Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

Abstract

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spaces $M_w^{p,\phi }(l_2)$. We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and ${\mathrm{g}}_{\lambda }^{\ast }$ -function, and their multilinear commutators on vector-valued generalized weighted Morrey spaces $M_w^{p,\phi }(l_2)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $\phi (x,r)$ without assuming any monotonicity property of $\phi (x,r)$ on r.

MSC:42B25, 42B35.

1 Introduction

It is well known that the commutator is an important integral operator and it plays a key role in harmonic analysis. In 1965, Calderon [1, 2] studied a kind of commutators, appearing in Cauchy integral problems of Lip-line. Let K be a Calderón-Zygmund singular integral operator and $b\in BMO({\mathbb{R}}^n)$. A well-known result of Coifman et al. [3] states that the commutator operator $[b,K]f=K(bf)-bKf$ is bounded on $L^p({\mathbb{R}}^n)$ for $1<p<\mathrm{\infty }$. The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [4–8]).

The classical Morrey spaces were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [7–10]. Recently, Komori and Shirai [11] first defined the weighted Morrey spaces $L^{p,\kappa }(w)$ and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [12, 13] introduced the generalized weighted Morrey spaces $M_w^{p,\phi }$ and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [14–16]).

The intrinsic square functions were first introduced by Wilson in [17, 18]. They are defined as follows. For $0<\alpha \le 1$, let $C_{\alpha }$ be the family of functions $\varphi :{\mathbb{R}}^n\to \mathbb{R}$ such that ϕ’s support is contained in $\{x:|x|\le 1\}$, ${\int }_{{\mathbb{R}}^n}\varphi (x)dx=0$, and for $x,x^{\prime }\in {\mathbb{R}}^n$,

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

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

$$A_{\alpha }f(t,y)\equiv \underset{\varphi \in C_{\alpha }}{sup}|f\ast {\varphi }_t(y)|,$$

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

$$G_{\alpha ,\beta }(f)(x)={\left({\iint }_{{\mathrm{\Gamma }}_{\beta }(x)}{(A_{\alpha }f(t,y))}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}},$$

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

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 $G_{\alpha ,\beta }(f)$ depends on kernels with uniform compact support, there is pointwise relation between $G_{\alpha ,\beta }(f)$ with different β:

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

We can see details in [17].

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

$$\begin{array}{c}{\mathrm{g}}_{\alpha }f(x)={\left({\int }_0^{\mathrm{\infty }}{(A_{\alpha }f(y,t))}^2\frac{dt}{t}\right)}^{\frac{1}{2}},\hfill \\ {\mathrm{g}}_{\lambda ,\alpha }^{\ast }f(x)={\left({\iint }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }f(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}.\hfill \end{array}$$

When we say that f maps into $l_2$, we mean that $\overrightarrow{f}(x)={(f_j)}_{j=1}^{\mathrm{\infty }}$, where each $f_j$ is Lebesgue measurable and, for almost every $x\in {\mathbb{R}}^n$

$${\parallel \overrightarrow{f}(x)\parallel }_{l_2}={\left(\sum _{j=1}^{\mathrm{\infty }}{|f_j(x)|}^2\right)}^{1/2}.$$

Let $\overrightarrow{f}=(f_1,f_2,\dots )$ be a sequence of locally integrable functions on ${\mathbb{R}}^n$. For any $x\in {\mathbb{R}}^n$, Wilson [18] also defined the vector-valued intrinsic square functions of $\overrightarrow{f}$ by ${\parallel G_{\alpha }\overrightarrow{f}(x)\parallel }_{l_2}$ and proved the following result.

Theorem A Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, and $w\in A_p$. Then the operators $G_{\alpha }$ and ${\mathrm{g}}_{\lambda ,\alpha }^{\ast }$ are bounded from $L_w^p(l_2)$ into itself for $p>1$ and from $L_w^1(l_2)$ to $W{L}_w^1(l_2)$.

Moreover, in [19], Lerner showed sharp $L_w^p$ norm inequalities for the intrinsic square functions in terms of the $A_p$ characteristic constant of w for all $1<p<\mathrm{\infty }$. Also Huang and Liu [20] studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In [21] and [22], Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In [23], Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j$, $j=1,\dots ,m$ be locally integrable function on ${\mathbb{R}}^n$. Setting

$$A_{\alpha ,\overrightarrow{b}}f(t,y)\equiv \underset{\varphi \in C_{\alpha }}{sup}|{\int }_{{\mathbb{R}}^n}\prod _{j=1}^m[b_j(x)-b_j(z)]{\varphi }_t(y-z)f(z)dz|,$$

the multilinear commutators are defined by

$$[\overrightarrow{b},G_{\alpha }]f(x)={\left({\iint }_{\mathrm{\Gamma }(x)}{(A_{\alpha ,\overrightarrow{b}}f(t,y))}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}},$$

$$[\overrightarrow{b},{\mathrm{g}}_{\alpha }]f(x)={\left({\int }_0^{\mathrm{\infty }}{(A_{\alpha ,\overrightarrow{b}}f(t,y))}^2\frac{dt}{t}\right)}^{\frac{1}{2}},$$

and

$$[\overrightarrow{b},{\mathrm{g}}_{\lambda ,\alpha }^{\ast }]f(x)={\left({\iint }_{{\mathbb{R}}_{+}^{n+1}}{\left(\frac{t}{t+|x-y|}\right)}^{\lambda n}{(A_{\alpha ,\overrightarrow{b}}f(t,y))}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}.$$

In [23], Wang proved the following result.

Theorem B Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $b\in BMO({\mathbb{R}}^n)$. Then the commutator operators $[b,G_{\alpha }]$ and $[b,{\mathrm{g}}_{\lambda ,\alpha }^{\ast }]$ are bounded from $L_w^p(l_2)$ into itself.

Analogously the following result may be proved.

Theorem B′ Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$. Let also $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j\in BMO({\mathbb{R}}^n)$, $j=1,\dots ,m$. Then the multilinear commutator operators $[\overrightarrow{b},G_{\alpha }]$ and $[\overrightarrow{b},{\mathrm{g}}_{\lambda ,\alpha }^{\ast }]$ are bounded from $L_w^p(l_2)$ into itself.

In this paper, we will consider the boundedness of the operators $G_{\alpha }$, ${\mathrm{g}}_{\alpha }$, ${\mathrm{g}}_{\lambda ,\alpha }^{\ast }$ and their multilinear commutators on vector-valued generalized weighted Morrey spaces. Let $\phi (x,r)$ be a positive measurable function on ${\mathbb{R}}^n\times {\mathbb{R}}_{+}$ and w be non-negative measurable function on ${\mathbb{R}}^n$. For any $\overrightarrow{f}\in L_w^{p,\mathrm{loc}}(l_2)$, we denote by $M_w^{p,\phi }(l_2)$ the vector-valued generalized weighted Morrey spaces, if

$${\parallel \overrightarrow{f}\parallel }_{M_w^{p,\phi }(l_2)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\phi {(x,r)}^{-1}w{(B(x,r))}^{-\frac{1}{p}}{\parallel {\parallel \overrightarrow{f}(\cdot )\parallel }_{l_2}\parallel }_{L_w^p(B(x,r))}<\mathrm{\infty }.$$

When $w\equiv 1$, then $M_w^{p,\phi }(l_2)$ coincide the vector-valued generalized Morrey spaces $M^{p,\phi }(l_2)$. There are many papers discussed the conditions on $\phi (x,r)$ to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [24] (see, also [10]), by Guliyev the following condition was imposed on the pair $({\phi }_1,{\phi }_2)$:

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

(1.1)

where $C>0$ does not depend on x and r. Under the above condition, they obtained the boundedness of Calderón-Zygmund singular integral operators from $M^{p,{\phi }_1}({\mathbb{R}}^n)$ to $M^{p,{\phi }_2}({\mathbb{R}}^n)$. Also, in [25] and [26], Guliyev et al. introduced a weaker condition: If $1\le p<\mathrm{\infty }$, there exists a constant $C>0$, such that, for any $x\in {\mathbb{R}}^n$ and $r>0$,

$${\int }_r^{\mathrm{\infty }}\frac{{ess\hspace{0.17em}inf}_{t<s<\mathrm{\infty }}{\phi }_1(x,s)s^{\frac{n}{p}}}{t^{\frac{n}{p}+1}}dt\le C{\phi }_2(x,r).$$

(1.2)

If the pair $({\phi }_1,{\phi }_2)$ satisfies condition (1.1), then $({\phi }_1,{\phi }_2)$ satisfied condition (1.2). But the opposite is not true. We can see Remark 4.7 in [26] for details.

