Endpoint boundedness for commutators of singular integral operators on weighted generalized Morrey spaces

Abstract

In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators.

1 Introduction and main results

The Morrey spaces were introduced by Morrey in [11] to investigate the local behavior of solutions to second order elliptic partial differential equations. Chiarenza and Frasca [2] showed the boundedness of the Hardy–Littlewood maximal operator, singular integral operators, and fractional integral operators on the Morrey spaces.

Let f be a measurable function on $\mathbb{R}^{n}$. The Hardy–Littlewood maximal function is defined by

$$ M(f) (x)=\sup_{B}\frac{1}{ \vert B \vert } \int _{B} \bigl\vert f(y) \bigr\vert \, {d}y, $$

where the supremum is taken over all balls B containing x.

We say that T is a singular integral operator if there exists a function K which satisfies the following conditions:

$$\begin{aligned}& Tf(x)={\mathrm{p.v.}} \int _{\mathbb{R}^{n}}K(x-y)f(y)\,dy, \\& \bigl\vert K(x) \bigr\vert \leq \frac{C}{ \vert x \vert ^{n}},\qquad \bigl\vert \nabla K(x) \bigr\vert \leq \frac{C}{ \vert x \vert ^{n+1}},\quad x\neq 0. \end{aligned}$$

The $\mathit{BMO}(\mathbb{R}^{n})$ space is defined by

$$ \mathit{BMO}\bigl(\mathbb{R}^{n}\bigr)=\biggl\{ b\in L_{\mathrm{loc}} \bigl(\mathbb{R}^{n}\bigr): \Vert b \Vert _{\mathit{BMO}}=\sup _{B} \frac{1}{ \vert B \vert } \int _{B} \bigl\vert b(x)-b_{B} \bigr\vert \,dx< \infty \biggr\} , $$

where $b_{B}=\frac{1}{|B|}\int _{B} b(y)\,dy$.

For the singular integral operator T and $b\in \mathit{BMO}$, the commutator $[b,T]$ is defined by

$$ [b,T]f(x)= \int _{\mathbb{R}^{n}}\bigl(b(x)-b(y)\bigr)K(x-y)f(y)\,dy. $$

For $1< p<\infty$, we say a weight $w\in A_{p}$ if

$$ [w]_{p} =\sup_{B} \biggl(\frac{1}{ \vert B \vert } \int _{B} w(y)\,dy \biggr) \biggl( \frac{1}{ \vert B \vert } \int _{B} w(y)^{-p'/p}\,dy \biggr)^{p/p'}< \infty . $$

For $p=1$, we write $w\in A_{1}$ if $Mw(y)\leq Cw(y)$, $a.e. y\in \mathbb{R}^{n}$.

It is a classical result that the operators T are bounded on $L^{p}(w)$ whenever $1< p<\infty$ and $w\in A_{p}$, and for $p=1$ and $w\in A_{1}$, we have the weak type result which can be found in [9]. Komori and Shirai extended them to the weighted Morrey spaces in [10].

Let f be a measurable function on $\mathbb{R}^{n}$ and $1\leq p<\infty$, $0\leq \kappa < 1$. For two weights w and u, the weighted Morrey space is defined by

$$ L^{p,\kappa }(w,u)=\bigl\{ f\in {L_{\mathrm{loc}}}^{p}(w): \Vert f \Vert _{L^{p,\kappa }(w,u)}< \infty \bigr\} , $$

where

$$ \Vert f \Vert _{L^{p,\kappa }(w,u)}=\sup_{B} \biggl( \frac{1}{u(B)^{\kappa }} \int _{B} \bigl\vert f(x) \bigr\vert ^{p}{w(x)} \,\mathrm{d}x \biggr)^{\frac{1}{p}}, $$

and the supremum is taken over all balls B in $\mathbb{R}^{n}$. When $w=u$, we write $L^{p,\kappa }(w,u)$ as $L^{p,\kappa }(w)$. Komori and Shirai in [10] proved that, for $1< p<\infty$ and $w\in A_{p}$, T and $[b,T]$ are bounded on $L^{p,\kappa }(w)$, and if $p=1$ and $w\in A_{1}$, then for all $t>0$ and any ball B,

$$ w\bigl(\bigl\{ x\in B: \bigl\vert Tf(x) \bigr\vert >t\bigr\} \bigr)\leq \frac{C}{t} \Vert f \Vert _{L^{1,\kappa }(w)}w(B)^{\kappa }. $$

Qi et al. [14] obtained the weighted endpoint estimates for the commutators of the singular integral operators with BMO functions and associated maximal operators on the weighted Morrey space $L^{1,\kappa }(w)$. They also gave similar results for the commutators of the fractional integral operators with BMO functions and associated maximal operators.

Let w and u be two weights and $1\leq q\leq \beta \leq p\leq \infty$. We define the generalized two-weight Morrey space $(L^{q}(w), L^{p}(u))^{\beta }:=(L^{q}(w), L^{p}(u))^{\beta }(\mathbb{R}^{n})$ as the space of all measurable functions f satisfying $\|f\|_{(L^{q}(w), L^{p}(u))^{\beta }}<\infty$, where

$$ \Vert f \Vert _{(L^{q}(w),L^{p}(u))^{\beta }}:=\sup_{r>0} {}_{r} \Vert f \Vert _{(L^{q}(w),L^{p}(u))^{\beta }}, $$

with

$$ _{r}{ \Vert f \Vert _{(L^{q}(w),L^{p}(u))^{\beta }}}= \biggl({ \int _{R^{n}}} {\bigl({u\bigl(B(y,r)\bigr)^{ \frac{1}{\beta }-\frac{1}{q}-\frac{1}{p}}} \Vert f{\chi }_{B(y,r)} \Vert _{L^{q}(w)}\bigr)^{p}}\,dy \biggr) ^{\frac{1}{p}}, $$

for any $r>0$, with the usual modification when $p=\infty$. In the case $w=u$, the spaces $(L^{q}(w), L^{p}(u))^{\beta }$ are the spaces $(L^{q}(w), L^{p})^{\beta }$ defined by Feuto in [7]. In the case $w=u\equiv 1$, the spaces $(L^{q}(w), L^{p}(u))^{\beta }$ are the spaces $(L^{q}, L^{p})^{\beta }$ defined in [8] by Fofana. For $q<\beta$ and $p=\infty$, the space $(L^{q}(w), L^{p})^{\beta }$ is the weighted Morrey space $L^{q,\kappa }(w)$ with $\kappa =\frac{1}{q}-\frac{1}{\beta }$.

