1. Introduction and main results

Variational inequalities originated from the intention of improving the well-known Doob maximal inequality. Relied upon the work of Lépingle [Reference Lépingle26], Bourgain [Reference Bourgain5] obtained the corresponding variational estimates for the Birkhoff ergodic averages and pointwise convergence results. This work has set up a new research subject in harmonic analysis and ergodic theory. Afterwards, the study of variational inequalities has been spilled over into harmonic analysis, probability and ergodic theory. Particularly, the classical work of $\rho$-variation operators for singular integrals was given in [Reference Campbell, Jones, Reinhold and Wierdl7], in which the authors obtained the $L^{p}$-bounds and weak type (1,1) bounds for $\rho$-variation operators of truncated Hilbert transform if $\rho >2$, and then extended to higher dimensional cases in [Reference Campbell, Jones, Reinhold and Wierdl8]. For further studies, we refer readers to [Reference Chen, Ding, Hong and Liu9, Reference Ding, Hong and Liu14, Reference Gillespie and Torrea17, Reference Ma, Torrea and Xu33, Reference Ma, Torrea and Xu34, Reference Wen, Wu and Zhang42], etc., for variation operators of singular integrals with rough kernels and weighted cases, [Reference Betancor, Fariña, Harbour and Rodríguez-Mesa3, Reference Chen, Ding, Hong and Liu10, Reference Hytönen, Lacey and Pérez22, Reference Liu and Wu31, Reference Wen, Guo and Wu41, Reference Wen, Wu and Zhang42], etc., for variation operators of commutators.

Here, we will focus on the variation operators associated with approximate identities. For the special case, the variation operators associated with heat and Poisson semigroups, Jones and Reinhold [Reference Jones and Reinhold25] and Crescimbeni et al. [Reference Crescimbeni, Macías, Menárguez, Torrea and Viviani12] independently established the $L^{p}$-bounds and weak type $(1,1)$ bounds with different approaches. Recently, Liu [Reference Liu32] generalized the results in [Reference Crescimbeni, Macías, Menárguez, Torrea and Viviani12, Reference Jones and Reinhold25] to the variation operators associated to approximate identities and obtained a variational characterization of Hardy spaces. In this paper, one of our main purposes is to extend the results in [Reference Liu32] to the weighted cases, and give a new characterization of weighted Hardy spaces via variation inequalities associated with approximate identities. Meanwhile, we will also consider the weighted variation inequalities associated with commutators of approximate identities and aim to provide new characterizations of certain weighted $BMO$ spaces. Before stating our results, we first recall some relevant notation and definitions.

Given a family of complex numbers $\mathfrak {a}:=\{a_t\}_{t\in I}$ with $I\subset (0, +\infty )$. For $\rho >1$, the $\rho$-variation of $\mathfrak {a}$ is given by

\[ \|\mathfrak{a}\|_{\mathcal{V}_\rho}:=\sup\left(\sum_{k\geq 1}|a_{t_k}-a_{t_{k+1}}|^{\rho}\right)^{1/\rho}, \]

where the supremum is taken over all finite decreasing sequences $\{t_k\}$ in $I$. From the definition of $\rho$-variation, it is easy to check that

(1.1)\begin{equation} \sup_{t\in I}|a_t|\leq|a_{t_0}|+\|\mathfrak{a}\|_{\mathcal{V}_\rho} \end{equation}

holds for arbitrary $t_0\in I$.