Recently, in [12, 13] (see, also [14–16]), Guliyev introduced a weighted condition: If $1\le p<\mathrm{\infty }$, there exists a constant $C>0$, such that, for any $x\in {\mathbb{R}}^n$ and $t>0$,

$${\int }_r^{\mathrm{\infty }}\frac{{ess\hspace{0.17em}inf}_{t<s<\mathrm{\infty }}{\phi }_1(x,s)w{(B(x,s))}^{\frac{1}{p}}}{w{(B(x,t))}^{\frac{1}{p}}}\frac{dt}{t}\le C{\phi }_2(x,r).$$

(1.3)

In this paper, we will obtain the boundedness of the vector-valued intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic $g_{\lambda }^{\ast }$ function and their multilinear commutators on vector-valued generalized weighted Morrey spaces when $w\in A_p$ and the pair $({\phi }_1,{\phi }_2)$ satisfies condition (1.3) or the following inequalities:

$${\int }_r^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r})\frac{{ess\hspace{0.17em}inf}_{t<s<\mathrm{\infty }}{\phi }_1(x,s)w{(B(x,s))}^{\frac{1}{p}}}{w{(B(x,t))}^{\frac{1}{p}}}\frac{dt}{t}\le C{\phi }_2(x,r),$$

(1.4)

where C does not depend on x and r. Our main results in this paper are stated as follows.

Theorem 1.1 Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.3). Then the operator $G_{\alpha }$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$ for $p>1$ and from $M_w^{1,{\phi }_1}(l_2)$ to $W{M}_w^{1,{\phi }_2}(l_2)$.

Theorem 1.2 Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, $\lambda >3+\frac{\alpha }{n}$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.3). Then the operator ${\mathrm{g}}_{\lambda ,\alpha }^{\ast }$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$ for $p>1$ and from $M_w^{1,{\phi }_1}(l_2)$ to $W{M}_w^{1,{\phi }_2}(l_2)$.

Theorem 1.3 Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.4). Let also $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j\in BMO({\mathbb{R}}^n)$, $j=1,\dots ,m$. Then $[\overrightarrow{b},G_{\alpha }]$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$.

Theorem 1.4 Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.4). Let also $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j\in BMO({\mathbb{R}}^n)$, $j=1,\dots ,m$. Then for $\lambda >3+\frac{\alpha }{n}$, $[\overrightarrow{b},{\mathrm{g}}_{\lambda ,\alpha }^{\ast }]$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$.

In [17], the author proved that the functions $G_{\alpha }f$ and ${\mathrm{g}}_{\alpha }f$ are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.

Corollary 1.5 Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.3), then ${\mathrm{g}}_{\alpha }$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$ for $p>1$ and from $M_w^{1,{\phi }_1}(l_2)$ to $W{M}_w^{1,{\phi }_2}(l_2)$.

Corollary 1.6 Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, and $({\phi }_1,{\phi }_2)$ satisfy condition (1.4). Let also $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j\in BMO({\mathbb{R}}^n)$, $j=1,\dots ,m$. Then $[\overrightarrow{b},{\mathrm{g}}_{\alpha }]$ is bounded from $M_w^{p,{\phi }_1}(l_2)$ to $M_w^{p,{\phi }_2}(l_2)$.

Remark 1.7 Note that, in the scalar valued case and for $m=1$, $w\equiv 1$ Theorems 1.1-1.4 and Corollaries 1.5-1.6 was proved in [27]. Also, in the scalar valued case and $m=1$, $w\equiv A_p$, and ${\phi }_1(x,r)={\phi }_2(x,r)\equiv w{(B(x,r))}^{\frac{\kappa -1}{p}}$, $0<\kappa <1$ Theorems 1.1-1.4 and Corollaries 1.5-1.6 were proved by Wang in [23, 28]. If $\phi (x,r)\equiv w{(B(x,r))}^{\frac{\kappa -1}{p}}$, then the vector-valued generalized weighed Morrey space $M_w^{p,\phi }(l_2)$ coincides with the vector-valued weighed Morrey space $L_w^{p,\kappa }(l_2)$ and the pair $(w{(B(x,r))}^{\frac{\kappa -1}{p}},w{(B(x,r))}^{\frac{\kappa -1}{p}})$ satisfies the two conditions (1.3) and (1.4). Indeed, by Lemma 3.1 there exist $C>0$ and $\delta >0$ such that for all $x\in {\mathbb{R}}^n$ and $t>r$:

$$w(B(x,t))\ge C{\left(\frac{t}{r}\right)}^{n\delta }w(B(x,r)).$$

Then

$$\begin{array}{rcl}{\int }_r^{\mathrm{\infty }}\frac{{ess\hspace{0.17em}inf}_{t<s<\mathrm{\infty }}w{(B(x,s))}^{\frac{\kappa }{p}}}{w{(B(x,t))}^{1/p}}\frac{dt}{t}& \le & {\int }_r^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r})\frac{{ess\hspace{0.17em}inf}_{t<s<\mathrm{\infty }}w{(B(x,s))}^{\frac{\kappa }{p}}}{w{(B(x,t))}^{1/p}}\frac{dt}{t}\\ =& {\int }_r^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r})w{(B(x,t))}^{\frac{\kappa -1}{p}}\frac{dt}{t}\\ \lesssim & {\int }_r^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r}){\left({\left(\frac{t}{r}\right)}^{n\delta }w(B(x,r))\right)}^{\frac{\kappa -1}{p}}\frac{dt}{t}\\ =& w{(B(x,r))}^{\frac{\kappa -1}{p}}{\int }_r^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r}){\left(\frac{t}{r}\right)}^{n\delta \frac{\kappa -1}{p}}\frac{dt}{t}\\ =& w{(B(x,r))}^{\frac{\kappa -1}{p}}{\int }_1^{\mathrm{\infty }}{ln}^m(e+\tau ){\tau }^{n\delta \frac{\kappa -1}{p}}\frac{d\tau }{\tau }\\ \approx & w{(B(x,r))}^{\frac{\kappa -1}{p}}.\end{array}$$

Throughout this paper, we use the notation $A\lesssim B$ to express that there is a positive constant C independent of all essential variables such that $A\le CB$. Moreover, C may be different from place to place.

2 Vector-valued generalized weighted Morrey spaces

The classical Morrey spaces $M^{p,\lambda }$ were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [29, 30].

We denote by $M^{p,\lambda }(l_2)\equiv M^{p,\lambda }({\mathbb{R}}^n,l_2)$ the vector-valued Morrey space, the space of all vector-valued functions $\overrightarrow{f}\in L^{p,\mathrm{loc}}(l_2)$ with finite quasinorm

$${\parallel \overrightarrow{f}\parallel }_{M^{p,\lambda }(l_2)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}{r}^{-\frac{\lambda }{p}}{\parallel \overrightarrow{f}\parallel }_{L^p(B(x,r),l_2)},$$

where $1\le p<\mathrm{\infty }$ and $0\le \lambda \le n$.

Note that $M^{p,0}(l_2)=L^p(l_2)$ and $M^{p,n}(l_2)=L^{\mathrm{\infty }}(l_2)$. If $\lambda <0$ or $\lambda >n$, then $M^{p,\lambda }(l_2)=\mathrm{\Theta }$, where Θ is the set of all vector-valued functions equivalent to 0 on ${\mathbb{R}}^n$.