Feuto [7] proved that the singular integral operators, the commutators of the singular integral operators with BMO functions, and other operators were bounded on these generalized weighted Morrey spaces $(L^{q}(w),L^{p})^{\beta }$ for $q>1$. Here we consider the boundedness of the commutators of the singular integral operators with BMO functions on the endpoint generalized weighted Morrey space $(L^{1}(w),L^{p})^{\beta }$. The weighted endpoint estimates for the commutators of the singular integral operators with BMO functions have many applications in partial differential equations. The BMO functions and the associated maximal operators can be applied in optimization problems, see [5, 6].

Let $\varPsi :[0,\infty )\rightarrow [0,\infty )$ be an increasing function. We define space $L^{\varPsi ,\infty }(w)$ as the space of all measurable functions f satisfying $\|f\|_{L^{\varPsi ,\infty }(w)} <\infty$, where

$$ \Vert f \Vert _{L^{\varPsi ,\infty }(w)}:= \sup_{t>0}t\varPsi \bigl(w \bigl\{ x\in \mathbb{R}^{n}: \bigl\vert f(x) \bigr\vert >t \bigr\} \bigr). $$

When $\varPsi (t)=t^{1/p}$ with $0< p<\infty$, then the space $L^{\varPsi ,\infty }(w)$ is the weak weighted Lebesgue space $L^{p,\infty }(w)$.

Let w, u be two weights, $\varPsi :[0,\infty )\rightarrow [0,\infty )$ be an increasing function and $1\leq \beta \leq p\leq \infty$. We define the generalized weak weighted Morrey space $(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }$ as the space of all measurable functions f satisfying $\|f\|_{(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }}<\infty$, where

$$ \Vert f \Vert _{(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }}:= \sup_{r>0}{}_{r} \Vert f \Vert _{(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }}, $$

with

$$ {}_{r}{ \Vert f \Vert _{(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }}}= \biggl({ \int _{ \mathbb{R}^{n}}} {\bigl({u\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1-\frac{1}{p}}} \Vert f{ \chi }_{B(y,r)} \Vert _{L^{\varPsi ,\infty }(w)}\bigr)^{p}}\,dy \biggr) ^{\frac{1}{p}}. $$

When $\varPsi (t)=t$, $w=u$, the space $(L^{\varPsi ,\infty }(w), L^{p}(u))^{\beta }$ is the generalized weak weighted Morrey space $(L^{1,\infty }(w), L^{p})^{\beta }$ defined in [7]. Feuto proved for the singular integral operator T, if $w\in A_{1}$, then

$$ \Vert Tf \Vert _{{(L^{1,\infty }(w),L^{p})}^{\beta }} \leq C \Vert f \Vert _{{(L^{1}(w),L^{p})}^{ \beta }}. $$

In this paper, we extend the methods used in [14] and obtain the endpoint boundedness for the commutators of the singular integral operators with BMO functions and the associated maximal operators on the generalized weighted Morrey spaces $(L^{1}(w), L^{p})^{\beta }$. The results are more general than [14] and have different forms. We also give similar results for the commutators of the fractional integral operators with BMO functions and the associated maximal operators.

In order to state our results, we need to recall some notations and facts about the Young functions and Orlicz spaces; for further information, see [1]. A function $\varPhi : [0,\infty )\rightarrow [0,\infty )$ is a Young function if it is convex and increasing, and if $\varPhi (0)=0$ and $\varPhi (t)\rightarrow \infty$ as $t\rightarrow \infty$.

Given a locally integrable function f and a Young function Φ, define the mean Luxemburg norm of f on a ball B by

$$ \Vert f \Vert _{\varPhi ,B}=\inf \biggl\{ \lambda >0: \frac{1}{ \vert B \vert } \int _{B}\varPhi \biggl(\frac{ \vert f(x) \vert }{\lambda } \biggr)\,dx\leq 1 \biggr\} . $$

For $\alpha , 0\leq \alpha < n$, and a Young function Φ, we define the Orlicz maximal operator

$$ M_{\alpha ,\varPhi }f(x)=\sup_{B\ni x} \vert B \vert ^{\frac{\alpha }{n}} \Vert f \Vert _{ \varPhi ,B}. $$

If $\alpha =0$, we write $M_{\alpha ,\varPhi }$ simply as $M_{\varPhi }$. If $\alpha =0$ and $\varPhi (t)=t$, $M_{\alpha ,\varPhi }$ is the Hardy–Littlewood maximal operator M. If $\varPhi _{\varepsilon }(t)=t\log (e+t)^{\varepsilon }$, $\varepsilon \geq 0$, we write $M_{\varPhi _{\varepsilon }}$ simply as $M_{L(\log L)^{\varepsilon }}$.

If $0<\alpha <n$ and $\varPhi (t)=t$, $M_{\alpha ,\varPhi }$ is a fractional maximal operator of order α, and we write it as $M_{\alpha }$. If $\varPhi _{\varepsilon }(t)=t\log (e+t)^{\varepsilon }$, we write $M_{\alpha ,\varPhi }$ simply as $M_{\alpha ,L(\log L)^{\varepsilon }}$.

Given $\alpha , 0<\alpha <n$, for an appropriate function f on $\mathbb{R}^{n}$, the fractional integral operator (or the Riesz potential) of order α is defined by

$$ I_{\alpha }f(x)= \int _{\mathbb{R}^{n}}\frac{f(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy. $$

For $b\in \mathit{BMO}({\mathbb{R}^{n}})$, we define the commutators of the operator $I_{\alpha }$ and b by

$$ [b,I_{\alpha }]f(x)= \int _{\mathbb{R}^{n}} \frac{(b(x)-b(y))f(y)}{ \vert x-y \vert ^{n-\alpha }}\,dy. $$

A weight w is said to belong to the class $A_{p,q}$ for $1< p,q<\infty$ if there exists a positive constant C such that, for any ball B in $\mathbb{R}^{n}$,

$$ \biggl(\frac{1}{ \vert B \vert } \int _{B}w(x)^{q}\,dx \biggr)^{1/q} \biggl(\frac{1}{ \vert B \vert } \int _{B}w(x)^{-p'}\,dx \biggr)^{1/p'}\leq C< \infty . $$

The following theorems are our main results.

Theorem 1.1

If$1< q\leq \beta < p<\infty$and$w\in A_{q}$, then the Hardy–Littlewood maximal operatorMand$M_{L(\log L)}$are bounded on$(L^{q}(w),L^{p})^{\beta }$.

