# Commutators of intrinsic square functions on generalized Morrey spaces

## Abstract

In this paper, we obtain the boundedness of intrinsic square functions and their commutators generated with BMO functions on generalized Morrey spaces. Our theorems extend some well-known results.

MSC:42B20, 42B35.

## 1 Introduction

The intrinsic square functions were first introduced by Wilson in [1, 2]. They are defined as follows. For $0<\alpha \le 1$, let ${\mathcal{C}}_{\alpha }$ be the family of functions $\varphi :{\mathbb{R}}^{n}↦\mathbb{R}$ such that ϕ’s support is contained in $\left\{x:|x|\le 1\right\}$, $\int \varphi \phantom{\rule{0.2em}{0ex}}dx=0$, and for $x,{x}^{\prime }\in {\mathbb{R}}^{n}$,

$|\varphi \left(x\right)-\varphi \left({x}^{\prime }\right)|\le {|x-{x}^{\prime }|}^{\alpha }.$

For $\left(y,t\right)\in {\mathbb{R}}_{+}^{n+1}$ and $f\in {L}_{\mathrm{loc}}^{1}\left({\mathbb{R}}^{n}\right)$, set

${A}_{\alpha }f\left(t,y\right)\equiv \underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|f\ast {\varphi }_{t}\left(y\right)|,$

where ${\varphi }_{t}\left(y\right)={t}^{-n}\varphi \left(\frac{y}{t}\right)$. Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula

${G}_{\alpha ,\beta }\left(f\right)\left(x\right)={\left(\int {\int }_{{\mathrm{\Gamma }}_{\beta }\left(x\right)}{\left({A}_{\alpha }f\left(t,y\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}},$

where ${\mathrm{\Gamma }}_{\beta }\left(x\right)=\left\{\left(y,t\right)\in {\mathbb{R}}_{+}^{n+1}:|x-y|<\beta t\right\}$. Denote ${G}_{\alpha ,1}\left(f\right)={G}_{\alpha }\left(f\right)$.

This function is independent of any particular kernel, such as Poisson kernel. It dominates pointwise the classical square function (Lusin area integral) and its real-variable generalizations. Although the function ${G}_{\alpha ,\beta }\left(f\right)$ depends on the kernels with uniform compact support, there is a pointwise relation between ${G}_{\alpha ,\beta }\left(f\right)$ with different β ($\beta \ge 1$):

${G}_{\alpha ,\beta }\left(f\right)\left(x\right)\le {\beta }^{\frac{3n}{2}+\alpha }{G}_{\alpha }\left(f\right)\left(x\right).$

We refer for details to [1].

The intrinsic Littlewood-Paley g-function and the intrinsic ${g}_{\lambda }^{\ast }$-function are defined, respectively, by

$\begin{array}{c}{g}_{\alpha }f\left(x\right)={\left({\int }_{0}^{\mathrm{\infty }}{\left({A}_{\alpha }f\left(t,y\right)\right)}^{2}\frac{dt}{t}\right)}^{\frac{1}{2}},\hfill \\ {g}_{\lambda ,\alpha }^{\ast }f\left(x\right)={\left(\int {\int }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{\left({A}_{\alpha }f\left(t,y\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}.\hfill \end{array}$

In [1], Wilson proved the following result.

Theorem A Let $1, $0<\alpha \le 1$, then ${G}_{\alpha }$ is bounded from ${L}^{p}\left({\mathbb{R}}^{n}\right)$ to itself.

After that, Huang and Liu [3] studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In [4] and [5], Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In [6] and [7], Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let b be a locally integrable function on ${\mathbb{R}}^{n}$. Setting

${A}_{\alpha ,b}f\left(t,y\right)\equiv \underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^{n}}\left[b\left(x\right)-b\left(z\right)\right]{\varphi }_{t}\left(y-z\right)f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz|,$

the commutators are defined by

$\begin{array}{c}\left[b,{G}_{\alpha }\right]f\left(x\right)={\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}{\left({A}_{\alpha ,b}f\left(t,y\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}},\hfill \\ \left[b,{g}_{\alpha }\right]f\left(x\right)={\left({\int }_{0}^{\mathrm{\infty }}{\left({A}_{\alpha ,b}f\left(t,y\right)\right)}^{2}\frac{dt}{t}\right)}^{\frac{1}{2}},\hfill \end{array}$

and

$\left[b,{g}_{\lambda ,\alpha }^{\ast }\right]f\left(x\right)={\left(\int {\int }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{\lambda n}{\left({A}_{\alpha ,b}f\left(t,y\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}.$

A function $f\in {L}_{\mathrm{loc}}^{1}\left({\mathbb{R}}^{n}\right)$ is said to be in $BMO\left({\mathbb{R}}^{n}\right)$ if

${\parallel f\parallel }_{\ast }=\underset{x\in {\mathbb{R}}^{n},r>0}{sup}\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}|f\left(y\right)-{f}_{B\left(x,r\right)}|\phantom{\rule{0.2em}{0ex}}dy<\mathrm{\infty },$

where ${f}_{B\left(x,r\right)}=\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$.