Let $\mathcal {F}:=\{F_t\}_{t>0}$ be a family of operators. Then the $\rho$-variation of the family $\mathcal {F}$ is defined as

\[ \mathcal{V}_\rho(\mathcal{F}f)(x):=\|\{F_tf(x)\}_{t>0}\|_{\mathcal{V}_\rho}. \]

In particular, let $\phi \in \mathcal {S}({{\mathbb {R}^{n}}})$ satisfying $\int _{{{\mathbb {R}^{n}}}}\phi (x)\,\textrm {d}x=1$, where $\mathcal {S}({{\mathbb {R}^{n}}})$ is the space of Schwartz functions. We consider the following family of operators

(1.2)\begin{equation} \Phi\star f(x):=\{\phi_t\ast f(x)\}_{t>0}, \end{equation}

where $\phi _t(x):=t^{-n}\phi (x/t)$. Then the $\rho$-variation of families $\Phi \star f$ is defined by

(1.3)\begin{equation} \mathcal{V}_\rho(\Phi\star f)(x)=\sup_{\{t_k\}\downarrow 0}\left(\sum_{k\geq 1}|\phi_{t_{k}}\ast f(x)-\phi_{t_{k+1}}\ast f(x)|^{\rho}\right)^{1/\rho}. \end{equation}

It is well-known that the $L^{p}$-boundedness of $\mathcal {V}_\rho (\Phi \star f)$ implies the almost everywhere convergence of $\{f\ast \phi _t\}_{t>0}$ as $t\rightarrow 0^{+}$ for every $f\in L^{p}({{\mathbb {R}^{n}}})$, without knowing the almost everywhere convergence property for $f$ in some dense subset in $L^{p}({{\mathbb {R}^{n}}})$, see [Reference Jones24, p. 90]. For $b\in L^{1}_\textrm {loc}(\mathbb {R}^{n})$ and $\phi$ being as above, we define $(\Phi \star f)_b$, the family of commutators associated with approximate identities, by

\[ (\Phi\star f)_b(x):=\{b(x)(\phi_t\ast f)(x)-\phi_t\ast (bf)(x)\}_{t>0}, \]

where

\[ b(x)(\phi_t\ast f)(x)-\phi_t\ast (bf)(x):=\int_{{{\mathbb{R}^{n}}}}\frac{1}{t^{n}}\phi\left(\frac{x-y}{t}\right)(b(x)-b(y))f(y)\,\textrm{d}y, \]

and the corresponding $\rho$-variation operator by

(1.4)\begin{align} \mathcal{V}_\rho((\Phi\star f)_b)(x)& :=\sup_{\{t_k\}\downarrow 0}\left(\sum_{k\geq 1}\left|b(x)(\phi_{t_k}\ast f)(x)-\phi_{t_k}\ast (bf)(x)\right.\right.\nonumber\\ & \quad\left.\left.\vphantom{\sum_{k\geq 1}}-b(x)(\phi_{t_{k+1}}\ast f)(x)+\phi_{t_{k+1}}\ast (bf)(x)\right|^{\rho}\right)^{1/\rho}. \end{align}

We now recall some relevent facts about weighted Hardy spaces; see, for example, [Reference García-Cuerva16, Reference Strömberg and Torchinsky40]. Let $\phi \in \mathcal {S}(\mathbb {R}^{n})$ with $\int _{\mathbb {R}^{n}}\phi (x)\,\textrm {d}x=1$. For $f\in \mathcal {S}'(\mathbb {R}^{n})$, define the maximal function $M_\phi$ by

(1.5)\begin{equation} M_\phi f(x):=\sup_{t>0}|f\ast\phi_t(x)|. \end{equation}

Then for $\omega \in A_\infty$, the Muckenhoupt class, and $0< p<\infty$, define the weighted Hardy spaces $H^{p}(\omega )$ by

\[ H^{p}(\omega):=\{f\in\mathcal{S}'(\mathbb{R}^{n}):M_\phi f\in L^{p}(\omega)\} \]

with the quasi-norm $\|f\|_{H^{p}(\omega )}:=\|M_\phi f\|_{L^{p}(\omega )}$. When $\omega \equiv 1$, we denote $H^{p}(\omega )$ by $H^{p}(\mathbb {R}^{n})$. It is well-known that the space $H^{p}(\omega )$ is independent of the choice of $\phi$ and $H^{p}(\omega )=L^{p}(\omega )$ for $p>1$ and $\omega \in A_p$.