If$q=1\leq \beta < p<\infty$and$w\in A_{1}$, then there exists a constant$C>0$independent offsuch that

$$ \bigl\Vert M(f) \bigr\Vert _{(L^{1,\infty }(w),L^{p})^{\beta }}\leq C \Vert f \Vert _{(L^{1}(w),L^{p})^{\beta }}. $$

Theorem 1.2

Let$1\leq \beta < p\leq \infty$, $w\in A_{1}$, $\varPhi (t)=t\log (e+t)$, and$\varPsi (t)=\frac{1}{\varPhi (1/t)}=\frac{t}{\log (e+t^{-1})}$, then there exists a constant$C>0$independent offsuch that

$$ \bigl\| \varPsi (M_{L(\log L)}f)\bigr\| _{(L^{1,\infty }(w),L^{p})^{\beta }}\leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }}. $$

Theorem 1.3

LetTbe any singular integral operator, $w\in A_{1}$, $\varPhi (t)=t\log (e+t)$, $\varPsi (t)=\frac{t}{\log (e+t^{-1})}$, and$b\in \mathit{BMO}$, $1\leq \beta < p\leq \infty$. Then there exists a constant$C>0$independent offsuch that

$$ \bigl\Vert \varPsi \bigl( \bigl\vert [b,T]f \bigr\vert \bigr) \bigr\Vert _{(L^{1,\infty }(w),L^{p})^{\beta }}\leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }}. $$

We also study similar estimates for the commutators of the fractional integral operators with BMO functions and the associated maximal operators and get the following results.

Theorem 1.4

Let$0<\alpha <n$, $w\in A_{1}$, $1/q=1-\alpha /n$, $1\leq \beta < p\leq \infty$, and$0<1+1/p-1/\beta <1/q$, $\varPhi (t)=t\log (e+t)$, $\varPsi (t)=\frac{t}{\log (e+t^{-1})}$, $\varGamma (t)=t^{1/q}\log (e+t)^{-1}$, and$\varTheta (t)=t^{1/q}\log (e+t^{-1})$. Then there exists a constant$C>0$independent offsuch that

$$\begin{aligned} \bigl\Vert \varPsi (M_{\alpha ,L(\log L)}f) \bigr\Vert _{(L^{\varGamma ,\infty }(w),L^{p})^{ \beta }} \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }}. \end{aligned}$$

Theorem 1.5

Let$0<\alpha <n$, $w\in A_{1}$, $b\in \mathit{BMO}$, $1/q=1-\alpha /n$, $1\leq \beta < p\leq \infty$, and$0<1+1/p-1/\beta <1/q$, $\varPhi (t)=t\log (e+t)$, $\varPsi (t)=\frac{t}{\log (e+t^{-1})}$, $\varGamma (t)=t^{1/q}\log (e+t)^{-1}$, and$\varTheta (t)=t^{1/q}\log (e+t^{-1})$. Then there exists a constant$C>0$independent offsuch that

$$\begin{aligned} \bigl\Vert \varPsi \bigl( \bigl\vert [b,I_{\alpha }]f \bigr\vert \bigr) \bigr\Vert _{(L^{\varGamma ,\infty }(w),L^{p})^{\beta }} \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }}. \end{aligned}$$

From these results, we see that the commutators of the fractional integral operators with the BMO functions and the associated maximal operators map the weighted Morrey spaces to some weighted Orlicz–Morrey spaces. Hence we can further consider the boundedness for these integral operators on general weighted Orlicz–Morrey spaces.

2 Proof of Theorem 1.1, Theorem 1.2, and Theorem 1.3

Lemma 2.1

([9])

Let$w\in A_{\infty }$, then there exists a constant$C>0$such that, for any cubeQ, $w(2Q)\leq Cw(Q)$.

Lemma 2.2

([9])

Let$1< p<\infty$and$w\in A_{p}$. Then there exists a constant$C>0$independent offsuch that

$$ \bigl\Vert M(f) \bigr\Vert _{L^{p}(w)}\leq C \Vert f \Vert _{L^{p}(w)}. $$

Let$w\in A_{1}$. Then there exists a constant$C>0$independent offsuch that

$$ \bigl\Vert M(f) \bigr\Vert _{L^{1,\infty }(w)}\leq C \Vert f \Vert _{L^{1}(w)}. $$

Lemma 2.3

([15])

There exists a constant$C>0$such that, for any ballBand all$x\in B$,

$$\begin{aligned} M(f\chi _{(2B)^{c}}) (x)\leq C\sum_{i=1}^{\infty } \frac{1}{ \vert 2^{i+1}B \vert } \int _{2^{i+1}B} \bigl\vert f(y) \bigr\vert \,dy \end{aligned}$$

for every locally integrable functionf.

Lemma 2.4

([12])

Let$\varPhi (t)=t\log (e+t)$, then there exists a positive constantCsuch that, for any weightwand all$t>0$,

$$ w\bigl(\bigl\{ x\in \mathbb{R}^{n}: M_{L(\log L)}f(x)>t \bigr\} \bigr)\leq C \int _{ \mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)Mw(x)\,dx $$

for every locally integrable functionf.

Lemma 2.5

([9])

Let$w\in A_{1}$, then there exists a constant$C>0$and$\eta >0$such that, for any ballBand a measurable subset$E\subset B$,

$$ \frac{w(E)}{w(B)}\leq C \biggl(\frac{ \vert E \vert }{ \vert B \vert } \biggr)^{\eta }. $$

Lemma 2.6

([7])

Let$1\leq s\leq q<\infty$, $w\in A_{q/s}$, and$T: L^{q}_{\mathrm{loc}}(w)\rightarrow L^{q}_{\mathrm{loc}}(w)$a sublinear operator which satisfies the following property: for all balls$B\subset \mathbb{R}^{n}$, $x\in B$,

$$ T(f\chi _{(2B)^{c}}) (x)\leq C\sum_{i=1}^{\infty }i \biggl( \frac{1}{ \vert 2^{i+1}B \vert } \int _{2^{i+1}B} \bigl\vert f(y) \bigr\vert ^{s}\,dy \biggr)^{1/s}. $$

Then

• (1) if$q>1$andTis bounded on$L^{q}(w)$, then it is also bounded on$(L^{q}(w),L^{p})^{\beta }$for$q\leq \beta < p\leq \infty$,

• (2) if for all$\lambda >0$,

  $$ w\bigl(\bigl\{ x\in \mathbb{R}^{n}: \bigl\vert T(f) (x) \bigr\vert >\lambda \bigr\} \bigr)\leq C \frac{1}{\lambda } \int _{\mathbb{R}^{n}} \bigl\vert f(y) \bigr\vert \, dyw(y)\,dy, $$

  then for$1\leq \beta < p\leq \infty$, Tis bounded on$(L^{1}(w),L^{p})^{\beta }$to$(L^{1,\infty }(w),L^{p})^{\beta }$.

