sup}{\phi }_2{(r)}^{-1}{\int }_r^{\mathrm{\infty }}{\parallel f\parallel }_{L^p(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}\\ =& C\underset{r>0}{sup}{\phi }_2{(r)}^{-1}{\int }_0^{r^{-\frac{n}{q}}}{\parallel f\parallel }_{L^p(B(0,t^{-\frac{q}{n}}))}dt\\ =& C\underset{r>0}{sup}{\phi }_2{\left(r^{-\frac{q}{n}}\right)}^{-1}{\int }_0^r{\parallel f\parallel }_{L^p(B(0,t^{-\frac{q}{n}}))}dt\\ \le & C\underset{r>0}{sup}{\phi }_1{\left(r^{-\frac{q}{n}}\right)}^{-1}{r}^{\frac{p}{q}}{\parallel f\parallel }_{L^p(B(0,r^{-\frac{q}{n}}))}=C{\parallel f\parallel }_{{\dot{B}}^{p,{\phi }_1}}.\end{array}$$

For the case of $q<s$, we can also use the same method to prove the desirable conclusion. □

$f\in L_{\mathrm{loc}}^1$, the rough fractional maximal function $M_{\mathrm{\Omega },\alpha }$ and the rough fractional integral $I_{\mathrm{\Omega },\alpha }$ are defined by

$$\begin{array}{c}{M}_{\mathrm{\Omega },\alpha }f(x)=\underset{t>0}{sup}\frac{1}{{|B(x,t)|}^{1+\alpha }}{\int }_{B(x,t)}|\mathrm{\Omega }(y)||f(y)|dy,\hfill \\ I_{\mathrm{\Omega },\alpha }f(x)=\underset{t>0}{sup}{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\Omega }(y)f(y)}{{|x-y|}^{n-\alpha }}dy\hfill \end{array}$$

for $0<\alpha <n$.

Corollary 2.6 Let $0<\alpha <n$, $1<p<\frac{n}{\alpha }$, $\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $q<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p}}}{t^{\frac{n}{q}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then $M_{\mathrm{\Omega },\alpha }$ and $I_{\mathrm{\Omega },\alpha }$ are both bounded from ${\dot{B}}^{p,{\phi }_1}$ to ${\dot{B}}^{q,{\phi }_2}$.

Remark 1 When $s=\mathrm{\infty }$, the comments in Theorem 2.2 and in Theorem 2.5 can be obtained from Lemmas 4.4 and 5.3 in [9].

3 The commutators of a linear operator with rough kernel

Let $\tilde{T}$ be a Calderón-Zygmund singular integral operator and $b\in \mathit{BMO}({\mathbb{R}}^n)$. The commutator operator $[\tilde{T},b]$ is defined by

$$[\tilde{T},b]f(x)=b(x)\tilde{T}f(x)-\tilde{T}(bf)(x).$$

A well-known result of Coifman et al. [24] states that the commutator $[\tilde{T},b]$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$ if and only if $b\in \mathit{BMO}({\mathbb{R}}^n)$.

Since $\mathit{BMO}\subset {\bigcap }_{p>1}{\mathit{CBMO}}^{p,\lambda }$ when $\lambda =0$, if we only assume $b\in {\mathit{CBMO}}^{p,\lambda }$, then $[\tilde{T},b]$ may not be a bounded operator on $L^p({\mathbb{R}}^n)$. However, it has some boundedness properties on other spaces. As a matter of fact, in [25] and [26], they considered the commutators with $b\in {\mathit{CBMO}}^{p,\lambda }$. Here we also obtain some boundedness of the commutators with $b\in {\mathit{CBMO}}^{p,\lambda }$ on generalized central Morrey spaces.

We need the following statement on the boundedness of the Hardy-type operator

$$H_{1}g(t):=\frac{1}{t}{\int }_0^{t}ln(1+\frac{t}{r})g(r)dr,0<t<\mathrm{\infty }.$$

Theorem F [27]

The inequality

$$ess\underset{t>0}{sup}{\nu }_2(t)H_{1}g(t)\le cess\underset{t>0}{sup}{\nu }_1(t)g(t)$$

holds for all non-negative and non-increasing g on $(0,\mathrm{\infty })$ if and only if

$$A:=\underset{t>0}{sup}\frac{{\nu }_2(t)}{t}{\int }_0^{t}ln(1+\frac{t}{r})\frac{dr}{ess{sup}_{0<\tau <r}{\nu }_1(\tau )}<\mathrm{\infty },$$

and $c\approx A$.