In [Reference Liu32], Liu showed that $\mathcal {V}_\rho (\Phi \star f)$ is bounded from the Hardy space $H^{p}(\mathbb {R}^{n})$ to $L^{p}(\mathbb {R}^{n})$, and obtained a characterization of $H^{p}({{\mathbb {R}^{n}}})$ via $\mathcal {V}_\rho (\Phi \star f)$. Now we formulate our first main result, which is the weighted version of the corresponding result in [Reference Liu32], as follows.

On the other hand, recalling the commutators

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

where $b$ is a given locally integrable function and $T$ is a linear operator, Coifman, Rochberg and Weiss [Reference Coifman, Rochberg and Weiss11] showed that $b\in BMO(\mathbb {R}^{n})$ if and only if $[b,R_j]$ is bounded on $L^{p}(\mathbb {R}^{n})$ for $1< p<\infty$, where $R_j$ is the $j$-th Riesz transform on $\mathbb {R}^{n}$, $j=1,\,\cdots,n$. For $n=1$, Bloom [Reference Bloom4] extended the above result to the following weighted case:

\[ \|[b,H]\|_{L^{p}(\mu)\to L^{p}(\lambda)}\simeq \|b\|_{ BMO_\nu(\mathbb{R})}\quad \text{for}\ 1< p<\infty,\,\mu,\,\lambda\in A_p, \]

where $H$ is the Hilbert transform, $\nu =(\mu /\lambda )^{1/p}$, and $BMO_\nu (\mathbb {R}^{n})$ is the following weighted $BMO$ space defined by

\begin{align*} {BMO}_\nu(\mathbb{R}^{n})& :=\{b\in L^{1}_\textrm{loc}(\mathbb{R}^{n}):\ \|b\|_{BMO_\nu(\mathbb{R}^{n})}\\ & :=\sup_{B\subset\mathbb{R}^{n}}\frac 1{\nu(B)}\int_B|b(y)-\langle b\rangle_B|dy<\infty\}. \end{align*}

Here the supremum is taken over all balls, $\langle b\rangle _B:=|B|^{-1}\int _B b(y)\,\textrm {d}y$. Subsequently, a lot of attention has been paid to this topic. We refer to [Reference Accomazzo1, Reference Accomazzo, Martínez-Perales and Rivera-Ríos2, Reference Guo, Lian and Wu18–Reference Hytönen21, Reference Lerner, Ombrosi and Rivera-Ríos29] and therein references for recent works.

In addition, it is well known that when $b\in BMO(\mathbb {R}^{n})$, and $T$ is a singular integral, $[b,T]$ may not map $H^{1}(\mathbb {R}^{n})$ boundedly into $L^{1}(\mathbb {R}^{n})$ (see [Reference Paluszyński38, remark]). To investigate the $H^{1}-L^{1}$ bound of $[b, T]$, Liang, Ky and Yang [Reference Liang, Ky and Yang30] introduced the following weighted BMO type space

\[ \mathcal{BMO}_\omega(\mathbb{R}^{n}):=\{b\in L^{1}_\textrm{loc}(\mathbb{R}^{n}):\ \|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}<\infty\}, \]

where

\[ \|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}:=\sup_{B\subset\mathbb{R}^{n}}\frac{1}{\omega(B)}\int_{B^{c}}\frac{\omega(x)}{|x-x_B|^{n}}\,\textrm{d}x\int_B|b(y)-\langle b\rangle_B|\,\textrm{d}y \]

for $\omega \in A_\infty$ and $\int _{\mathbb {R}^{n}}\frac {\omega (x)}{1+|x|^{n}}\,\textrm {d}x<\infty$, the supremum is taken over all balls $B:=B(x_B,r)$. It is clear that $\mathcal {BMO}_\omega (\mathbb {R}^{n})\subsetneq BMO(\mathbb {R}^{n})$ (see [Reference Liang, Ky and Yang30]). And in [Reference Liang, Ky and Yang30], the authors showed that for $\delta$-Calderón-Zygmund operator $T$ and $\omega \in A_{(n+\delta )/n}$, $b\in \mathcal {BMO}_\omega (\mathbb {R}^{n})$ if and only if $[b,T]$ is bounded from $H^{1}(\omega )$ to $L^{1}(\omega )$.