Proof of Theorem 1.1

By Lemma 2.2, Lemma 2.3, and Lemma 2.6, we obtain that the Hardy–Littlewood maximal operator M is bounded on $(L^{q}(w),L^{p})^{\beta }$ for $w\in A_{q}$, and for $w\in A_{1}$, then there exists a constant $C>0$ independent of f such that

$$ \bigl\Vert M(f) \bigr\Vert _{(L^{1,\infty }(w),L^{p})^{\beta }}\leq C \Vert f \Vert _{(L^{1}(w),L^{p})^{\beta }}. $$

Because $M_{L(\log L)}\approx M^{2}$, which was obtained by Perez in [12], we have $M_{L(\log L)}$ is bounded on $(L^{q}(w),L^{p})^{\beta }$. This ends the proof. □

Proof of Theorem 1.2

Fix $y\in \mathbb{R}^{n}$ and $r>0$, let $B=B(y,r)$ be a ball centered at y with radius r. By Lemma 2.4, we have

$$\begin{aligned} w\bigl(\bigl\{ x\in B: M_{L(\log L)}f(x)>t\bigr\} \bigr) =& \int _{\{x\in \mathbb{R}^{n}: M_{L( \log L)}f(x)>t\}}\chi _{B}(x) w(x)\,dx \\ \leq & C \int _{\mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)M( \chi _{B}w) (x)\,dx \\ \leq & C \biggl( \int _{3B} + \int _{(3B)^{c}} \biggr)\varPhi \biggl( \frac{ \vert f(x) \vert }{t} \biggr)M( \chi _{B}w) (x)\,dx \\ \leq &{\mathrm{I}}+\mathrm{II}. \end{aligned}$$

To estimate the term I, since $w\in A_{1}$, we have

$$\begin{aligned} {\mathrm{I}} \leq & C \int _{3B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)w(x)\,dx \\ \leq & C \varPhi \biggl(\frac{1}{t} \biggr) \int _{3B}\varPhi \bigl( \bigl\vert f(x) \bigr\vert \bigr)w(x)\,dx \\ \leq & C \varPhi \biggl(\frac{1}{t} \biggr) \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(w)}. \end{aligned}$$

For the term II, observe that for $x\in (3B)^{c}$, $x\in B^{\prime }$, $B'$ is a ball and $B^{\prime }\cap B\neq \emptyset$. We have

$$\begin{aligned} \frac{1}{ \vert B^{\prime } \vert } \int _{B^{\prime }}\chi _{B}(z)w(z)\,dz =&\frac{1}{ \vert B^{\prime } \vert } \int _{B^{\prime }\cap B}w(z)\,dz \\ \leq &\frac{C}{ \vert x-y \vert ^{n}} \int _{B} w(z)\,dz=\frac{C}{ \vert x-y \vert ^{n}}w(B). \end{aligned}$$

Therefore we obtain

$$ M(\chi _{B}w) (x)\leq C \vert x-y \vert ^{-n}w(B). $$

Since $w\in A_{1}$, we get

$$\begin{aligned} {\mathrm{II}} \leq & C \int _{(3B)^{c}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \vert x-y \vert ^{-n}w(B)\,dx \\ \leq & C\sum_{j=1}^{\infty } \int _{3^{j+1}B\backslash 3^{j}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \frac{w(B)}{ \vert 3^{j+1}B \vert }\,dx \\ \leq & C\sum_{j=1}^{\infty }\frac{w(B)}{w(3^{j+1}B)} \int _{3^{j+1}B} \varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)w(x)\,dx \\ \leq & C\varPhi \biggl(\frac{1}{t} \biggr)\sum _{j=1}^{\infty}\frac{w(B)}{w(3^{j+1}B)} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(w)}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \bigl\Vert \varPsi (M_{L(\log L)}f)\chi _{B} \bigr\Vert _{L^{1,\infty }(w)} =&\sup_{t>0}tw \bigl\{ x\in B: \varPsi (M_{L(\log L)}f) (x)>t\bigr\} \\ =& \sup_{t>0}tw\bigl\{ x\in B: M_{L(\log L)}f(x)>\varPsi ^{-1}(t)\bigr\} \\ =& \sup_{t>0}\varPsi (t)w\bigl\{ x\in B: M_{L(\log L)}f(x)>t \bigr\} \\ \leq &C \Biggl( \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(w)} \\ &{}+ \sum_{j=1}^{\infty }\frac{w(B)}{w(3^{j+1}B)} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(w)} \Biggr). \end{aligned}$$

Thus, for any $r>0$, by Lemma 2.1 and Lemma 2.5, we have

$$\begin{aligned} &{}_{r}{ \bigl\Vert \varPsi (M_{L(\log L)}f) \bigr\Vert _{(L^{1,\infty }(w),L^{p})^{\beta }}} \\ &\quad = \biggl({ \int _{\mathbb{R}^{n}}} {\bigl({w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}}} \bigl\Vert \varPsi (M_{L(\log L)}f)\bigr){\chi }_{B(y,r)} \bigr\Vert _{L^{1,\infty }(w)}\biggr)^{p}}\,dy ) ^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(w)}\bigr)^{p}\,dy \biggr)^{ \frac{1}{p}} \\ &\qquad {}+ C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty} \frac{w(B(y,r))}{w(B(y,3^{j+1}r))}w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(w)} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,3r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(w)}\bigr)^{p}\,dy \biggr)^{ \frac{1}{p}} \\ &\qquad {}+C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty } \biggl( \frac{w(B(y,r))}{w(B(y,3^{j+1}r))} \biggr)^{\frac{1}{\beta }-\frac{1}{p}}w\bigl(B\bigl(y,3^{j+1}r \bigr)\bigr)^{\frac{1}{\beta }-1-\frac{1}{p}}\\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(w)} \Biggr)^{p}\,dy \Biggr)^{ \frac{1}{p}} \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }} \Biggl(1+\sum_{j=1}^{\infty } \frac{1}{3^{jn\eta (\frac{1}{\beta }-\frac{1}{p})}} \Biggr) \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }}. \end{aligned}$$

This ends the proof. □

Lemma 2.7

([13])