We define the vector-valued generalized weighed Morrey spaces as follows.

Definition 2.1 Let $1\le p<\mathrm{\infty }$, φ be a positive measurable vector-valued function on ${\mathbb{R}}^n\times (0,\mathrm{\infty })$ and w be non-negative measurable function on ${\mathbb{R}}^n$. We denote by $M_w^{p,\phi }(l_2)$ the vector-valued generalized weighted Morrey space, the space of all vector-valued functions $\overrightarrow{f}\in L_w^{p,\mathrm{loc}}(l_2)$ with finite norm

$${\parallel \overrightarrow{f}\parallel }_{M_w^{p,\phi }(l_2)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\phi {(x,r)}^{-1}w{(B(x,r))}^{-\frac{1}{p}}{\parallel f\parallel }_{L_w^p(B(x,r),l_2)},$$

where $L_w^p(B(x,r),l_2)$ denotes the vector-valued weighted $L^p$-space of measurable functions f for which

$${\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x,r))}\equiv {\parallel \overrightarrow{f}{\chi }_{{}_{B(x,r)}}\parallel }_{L_w^p({\mathbb{R}}^n)}={({\int }_{B(x,r)}{\parallel \overrightarrow{f}(y)\parallel }_{l_2}^{p}w(y)dy)}^{\frac{1}{p}}.$$

Furthermore, by $W{M}_w^{p,\phi }(l_2)$ we denote the vector-valued weak generalized weighted Morrey space of all functions $f\in W{L}_w^{p,\mathrm{loc}}(l_2)$ for which

$${\parallel \overrightarrow{f}\parallel }_{W{M}_w^{p,\phi }(l_2)}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\phi {(x,r)}^{-1}w{(B(x,r))}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{W{L}_w^p(B(x,r),l_2)}<\mathrm{\infty },$$

where $W{L}_w^p(B(x,r),l_2)$ denotes the weak $L_w^p$-space of measurable functions f for which

$${\parallel \overrightarrow{f}\parallel }_{W{L}_w^p(B(x,r),l_2)}\equiv {\parallel \overrightarrow{f}{\chi }_{{}_{B(x,r)}}\parallel }_{W{L}_w^p(l_2)}=\underset{t>0}{sup}t{({\int }_{\{y\in B(x,r):{\parallel \overrightarrow{f}(y)\parallel }_{l_2}>t\}}w(y)dy)}^{\frac{1}{p}}.$$

Remark 2.2

• (1) If $w\equiv 1$, then $M_1^{p,\phi }(l_2)=M^{p,\phi }(l_2)$ is the vector-valued generalized Morrey space.

• (2) If $\phi (x,r)\equiv w{(B(x,r))}^{\frac{\kappa -1}{p}}$, then $M_w^{p,\phi }(l_2)=L_w^{p,\kappa }(l_2)$ is the vector-valued weighted Morrey space.

• (3) If $\phi (x,r)\equiv v{(B(x,r))}^{\frac{\kappa }{p}}w{(B(x,r))}^{-\frac{1}{p}}$, then $M_w^{p,\phi }(l_2)=L_{v,w}^{p,\kappa }(l_2)$ is the vector-valued two weighted Morrey space.

• (4) If $w\equiv 1$ and $\phi (x,r)=r^{\frac{\lambda -n}{p}}$ with $0<\lambda <n$, then $M_w^{p,\phi }(l_2)=L^{p,\lambda }(l_2)$ is the vector-valued Morrey space and $W{M}_w^{p,\phi }(l_2)=W{L}^{p,\lambda }(l_2)$ is the vector-valued weak Morrey space.

• (5) If $\phi (x,r)\equiv w{(B(x,r))}^{-\frac{1}{p}}$, then $M_w^{p,\phi }(l_2)=L_w^p(l_2)$ is the vector-valued weighted Lebesgue space.

3 Preliminaries and some lemmas

By a weight function, briefly weight, we mean a locally integrable function on ${\mathbb{R}}^n$ which takes values in $(0,\mathrm{\infty })$ almost everywhere. For a weight w and a measurable set E, we define $w(E)={\int }_{E}w(x)dx$, and denote the Lebesgue measure of E by $|E|$ and the characteristic function of E by ${\chi }_{{}_E}$. Given a weight w, we say that w satisfies the doubling condition if there exists a constant $D>0$ such that for any ball B, we have $w(2B)\le Dw(B)$. When w satisfies this condition, we write for brevity $w\in {\mathrm{\Delta }}_2$.

If w is a weight function, we denote by $L_w^p(l_2)\equiv L_w^p({\mathbb{R}}^n,l_2)$ the vector-valued weighted Lebesgue space defined by finiteness of the norm

$${\parallel \overrightarrow{f}\parallel }_{L_w^p(l_2)}={({\int }_{{\mathbb{R}}^n}{\parallel \overrightarrow{f}(x)\parallel }_{l_2}^{p}w(x)dx)}^{\frac{1}{p}}<\mathrm{\infty },\text{if }1\le p<\mathrm{\infty }$$

and by ${\parallel \overrightarrow{f}\parallel }_{L_w^{\mathrm{\infty }}(l_2)}={ess\hspace{0.17em}sup}_{x\in {\mathbb{R}}^n}{\parallel \overrightarrow{f}(x)\parallel }_{l_2}w(x)$ if $p=\mathrm{\infty }$.

We recall that a weight function w is in the Muckenhoupt class $A_p$ [31], $1<p<\mathrm{\infty }$, if

$$\begin{array}{rcl}{[w]}_{A_p}& :=& \underset{B}{sup}{[w]}_{A_p(B)}\\ =& \underset{B}{sup}(\frac{1}{|B|}{\int }_{B}w(x)dx){\left(\frac{1}{|B|}{\int }_{B}w{(x)}^{1-p^{\prime }}dx\right)}^{p-1}<\mathrm{\infty },\end{array}$$

where the sup is taken with respect to all the balls B and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. Note that, for all balls B, by Hölder’s inequality

$${[w]}_{A_p(B)}^{1/p}={|B|}^{-1}{\parallel w\parallel }_{L^1(B)}^{1/p}{\parallel w^{-1/p}\parallel }_{L^{p^{\prime }}(B)}\ge 1.$$

For $p=1$, the class $A_1$ is defined by the condition $Mw(x)\le Cw(x)$ with ${[w]}_{A_1}={sup}_{x\in {\mathbb{R}}^n}\frac{Mw(x)}{w(x)}$, and for $p=\mathrm{\infty }$, $A_{\mathrm{\infty }}={\bigcup }_{1\le p<\mathrm{\infty }}{A}_p$ and ${[w]}_{A_{\mathrm{\infty }}}={inf}_{1\le p<\mathrm{\infty }}{[w]}_{A_p}$.

Lemma 3.1 ([32])

• (1) If $w\in A_p$ for some $1\le p<\mathrm{\infty }$, then $w\in {\mathrm{\Delta }}_2$. Moreover, for all $\lambda >1$

  $$w(\lambda B)\le {\lambda }^{np}{[w]}_{A_p}w(B).$$

• (2) If $w\in A_{\mathrm{\infty }}$, then $w\in {\mathrm{\Delta }}_2$. Moreover, for all $\lambda >1$

  $$w(\lambda B)\le 2^{{\lambda }^n}{[w]}_{A_{\mathrm{\infty }}}w(B).$$

• (3) If $w\in A_p$ for some $1\le p\le \mathrm{\infty }$, then there exist $C>0$ and $\delta >0$ such that for any ball B and a measurable set $S\subset B$,

  $$\frac{w(S)}{w(B)}\le C{\left(\frac{|S|}{|B|}\right)}^{\delta }.$$

We are going to use the following result on the boundedness of the Hardy operator:

$$(Hg)(t):=\frac{1}{t}{\int }_0^{t}g(r)d\mu (r),0<t<\mathrm{\infty },$$

where μ is a non-negative Borel measure on $(0,\mathrm{\infty })$.