Inspired by the results above, it is natural to ask whether the corresponding characterizations can be given via variation operators of commutators associated with approximate identities. Our next theorems will give a positive answer to this question.

Theorem 1.2 Let $\phi \in \mathcal {S}(\mathbb {R}^{n})$ with $\int _{{{\mathbb {R}^{n}}}}\phi (x)\,{\rm d}x=1$, $1< p<\infty$, $\mu,\lambda \in A_{p}$ and $\nu =(\mu \lambda ^{-1})^{1/p}$. If $\phi$ or $b$ is a real-valued function, then for $\rho >2$, the following statements are equivalent:

1. (1) $f\mapsto \mathcal {V}_\rho ((\Phi \star f)_b)$ is bounded from $L^{p}(\mu )$ to $L^{p}(\lambda );$

2. (2) $b\in BMO_\nu (\mathbb {R}^{n})$.

Moreover, we have the estimate

\[ \|\mathcal{V}_\rho((\Phi\star f)_b)\|_{L^{p}(\lambda)}\lesssim\|b\|_{BMO_\nu({{\mathbb{R}^{n}}})}([\mu]_{A_p}[\lambda]_{A_p})^{\max\{1,\frac{1}{p-1}\}}\|f\|_{L^{p}(\mu)}. \]

The rest of the paper is organized as follows. After providing the weighted estimates of variation operators in §2, we will prove theorem 1.1 in § 3. The proofs of theorems 1.2 and 1.3 will be given in §4 and 5, respectively.

We end this section by making some conventions. Denote $f\lesssim g$, $f\thicksim g$ if $f\leq Cg$ and $f\lesssim g \lesssim f$, respectively. For any ball $B:=B(x_0,r)\subset \mathbb {R}^{n}$, $\langle f\rangle _B$ means the mean value of $f$ over $B$, $\chi _B$ represents the characteristic function of $B$, $\int _B\omega (y)dy$ is denoted by $\omega (B)$. For $a\in \mathbb {R}$, $\lfloor a\rfloor$ is the largest integer no more than $a$.

2. The weighted estimate of variation operators

In this section, we establish the weighted estimate of variation operators associated with approximations to the identity, which is useful in the proof of theorem 1.1. We begin with the following definition of $A_p$ weights.

A weight $\omega$ is a non-negative locally integrable function on $\mathbb {R}^{n}$. Let $1< p<\infty$. We say that $\omega \in A_p$ if there exists a positive constant $C$ such that

\[ [\omega]_{A_p}:=\sup_{Q}\left(\frac{1}{|Q|}\int_{Q}\omega(y)\,\textrm{d}y\right)\left(\frac{1}{|Q|} \int_{Q}\omega(y)^{1-p'}\,\textrm{d}y\right)^{p-1}\leq C, \]

where $1/p+1/p'=1$ and the supremum is taken over all cubes $Q\subset \mathbb {R}^{n}$. We recall that if $\omega \in A_p$, then for any $\lambda >1$ and all balls $B$, we have $\omega (\lambda B)\leq c_n \lambda ^{np}[\omega ]_{A_p}\omega (B)$. When $p=1$, we say that $\omega \in A_1$ if

\[ [\omega]_{A_1}:=\left\|\frac{M\omega}{\omega}\right\|_{L^{\infty}({{\mathbb{R}^{n}}})}<\infty, \]

where $M$ is the Hardy-Littlewood maximal operator. For $\omega \in A_\infty :=\cup _{1\leq p<\infty }A_p$, the $A_\infty$ constant is given by

\[ [\omega]_{A_\infty}:=\sup_{Q\subset {{\mathbb{R}^{n}}}}\frac{1}{\omega(Q)}\int_QM(\chi_Q\omega)(x)\,\textrm{d}x. \]

Now we recall the definitions of dyadic lattice, sparse family and sparse operator; see, for example, [Reference Lerner27, Reference Lerner, Ombrosi and Rivera-Ríos28, Reference Pereyra39]. Given a cube $Q\subset \mathbb {R}^{n}$, let $\mathcal {D}(Q)$ be the set of cubes obtained by repeatedly subdividing $Q$ and its descendants into $2^{n}$ congruent subcubes.