Lemma 3.1 Let $1<p<\mathrm{\infty }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, $T_{\mathrm{\Omega }}$ is a sublinear operator and satisfies (1.1) with $\mathrm{\Omega }\in L^s(S^{n-1})$.

When $s^{\prime }\le p$ and $T_{\mathrm{\Omega }}$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$, then the inequality

$${\parallel [T_{\mathrm{\Omega }},b]f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^{p_1}({\mathbb{R}}^n)$; or $p_1<s$ and $T_{\mathrm{\Omega }}$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$, then the inequality

$${\parallel [T_{\mathrm{\Omega }},b]f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}$$

holds for any ball $B(0,r)$ and for all $f\in L_{\mathrm{loc}}^{p_1}({\mathbb{R}}^n)$.

Proof Let $1<p<\mathrm{\infty }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. For any $r>0$, set $B=B(0,r)$ and $2B=B(0,2r)$. We can write

$$f(x)=f(x){\chi }_{2B}(x)+f(x){\chi }_{{(2B)}^c}(x):=f_1(x)+f_2(x)$$

and

$$\begin{array}{rcl}[T_{\mathrm{\Omega }},b]f(x)& =& (b(x)-b_B)T_{\mathrm{\Omega }}{f}_1(x)-T_{\mathrm{\Omega }}\left((b(\cdot )-b_B)f_1\right)(x)\\ +(b(x)-b_B)T_{\mathrm{\Omega }}{f}_2(x)-T_{\mathrm{\Omega }}\left((b(\cdot )-b_B)f_2\right)(x)\\ =:& I_1+I_2+I_3+I_4.\end{array}$$

Hence, we have

$${\parallel [T_{\mathrm{\Omega }},b]f\parallel }_{L^p(B)}\le {\parallel I_1\parallel }_{L^p(B)}+{\parallel I_2\parallel }_{L^p(B)}+{\parallel I_3\parallel }_{L^p(B)}+{\parallel I_4\parallel }_{L^p(B)}.$$

Since $T_{\mathrm{\Omega }}$ is bounded on $L^p({\mathbb{R}}^n)$, it follows that

$$\begin{array}{rcl}{\parallel I_1\parallel }_{L^p(B)}& =& {\left({\int }_B{|b(x)-b_B|}^p{|T_{\mathrm{\Omega }}{f}_1(x)|}^{p}dx\right)}^{\frac{1}{p}}\\ \le & {\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|T_{\mathrm{\Omega }}{f}_1(x)|}^{p_1}dx\right)}^{\frac{1}{p_1}}\\ \le & C{r}^{\frac{n}{p_2}+n\lambda }{\parallel b\parallel }_{{\mathit{CBMO}}^{p_2,\lambda }}{\parallel f\parallel }_{L^{p_1}(2B)},\end{array}$$

where the constant $C>0$ is independent of f.

For $I_2$, we have

$$\begin{array}{rl}{\parallel I_2\parallel }_{L^p(B)}& ={\left({\int }_B{|T_{\mathrm{\Omega }}\left((b(\cdot )-b_B)f_1\right)(x)|}^{p}dx\right)}^{\frac{1}{p}}\\ \le C{\left({\int }_{{\mathbb{R}}^n}{|b(x)-b_B|}^p{|f_1(x)|}^{p}dx\right)}^{\frac{1}{p}}\\ \le C{\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|f(x)|}^{p_1}dx\right)}^{\frac{1}{p_1}}\\ \le C{r}^{\frac{n}{p_2}+n\lambda }{\parallel b\parallel }_{{\mathit{CBMO}}^{p_2,\lambda }}{\parallel f\parallel }_{L^{p_1}(2B)}.\end{array}$$

For $I_3$, it is known that $x\in B$, $y\in {(2B)}^c$, which implies $\frac{1}{2}|y|\le |x-y|\le \frac{3}{2}|y|$.