LetTbe any Calderón–Zygmund singular integral operator, $\varPhi (t)=t\log (e+t)$, $\varepsilon >0$, and$b\in \mathit{BMO}$. Then there exists a positive constantCsuch that, for all weightsw,

$$ w\bigl(\bigl\{ x\in \mathbb{R}^{n}: \bigl\vert [b,T]f(x) \bigr\vert >t\bigr\} \bigr)\leq C \int _{\mathbb{R}^{n}} \varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)M_{L(\log L)^{1+\varepsilon }}w(x)\,dx. $$

Lemma 2.8

([9])

Let$w\in A_{1}$, then there exist a constant$C>0$and$\theta >0$such that, for any ballB,

$$ \biggl(\frac{1}{ \vert B \vert } \int _{B} w(y)^{1+\theta }\,dy \biggr)^{ \frac{1}{1+\theta }}\leq C \frac{1}{ \vert B \vert } \int _{B} w(y)\,dy. $$

Proof of Theorem 1.3

Fix $y\in \mathbb{R}^{n}$ and $r>0$, let $B=B(y,r)$. By Lemma 2.7, we have

$$\begin{aligned} w\bigl(\bigl\{ x\in B: \bigl\vert [b,T]f(x) \bigr\vert >t\bigr\} \bigr) =& \int _{\{x\in \mathbb{R}^{n}: \vert [b,T]f(x) \vert >t \}}w(x)\chi _{B}(x) \,dx \\ \leq & C \int _{\mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)M_{L( \log L)^{1+\varepsilon }}(w\chi _{B}) (x)\,dx \\ \leq & C \biggl( \int _{3B} + \int _{(3B)^{c}} \biggr)\varPhi \biggl( \frac{ \vert f(x) \vert }{t} \biggr)M_{L(\log L)^{1+\varepsilon }}(w\chi _{B}) (x)\,dx \\ \leq &{\mathrm{I}}+\mathrm{II}. \end{aligned}$$

To estimate the term I, since $w\in A_{1}$, it is easy to prove that $M_{L(\log L)^{1+\varepsilon }}(w\chi _{B})(x)\leq Cw(x)$, $x\in 3B$, we have

$$\begin{aligned} {\mathrm{I}}\leq C \int _{3B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)w(x)\,dx \leq C \varPhi \biggl(\frac{1}{t} \biggr) \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(w)}. \end{aligned}$$

For the term II, observe that for $x\in (3B)^{c}$, $x\in B'$, $B'$ is a ball and $B'\cap B\neq \emptyset$, by Lemma 2.8, for any $\delta : 0<\delta \leq \theta$, we have

$$\begin{aligned} \biggl(\frac{1}{ \vert B' \vert } \int _{B'}\bigl(w(z)\chi _{B}(z) \bigr)^{1+\delta }\,dz \biggr)^{ \frac{1}{1+\delta }} \leq & \biggl(\frac{1}{ \vert B' \vert } \int _{B} w(z)^{1+ \delta }\,dz \biggr)^{\frac{1}{1+\delta }} \\ =& \biggl(\frac{ \vert B \vert }{ \vert B' \vert } \biggr)^{\frac{1}{1+\delta }} \biggl( \frac{1}{ \vert B \vert } \int _{B} w(z)^{1+\delta }\,dz \biggr)^{\frac{1}{1+\delta }} \\ \leq &C \biggl(\frac{ \vert B \vert }{ \vert B' \vert } \biggr)^{\frac{1}{1+\delta }} \biggl( \frac{1}{ \vert B \vert } \int _{B} w(z)\,dz \biggr) \\ \leq &C \biggl(\frac{ \vert B \vert }{ \vert B' \vert } \biggr)^{\frac{1}{1+\delta }} \frac{w(B)}{ \vert B \vert }. \end{aligned}$$

Noticing the definition of the maximal function M, we obtain

$$\begin{aligned} M_{L(\log L)^{1+\varepsilon }}(w\chi _{B}) (x) \leq & \bigl(M\bigl(w^{1+\delta } \chi _{B}\bigr) (x)\bigr)^{\frac{1}{1+\delta }} \\ \leq & C \biggl(\frac{ \vert B \vert }{ \vert x-y \vert ^{n}} \biggr)^{\frac{1}{1+\delta }} \frac{w(B)}{ \vert B \vert } \end{aligned}$$

and

$$\begin{aligned} {\mathrm{II}} \leq & C \int _{(3B)^{c}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \biggl( \frac{ \vert B \vert }{ \vert x-y \vert ^{n}} \biggr)^{\frac{1}{1+\delta }}\frac{w(B)}{ \vert B \vert }\,dx \\ \leq & C \sum_{j=1}^{\infty } \int _{3^{j+1}B\setminus 3^{j}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{ \frac{1}{1+\delta }}\frac{w(B)}{ \vert B \vert }\,dx \\ \leq & C \sum_{j=1}^{\infty } \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{ \frac{1}{1+\delta }}{\frac{w(B)}{ \vert B \vert }} {\frac{ \vert 3^{j+1}B \vert }{w(3^{j+1}B)}} \int _{3^{j+1}B} \varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)w(x)\,dx \\ \leq & C\varPhi \biggl(\frac{1}{t}\biggr) \sum _{j=1}^{\infty } \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\frac{-\delta }{1+\delta }}{ \frac{w(B)}{w(3^{j+1}B)}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(w)}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \bigl\Vert \varPsi \bigl( \bigl\vert [b,T]f \bigr\vert \bigr)\chi _{B} \bigr\Vert _{L^{1,\infty }(w)} =&\sup_{t>0}tw \bigl\{ x \in B: \varPsi \bigl( \bigl\vert [b,T]f \bigr\vert \bigr) (x))>t\bigr\} \\ =& \sup_{t>0}\varPsi (t)w\bigl\{ x\in B: \bigl\vert [b,T]f(x) \bigr\vert >t\bigr\} \\ \leq &C \Biggl( \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(w)} \\ &{}+ \sum_{j=1}^{\infty } \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{ \frac{-\delta }{1+\delta }}\frac{w(B)}{w(3^{j+1}B)} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(w)} \Biggr). \end{aligned}$$