Let $\mathcal {S}$ be a sparse family. Define the sparse operator $\mathcal {T}_{\mathcal {S}}$ by

(2.1)\begin{equation} \mathcal{T}_{\mathcal{S}}f(x):=\sum_{Q\in \mathcal{S}}\langle|f|\rangle_Q\chi_Q(x). \end{equation}

Then for $\omega \in A_p$ with $1< p<\infty$

(2.2)\begin{equation} \|\mathcal{T}_{\mathcal{S}}f\|_{L^{p}(\omega)}\lesssim[\omega]_{A_p}^{\max\{1,\frac{1}{p-1}\}}\|f\|_{L^{p}(\omega)}; \end{equation}

see [Reference Cruz-Uribe, Martell and Pérez13] for $p=2$ and [Reference Lerner27, Reference Moen35] for $p>1$. Moreover, for $\omega \in A_1$ and $\alpha >0$,

(2.3)\begin{equation} \alpha\omega(\{x\in\mathbb{ R}^{n}:\mathcal{T}_{\mathcal{S}}f(x)>\alpha\})\lesssim[\omega]_{A_1} \log(e+[\omega]_{A_\infty})\|f\|_{L^{1}(\omega)}; \end{equation}

see [Reference Domingo-Salazar, Lacey and Rey15, Reference Hytönen and Pérez23].

We now give the following pointwise estimate of variation operators in terms of sparse operators.

Lemma 2.3 Let $\rho >2$. Then for each $f\in L_c^{\infty }(\mathbb {R}^{n})$, there exist $3^{n}$ dyadic lattices $\mathcal {D}^{j}$ and sparse families $\mathcal {S}_j\subset \mathcal {D}^{j}$ such that for a.e. $x\in \mathbb {R}^{n}$,

\[ \mathcal{V}_\rho(\Phi\star f)(x)\lesssim\sum_{j=1}^{3^{n}}\mathcal{T}_{\mathcal{S}_j}(f)(x). \]

To prove lemma 2.3, we introduce the maximal function $\mathcal {M}_{\mathcal {V}_\rho (\Phi )}$ by

\[ \mathcal{M}_{\mathcal{V}_\rho(\Phi)}f(x):=\sup_{Q\ni x} \mathop {ess\,sup}\limits_{\xi\in Q}\mathcal{V}_\rho(\Phi\star (f\chi_{{{\mathbb{R}^{n}}}\backslash 3Q}))(\xi). \]

Since $\mathcal {V}_\rho (\Phi \star f)$ is of weak type (1,1) (see [Reference Liu32, theorem 2.6]), then arguing as in [Reference Lerner27, theorem 3.1] (see also [Reference Lerner27, remark 4.3]), lemma 2.3 readily follows if $\mathcal {M}_{\mathcal {V}_\rho (\Phi )}$ is of weak type $(1,1)$. Therefore, we need only to settle the following result.

Proof. For any $f\in L^{1}(\mathbb {R}^{n})$ and $x\in \mathbb {R}^{n}$, we first show that for any cube $Q\ni x$,

(2.4)\begin{equation} \mathcal{V}_\rho(\Phi\star (f\chi_{\mathbb{R}^{n}\backslash 3Q}))(\xi)\lesssim Mf(x)+M_{1/2}(\mathcal{V}_\rho(\Phi\star f))(x)\quad \text{for a.e.}\ \xi\in Q, \end{equation}

where $M_{1/2}(f):=(M(|f|^{1/2}))^{2}$, and the implicit constant is independent of $f$, $x$ and $\xi$.