Theorem 3.2 ([33])

The inequality

$$\underset{t>0}{ess\hspace{0.17em}sup}\omega (t)Hg(t)\le c\underset{t>0}{ess\hspace{0.17em}sup}v(t)g(t)$$

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

$$A:=\underset{t>0}{sup}\frac{\omega (t)}{t}{\int }_0^t\frac{d\mu (r)}{{ess\hspace{0.17em}sup}_{0<s<r}v(s)}<\mathrm{\infty },$$

and $c\approx A$.

We also need the following statement on the boundedness of the Hardy type operator:

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

where μ is a non-negative Borel measure on $(0,\mathrm{\infty })$.

Theorem 3.3 The inequality

$$\underset{t>0}{ess\hspace{0.17em}sup}\omega (t)H_{1}g(t)\le c\underset{t>0}{ess\hspace{0.17em}sup}v(t)g(t)$$

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

$$A_1:=\underset{t>0}{sup}\frac{\omega (t)}{t}{\int }_0^t{ln}^m(e+\frac{t}{r})\frac{d\mu (r)}{{ess\hspace{0.17em}sup}_{0<s<r}v(s)}<\mathrm{\infty },$$

and $c\approx A_1$.

Note that Theorem 3.3 can be proved analogously to Theorem 4.3 in [34].

Definition 3.4 $BMO({\mathbb{R}}^n)$ is the Banach space modulo constants with the norm ${\parallel \cdot \parallel }_{\ast }$ defined by

$${\parallel b\parallel }_{\ast }=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\frac{1}{|B(x,r)|}{\int }_{B(x,r)}|b(y)-b_{B(x,r)}|dy<\mathrm{\infty },$$

where $b\in L_1^{\mathrm{loc}}({\mathbb{R}}^n)$ and

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

Lemma 3.5 ([35], Theorem 5, p.236)

Let $w\in A_{\mathrm{\infty }}$. Then the norm ${\parallel \cdot \parallel }_{\ast }$ is equivalent to the norm

$${\parallel b\parallel }_{\ast ,w}=\underset{x\in {\mathbb{R}}^n,r>0}{sup}\frac{1}{w(B(x,r))}{\int }_{B(x,r)}|b(y)-b_{B(x,r),w}|w(y)dy,$$

where

$$b_{B(x,r),w}=\frac{1}{w(B(x,r))}{\int }_{B(x,r)}b(y)w(y)dy.$$

Remark 3.6 (1) The John-Nirenberg inequality: there are constants $C_1,C_2>0$, such that for all $b\in BMO({\mathbb{R}}^n)$ and $\beta >0$

$$\left|\{x\in B:|b(x)-b_B|>\beta \}\right|\le C_1|B|e^{-C_2\beta /{\parallel b\parallel }_{\ast }},\mathrm{\forall }B\subset {\mathbb{R}}^n.$$

(2) For $1\le p<\mathrm{\infty }$ the John-Nirenberg inequality implies that

$${\parallel b\parallel }_{\ast }\approx \underset{B}{sup}{\left(\frac{1}{|B|}{\int }_B{|b(y)-b_B|}^{p}dy\right)}^{\frac{1}{p}}$$

(3.1)

and for $1\le p<\mathrm{\infty }$ and $w\in A_{\mathrm{\infty }}$

$${\parallel b\parallel }_{\ast }\approx \underset{B}{sup}{(\frac{1}{w(B)}{\int }_B{|b(y)-b_B|}^{p}w(y)dy)}^{\frac{1}{p}}.$$

(3.2)

Note that by the John-Nirenberg inequality and Lemma 3.1 (part 3) it follows that

$$w\left(\{x\in B:|b(x)-b_B|>\beta \}\right)\le C_1^{\delta }w(B)e^{-C_2\beta \delta /{\parallel b\parallel }_{\ast }}$$

for some $\delta >0$. Hence

$$\begin{array}{rcl}{\int }_B{|b(y)-b_B|}^{p}w(y)dy& =& p{\int }_0^{\mathrm{\infty }}{\beta }^{p-1}w\left(\{x\in B:|b(x)-b_B|>\beta \}\right)d\beta \\ \le & p{C}_1^{\delta }w(B){\int }_0^{\mathrm{\infty }}{\beta }^{p-1}{e}^{-C_2\beta \delta /{\parallel b\parallel }_{\ast }}d\beta =C_{3}w(B){\parallel b\parallel }_{\ast }^p,\end{array}$$

where $C_3>0$ depends only on $C_1^{\delta }$, $C_2$, p, and δ, which implies (3.2).

Also (3.1) is a particular case of (3.2) with $w\equiv 1$.

The following lemma was proved in [13].

Lemma 3.7 (i) Let $w\in A_{\mathrm{\infty }}$ and $b\in BMO({\mathbb{R}}^n)$. Let also $1\le p<\mathrm{\infty }$, $x\in {\mathbb{R}}^n$, $k>0$, and $r_1,r_2>0$. Then

$${(\frac{1}{w(B(x,r_1))}{\int }_{B(x,r_1)}{|b(y)-b_{B(x,r_2),w}|}^{kp}w(y)dy)}^{\frac{1}{p}}\le C{(1+|ln\frac{r_1}{r_2}|)}^k{\parallel b\parallel }_{\ast }^k,$$

where $C>0$ is independent of f, w, x, $r_1$, and $r_2$.

(ii) Let $w\in A_p$ and $b\in BMO({\mathbb{R}}^n)$. Let also $1<p<\mathrm{\infty }$, $x\in {\mathbb{R}}^n$, $k>0$, and $r_1,r_2>0$. Then

$${\left(\frac{1}{w^{1-p^{\prime }}(B(x,r_1))}{\int }_{B(x,r_1)}{|b(y)-b_{B(x,r_2),w}|}^{k{p}^{\prime }}w{(y)}^{1-p^{\prime }}dy\right)}^{\frac{1}{p^{\prime }}}\le C{(1+|ln\frac{r_1}{r_2}|)}^k{\parallel b\parallel }_{\ast }^k,$$

where $C>0$ is independent of f, w, x, $r_1$, and $r_2$.

4 Proofs of main theorems

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

Lemma 4.1 [23]

For $j\in {\mathbb{Z}}_{+}$, denote

$$G_{\alpha ,2^j}(f)(x)={\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{(A_{\alpha }f(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}.$$

Let $0<\alpha \le 1$, $1<p<\mathrm{\infty }$, and $w\in A_p$. Then any $j\in {\mathbb{Z}}_{+}$, we have

$${\parallel G_{\alpha ,2^j}(f)\parallel }_{L_w^p}\lesssim 2^{j(\frac{3n}{2}+\alpha )}{\parallel G_{\alpha }(f)\parallel }_{L_w^p}.$$

This lemma is easy by the following inequality, which is proved in [17]:

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

By a similar argument to [2], we can get the following lemma.

Lemma 4.2 Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, and $w\in A_p$, then the multilinear commutator $[\overrightarrow{b},G_{\alpha }]$ is bounded from $L_w^p(l_2)$ to itself whenever $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $b_j\in BMO({\mathbb{R}}^n)$, $j=1,\dots ,m$.

Now we are in a position to prove the theorems.

Lemma 4.3 Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, and $w\in A_p$.

Then, for $p>1$, the inequality

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}\lesssim {(w(B))}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}$$

holds for any ball $B=B(x_0,r)$ and for all $\overrightarrow{f}\in L_w^{p,\mathrm{loc}}(l_2)$.

Moreover, for $p=1$ the inequality

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{W{L}_w^1(B,l_2)}\lesssim w(B){\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-1}\frac{dt}{t}$$

holds for any ball $B=B(x_0,r)$ and for all $\overrightarrow{f}\in L_w^{1,\mathrm{loc}}(l_2)$.