In this paper, we will consider ${G}_{\alpha }$, ${g}_{\alpha }$, ${g}_{\lambda ,\alpha }^{\ast }$ and their commutators on generalized Morrey spaces. Let $\phi \left(x,r\right)$ be a positive measurable function on ${\mathbb{R}}^{n}×{\mathbb{R}}^{+}$. For any $f\in {L}_{\mathrm{loc}}^{p}\left({\mathbb{R}}^{n}\right)$, we denote by ${L}^{p,\phi }\left({\mathbb{R}}^{n}\right)$ the generalized Morrey spaces, if

${\parallel f\parallel }_{{L}^{p,\phi }\left({\mathbb{R}}^{n}\right)}=\underset{x\in {\mathbb{R}}^{n},r>0}{sup}\phi {\left(x,r\right)}^{-1}{\left({\int }_{B\left(x,r\right)}{|f\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}<\mathrm{\infty }.$

In [8], Mizuhara introduced these generalized Morrey spaces ${L}^{p,\phi }\left({\mathbb{R}}^{n}\right)$ and discussed the boundedness of the Calderón-Zygmund singular integral operators. Note that the generalized Morrey spaces ${L}^{p,\omega }\left({\mathbb{R}}^{n}\right)$ with normalized norm

${\parallel f\parallel }_{{L}^{p,\omega }\left({\mathbb{R}}^{n}\right)}=\underset{x\in {\mathbb{R}}^{n},r>0}{sup}\omega {\left(x,r\right)}^{-1}{|B\left(x,r\right)|}^{-\frac{1}{p}}{\left({\int }_{B\left(x,r\right)}{|f\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}},$

were first defined by Guliyev in [9]. When $\omega \left(x,r\right)={r}^{\frac{\lambda -n}{p}}$, ${L}^{p,\omega }\left({\mathbb{R}}^{n}\right)={L}^{p,\lambda }\left({\mathbb{R}}^{n}\right)$. It is the classical Morrey space which was first introduced by Morrey in [10]. There are many papers discussed the conditions on $\omega \left(x,r\right)$ to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [8], the function φ is supposed to be a positively growth function and satisfy the double condition: for all $r>0$, $\phi \left(2r\right)\le D\phi \left(r\right)$, where $D\ge 1$ is a constant independent of r. This type of conditions on φ is studied by many authors; see, for example, [11, 12]. In [13], the following statement was proved by Nakai for the Calderón-Zygmund singular integral operators T.

Theorem B Let $1\le p<\mathrm{\infty }$ and let $\omega \left(x,r\right)$ satisfy the conditions

${c}^{-1}\omega \left(x,r\right)\le \omega \left(x,t\right)\le c\omega \left(x,r\right),$

whenever $r\le t\le 2r$, where c (≥1) does not depend on $t,r,x\in {\mathbb{R}}^{n}$ and

${\int }_{r}^{\mathrm{\infty }}\omega {\left(x,t\right)}^{p}\frac{dt}{t}\le c\omega {\left(x,r\right)}^{p},$

where c does not depend on x and r. Then the operator T is bounded on ${L}^{p,\omega }\left({\mathbb{R}}^{n}\right)$ for $p>1$ and from ${L}^{1,\omega }\left({\mathbb{R}}^{n}\right)$ to $\mathrm{W}{L}^{1,\omega }\left({\mathbb{R}}^{n}\right)$ for $p=1$.

The following statement, containing some results which were obtained in [8] and [13], was proved by Guliyev in [14, 15] (also see [16]).

Theorem C Let $1\le p<\mathrm{\infty }$ and let the pair $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy the condition

${\int }_{t}^{\mathrm{\infty }}{\omega }_{1}\left(x,r\right)\frac{dr}{r}\le c{\omega }_{2}\left(x,t\right),$
(1)

where c does not depend on x and t. Then the operator T is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$ for $p>1$ and from ${L}^{1,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to $\mathrm{W}{L}^{1,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$ for $p=1$.