Thus, for any $r>0$, by Lemma 2.1 and Lemma 2.5, we have

$$\begin{aligned} &{}_{r}{ \bigl\Vert \varPsi \bigl( \bigl\vert [b,T]f \bigr\vert \bigr) \bigr\Vert _{(L^{1,\infty }(w),L^{p})^{\beta }}} \\ &\quad = \biggl({ \int _{\mathbb{R}^{n}}} {\bigl({w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}}} \bigl\Vert \varPsi \bigl( \bigl\vert [b,T]f \bigr\vert \bigr){\chi }_{B(y,r)} \bigr\Vert _{L^{1,\infty }(w)}\bigr)^{p}}\,dy \biggr) ^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(w)}\bigr)^{p}\,dy \biggr)^{ \frac{1}{p}} \\ &\qquad {}+ C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty } \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\frac{-\delta }{1+\delta }} \frac{w(B(y,r))^{\frac{1}{\beta }-\frac{1}{p}}}{w(B(y,3^{j+1}r))} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(w)} \Biggr)^{p}\,dy \Biggr)^{ \frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,3r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(w)}\bigr)^{p}\,dy \biggr)^{ \frac{1}{p}} \\ &\qquad {}+C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty } \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\eta (\frac{1}{\beta }-\frac{1}{p})-\frac{\delta }{1+\delta }}w\bigl(B\bigl(y,3^{j+1}r \bigr)\bigr)^{ \frac{1}{\beta }-1-\frac{1}{p}}\\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(w)} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }} \Biggl(1+\sum_{j=1}^{\infty } \biggl(\frac{1}{3^{(j+1)n}} \biggr)^{\eta (\frac{1}{\beta }- \frac{1}{p})-\frac{\delta }{1+\delta }} \Biggr) \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(w),L^{p})^{\beta }}, \end{aligned}$$

in which we take $\delta >0$ small enough such that $\eta (\frac{1}{\beta }-\frac{1}{p})-\frac{\delta }{1+\delta }>0$. This ends the proof. □

3 Proof of Theorem 1.4 and Theorem 1.5

Given an increasing function $\varphi : [0,\infty )\rightarrow [0,\infty )$, as in [3], we define the function $h_{\varphi }$ by

$$\begin{aligned} h_{\varphi }(s)=\sup_{t>0} \frac{\varphi (st)}{\varphi (t)},\quad 0 \leq s< \infty . \end{aligned}$$

If φ is submultiplicative, then $h_{\varphi }\approx \varphi$. Also, for all $s,t>0$, $\varphi (st)\leq h_{\varphi }(s)\varphi (t)$.

In this section, we set $\varPhi (t)=t\log (e+t)$, it is submultiplicative and so $h_{\varPhi }\approx \varPhi$. Let $0<\alpha <n$, and q be a number $1/q=1-\alpha /n$. Denote

$$ \varGamma (t)= \textstyle\begin{cases} 0, & t=0, \\ \frac{t}{\varPhi (t^{\alpha /n})}, & t>0. \end{cases} $$

So

$$ \varGamma (t)\approx t^{1/q}\log (e+t)^{-1}. $$

The function Γ is invertible with

$$ \varGamma ^{-1}(t)\approx \bigl[t\log (e+t)\bigr]^{q}= \varPhi (t)^{q} . $$

Lemma 3.1

([3])

If$\varphi (t)/t$is decreasing, then for any positive sequence$\{t_{j}\}$,

$$ \varphi \biggl(\sum_{j} t_{j}\biggr) \leq \sum_{j}\varphi (t_{j}). $$

Lemma 3.2

([14])

Let$0<\alpha <n$, $1/q=1-\alpha /n$. Then there exists a constant$C>0$such that, for any$t>0$, for any weightw, we have

$$\begin{aligned} &\varGamma \bigl(w\bigl(\bigl\{ x\in \mathbb{R}^{n}: M_{\alpha ,L\log L}(f) (x)>t\bigr\} \bigr)\bigr) \\ &\quad \leq C \int _{\mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(y) \vert }{t} \biggr)h_{\varPsi }\bigl(Mw(y)\bigr)\,dy. \end{aligned}$$

Proof of Theorem 1.4

Fix $y\in \mathbb{R}^{n}$ and $r>0$, let $B=B(y,r)$. By Lemma 3.2, we have

$$\begin{aligned} \varGamma \bigl(w\bigl(\bigl\{ x\in B: M_{\alpha ,L(\log L)}f(x)>t\bigr\} \bigr) \bigr) =& \varGamma \biggl( \int _{\{x\in \mathbb{R}^{n}: M_{\alpha ,L(\log L)}f(x)>t\}} w(x) \chi _{B}(x)\,dx \biggr) \\ \leq & C \int _{\mathbb{R}^{n}}\varGamma \biggl(\frac{ \vert f(x) \vert }{t} \biggr)h_{\varGamma }\bigl(M(w\chi _{B})\bigr) (x)\,dx \\ \leq & C \biggl( \int _{3B} + \int _{(3B)^{c}} \biggr)\varGamma \biggl( \frac{ \vert f(x) \vert }{t} \biggr)h_{\varGamma }\bigl(M(w\chi _{B})\bigr) (x)\,dx \\ \leq & {\mathrm{I}}+\mathrm{II}. \end{aligned}$$

Now we estimate the term I. Noticing that, for $s>0$, we have

$$\begin{aligned} h_{\varGamma }(s)=\sup_{t>0}\frac{\varGamma (st)}{\varGamma (t)}=s \sup _{t>0} \frac{\varPhi (t^{\alpha /n})}{\varPhi (((st)^{\alpha /n})}\leq C\varTheta (s). \end{aligned}$$

Since $w\in A_{1}$, we get

$$\begin{aligned} {\mathrm{I}} \leq & C \int _{3B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)h_{\varGamma }\bigl(w(x)\bigr)\,dx \\ \leq & C \int _{3B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \bigl(w(x)\bigr)\,dx \\ \leq & C \varPhi (1/t) \bigl\Vert \varPhi \bigl( \vert f \vert \chi _{3B}\bigr) \bigr\Vert _{L^{1}(\varTheta (w))}. \end{aligned}$$