Proof The main ideas of these proofs come from [13]. For arbitrary $x\in {\mathbb{R}}^n$, set $B=B(x_0,r)$, $2B\equiv B(x_0,2r)$. We decompose $\overrightarrow{f}={\overrightarrow{f}}_0+{\overrightarrow{f}}_{\mathrm{\infty }}$, where ${\overrightarrow{f}}_0(y)=\overrightarrow{f}(y){\chi }_{2B}(y)$, ${\overrightarrow{f}}_{\mathrm{\infty }}(y)=\overrightarrow{f}(y)-{\overrightarrow{f}}_0(y)$. Then

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B(x_0,r),l_2)}\le {\parallel G_{\alpha }{\overrightarrow{f}}_0\parallel }_{L_w^p(B(x_0,r),l_2)}+{\parallel G_{\alpha }{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L^p(B(x_0,r),l_2)}:=I+II.$$

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

$$I\le {\parallel G_{\alpha }{\overrightarrow{f}}_0\parallel }_{L_w^p(l_2)}\lesssim {\parallel {\overrightarrow{f}}_0\parallel }_{L_w^p(l_2)}={\parallel \overrightarrow{f}\parallel }_{L_w^p(2B,l_2)}.$$

(4.1)

On the other hand,

$$\begin{array}{rcl}{\parallel \overrightarrow{f}\parallel }_{L_w^p(2B,l_2)}& \approx & |B|{\parallel \overrightarrow{f}\parallel }_{L_w^p(2B,l_2)}{\int }_{2r}^{\mathrm{\infty }}\frac{dt}{t^{n+1}}\\ \le & |B|{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}\frac{dt}{t^{n+1}}\\ \lesssim & w{(B)}^{\frac{1}{p}}{\parallel w^{-1/p}\parallel }_{L_{p^{\prime }}(B)}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}\frac{dt}{t^{n+1}}\\ \lesssim & w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\parallel w^{-1/p}\parallel }_{L_{p^{\prime }}(B(x_0,t))}\frac{dt}{t^{n+1}}\\ \lesssim & {[\omega ]}_{A_p}^{1/p}w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.\end{array}$$

(4.2)

Therefore from (4.1) and (4.2) we get

$$I\lesssim w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

(4.3)

Then let us estimate II:

$${\parallel \overrightarrow{f}\ast {\varphi }_t(y)\parallel }_{l_2}={\parallel t^{-n}{\int }_{|y-z|\le t}\varphi \left(\frac{y-z}{t}\right){\overrightarrow{f}}_{\mathrm{\infty }}(z)dz\parallel }_{l_2}\le t^{-n}{\int }_{|y-z|\le t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz.$$

Since $x\in B(x_0,r)$, $(y,t)\in \mathrm{\Gamma }(x)$, we have $|z-x|\le |z-y|+|y-x|\le 2t$, and

$$r\le |z-x_0|-|x_0-x|\le |x-z|\le |x-y|+|y-z|\le 2t.$$

So, we obtain

$$\begin{array}{rcl}{\parallel G_{\alpha }{\overrightarrow{f}}_{\mathrm{\infty }}(x)\parallel }_{l_2}& \le & {\left({\iint }_{\mathrm{\Gamma }(x)}{\left(t^{-n}{\int }_{|y-z|\le t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}\\ \le & {\left({\int }_{t>r/2}{\int }_{|x-y|<t}{\left({\int }_{|x-z|\le 2t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{dydt}{t^{3n+1}}\right)}^{\frac{1}{2}}\\ \lesssim & {\left({\int }_{t>r/2}{\left({\int }_{|z-x|\le 2t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{dt}{t^{2n+1}}\right)}^{\frac{1}{2}}.\end{array}$$

By Minkowski’s and Hölder’s inequalities and $|z-x|\ge |z-x_0|-|x_0-x|\ge \frac{1}{2}|z-x_0|$, we have

$$\begin{array}{rcl}{\parallel G_{\alpha }{\overrightarrow{f}}_{\mathrm{\infty }}(x)\parallel }_{l_2}& \lesssim & {\int }_{{\mathbb{R}}^n}{\left({\int }_{t>\frac{|z-x|}{2}}\frac{dt}{t^{2n+1}}\right)}^{\frac{1}{2}}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\\ \lesssim & {\int }_{|z-x_0|>2r}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|z-x|}^n}dz\lesssim {\int }_{|z-x_0|>2r}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|z-x_0|}^n}dz\\ =& {\int }_{|z-x_0|>2r}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}{\int }_{|z-x_0|}^{+\mathrm{\infty }}\frac{dt}{t^{n+1}}dz\\ =& {\int }_{2r}^{+\mathrm{\infty }}{\int }_{2r<|z-x_0|<t}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{dt}{t^{n+1}}\\ \lesssim & {\int }_{2r}^{\mathrm{\infty }}{\parallel {\parallel \overrightarrow{f}(z)\parallel }_{l_2}\parallel }_{L_w^p(B(x_0,t))}{\parallel w^{-1}\parallel }_{L_{p^{\prime }}(B(x_0,t))}\frac{dt}{t^{n+1}}\\ \lesssim & {\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.\end{array}$$

Thus,

$${\parallel G_{\alpha }{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L_w^p(B,l_2)}\lesssim w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

(4.4)

By combining (4.3) and (4.4), we have

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}\lesssim w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

 □

Proof of Theorem 1.1 By Lemma 4.3 and Theorem 3.2 we have for $p>1$

$$\begin{array}{rcl}{\parallel G_{\alpha }\overrightarrow{f}\parallel }_{M_w^{p,{\phi }_2}(l_2)}& \lesssim & \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_r^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_0^{r^{-1}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t^{-1}),l_2)}{\left(w\left(B(x_0,t^{-1})\right)\right)}^{-\frac{1}{p}}\frac{dt}{t}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r^{-1})}^{-1}r\frac{1}{r}{\int }_0^r{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t^{-1}),l_2)}{\left(w\left(B(x_0,t^{-1})\right)\right)}^{-\frac{1}{p}}\frac{dt}{t}\\ \lesssim & \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r^{-1})}^{-1}{\left(w\left(B(x_0,r^{-1})\right)\right)}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,r^{-1}),l_2)}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r)}^{-1}{\left(w(B(x_0,r))\right)}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,r),l_2)}={\parallel \overrightarrow{f}\parallel }_{M_w^{p,{\phi }_1}(l_2)}\end{array}$$

and for $p=1$

$$\begin{array}{rcl}{\parallel G_{\alpha }\overrightarrow{f}\parallel }_{W{M}_w^{1,{\phi }_2}(l_2)}& \lesssim & \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_r^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-1}\frac{dt}{t}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_0^{r^{-1}}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,t^{-1}),l_2)}{\left(w\left(B(x_0,t^{-1})\right)\right)}^{-1}\frac{dt}{t}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r^{-1})}^{-1}r\frac{1}{r}{\int }_0^r{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,t^{-1}),l_2)}{\left(w\left(B(x_0,t^{-1})\right)\right)}^{-1}\frac{dt}{t}\\ \lesssim & \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r^{-1})}^{-1}{\left(w\left(B(x_0,r^{-1})\right)\right)}^{-1}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,r^{-1}),l_2)}\\ =& \underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r)}^{-1}{\left(w(B(x_0,r))\right)}^{-1}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,r),l_2)}={\parallel \overrightarrow{f}\parallel }_{M_w^{1,{\phi }_1}(l_2)}.\end{array}$$

 □

Lemma 4.4 Let $1\le p<\mathrm{\infty }$, $0<\alpha \le 1$, $\lambda >3+\frac{\alpha }{n}$, and $w\in A_p$. Then, for $p>1$, the inequality

$${\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}\lesssim {(w(B))}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}$$

holds for any ball $B=B(x_0,r)$ and for all $\overrightarrow{f}\in L_w^{p,\mathrm{loc}}(l_2)$.

Moreover, for $p=1$ the inequality

$${\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})\parallel }_{W{L}_w^1(B,l_2)}\lesssim w(B){\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^1(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-1}\frac{dt}{t}$$

holds for any ball $B=B(x_0,r)$ and for all $\overrightarrow{f}\in L_w^{1,\mathrm{loc}}(l_2)$.