Recently, in [17] and [9], Guliyev et al. introduced a weaker condition for the boundedness of Calderón-Zygmund singular integral operators from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$: If $1\le p<+\mathrm{\infty }$, for any $x\in {\mathbb{R}}^{n}$ and $t>0$, there exists a constant $c>0$, such that

${\int }_{t}^{\mathrm{\infty }}\frac{ess{inf}_{r
(2)

By an easy computation, we can check that if the pair $\left({\omega }_{1},{\omega }_{2}\right)$ satisfies double condition, then it will satisfy condition (1). Moreover, if $\left({\omega }_{1},{\omega }_{2}\right)$ satisfies condition (1), it will also satisfy condition (2). But the opposite is not true. We refer to [13] and Remark 4.7 in [9] for details.

In this paper, we will obtain the boundedness of the intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic ${g}_{\lambda }^{\ast }$ function and their commutators on generalized Morrey spaces when the pair $\left({\omega }_{1},{\omega }_{2}\right)$ satisfies condition (2) or the following inequality:

${\int }_{t}^{\mathrm{\infty }}\left(1+ln\frac{r}{t}\right)\frac{ess{inf}_{r
(3)

Our main results in this paper are stated as follows.

Theorem 1.1 Let $1, $0<\alpha \le 1$, let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (2), then ${G}_{\alpha }$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

Theorem 1.2 Let $1, $0<\alpha \le 1$, let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (2), then for $\lambda >3+\frac{2\alpha }{n}$, we have ${g}_{\lambda ,\alpha }^{\ast }$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

Theorem 1.3 Let $1, $0<\alpha \le 1$, $b\in BMO$, let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (3), then $\left[b,{G}_{\alpha }\right]$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

Theorem 1.4 Let $1, $0<\alpha \le 1$, $b\in BMO$, let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (3), then for $\lambda >3+\frac{2\alpha }{n}$, $\left[b,{g}_{\lambda ,\alpha }^{\ast }\right]$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

In [1], the author proved that the functions ${G}_{\alpha }$ and ${g}_{\alpha }$ are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.

Corollary 1.5 Let $1, $0<\alpha \le 1$, let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (2), then ${g}_{\alpha }$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

Corollary 1.6 Let $1, $0<\alpha \le 1$, $b\in BMO$, and let $\left({\omega }_{1},{\omega }_{2}\right)$ satisfy condition (3), then $\left[b,{g}_{\alpha }\right]$ is bounded from ${L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)$ to ${L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)$.

Throughout this paper, we use the notation $A⪯B$ to mean that there is a positive constant C (≥1) independent of all essential variables such that $A\le CB$. Moreover, C maybe different from place to place.

## 2 Proofs of main theorems

Before proving the main theorems, we need the following lemmas.

Lemma 2.1 ([18])

The inequality $ess{sup}_{t>0}\omega \left(t\right)Hg\left(t\right)⪯ess{sup}_{t>0}v\left(t\right)g\left(t\right)$ holds for all non-negative and non-increasing g on $\left(0,\mathrm{\infty }\right)$ if and only if

$A:=\underset{t>0}{sup}\frac{\omega \left(t\right)}{t}{\int }_{0}^{t}\frac{dr}{ess{sup}_{0
(4)

where $Hg\left(t\right)$ is the Hardy operator $Hg\left(t\right):=\frac{1}{t}{\int }_{0}^{t}g\left(r\right)\phantom{\rule{0.2em}{0ex}}dr$, $0.

Lemma 2.2 ([19])

1. (1)

For $1,

${\parallel f\parallel }_{\ast }\approx \underset{x\in {\mathbb{R}}^{n},r>0}{sup}{\left(\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}{|f\left(y\right)-{f}_{B\left(x,r\right)}|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}.$
2. (2)

Let $f\in BMO\left({\mathbb{R}}^{n}\right)$, $0<2r, then

$|{f}_{B\left(x,r\right)}-{f}_{B\left(x,t\right)}|⪯{\parallel f\parallel }_{\ast }ln\frac{t}{r}.$

Lemma 2.3 For $j\in {\mathbb{Z}}^{+}$, denote

${G}_{\alpha ,{2}^{j}}\left(f\right)\left(x\right)={\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left({A}_{\alpha }f\left(y,t\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}.$

Let $1, $0<\alpha \le 1$, then we have

${\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}{\parallel {G}_{\alpha }\left(f\right)\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}.$

From [1], we know that

${G}_{\alpha ,\beta }\left(f\right)\left(x\right)\le {\beta }^{\frac{3n}{2}+\alpha }{G}_{\alpha }\left(f\right)\left(x\right).$

Then, by an easy computation, we get Lemma 2.3.