Indeed, for any cube $Q\ni x$, a.e. $\xi \in Q$ and $z\in Q$, we write

\begin{align*} \mathcal{V}_\rho(\Phi\star (f\chi_{\mathbb{R}^{n}\backslash 3Q}))(\xi)& \le\sup_{\{t_k\}\downarrow 0}\left(\sum_k\left|\int_{\mathbb{R}^{n}\backslash 3Q} [(\phi_{t_k}(\xi-y)-\phi_{t_{k+1}}(\xi-y))\right.\right.\\ & \quad\left.\left.\vphantom{\sum_k}-(\phi_{t_k}(z-y)-\phi_{t_{k+1}}(z-y))]f(y)\,\textrm{d}y\right|^{\rho}\right)^{1/\rho}\\ & \quad+\mathcal{V}_\rho(\Phi\star (f\chi_{3Q}))(z)+\mathcal{V}_\rho(\Phi\star f)(z)\\ & =:J(\xi,z)+\mathcal{V}_\rho(\Phi\star (f\chi_{3Q}))(z)+\mathcal{V}_\rho(\Phi\star f)(z). \end{align*}

By the Minkowski inequality and mean value theorem, we see that for given $\xi,z\in Q$, $y\in {{\mathbb {R}^{n}}}\backslash 3Q$ and some $\theta \in (0,1)$,

(2.5)\begin{align} & \|\{\phi_t(\xi-y)-\phi_t(z-y)\}_{t>0}\|_{\mathcal{V}_\rho}\nonumber\\ & \quad\leq\sup_{\{t_k\}\downarrow 0}\sum_k\left|\int_{t_{k+1}}^{t_k}\frac{\partial}{\partial t}(\phi_t(\xi-y)-\phi_t(z-y))\,\textrm{d}t\right|\nonumber\\ & \quad\leq \int_{0}^{\infty}\left|\frac{\partial}{\partial t}(\phi_t(\xi-y)-\phi_t(z-y))\right|\textrm{d}t \nonumber\\ & \quad\lesssim\int_{0}^{\infty}\frac{|\xi-z|}{t^{n+1}}\frac{|(\xi-y)+\theta(z-\xi)|}{t^{2}}\left(1+\frac{|(\xi-y)+\theta(z-\xi)|}{t}\right)^{{-}n-3}\,\textrm{d}t\nonumber\\ & \qquad+\int_{0}^{\infty}\frac{|\xi-z|}{t^{n+2}}\left(1+\frac{|(\xi-y)+\theta(z-\xi)|}{t}\right)^{{-}n-2}\,\textrm{d}t\nonumber\\ & \quad\lesssim\int_0^{\infty}\frac{|\xi-z|}{t^{n+1}}\frac{|\xi-y|}{t^{2}}\frac{t}{|\xi-y|}(1+|\xi-y|/t)^{{-}n-2}\,\textrm{d}t\nonumber\\ & \qquad+|z-\xi|\int_{0}^{\infty}t^{{-}n-2}(1+|\xi-y|/t)^{{-}n-2}\,\textrm{d}t\nonumber\\ & \quad\lesssim |z-\xi|\int_{0}^{\infty}t^{{-}n-2}(1+|\xi-y|/t)^{{-}n-2}\,\textrm{d}t\lesssim \frac{|z-\xi|}{|\xi-y|^{n+1}}. \end{align}

Consequently,

\begin{align*} J(\xi,z)& \leq\int_{\mathbb{R}^{n}\backslash 3Q}\|\{\phi_t(\xi-y)-\phi_t(z-y)\}_{t>0}\|_{\mathcal{V}_\rho}|f(y)|\,\textrm{d}y\\ & \lesssim\int_{\mathbb{R}^{n}\backslash 3Q}\frac{|z-\xi|}{|z-y|^{n+1}}|f(y)|\,\textrm{d}y\lesssim Mf(x). \end{align*}