• (i) When $s^{\prime }\le p$ and by Fubini’s theorem, we have

  $$\begin{array}{rcl}|T_{\mathrm{\Omega }}{f}_2(x)|& \le & C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^n}\\ =& C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|{\int }_{|y|}^{\mathrm{\infty }}\frac{dt}{t^{n+1}}dy\\ =& C{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |y|<t}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p_1}-\frac{1}{s}}\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}},\end{array}$$

thus

$$\begin{array}{rcl}{\parallel I_3\parallel }_{L^p(B)}& \le & C{\left({\int }_B{|b(x)-b_B|}^{p}dx\right)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}\\ \le & C{r}^{\frac{n}{p}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}\\ \le & C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}.\end{array}$$

• (ii) When $p_1<s$, by Fubini’s theorem and the Minkowski inequality, we get

  $$\begin{array}{rcl}{\parallel I_3\parallel }_{L^p(B)}& \le & {\left({\int }_B{\left|{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(x)-b_B||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\right|}^{p}dx\right)}^{\frac{1}{p}}\\ \le & {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|b(x)-b_B|}^p{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n+1}}\\ \le & {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p_1}dx\right)}^{\frac{1}{p_1}}dy\frac{dt}{t^{n+1}}\\ \le & C{r}^{\frac{n}{p_2}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dx\right)}^{\frac{1}{s}}{|B|}^{\frac{1}{p_1}-\frac{1}{s}}dy\frac{dt}{t^{n+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}.\end{array}$$

On the other hand, for $I_4$, by Fubini’s theorem, we have

$$\begin{array}{rcl}|[T_{\mathrm{\Omega }},b]f_2(x)|& \le & C{\int }_{{(2B)}^c}|b(y)-b_B||f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^n}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_B||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ +C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b_{B(0,r)}-b_{B(0,t)}||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ =:& I_{41}+I_{42}.\end{array}$$

• (i) When $s^{\prime }\le p$, we obtain

  $$\begin{array}{rcl}{I}_{41}& \le & C{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^p{|f(y)|}^{p}dy\right)}^{\frac{1}{p}}\\ \cdot {\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p}-\frac{1}{s}}\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^{p_2}dy\right)}^{\frac{1}{p_2}}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p}+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}},\end{array}$$

then

$${\parallel I_{41}\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}.$$

Moreover,

$$\begin{array}{rcl}{I}_{42}& \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\left({\int }_{B(0,t)}{|f(y)|}^{p_1}dy\right)}^{\frac{1}{p_1}}\\ \cdot {\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p_1}-\frac{1}{s}}\frac{dt}{t^{n+1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}},\end{array}$$

then

$${\parallel I_{42}\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}.$$

By estimating $I_{41}$ and $I_{42}$, we obtain

$${\parallel I_4\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}+1}}.$$

• (ii) When $p_1<s$, by the Minkowski inequality, we get

  $$\begin{array}{rcl}{\parallel I_{41}\parallel }_{L^p(B)}& \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)|dy\frac{dt}{t^{n-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^{p_2}dy\right)}^{\frac{1}{p_2}}{\parallel f\parallel }_{L^{p_1}(B(0,t))}{t}^{n-\frac{n}{p}}\frac{dt}{t^{n-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}\end{array}$$

and

$$\begin{array}{rcl}{\parallel I_{42}\parallel }_{L^p(B)}& \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}.\end{array}$$

Hence, we have

$${\parallel I_4\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}.$$

Moreover, for any $q>0$, we have

$$\begin{array}{rcl}{r}^{\frac{n}{p_2}+n\lambda }{\parallel f\parallel }_{L^{p_1}(2B)}& =& C{r}^{\frac{n}{p_2}+n\lambda +\frac{n}{q}}{\parallel f\parallel }_{L^{p_1}(2B)}{\int }_{2r}^{\mathrm{\infty }}\frac{dt}{t^{\frac{n}{q}+1}}\\ \le & C{r}^{\frac{n}{p_2}+\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}\\ \le & C{r}^{\frac{n}{p_2}+\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.\end{array}$$

Now combining all the above estimates, we end the proof. □

Then we have the following conclusions.

Theorem 3.2 Let $1<p<\mathrm{\infty }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. Let $T_{\mathrm{\Omega }}$ be a linear operator satisfying (1.1) and bounded on $L^p({\mathbb{R}}^n)$ for $p>1$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{p_1}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $p_1<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $[T_{\mathrm{\Omega }},b]$ is bounded from ${\dot{B}}^{p_1,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Corollary 3.3 Let $1<p<\mathrm{\infty }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $\mathrm{\Omega }\in L^s(S^{n-1})$. If either of the two conditions

• (i) when $s^{\prime }\le p$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{p_1}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $p_1<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{p_1}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $[{\tilde{T}}_{\mathrm{\Omega }},b]$ is bounded from ${\dot{B}}^{p_1,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

About the commutator of linear operator $T_{\mathrm{\Omega },\alpha }$ satisfying (1.2), we get the following corresponding results.