For the term II, observe that for $x\in (3B)^{c}$, $x\in B'$, $B'$ is a ball and $B'\cap B\neq \emptyset$. As in the proof of Theorem 1.2, we have

$$ M(\chi _{B}w) (x)\leq C \vert x-y \vert ^{-n}w(B). $$

Since $w\in A_{1}$, Θ is submultiplicative, we get

$$\begin{aligned} {\mathrm{II}} \leq & C \int _{(3B)^{c}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)h_{\varPsi }\bigl( \vert x-y \vert ^{-n}w(B)\bigr)\,dx \\ \leq & C\sum_{j=1}^{\infty } \int _{3^{j+1}B\backslash 3^{j}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \biggl(\frac{w(B)}{ \vert 3^{j+1}B \vert }\biggr)\,dx \\ \leq &C\sum_{j=1}^{\infty } \int _{3^{j+1}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \biggl(\frac{w(3^{j+1}B)}{ \vert 3^{j+1}B \vert }\frac{w(B)}{w(3^{j+1}B)}\biggr)\,dx \\ \leq &C\sum_{j=1}^{\infty } \int _{3^{j+1}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \biggl(w(x)\frac{w(B)}{w(3^{j+1}B)} \biggr)\,dx \\ \leq & C\sum_{j=1}^{\infty }\varTheta \biggl( \frac{w(B)}{w(3^{j+1}B)} \biggr) \int _{3^{j+1}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \bigl(w(x)\bigr)\,dx \\ \leq & C\varPhi (1/t)\sum_{j=1}^{\infty } \varTheta \biggl( \frac{w(B)}{w(3^{j+1}B)} \biggr) \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}( \varTheta (w))}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} & \bigl\Vert \varPsi (M_{\alpha ,L(\log L)}f)\chi _{B} \bigr\Vert _{L^{\varGamma ,\infty }(w)} \\ &\quad =\sup_{t>0}t\varGamma \bigl(w\bigl\{ x\in B: \varPsi (M_{\alpha ,L(\log L)}f) (x)>t \bigr\} \bigr) \\ &\quad = \sup_{t>0}t\varGamma \bigl(w\bigl\{ x\in B: M_{\alpha ,L(\log L)}f(x)>\varPsi ^{-1}(t) \bigr\} \bigr) \\ &\quad = \sup_{t>0}\varPsi (t)\varGamma \bigl(w\bigl\{ x\in B: M_{\alpha ,L(\log L)}f(x)>t \bigr\} \bigr) \\ &\quad \leq C \Biggl( \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(\varTheta (w))} \\ &\qquad{} + \sum_{j=1}^{\infty }\varTheta \biggl(\frac{w(B)}{w(3^{j+1}B)} \biggr) \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(\varTheta (w))} \Biggr). \end{aligned}$$

Thus, for any $r>0$, we have

$$\begin{aligned} &{}_{r}{ \bigl\Vert \varPsi (M_{\alpha ,L(\log L)}f) \bigr\Vert _{(L^{\varGamma ,\infty }(w),L^{p})^{ \beta }}} \\ &\quad = \biggl({ \int _{\mathbb{R}^{n}}} {\bigl({w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}}} \bigl\Vert \varPsi (M_{\alpha ,L(\log L)}f)\bigr){\chi }_{B(y,r)} \bigr\Vert _{L^{ \varGamma ,\infty }(w)}\biggr)^{p}}\,dy ) ^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(\varTheta (w))}\bigr)^{p}\,dy \biggr)^{\frac{1}{p}} \\ &\qquad {}+ C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty } \varTheta \biggl( \frac{w(B(y,r))}{w(B(y,3^{j+1}r))} \biggr)w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}}\\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(\varTheta (w))} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,3r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(\varTheta (w))}\bigr)^{p}\,dy \biggr)^{\frac{1}{p}} \\ &\qquad {}+C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty} \frac{\log (e+3^{jn\eta })}{3^{jn\eta (\frac{1}{\beta }-1-\frac{1}{p}+\frac{1}{q})}}w\bigl(B\bigl(y,3^{j+1}r\bigr)\bigr)^{ \frac{1}{\beta }-1-\frac{1}{p}}\\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}( \varTheta (w))} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }} \Biggl(1+ \sum_{j=1}^{\infty} \frac{\log (e+3^{jn\eta })}{3^{jn\eta (\frac{1}{\beta }-1-\frac{1}{p}+\frac{1}{q})}} \Biggr) \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }}. \end{aligned}$$

This ends the proof. □

Lemma 3.3

([4])

Let$0<\alpha <n$, $1/q=1-\alpha /n$, $w\in A_{1}$, and$b\in \mathit{BMO}$. Then there exists a constant$C>0$such that, for any$t>0$,

$$\begin{aligned} &\varGamma \bigl(w\bigl(\bigl\{ x\in \mathbb{R}^{n}: \bigl\vert [b,I_{\alpha }](f) (x) \bigr\vert >t\bigr\} \bigr)\bigr) \\ &\quad \leq C \int _{\mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(y) \vert }{t} \biggr) \varTheta \bigl(w(y)\bigr)\,dy. \end{aligned}$$

Lemma 3.4

([9])

Let$f(x)\geq 0$, $f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$, and$0<\delta <1$, then$M(f)^{\delta }\in A_{1}$.

Proof of Theorem 1.5

Fix $y\in \mathbb{R}^{n}$ and $r>0$, let $B=B(y,r)$. For any $w\in A_{1}$ and $\delta : 0<\delta \leq \theta$, by Lemma 3.4, we have $M(w^{1+\delta }\chi _{B})^{1/(1+\delta )}\in A_{1}$. By Lemma 3.3, we obtain

$$\begin{aligned} &\varGamma \bigl(w\bigl(\bigl\{ x\in B: \bigl\vert [b,I_{\alpha }]f(x) \bigr\vert >t\bigr\} \bigr)\bigr) \\ &\quad = \varGamma \biggl( \int _{\{x\in \mathbb{R}^{n}: \vert [b,I_{\alpha }]f(x) \vert >t\}}w(x) \chi _{B}(x)\,dx \biggr) \\ &\quad \leq C\varGamma \biggl( \int _{\{x\in \mathbb{R}^{n}: \vert [b,I_{\alpha }]f(x) \vert >t \}}M(w\chi _{B}) (x)\,dx \biggr) \\ &\quad \leq C \varGamma \biggl( \int _{\{x\in \mathbb{R}^{n}: \vert [b,I_{\alpha }]f(x) \vert >t \}}\bigl(M\bigl(w^{1+\delta }\chi _{B} \bigr) (x)\bigr)^{1/(1+\delta )} \,dx \biggr) \\ &\quad \leq C \int _{\mathbb{R}^{n}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \varTheta \bigl(\bigl(M\bigl(w^{1+\delta }\chi _{B}\bigr) (x) \bigr)^{1/(1+\delta )}\bigr)\,dx \\ &\quad \leq C \biggl( \int _{3B} + \int _{(3B)^{c}} \biggr)\varPhi \biggl( \frac{ \vert f(x) \vert }{t} \biggr) \varTheta \bigl(\bigl(M\bigl(w^{1+\delta }\chi _{B}\bigr) (x) \bigr)^{1/(1+ \delta )}\bigr)\,dx \\ &\quad \leq {\mathrm{I}}+\mathrm{II}. \end{aligned}$$