By a similar argument as in [20], we can easily get the following lemma.

Lemma 2.4 Let $1, $0<\alpha \le 1$, then the commutators $\left[b,{G}_{\alpha }\right]$ is bounded from ${L}^{p}\left({\mathbb{R}}^{n}\right)$ to itself whenever $b\in BMO$.

Now we are in a position to prove the theorems.

Proof of Theorem 1.1 The main ideas of these proofs come from [9]. We decompose $f={f}_{1}+{f}_{2}$, where ${f}_{1}\left(y\right)=f\left(y\right){\chi }_{2B}\left(y\right)$, ${f}_{2}\left(y\right)=f\left(y\right)-{f}_{1}\left(y\right)$, $B:=B\left({x}_{0},r\right)$. Then

${\parallel {G}_{\alpha }f\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}\le {\parallel {G}_{\alpha }{f}_{1}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}+{\parallel {G}_{\alpha }{f}_{2}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}:=I+\mathit{II}.$

First, let us estimate I. By Theorem A, we obtain

$I\le {\parallel {G}_{\alpha }{f}_{1}\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}⪯{\parallel {f}_{1}\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}={\parallel f\parallel }_{{L}^{p}\left(2B\right)}⪯{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}{t}^{-\frac{n}{p}-1}\phantom{\rule{0.2em}{0ex}}dt.$
(5)

Then let us estimate II. Recalling the properties of function ϕ, we know that

$|{f}_{2}\ast {\varphi }_{t}\left(y\right)|=|{t}^{-n}{\int }_{|y-z|\le t}\varphi \left(\frac{y-z}{t}\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz|⪯{t}^{-n}{\int }_{|y-z|\le t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz.$

Since $x\in B\left({x}_{0},r\right)$, $\left(y,t\right)\in \mathrm{\Gamma }\left(x\right)$ and $|z-{x}_{0}|\ge 2r$, we have

$r\le |z-{x}_{0}|-|{x}_{0}-x|\le |x-z|\le |x-y|+|y-z|\le 2t.$

So, we obtain

$\begin{array}{rcl}{G}_{\alpha }{f}_{2}\left(x\right)& ⪯& {\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}|{t}^{-n}{\int }_{|y-z|\le t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ \le & {\left({\int }_{t>r/2}{\int }_{|x-y|r/2}{\left({\int }_{|z-x|\le 2t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{dt}{{t}^{2n+1}}\right)}^{\frac{1}{2}}.\end{array}$

By Minkowski’s inequality and $|z-x|\ge |z-{x}_{0}|-|{x}_{0}-x|\ge \frac{1}{2}|z-{x}_{0}|$, we have

$\begin{array}{rcl}{G}_{\alpha }{f}_{2}\left(x\right)& ⪯& {\int }_{{\mathbb{R}}^{n}}{\left({\int }_{t>\frac{|z-x|}{2}}\frac{dt}{{t}^{2n+1}}\right)}^{\frac{1}{2}}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\\ ⪯& {\int }_{|z-{x}_{0}|>2r}\frac{|f\left(z\right)|}{{|z-x|}^{n}}\phantom{\rule{0.2em}{0ex}}dz⪯{\int }_{|z-{x}_{0}|>2r}\frac{|f\left(z\right)|}{{|z-{x}_{0}|}^{n}}\phantom{\rule{0.2em}{0ex}}dz\\ ⪯& {\int }_{|z-{x}_{0}|>2r}|f\left(z\right)|{\int }_{|z-{x}_{0}|}^{+\mathrm{\infty }}\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dz\\ =& {\int }_{2r}^{+\mathrm{\infty }}{\int }_{2r<|z-{x}_{0}|

The last inequality is due to Hölder’s inequality. Thus,

${\parallel {G}_{\alpha }{f}_{2}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}{t}^{-\frac{n}{p}-1}\phantom{\rule{0.2em}{0ex}}dt.$
(6)

By combining (5) and (6), we have

${\parallel {G}_{\alpha }f\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}{t}^{-\frac{n}{p}-1}\phantom{\rule{0.2em}{0ex}}dt.$