Lemma 3.4 Let $0<\alpha <n$, $1<p_1<\frac{n}{\alpha }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$, $p_1^{\prime }<p_2<\mathrm{\infty }$ and let

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}-\frac{\alpha }{n},\frac{1}{q}=\frac{1}{p_1}-\frac{\alpha }{n},\frac{1}{h}=\frac{1}{p_1}+\frac{1}{p_2}.$$

$T_{\mathrm{\Omega }-\alpha }$ is a sublinear operator and satisfies (1.2) with $\mathrm{\Omega }\in L^s(S^{n-1})$.

When $s^{\prime }\le h$, $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^t({\mathbb{R}}^n)$ to $L^m({\mathbb{R}}^n)$ for any $1<t<\frac{n}{\alpha }$ and $\frac{1}{m}=\frac{1}{t}-\frac{\alpha }{n}$, then the inequality

$${\parallel [T_{\mathrm{\Omega },\alpha },b]f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}$$

holds for any ball $B(0,r)$ and all $f\in L_{\mathrm{loc}}^{p_1}({\mathbb{R}}^n)$; or $q<s$, $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^t({\mathbb{R}}^n)$ to $L^m({\mathbb{R}}^n)$ for any $1<t<\frac{n}{\alpha }$ and $\frac{1}{m}=\frac{1}{t}-\frac{\alpha }{n}$, then the inequality

$${\parallel [T_{\mathrm{\Omega },\alpha },b]f\parallel }_{L^p(B(0,r))}\le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}$$

holds for any ball $B(0,r)$ and all $f\in L_{\mathrm{loc}}^{p_1}({\mathbb{R}}^n)$.

Proof Let $0<\alpha <n$, $1<p_1<\frac{n}{\alpha }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$. For any $r>0$, set $B=B(0,r)$ and $2B=B(0,2r)$. We also write

$$f(x)=f(x){\chi }_{2B}(x)+f(x){\chi }_{{(2B)}^c}(x):=f_1(x)+f_2(x)$$

and

$$\begin{array}{rcl}[T_{\mathrm{\Omega },\alpha },b]f(x)& =& (b(x)-b_B)T_{\mathrm{\Omega },\alpha }{f}_1(x)-T_{\mathrm{\Omega },\alpha }\left((b(\cdot )-b_B)f_1\right)(x)\\ +(b(x)-b_B)T_{\mathrm{\Omega },\alpha }{f}_2(x)-T_{\mathrm{\Omega },\alpha }\left((b(\cdot )-b_B)f_2\right)(x)\\ =:& {\mathit{II}}_1+{\mathit{II}}_2+{\mathit{II}}_3+{\mathit{II}}_4.\end{array}$$

Hence, we have

$${\parallel [T_{\mathrm{\Omega },\alpha },b]f\parallel }_{L^p(B)}\le {\parallel {\mathit{II}}_1\parallel }_{L^p(B)}+{\parallel {\mathit{II}}_2\parallel }_{L^p(B)}+{\parallel {\mathit{II}}_3\parallel }_{L^p(B)}+{\parallel {\mathit{II}}_4\parallel }_{L^p(B)}.$$

Since $T_{\mathrm{\Omega },\alpha }$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ and $\frac{1}{q}=\frac{1}{p_1}-\frac{\alpha }{n}$, it follows that

$$\begin{array}{rcl}{\parallel {\mathit{II}}_1\parallel }_{L^p(B)}& =& {\left({\int }_B{|b(x)-b_B|}^p{|T_{\mathrm{\Omega },\alpha }{f}_1(x)|}^{p}dx\right)}^{\frac{1}{p}}\\ \le & {\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|T_{\mathrm{\Omega },\alpha }{f}_1(x)|}^{q}dx\right)}^{\frac{1}{q}}\\ \le & C{r}^{\frac{n}{p_2}+n\lambda }{\parallel b\parallel }_{{\mathit{CBMO}}^{p_2,\lambda }}{\parallel f\parallel }_{L^{p_1}(2B)},\end{array}$$

where the constant $C>0$ is independent of f.