Proof From the definition of ${\mathrm{g}}_{\lambda ,\alpha }^{\ast }(f)$, we readily see that

$$\begin{array}{rcl}{\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})(x)\parallel }_{l_2}& =& {\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{{\mathbb{R}}^n}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ \le & {\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|<t}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ +{\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\ge t}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ :=& III+IV.\end{array}$$

First, let us estimate III:

$$III\le {\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|<t}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\le {\parallel G_{\alpha }\overrightarrow{f}(x)\parallel }_{l_2}.$$

Now, let us estimate IV:

$$\begin{array}{rcl}IV& \le & {\parallel {\left(\sum _{j=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_{2^{j-1}t\le |x-y|\le 2^{j}t}{\left(\frac{t}{t+|x-y|}\right)}^{n\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ \lesssim & {\parallel {\left(\sum _{j=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_{2^{j-1}t\le |x-y|\le 2^{j}t}{2}^{-jn\lambda }{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ \lesssim & \sum _{j=1}^{\mathrm{\infty }}{2}^{-jn\lambda }{\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}:=\sum _{j=1}^{\mathrm{\infty }}{2}^{-jn\lambda }{\parallel G_{\alpha ,2^j}(\overrightarrow{f})(x)\parallel }_{l_2}.\end{array}$$

Thus,

$${\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}\le {\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}+\sum _{j=1}^{\mathrm{\infty }}{2}^{-\frac{jn\lambda }{2}}{\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}.$$

(4.5)

By Lemma 4.3, we have

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}\lesssim {(w(B))}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t))}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

(4.6)

In the following, we will estimate ${\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}$. We divide ${\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}$ into two parts,

$${\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}\le {\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_0)\parallel }_{L_w^p(B,l_2)}+{\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_{\mathrm{\infty }})\parallel }_{L_w^p(B,l_2)},$$

(4.7)

where ${\overrightarrow{f}}_0(y)=\overrightarrow{f}(y){\chi }_{2B}(y)$, ${\overrightarrow{f}}_{\mathrm{\infty }}(y)=\overrightarrow{f}(y)-{\overrightarrow{f}}_{\mathrm{\infty }}(y)$. For the first part, by Lemma 4.1,

$$\begin{array}{rcl}{\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_0)\parallel }_{L_w^p(B,l_2)}& \lesssim & 2^{j(\frac{3n}{2}+\alpha )}{\parallel G_{\alpha }({\overrightarrow{f}}_0)\parallel }_{L_w^p(l_2)}\lesssim 2^{j(\frac{3n}{2}+\alpha )}{\parallel f\parallel }_{L_w^p(B,l_2)}\\ \lesssim & 2^{j(\frac{3n}{2}+\alpha )}w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.\end{array}$$

(4.8)

For the second part,

$$\begin{array}{rcl}{\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_{\mathrm{\infty }})(x)\parallel }_{l_2}& =& {\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{(A_{\alpha }\overrightarrow{f}(y,t))}^2\frac{dydt}{t^{n+1}}\right)}^{l/2}\parallel }_{l_2}\\ =& {\parallel {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{\left(\underset{\varphi \in C_{\alpha }}{sup}|\overrightarrow{f}\ast {\varphi }_t(y)|\right)}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}\parallel }_{l_2}\\ \le & {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{\left({\int }_{|z-y|\le t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{dydt}{t^{3n+1}}\right)}^{\frac{1}{2}}.\end{array}$$

Since $|x-z|\le |y-z|+|x-y|\le 2^{j+1}t$, we get

$$\begin{array}{rcl}{\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_{\mathrm{\infty }})(x)\parallel }_{l_2}& \le & {\left({\int }_0^{\mathrm{\infty }}{\int }_{|x-y|\le 2^{j}t}{\left({\int }_{|x-z|\le 2^{j+1}t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{dydt}{t^{3n+1}}\right)}^{\frac{1}{2}}\\ \le & {\left({\int }_0^{\mathrm{\infty }}{\left({\int }_{|z-x|\le 2^{j+1}t}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}dz\right)}^2\frac{2^{jn}dt}{t^{2n+1}}\right)}^{\frac{1}{2}}\\ \le & 2^{\frac{jn}{2}}{\int }_{{\mathbb{R}}^n}{\left({\int }_{t\ge \frac{|x-z|}{2^{j+1}}}{\parallel {\overrightarrow{f}}_{\mathrm{\infty }}(z)\parallel }_{l_2}^2\frac{dt}{t^{2n+1}}\right)}^{\frac{1}{2}}dz\\ \le & 2^{\frac{3jn}{2}}{\int }_{|x_0-z|>2r}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x-z|}^n}dz.\end{array}$$

For $|z-x|\ge |x_0-z|-|x-x_0|\ge |x_0-z|-\frac{1}{2}|x_0-z|=\frac{1}{2}|x_0-z|$, so by Fubini’s theorem and Hölder’s inequality, we obtain

$$\begin{array}{rcl}{\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_{\mathrm{\infty }})(x)\parallel }_{l_2}& \le & 2^{\frac{3jn}{2}}{\int }_{|x_0-z|>2r}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\\ =& 2^{\frac{3jn}{2}}{\int }_{|x_0-z|>2r}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}{\int }_{|x_0-z|}^{\mathrm{\infty }}\frac{dt}{t^{n+1}}dz\\ \le & 2^{\frac{3jn}{2}}{\int }_{2r}^{\mathrm{\infty }}{\int }_{|x_0-z|<t}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{dt}{t^{n+1}}\\ \le & 2^{\frac{3jn}{2}}{\int }_{2r}^{\mathrm{\infty }}{\parallel {\parallel \overrightarrow{f}(\cdot )\parallel }_{l_2}\parallel }_{L^1(B(x_0,t))}\frac{dt}{t^{n+1}}\\ \le & 2^{\frac{3jn}{2}}{\int }_{2r}^{\mathrm{\infty }}{\parallel {\parallel \overrightarrow{f}(\cdot )\parallel }_{l_2}\parallel }_{L_w^p(B(x_0,t))}{\parallel w^{-1}\parallel }_{L_{p^{\prime }}(B(x_0,t))}\frac{dt}{t^{n+1}}\\ \le & 2^{\frac{3jn}{2}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.\end{array}$$

So,

$${\parallel G_{\alpha ,2^j}({\overrightarrow{f}}_{\mathrm{\infty }})\parallel }_{L_w^p(B,l_2)}\le 2^{\frac{3jn}{2}}w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

(4.9)

Combining (4.7), (4.8), and (4.9), we have

$${\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}\lesssim 2^{j(\frac{3n}{2}+\alpha )}w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}.$$

(4.10)

Thus,

$${\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}\le {\parallel G_{\alpha }\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}+\sum _{j=1}^{\mathrm{\infty }}{2}^{-\frac{jn\lambda }{2}}{\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{L_w^p(B,l_2)}.$$

(4.11)

Since $\lambda >3+\frac{\alpha }{n}$, by (4.6), (4.10), and (4.11), we have the desired lemma. □

Proof of Theorem 1.2 From inequality (4.5) we have

$${\parallel {\mathrm{g}}_{\lambda ,\alpha }^{\ast }(\overrightarrow{f})\parallel }_{M_w^{p,{\phi }_2}(l_2)}\le {\parallel G_{\alpha }\overrightarrow{f}\parallel }_{M_w^{p,{\phi }_2}(l_2)}+\sum _{j=1}^{\mathrm{\infty }}{2}^{-\frac{jn\lambda }{2}}{\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{M_w^{p,{\phi }_2}(l_2)}.$$

(4.12)

By Theorem 1.1, we have

$${\parallel G_{\alpha }\overrightarrow{f}\parallel }_{M_w^{p,{\phi }_2}(l_2)}\lesssim {\parallel \overrightarrow{f}\parallel }_{M_w^{p,{\phi }_1}(l_2)}.$$

(4.13)