So, let $t={s}^{-\frac{p}{n}}$; we have

$\begin{array}{rcl}{\parallel {G}_{\alpha }f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}& ⪯& \underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},r\right)}^{-1}{|B\left({x}_{0},r\right)|}^{-\frac{1}{p}}{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\phantom{\rule{0.2em}{0ex}}dt\\ ⪯& \underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},r\right)}^{-1}{\int }_{0}^{{r}^{-\frac{n}{p}}}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\phantom{\rule{0.2em}{0ex}}ds\\ =& \underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}{\int }_{0}^{r}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

Take $w\left(t\right)={\omega }_{2}{\left({x}_{0},{t}^{-\frac{p}{n}}\right)}^{-1}t$, $v\left(t\right)={\omega }_{1}{\left({x}_{0},{t}^{-\frac{p}{n}}\right)}^{-1}t$. Since $\left({\omega }_{1},{\omega }_{2}\right)$ satisfies condition (2), we can verify that $w\left(t\right)$, $v\left(t\right)$ satisfy condition (4). Let $g\left(s\right)={\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}$. Obviously, it is decreasing on variable s. So, by Lemma 2.1, we can conclude the following estimates:

${\parallel {G}_{\alpha }f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}⪯\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{1}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}r{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{r}^{-\frac{p}{n}}\right)\right)}={\parallel f\parallel }_{{L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)}.$

□

Proof of Theorem 1.2

$\begin{array}{rcl}{\left[{g}_{\lambda ,\alpha }^{\ast }\left(f\right)\left(x\right)\right]}^{2}& =& {\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|

First, let us estimate III:

$\mathit{III}\le {\int }_{0}^{+\mathrm{\infty }}{\int }_{|x-y|

Then let us estimate IV:

$\begin{array}{rcl}\mathit{IV}& \le & \sum _{j=1}^{\mathrm{\infty }}{\int }_{0}^{\mathrm{\infty }}{\int }_{{2}^{j-1}t\le |x-y|\le {2}^{j}t}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{\left({A}_{\alpha }f\left(y,t\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\\ ⪯& \sum _{j=1}^{\mathrm{\infty }}{\int }_{0}^{\mathrm{\infty }}{\int }_{{2}^{j-1}t\le |x-y|\le {2}^{j}t}{2}^{-jn\lambda }{\left({A}_{\alpha }f\left(y,t\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\\ ⪯& \sum _{j=1}^{\mathrm{\infty }}{2}^{-jn\lambda }{\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left({A}_{\alpha }f\left(y,t\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\\ :=& \sum _{j=1}^{\mathrm{\infty }}{2}^{-jn\lambda }{\left({G}_{\alpha ,{2}^{j}}\left(f\right)\left(x\right)\right)}^{2}.\end{array}$

Thus,

${\parallel {g}_{\lambda ,\alpha }^{\ast }\left(f\right)\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}\le {\parallel {G}_{\alpha }f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}+\sum _{j=1}^{\mathrm{\infty }}{2}^{-\frac{jn\lambda }{2}}{\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}.$
(7)

By Theorem 1.1, we have

${\parallel {G}_{\alpha }f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}⪯{\parallel f\parallel }_{{L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)}.$
(8)

To complete the proof, it suffices to estimate ${\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}$. Take ${f}_{1}\left(y\right)=f\left(y\right){\chi }_{2B}\left(y\right)$, ${f}_{2}\left(y\right)=f\left(y\right)-{f}_{1}\left(y\right)$, $2B=B\left({x}_{0},2r\right)$. Then

${\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}\le {\parallel {G}_{\alpha ,{2}^{j}}\left({f}_{1}\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}+{\parallel {G}_{\alpha ,{2}^{j}}\left({f}_{2}\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}.$
(9)

For the first part, by Lemma 2.3, we obtain

$\begin{array}{rl}{\parallel {G}_{\alpha ,{2}^{j}}\left({f}_{1}\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}& ⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}{\parallel {G}_{\alpha }\left({f}_{1}\right)\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}{\parallel f\parallel }_{{L}^{p}\left(2B\right)}\\ ⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\phantom{\rule{0.2em}{0ex}}dt.\end{array}$
(10)

For the other part, we know

$\begin{array}{rcl}{G}_{\alpha ,{2}^{j}}\left({f}_{2}\right)\left(x\right)& =& {\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left({A}_{\alpha }{f}_{2}\left(y,t\right)\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ =& {\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left(\underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{f}_{2}\ast {\varphi }_{t}\left(y\right)|\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ ⪯& {\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left({\int }_{|z-y|\le t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{3n+1}}\right)}^{\frac{1}{2}}.\end{array}$