For ${\mathit{II}}_2$, $\frac{1}{p}=\frac{1}{h}-\frac{\alpha }{n}$ and $\frac{1}{h}=\frac{1}{p_1}+\frac{1}{p_2}$,

$$\begin{array}{rcl}{\parallel {\mathit{II}}_2\parallel }_{L^p(B)}& =& {\left({\int }_B{|T_{\mathrm{\Omega },\alpha }\left((b(\cdot )-b_B)f_1\right)(x)|}^{p}dx\right)}^{\frac{1}{p}}\\ \le & {\left({\int }_{{\mathbb{R}}^n}{|b(x)-b_B|}^h{|f_1(x)|}^{h}dx\right)}^{\frac{1}{h}}\\ \le & {\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|f(x)|}^{p_1}dx\right)}^{\frac{1}{p_1}}\\ \le & C{r}^{\frac{n}{p_2}+n\lambda }{\parallel b\parallel }_{{\mathit{CBMO}}^{p_2,\lambda }}{\parallel f\parallel }_{L^{p_1}(2B)}.\end{array}$$

For $I_3$,

• (i) when $s^{\prime }\le h$, by Fubini’s theorem, since $x\in B$, we have

  $$\begin{array}{rcl}|T_{\mathrm{\Omega },\alpha }{f}_2(x)|& \le & C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^{n-\alpha }}\\ =& C{\int }_{{(2B)}^c}|f(y)||\mathrm{\Omega }(x-y)|{\int }_{|y|}^{\mathrm{\infty }}\frac{dt}{t^{n-\alpha +1}}dy\\ =& C{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |y|<t}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p_1}-\frac{1}{s}}\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}},\end{array}$$

thus

$$\begin{array}{rcl}{\parallel I_3\parallel }_{L^p(B)}& \le & C{\left({\int }_B{|b(x)-b_B|}^{p}dx\right)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}\\ \le & C{\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{r}^{\frac{n}{q}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}\\ \le & C{r}^{\frac{n}{p}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}\\ \le & C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}};\end{array}$$

• (ii) when $q<s$, by Fubini’s theorem and the Minkowski inequality, we get

  $$\begin{array}{r}{\parallel I_3\parallel }_{L^p(B)}\\ \le {\left({\int }_B{\left|{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(x)-b_B||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\right|}^{p}dx\right)}^{\frac{1}{p}}\\ \le {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|b(x)-b_B|}^p{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n-\alpha +1}}\\ \le {\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|b(x)-b_B|}^{p_2}dx\right)}^{\frac{1}{p_2}}{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{q}dx\right)}^{\frac{1}{q}}dy\frac{dt}{t^{n-\alpha +1}}\\ \le C{r}^{\frac{n}{p_2}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|{\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dx\right)}^{\frac{1}{s}}{|B|}^{\frac{1}{q}-\frac{1}{s}}dy\frac{dt}{t^{n-\alpha +1}}\\ \le C{r}^{\frac{n}{p}-\frac{n}{s}+n\lambda }{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}-\alpha +1}}\\ \le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}.\end{array}$$

On the other hand, for $I_4$, by Fubini’s theorem, we have

$$\begin{array}{rcl}|[T_{\mathrm{\Omega },\alpha },b]f_2(x)|& \le & C{\int }_{{(2B)}^c}|b(y)-b_B||f(y)||\mathrm{\Omega }(x-y)|\frac{dy}{{|y|}^{n-\alpha }}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_B||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\\ +C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b_{B(0,r)}-b_{B(0,t)}||f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\\ =:& I_{41}+I_{42}.\end{array}$$

• (i) When $s^{\prime }\le h$, we obtain

  $$\begin{array}{rcl}{I}_{41}& \le & C{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^h{|f(y)|}^{h}dy\right)}^{\frac{1}{h}}\\ \cdot {\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{h}-\frac{1}{s}}\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^{p_2}dy\right)}^{\frac{1}{p_2}}{\parallel f\parallel }_{L^{p_1}(B(0,t))}{t}^{n-\frac{n}{h}}\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}},\end{array}$$

then

$${\parallel I_{41}\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.$$

For $I_{42}$, we have

$$\begin{array}{rcl}{I}_{42}& \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)||\mathrm{\Omega }(x-y)|dy\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\left({\int }_{B(0,t)}{|f(y)|}^{p_1}dy\right)}^{\frac{1}{p_1}}\\ \cdot {\left({\int }_{B(0,t)}{|\mathrm{\Omega }(x-y)|}^{s}dy\right)}^{\frac{1}{s}}{|B(0,t)|}^{1-\frac{1}{p_1}-\frac{1}{s}}\frac{dt}{t^{n-\alpha +1}}\\ \le & C{\int }_{2r}^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}},\end{array}$$