In the following, we will estimate ${\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{M_w^{p,{\phi }_2}(l_2)}$. Thus, by substitution of variables and Theorem 3.2, we get

$$\begin{array}{c}{\parallel G_{\alpha ,2^j}(\overrightarrow{f})\parallel }_{M_w^{p,{\phi }_2}(l_2)}\hfill \\ \lesssim 2^{j(\frac{3n}{2}+\alpha )}\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_r^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t),l_2)}{\left(w(B(x_0,t))\right)}^{-\frac{1}{p}}\frac{dt}{t}\hfill \\ =2^{j(\frac{3n}{2}+\alpha )}\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r^{-1})}^{-1}r\frac{1}{r}{\int }_0^r{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,t^{-1}),l_2)}{\left(w\left(B(x_0,t^{-1})\right)\right)}^{-\frac{1}{p}}\frac{dt}{t}\hfill \\ \lesssim 2^{j(\frac{3n}{2}+\alpha )}\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r^{-1})}^{-1}{\left(w\left(B(x_0,r^{-1})\right)\right)}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,r^{-1}),l_2)}\hfill \\ =2^{j(\frac{3n}{2}+\alpha )}{\parallel \overrightarrow{f}\parallel }_{M_w^{p,{\phi }_1}(l_2)}.\hfill \end{array}$$

(4.14)

Since $\lambda >3+\frac{\alpha }{n}$, by (4.12), (4.13), and (4.14), we have the desired theorem. □

Lemma 4.5 Let $1<p<\mathrm{\infty }$, $0<\alpha \le 1$, $w\in A_p$, $\overrightarrow{b}=(b_1,\dots ,b_m)$, and $b_i\in BMO({\mathbb{R}}^n)$, $i=1,\dots ,m$. Then the inequality

$${\parallel [\overrightarrow{b},G_{\alpha }]\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}\lesssim {(w(B))}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{ln}^m(e+\frac{t}{r}){\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}{\left(w(B(x_0,\tau ))\right)}^{-\frac{1}{p}}\frac{d\tau }{\tau }$$

holds for any ball $B=B(x_0,r)$ and for all $f\in L_w^{p,\mathrm{loc}}(l_2)$.

Proof We decompose $\overrightarrow{f}={\overrightarrow{f}}_0+{\overrightarrow{f}}_{\mathrm{\infty }}$, where ${\overrightarrow{f}}_0=\overrightarrow{f}{\chi }_{2B}$ and ${\overrightarrow{f}}_{\mathrm{\infty }}=\overrightarrow{f}-{\overrightarrow{f}}_0$. Then

$${\parallel [\overrightarrow{b},G_{\alpha }]\overrightarrow{f}\parallel }_{L_w^p(B,l_2)}\le {\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_0\parallel }_{L_w^p(B,l_2)}+{\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L_w^p(B,l_2)}.$$

Denote by ${\parallel \overrightarrow{b}\parallel }_{\ast }={\prod }_{i=1}^m{\parallel b_j\parallel }_{\ast }$. By Lemma 4.2, we have

$$\begin{array}{rcl}{\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_0\parallel }_{L_w^p(B,l_2)}& \lesssim & {\parallel \overrightarrow{b}\parallel }_{\ast }{\parallel {\overrightarrow{f}}_0\parallel }_{L_w^p(l_2)}={\parallel \overrightarrow{b}\parallel }_{\ast }{\parallel \overrightarrow{f}\parallel }_{L_w^p(2B,l_2)}\\ \lesssim & {\parallel \overrightarrow{b}\parallel }_{\ast }w{(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}{\left(w(B(x_0,\tau ))\right)}^{-\frac{1}{p}}\frac{d\tau }{\tau }.\end{array}$$

(4.15)

For the term ${\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L_w^p(B,l_2)}$, without loss of generality, we can assume $m=2$. Thus, the operator $[\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}$ can be divided into four parts:

$$\begin{array}{c}|[\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}(x)|\hfill \\ \le {\left({\iint }_{\mathrm{\Gamma }(x)}\underset{\varphi \in C_{\alpha }}{sup}{|{\int }_{{\mathbb{R}}^n}{\varphi }_t(y-z)(b_1(z)-{(b_1)}_{B,\omega })(b_2(z)-{(b_2)}_{B,\omega }){\overrightarrow{f}}_{\mathrm{\infty }}(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\hfill \\ +|b_1(x)-{(b_1)}_{B,\omega }|{\left({\iint }_{\mathrm{\Gamma }(x)}\underset{\varphi \in C_{\alpha }}{sup}{|{\int }_{{\mathbb{R}}^n}{\varphi }_t(y-z)(b_2(z)-{(b_2)}_{B,\omega }){\overrightarrow{f}}_{\mathrm{\infty }}(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\hfill \\ +|b_2(x)-{(b_2)}_{B,\omega }|{\left({\iint }_{\mathrm{\Gamma }(x)}\underset{\varphi \in C_{\alpha }}{sup}{|{\int }_{{\mathbb{R}}^n}{\varphi }_t(y-z)(b_1(z)-{(b_1)}_{B,\omega }){\overrightarrow{f}}_{\mathrm{\infty }}(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\hfill \\ \le \left|(b_1(x)-{(b_1)}_{B,\omega })(b_2(x)-{(b_2)}_{B,\omega })\right|{\left({\iint }_{\mathrm{\Gamma }(x)}\underset{\varphi \in C_{\alpha }}{sup}{|{\int }_{{\mathbb{R}}^n}{\varphi }_t(y-z){\overrightarrow{f}}_{\mathrm{\infty }}(z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{1/2}\hfill \\ \equiv I_1(x)+I_2(x)+I_3(x)+I_4(x).\hfill \end{array}$$

For $x\in B$ we have

$$\begin{array}{rcl}{\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}(x)\parallel }_{l_2}& \le & {\parallel I_1(x)\parallel }_{l_2}+{\parallel I_2(x)\parallel }_{l_2}+{\parallel I_3(x)\parallel }_{l_2}+{\parallel I_4(x)\parallel }_{l_2}\\ \le & {\int }_{{}^C(2B)}|b_1(z)-{(b_1)}_{B,\omega }||b_2(z)-{(b_2)}_{B,\omega }|\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\\ +|b_1(x)-{(b_1)}_{B,\omega }|{\int }_{{}^C(2B)}|b_2(z)-{(b_2)}_{B,\omega }|\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\\ +|b_2(x)-{(b_2)}_{B,\omega }|{\int }_{{}^C(2B)}|b_1(z)-{(b_1)}_{B,\omega }|\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\\ +\left|(b_1(x)-{(b_1)}_{B,\omega })\right|\left|(b_2(x)-{(b_2)}_{B,\omega })\right|{\int }_{{}^C(2B)}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz.\end{array}$$

Then

$$\begin{array}{c}{\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L_w^p(B,l_2)}\hfill \\ \le {({\int }_B{\left({\int }_{{}^C(2B)}\frac{{\prod }_{i=1}^2|b_i(z)-{(b_i)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\right)}^p\omega (x)dx)}^{1/p}\hfill \\ +{({\int }_B|b_1(x)-{(b_1)}_{B,\omega }|{\left({\int }_{{}^C(2B)}\frac{|b_2(z)-{(b_2)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\right)}^p\omega (x)dx)}^{1/p}\hfill \\ +{({\int }_B|b_2(x)-{(b_2)}_{B,\omega }|{\left({\int }_{{}^C(2B)}\frac{|b_1(z)-{(b_1)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\right)}^p\omega (x)dx)}^{1/p}\hfill \\ +{({\int }_B{\left({\int }_{{}^C(2B)}\frac{{\prod }_{i=1}^2|b_i(x)-{(b_i)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\right)}^p\omega (x)dx)}^{1/p}\hfill \\ \equiv I_1+I_2+I_3+I_4.\hfill \end{array}$$