Since $|z-x|\le |z-y|+|y-x|\le {2}^{j+1}t$, by Minkowski’s inequality, we get

$\begin{array}{rcl}{G}_{\alpha ,{2}^{j}}\left({f}_{2}\right)\left(x\right)& ⪯& {\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le {2}^{j}t}{\left({\int }_{|z-x|\le {2}^{j+1}t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{3n+1}}\right)}^{\frac{1}{2}}\\ ⪯& {\left({\int }_{0}^{\mathrm{\infty }}{\left({\int }_{|z-x|\le {2}^{j+1}t}|{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{{2}^{jn}\phantom{\rule{0.2em}{0ex}}dt}{{t}^{2n+1}}\right)}^{\frac{1}{2}}\\ \le & {2}^{\frac{jn}{2}}{\int }_{{\mathbb{R}}^{n}}{\left({\int }_{t\ge \frac{|z-x|}{{2}^{j+1}}}{|{f}_{2}\left(z\right)|}^{2}\frac{1}{{t}^{2n+1}}\phantom{\rule{0.2em}{0ex}}dt\right)}^{\frac{1}{2}}\phantom{\rule{0.2em}{0ex}}dz\\ ⪯& {2}^{\frac{3jn}{2}}{\int }_{|z-{x}_{0}|>2r}\frac{|f\left(z\right)|}{{|z-x|}^{n}}\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

For $x\in B\left({x}_{0},r\right)$, we have $|z-x|\ge |z-{x}_{0}|-|{x}_{0}-x|\ge |z-{x}_{0}|-\frac{1}{2}|z-{x}_{0}|=\frac{1}{2}|z-{x}_{0}|$. So by Fubini’s theorem and Hölder’s inequality, we obtain

$\begin{array}{rcl}{G}_{\alpha ,{2}^{j}}\left({f}_{2}\right)\left(x\right)& ⪯& {2}^{\frac{3jn}{2}}{\int }_{|z-{x}_{0}|>2r}\frac{|f\left(z\right)|}{{|z-{x}_{0}|}^{n}}\phantom{\rule{0.2em}{0ex}}dz\\ ⪯& {2}^{\frac{3jn}{2}}{\int }_{|z-{x}_{0}|>2r}|f\left(z\right)|{\int }_{|z-{x}_{0}|}^{\mathrm{\infty }}\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dz\\ =& {2}^{\frac{3jn}{2}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{|z-{x}_{0}|

Thus,

${\parallel {G}_{\alpha ,{2}^{j}}\left({f}_{2}\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{2}^{\frac{3jn}{2}}{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\phantom{\rule{0.2em}{0ex}}dt.$
(11)

Combining by (9), (10), and (11), we have

${\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\phantom{\rule{0.2em}{0ex}}dt.$

Thus, by substitution of variables and Lemma 2.1, we get

$\begin{array}{rl}{\parallel {G}_{\alpha ,{2}^{j}}\left(f\right)\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}& ⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left(B\left({x}_{0},r\right)\right)}^{-1}{|B\left({x}_{0},r\right)|}^{-\frac{1}{p}}{\int }_{0}^{{r}^{-\frac{n}{p}}}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\phantom{\rule{0.2em}{0ex}}ds\\ ={2}^{j\left(\frac{3n}{2}+\alpha \right)}\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}{\int }_{0}^{r}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\phantom{\rule{0.2em}{0ex}}ds\\ ⪯{2}^{j\left(\frac{3n}{2}+\alpha \right)}\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{1}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}r{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{r}^{-\frac{p}{n}}\right)\right)}\\ ={2}^{j\left(\frac{3n}{2}+\alpha \right)}{\parallel f\parallel }_{{L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)}.\end{array}$
(12)

Since $\lambda >3+\frac{2\alpha }{n}$, by (7), (8) and (12), we have the desired theorem. □

Proof of Theorem 1.3 We decompose $f={f}_{1}+{f}_{2}$ as in the proof of Theorem 1.2, where ${f}_{1}=f{\chi }_{2B}$ and ${f}_{2}=f-{f}_{1}$. Then

${\parallel \left[b,{G}_{\alpha }\right]f\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}\le {\parallel \left[b,{G}_{\alpha }\right]{f}_{1}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}+{\parallel \left[b,{G}_{\alpha }\right]{f}_{2}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}.$