then

$${\parallel I_{42}\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.$$

Then, by estimating $I_{41}$ and $I_{42}$, we obtain

$${\parallel I_4\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}+1}}.$$

• (ii) When $q<s$, by the Minkowski inequality, we get

  $$\begin{array}{rcl}{\parallel I_{41}\parallel }_{L^p(B)}& \le & C{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n-\alpha +1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(0,t)}|b(y)-b_{B(0,t)}||f(y)|dy\frac{dt}{t^{n-\frac{n}{s}-\alpha +1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(0,t)}{|b(y)-b_{B(0,t)}|}^{p_2}dy\right)}^{\frac{1}{p_2}}{\parallel f\parallel }_{L^{p_1}(B(0,t))}{t}^{n-\frac{n}{h}}\frac{dt}{t^{n-\frac{n}{s}-\alpha +1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}\end{array}$$

and

$$\begin{array}{rcl}{\parallel I_{42}\parallel }_{L^p(B)}& \le & C{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)|{\left({\int }_B{|\mathrm{\Omega }(x-y)|}^{p}dx\right)}^{\frac{1}{p}}dy\frac{dt}{t^{n-\alpha +1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\int }_{B(0,t)}|f(y)|dy\frac{dt}{t^{n-\frac{n}{s}-\alpha +1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}|b_{B(0,r)}-b_{B(0,t)}|{\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}\\ \le & C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}.\end{array}$$

Hence, we have

$${\parallel I_4\parallel }_{L^p(B)}\le C{r}^{\frac{n}{p}-\frac{n}{s}}{\int }_{2r}^{\mathrm{\infty }}{t}^{n\lambda }(1+ln\frac{t}{r}){\parallel f\parallel }_{L^{p_1}(B(0,t))}\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}.$$

Then we end this proof. □

Theorem 3.5 Let $0<\alpha <n$, $1<p_1<\frac{n}{\alpha }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$, $p_1^{\prime }<p_2<\mathrm{\infty }$ and

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}-\frac{\alpha }{n},\frac{1}{q}=\frac{1}{p_1}-\frac{\alpha }{n},\frac{1}{h}=\frac{1}{p_1}+\frac{1}{p_2}.$$

Let $T_{\mathrm{\Omega },\alpha }$ be a linear operator satisfying (1.2) with $\mathrm{\Omega }\in L^s(S^{n-1})$, which is bounded from $L^t({\mathbb{R}}^n)$ to $L^m({\mathbb{R}}^n)$ for any $1<t<\frac{n}{\alpha }$, $\frac{1}{m}=\frac{1}{t}-\frac{\alpha }{n}$. If either of the two conditions

• (i) when $s^{\prime }\le h$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $q<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{q}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $[T_{\mathrm{\Omega },\alpha },b]$ is bounded from ${\dot{B}}^{p_1,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Corollary 3.6 Let $0<\alpha <n$, $1<p_1<\frac{n}{\alpha }$, $b\in {\mathit{CBMO}}^{p_2,\lambda }$, $0<\lambda <\frac{1}{n}$, $p_1^{\prime }<p_2<\mathrm{\infty }$, $\mathrm{\Omega }\in L^s(S^{n-1})$ and

$$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}-\frac{\alpha }{n},\frac{1}{q}=\frac{1}{p_1}-\frac{\alpha }{n},\frac{1}{h}=\frac{1}{p_1}+\frac{1}{p_2}.$$

If either of the two conditions

• (i) when $s^{\prime }\le h$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{q}+1}}dt\le C{\phi }_2(r),$$

• (ii) when $q<s$, $({\phi }_1,{\phi }_2)$ satisfies the condition

  $${\int }_r^{\mathrm{\infty }}(1+ln\frac{t}{r})t^{n\lambda }\frac{ess{inf}_{t<\tau <\mathrm{\infty }}{\phi }_1(\tau ){\tau }^{\frac{n}{p_1}}}{t^{\frac{n}{q}-\frac{n}{s}+1}}dt\le C{\phi }_2(r)r^{\frac{n}{s}}$$

is satisfied, then the operator $[I_{\mathrm{\Omega },\alpha },b]$ is bounded from ${\dot{B}}^{p_1,{\phi }_1}$ to ${\dot{B}}^{p,{\phi }_2}$.

Remark 2 In our main results, if we let ${\phi }_1=r^{n{\lambda }_1}$ and ${\phi }_2=r^{n{\lambda }_2}$, then by calculating we can recover some known results in [7] and [25].

Author’s contributions

The author contributed to all the main results in this paper.