Now we estimate the term I. Noticing that $w\in A_{1}$, Lemma 2.8, we have $\varTheta ((M(w^{1+\delta }\chi _{B})(x))^{1/(1+\delta )})\leq C\varTheta (Mw(x)) \leq C\varTheta (w(x))$. Then

$$\begin{aligned} {\mathrm{I}}\leq C \int _{3B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr)\varTheta \bigl(w(x)\bigr)\,dx \leq C\varPhi (1/t) \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(\varTheta (w))}. \end{aligned}$$

For the term II, as the proof of Theorem 1.3, for $x\in (3B)^{c}$,

$$ \bigl(M\bigl(w^{1+\delta }\chi _{B}\bigr) (x)\bigr)^{\frac{1}{1+\delta }} \leq C \biggl( \frac{ \vert B \vert }{ \vert x-y \vert ^{n}} \biggr)^{\frac{1}{1+\delta }}\frac{w(B)}{ \vert B \vert }. $$

By Lemma 2.2, we get

$$\begin{aligned} {\mathrm{II}} \leq & C \int _{(3B)^{c}}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \varTheta \biggl( \biggl(\frac{ \vert B \vert }{ \vert x-y \vert ^{n}} \biggr)^{\frac{1}{1+\delta }} \frac{w(B)}{ \vert B \vert } \biggr)\,dx \\ \leq & C \sum_{j=1}^{\infty } \int _{3^{j+1}B\setminus 3^{j}B}\varPhi \biggl(\frac{ \vert f(x) \vert }{t} \biggr) \varTheta \biggl( \biggl(\frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\eta -\frac{\delta }{1+\delta }}w(x) \biggr)\,dx \\ \leq & C \varPhi (1/t)\sum_{j=1}^{\infty } \varTheta \biggl( \frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\eta -\frac{\delta }{(1+\delta )}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}(\varTheta (w))}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} & \bigl\Vert \varPsi \bigl( \bigl\vert [b,I_{\alpha }]f \bigr\vert \bigr)\chi _{B} \bigr\Vert _{L^{\varGamma ,\infty }(w)} \\ &\quad =\sup_{t>0}t\varGamma \bigl(w\bigl\{ x\in B: \varPsi \bigl( \bigl\vert [b,I_{\alpha }]f \bigr\vert \bigr) (x)>t\bigr\} \bigr) \\ &\quad = \sup_{t>0}t\varGamma \bigl(w\bigl\{ x\in B: \bigl\vert [b,I_{\alpha }]f(x) \bigr\vert >\varPsi ^{-1}(t) \bigr\} \bigr) \\ &\quad = \sup_{t>0}\varPsi (t)\varGamma \bigl(w\bigl\{ x\in B: \bigl\vert [b,I_{\alpha }]f(x) \bigr\vert >t\bigr\} \bigr) \\ &\quad \leq C \Biggl( \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3B} \bigr\Vert _{L^{1}(\varTheta (w))} \\ &\qquad {}+ \sum_{j=1}^{\infty }\varTheta \biggl(\frac{ \vert B \vert }{ \vert 3^{j+1}B \vert } \biggr)^{\eta - \frac{\delta }{(1+\delta )}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{3^{j+1}B} \bigr\Vert _{L^{1}( \varTheta (w))} \Biggr). \end{aligned}$$

Thus, for any $r>0$, we have

$$\begin{aligned} &{}_{r}{ \bigl\Vert \varPsi \bigl( \bigl\vert [b,I_{\alpha }]f \bigr\vert \bigr)\chi _{B} \bigr\Vert _{(L^{\varGamma ,\infty }(w),L^{p})^{ \beta }}} \\ &\quad = \biggl({ \int _{\mathbb{R}^{n}}} {\bigl({w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}}} \bigl\Vert \varPsi \bigl( \bigl\vert [b,I_{\alpha }]f \bigr\vert \bigr)\chi _{B(y,r)} \bigr\Vert _{L^{\varGamma , \infty }(w)}\bigr)^{p}}\,dy \biggr) ^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(\varTheta (w))}\bigr)^{p}\,dy \biggr)^{\frac{1}{p}} \\ &\qquad {}+C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty } \varTheta \biggl( \frac{ \vert B(y,r) \vert }{ \vert B(y,3^{j+1}r) \vert } \biggr)^{\eta - \frac{\delta }{(1+\delta )}}w\bigl(B(y,r) \bigr)^{\frac{1}{\beta }-1-\frac{1}{p}}\\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(\varTheta (w))} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \biggl( \int _{\mathbb{R}^{n}}\bigl(w\bigl(B(y,3r)\bigr)^{\frac{1}{\beta }-1- \frac{1}{p}} \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3r)} \bigr\Vert _{L^{1}(\varTheta (w))}\bigr)^{p}\,dy \biggr)^{\frac{1}{p}} \\ &\qquad {}+C \Biggl( \int _{\mathbb{R}^{n}} \Biggl(\sum_{j=1}^{\infty} \frac{\log (e+3^{jn(\eta -\frac{\delta }{1+\delta })})}{3^{jn(\eta (\frac{1}{\beta }-1-\frac{1}{p}+\frac{1}{q})-\frac{\delta }{\delta (1+q)})}} w\bigl(B\bigl(y,3^{j+1}r\bigr)\bigr)^{\frac{1}{\beta }-1-\frac{1}{p}} \\ &\qquad {}\times \bigl\Vert \varPhi \bigl( \vert f \vert \bigr)\chi _{B(y,3^{j+1}r)} \bigr\Vert _{L^{1}(\varTheta (w))} \Biggr)^{p}\,dy \Biggr)^{\frac{1}{p}} \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }} \Biggl(1+ \sum_{j=1}^{\infty} \frac{\log (e+3^{jn(\eta -\frac{\delta }{1+\delta })})}{3^{jn(\eta (\frac{1}{\beta }-1-\frac{1}{p}+\frac{1}{q})-\frac{\delta }{\delta (1+q)})}} \Biggr) \\ &\quad \leq C \bigl\Vert \varPhi \bigl( \vert f \vert \bigr) \bigr\Vert _{(L^{1}(\varTheta (w)),L^{p}(w))^{\beta }}, \end{aligned}$$

in which we take $\delta >0$ small enough such that $\eta (\frac{1}{q}-1-\frac{1}{p}+\frac{1}{\beta })- \frac{\delta }{q(1+\delta )}>0$ and $\eta -\frac{\delta }{1+\delta }>0$. This ends the proof. □