By Lemma 2.4, we have

${\parallel \left[b,{G}_{\alpha }\right]{f}_{1}\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{\parallel {f}_{1}\parallel }_{{L}^{p}\left({\mathbb{R}}^{n}\right)}={\parallel f\parallel }_{{L}^{p}\left(2B\right)}⪯{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\phantom{\rule{0.2em}{0ex}}dt.$

Next, we estimate the second part. We divide it into two parts. We have

$\begin{array}{rcl}\left[b,{G}_{\alpha }\right]{f}_{2}\left(x\right)& =& {\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}\underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^{n}}\left[b\left(x\right)-b\left(z\right)\right]{\varphi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ \le & {\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}\underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^{n}}\left[b\left(x\right)-{b}_{B}\right]{\varphi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ +{\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}\underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^{n}}\left[{b}_{B}-b\left(z\right)\right]{\varphi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}\\ :=& V+\mathit{VI}.\end{array}$

First, for V, we find that

$V=|b\left(x\right)-{b}_{B}|{\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}\underset{\varphi \in {\mathcal{C}}_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^{n}}{\varphi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{\frac{1}{2}}=|b\left(x\right)-{b}_{B}|{G}_{\alpha }{f}_{2}\left(x\right).$

Following the proof in Theorem 1.1, we get

$\begin{array}{r}{\left({\int }_{B\left({x}_{0},r\right)}{|b\left(x\right)-{b}_{B}|}^{p}{|{G}_{\alpha }{f}_{2}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}\\ \phantom{\rule{1em}{0ex}}⪯{\left({\int }_{B\left({x}_{0},r\right)}{|b\left(x\right)-{b}_{B}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{dt}{{t}^{\frac{n}{p}+1}}\\ \phantom{\rule{1em}{0ex}}⪯{\parallel b\parallel }_{\ast }{r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{dt}{{t}^{\frac{n}{p}+1}}.\end{array}$

For VI, since $|y-x|, we get $|x-z|<2t$. Thus, by Minkowski’s inequality, we obtain

$\begin{array}{rcl}\mathit{VI}& ⪯& {\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}|{\int }_{|x-z|<2t}|{b}_{B}-b\left(z\right)||{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{3n+1}}\right)}^{\frac{1}{2}}\\ ⪯& {\left({\int }_{0}^{\mathrm{\infty }}|{\int }_{|x-z|<2t}|{b}_{B}-b\left(z\right)||{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz{|}^{2}\frac{dt}{{t}^{2n+1}}\right)}^{\frac{1}{2}}\\ ⪯& {\int }_{|{x}_{0}-z|>2r}|{b}_{B}-b\left(z\right)||f\left(z\right)|\frac{1}{{|x-z|}^{n}}\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

Since $|z-x|\ge \frac{1}{2}|z-{x}_{0}|$, by Fubini’s theorem, we get

$\begin{array}{rcl}{\left({\int }_{B\left({x}_{0},r\right)}{|\mathit{VI}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}& ⪯& {\left({\int }_{B\left({x}_{0},r\right)}|{\int }_{|{x}_{0}-z|>2r}|{b}_{B}-b\left(z\right)||f\left(z\right)|\frac{1}{{|x-z|}^{n}}\phantom{\rule{0.2em}{0ex}}dz{|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}\\ ⪯& {r}^{\frac{n}{p}}{\int }_{|{x}_{0}-z|>2r}|{b}_{B}-b\left(z\right)||f\left(z\right)|\frac{1}{{|{x}_{0}-z|}^{n}}\phantom{\rule{0.2em}{0ex}}dz\\ ⪯& {r}^{\frac{n}{p}}{\int }_{|{x}_{0}-z|>2r}|{b}_{B}-b\left(z\right)||f\left(z\right)|{\int }_{|{x}_{0}-z|}^{+\mathrm{\infty }}\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dz\\ \le & {r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\int }_{B\left({x}_{0},t\right)}|{b}_{B}-b\left(z\right)||f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\\ \le & {r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\int }_{B\left({x}_{0},t\right)}|{b}_{B}-{b}_{B\left({x}_{0},t\right)}||f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\\ +{r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\int }_{B\left({x}_{0},t\right)}|b\left(z\right)-{b}_{B\left({x}_{0},t\right)}||f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\frac{1}{{t}^{n+1}}\phantom{\rule{0.2em}{0ex}}dt\\ :=& A+B.\end{array}$