Let us estimate $I_1$:

$$\begin{array}{rcl}{I}_1& =& \omega {(B)}^{1/p}{\int }_{{}^C(2B)}\frac{{\prod }_{i=1}^2|b_i(z)-{(b_i)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\\ \approx & \omega {(B)}^{1/p}{\int }_{{}^C(2B)}\prod _{i=1}^2|b_i(z)-{(b_i)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}{\int }_{|x_0-z|}^{\mathrm{\infty }}\frac{d\tau }{{\tau }^{n+1}}dz\\ \approx & \omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |x_0-z|<\tau }\prod _{i=1}^2|b_i(z)-{(b_i)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{d\tau }{{\tau }^{n+1}}\\ \lesssim & \omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(x_0,\tau )}\prod _{i=1}^2|b_i(z)-{(b_i)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{d\tau }{{\tau }^{n+1}}.\end{array}$$

Applying Hölder’s inequality and by Lemma 3.7, we get

$$\begin{array}{rcl}{I}_1& \le & \omega {(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}\prod _{i=1}^2{\left({\int }_{B(x_0,\tau )}{|b_i(z)-{(b_i)}_{B,\omega }|}^{2p\prime }\omega {(z)}^{1-2p\prime }dz\right)}^{\frac{1}{2p\prime }}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\frac{d\tau }{{\tau }^{n+1}}\\ \le & \prod _{i=1}^2{\parallel b_j\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{(1+ln\frac{\tau }{r})}^2{\parallel {\omega }^{-1/p}\parallel }_{L_w^{p\prime }(B(x_0,\tau ),l_2)}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\frac{d\tau }{{\tau }^{n+1}}\\ \le & {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{ln}^2(e+\frac{\tau }{r}){\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.\end{array}$$

Let us estimate $I_2$:

$$\begin{array}{rcl}{I}_2& =& {({\int }_B{|b_1(x)-{(b_1)}_{B,\omega }|}^p\omega (x)dx)}^{1/p}{\int }_{{}^C(2B)}\frac{|b_2(z)-{(b_2)}_{B,\omega }|}{{|x_0-z|}^n}{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\\ \lesssim & {\parallel b_1\parallel }_{\ast }\omega {(B)}^{1/p}\left[{\int }_{{}^C(2B)}|b_2(z)-{(b_2)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}{\int }_{|x_0-z|}^{\mathrm{\infty }}\frac{d\tau }{{\tau }^{n+1}}dz\right]\\ \approx & {\parallel b_1\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\int }_{2r\le |x_0-z|\le \tau }|b_2(z)-{(b_2)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{d\tau }{{\tau }^{n+1}}\\ \lesssim & {\parallel b_1\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\int }_{B(x_0,\tau )}|b_2(z)-{(b_2)}_{B,\omega }|{\parallel \overrightarrow{f}(z)\parallel }_{l_2}dz\frac{d\tau }{{\tau }^{n+1}}.\end{array}$$

Applying Hölder’s inequality and by Lemma 3.7, we get

$$\begin{array}{rcl}{I}_2& \le & {\parallel b_1\parallel }_{\ast }\omega {(B)}^{\frac{1}{p}}{\int }_{2r}^{\mathrm{\infty }}{\left({\int }_{B(x_0,\tau )}{|b_2(z)-{(b_2)}_{B,\omega }|}^{p\prime }\omega {(z)}^{1-p\prime }dz\right)}^{\frac{1}{p\prime }}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\frac{d\tau }{{\tau }^{n+1}}\\ \le & \prod _{i=1}^2{\parallel b_j\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}(1+ln\frac{\tau }{r}){\parallel {\omega }^{-1/p}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\frac{d\tau }{{\tau }^{n+1}}\\ \le & {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{ln}^2(e+\frac{\tau }{r}){\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.\end{array}$$

In the same way, we shall get the result of $I_3$:

$$I_3\le {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{ln}^2(e+\frac{\tau }{r}){\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.$$

In order to estimate $I_4$ note that

$$\begin{array}{rcl}{I}_4& =& {({\int }_B\prod _{i=1}^2{|b_i(x)-{(b_i)}_{B,\omega }|}^p\omega (x)dx)}^{1/p}{\int }_{{}^C(2B)}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\\ \le & \prod _{i=1}^2{({\int }_B{|b_i(x)-{(b_i)}_{B,\omega }|}^{2p}\omega (x)dx)}^{1/2p}{\int }_{{}^C(2B)}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz.\end{array}$$

By Lemma 3.7, we get

$$I_4\le {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{{}^C(2B)}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz.$$

Applying Hölder’s inequality, we get

$$\begin{array}{c}{\int }_{{}^C(2B)}\frac{{\parallel \overrightarrow{f}(z)\parallel }_{l_2}}{{|x_0-z|}^n}dz\hfill \\ \le {\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}{\parallel {\omega }^{-1/p}\parallel }_{L_w^{p\prime }(B(x_0,\tau ),l_2)}\frac{d\tau }{{\tau }^{n+1}}\hfill \\ \le {[\omega ]}_{A_p}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.\hfill \end{array}$$

(4.16)

Thus by (4.16)

$$I_4\le {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.$$

Summing up $I_1$ and $I_4$, for all $p\in [1,\mathrm{\infty })$ we get

$$\begin{array}{rcl}{\parallel [\overrightarrow{b},G_{\alpha }]{\overrightarrow{f}}_{\mathrm{\infty }}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}& \le & {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{ln}^2(e+\frac{\tau }{r}){\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\\ \times \omega (B(x_0,\tau ))\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.\end{array}$$

(4.17)

Finally, from (4.2), (4.15), and (4.17) we get

$$\begin{array}{c}{\parallel [\overrightarrow{b},G_{\alpha }]\overrightarrow{f}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\hfill \\ \lesssim {\parallel \overrightarrow{b}\parallel }_{\ast }\omega {(B)}^{1/p}{\int }_{2r}^{\mathrm{\infty }}{ln}^2(e+\frac{\tau }{r}){\parallel f\parallel }_{L_w^p(B(x_0,\tau ),l_2)}\omega {(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }.\hfill \end{array}$$

 □

Proof of Theorem 1.3 By substitution of variables, we obtain

$$\begin{array}{c}{\parallel [\overrightarrow{b},G_{\alpha }]\overrightarrow{f}\parallel }_{M_{\omega }^{p,{\phi }_2}(l_2)}\hfill \\ \lesssim {\parallel \overrightarrow{b}\parallel }_{\ast }\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_{2r}^{\mathrm{\infty }}{ln}^m(e+\frac{\tau }{r}){\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,\tau ),l_2)}w{(B(x_0,\tau ))}^{-1/p}\frac{d\tau }{\tau }\hfill \\ \lesssim {\parallel \overrightarrow{b}\parallel }_{\ast }\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r)}^{-1}{\int }_0^{r^{-1}}{ln}^m(e+\frac{1}{\tau r}){\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,{\tau }^{-1}),l_2)}w{\left(B(x_0,{\tau }^{-1})\right)}^{-\frac{1}{p}}\frac{d\tau }{\tau }\hfill \\ ={\parallel \overrightarrow{b}\parallel }_{\ast }\underset{x\in {\mathbb{R}}^n,r>0}{sup}{\phi }_2{(x_0,r^{-1})}^{-1}r\frac{1}{r}{\int }_0^r{ln}^m(e+\frac{r}{\tau }){\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,{\tau }^{-1}),l_2)}w{\left(B(x_0,{\tau }^{-1})\right)}^{-\frac{1}{p}}\frac{d\tau }{\tau }\hfill \\ \lesssim {\parallel \overrightarrow{b}\parallel }_{\ast }\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r^{-1})}^{-1}w{\left(B(x_0,r^{-1})\right)}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,r^{-1}),l_2)}\hfill \\ ={\parallel \overrightarrow{b}\parallel }_{\ast }\underset{x_0\in {\mathbb{R}}^n,r>0}{sup}{\phi }_1{(x_0,r)}^{-1}w{(B(x_0,r))}^{-\frac{1}{p}}{\parallel \overrightarrow{f}\parallel }_{L_w^p(B(x_0,r),l_2)}\hfill \\ ={\parallel \overrightarrow{b}\parallel }_{\ast }{\parallel \overrightarrow{f}\parallel }_{M_w^{p,{\phi }_1}(l_2)}.\hfill \end{array}$$

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