For A, using Lemma 2.2 and Hölder’s inequality, we have

$\begin{array}{rcl}A& ⪯& {\parallel b\parallel }_{\ast }{r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\int }_{B\left({x}_{0},t\right)}|f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\frac{1}{{t}^{n+1}}ln\frac{t}{r}\phantom{\rule{0.2em}{0ex}}dt\\ ⪯& {r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}ln\frac{t}{r}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{dt}{{t}^{\frac{n}{p}+1}}.\end{array}$

For B, we denote $D={\int }_{B\left({x}_{0},t\right)}|f\left(z\right)||{b}_{B\left({x}_{0},t\right)}-b\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz$. Then, by Hölder’s inequality and Lemma 2.2, we get

$\begin{array}{rcl}D& \le & {\left({\int }_{B\left({x}_{0},t\right)}{|f\left(z\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dz\right)}^{\frac{1}{p}}{\left({\int }_{B\left({x}_{0},t\right)}{|{b}_{B\left({x}_{0},t\right)}-b\left(z\right)|}^{{p}^{\prime }}\phantom{\rule{0.2em}{0ex}}dz\right)}^{\frac{1}{{p}^{\prime }}}\\ ⪯& {t}^{\frac{n}{{p}^{\prime }}}{\parallel b\parallel }_{\ast }{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}.\end{array}$

This yields $B⪯{r}^{\frac{n}{p}}{\int }_{2r}^{+\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{dt}{{t}^{\frac{n}{p}+1}}$. Thus,

${\parallel \left[b,{G}_{\alpha }\right]f\parallel }_{{L}^{p}\left(B\left({x}_{0},r\right)\right)}⪯{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\left(1+ln\frac{t}{r}\right)\phantom{\rule{0.2em}{0ex}}dt.$

By a change of variables, we obtain

$\begin{array}{c}{\parallel \left[b,{G}_{\alpha }\right]f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}\hfill \\ \phantom{\rule{1em}{0ex}}⪯\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},r\right)}^{-1}{|B\left({x}_{0},r\right)|}^{-\frac{1}{p}}{r}^{\frac{n}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},t\right)\right)}\frac{1}{{t}^{\frac{n}{p}+1}}\left(1+ln\frac{t}{r}\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ \phantom{\rule{1em}{0ex}}⪯\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},r\right)}^{-1}{\int }_{0}^{{r}^{-\frac{n}{p}}}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\left(1+ln\frac{{s}^{-\frac{p}{n}}}{r}\right)\phantom{\rule{0.2em}{0ex}}ds\hfill \\ \phantom{\rule{1em}{0ex}}=\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{2}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}{\int }_{0}^{r}{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{s}^{-\frac{p}{n}}\right)\right)}\left(1+\frac{p}{n}ln\frac{r}{s}\right)\phantom{\rule{0.2em}{0ex}}ds.\hfill \end{array}$

Let $w\left(t\right)={\omega }_{2}{\left({x}_{0},{t}^{-\frac{p}{n}}\right)}^{-1}t$, $v\left(t\right)={\omega }_{1}{\left({x}_{0},{t}^{-\frac{p}{n}}\right)}^{-1}t$. Since $\left({\omega }_{1},{\omega }_{2}\right)$ satisfies condition (3), by a similarly argument with Theorem 1.1, we conclude the following estimates:

${\parallel \left[b,{G}_{\alpha }\right]f\parallel }_{{L}^{p,{\omega }_{2}}\left({\mathbb{R}}^{n}\right)}⪯\underset{{x}_{0}\in {\mathbb{R}}^{n},r>0}{sup}{\omega }_{1}{\left({x}_{0},{r}^{-\frac{p}{n}}\right)}^{-1}r{\parallel f\parallel }_{{L}^{p}\left(B\left({x}_{0},{r}^{-\frac{p}{n}}\right)\right)}={\parallel f\parallel }_{{L}^{p,{\omega }_{1}}\left({\mathbb{R}}^{n}\right)}.$

Using an argument similar to the above proofs and that of Theorem 1.2, we can also show the boundedness of $\left[b,{g}_{\lambda ,\alpha }^{\ast }\right]$. □

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## Acknowledgements

This work was completed with the support of Scientific Research Fund of Zhejiang Provincial Education Department No. Y201225707.

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Wu, X., Zheng, T. Commutators of intrinsic square functions on generalized Morrey spaces. J Inequal Appl 2014, 128 (2014). https://doi.org/10.1186/1029-242X-2014-128