Then

\begin{align*} \mathcal{V}_\rho(\Phi\star (f\chi_{\mathbb{R}^{n}\backslash 3Q}))(\xi)& \lesssim Mf(x)+ \inf_{z\in Q}\left[\mathcal{V}_\rho(\Phi\star (f\chi_{3Q}))(z)+\mathcal{V}_\rho(\Phi\star f)(z)\right]\\ & \lesssim Mf(x)+\left[\frac{1}{|Q|}\int_Q\mathcal{V}_\rho(\Phi\star (f\chi_{3Q}))(z)^{1/2}\,\textrm{d}z\right]^{2}\\ & \quad+ \left[\frac{1}{|Q|}\int_Q\mathcal{V}_\rho(\Phi\star f)(z)^{1/2}\,\textrm{d}z\right]^{2}\\ & \lesssim Mf(x)+\frac{1}{|Q|}\int_{3Q}|f(y)|\,\textrm{d}y +M_{1/2}(\mathcal{V}_\rho(\Phi\star f))(x)\\ & \lesssim Mf(x)+M_{1/2}(\mathcal{V}_\rho(\Phi\star f))(x), \end{align*}

where in the last-to-second inequality, we used the Kolmogorov inequality and the weak type $(1,1)$ of $\mathcal {V}_\rho (\Phi \star f)$ (see [Reference Liu32]). Therefore, (2.4) holds.

Recall that $M_{1/2}(f)$ is bounded on $L^{1,\infty }(\mathbb {R}^{n})$ (see [Reference Nazarov, Treil and Volberg37]). Then for $f\in L^{1}(\mathbb {R}^{n})$, we have $\mathcal {V}_\rho (\Phi \star f)\in L^{1,\infty }(\mathbb {R}^{n})$ and

\begin{align*} \|M_{1/2}(\mathcal{V}_\rho(\Phi\star f))\|_{L^{1,\infty}(\mathbb{R}^{n})}& \lesssim\|\mathcal{V}_\rho(\Phi\star f)\|_{L^{1,\infty}(\mathbb{R}^{n})} \\ & \lesssim\|\mathcal{V}_\rho(\Phi)\|_{L^{1}(\mathbb{R}^{n})\rightarrow L^{1,\infty}(\mathbb{R}^{n})}\|f\|_{L^{1}(\mathbb{R}^{n})}. \end{align*}

This, together with (2.4) and the weak type $(1,1)$ of $M$, implies lemma 2.4.

By lemmas 2.3, (2.2) and (2.3), we now obtain the following weighted estimate of variation operators associated with approximations to the identity.

Theorem 2.5 Let $\rho >2$, $1< p<\infty$ and $\omega \in A_p$. Then for any $f\in L^{p}(\omega )$,

\[ \|\mathcal{V}_\rho(\Phi\star f)\|_{L^{p}(\omega)}\lesssim[\omega]_{A_p}^{\max\{1,\frac{1}{p-1}\}}\|f\|_{L^{p}(\omega)}. \]

If $\omega \in A_1$, then for any $f\in L^{1}(\omega )$,

\[ \|\mathcal{V}_\rho(\Phi\star f)\|_{L^{1,\infty}(\omega)}\lesssim[\omega]_{A_1}\log(e+[\omega]_{A_\infty})\|f\|_{L^{1}(\omega)}. \]

3. Characterization of weighted Hardy spaces

In this section, we give the proof of theorem 1.1. Let us begin recalling a definition and a lemma that will be useful for us.

We now recall the following useful lemma on the boundedness criterion of an operator from $H^{p}(\omega )$ to $L^{p}(\omega )$ with $p\in (0,1]$, established by Bownik, Li, Yang and Zhou [Reference Bownik, Li, Yang and Zhou6].

Now, we are in the position to prove theorem 1.1.

4. The characterization of $BMO_\nu (\mathbb {R}^{n})$

This section is concerned with the proof of theorem 1.2. We first recall the following relevant notation and the equivalent definition of $BMO_\nu (\mathbb {R}^{n})$.

Definition 4.1 (cf. [Reference Lerner, Ombrosi and Rivera-Ríos29]) By a median value of a real-valued measurable function $f$ over a measure set $E$ of positive finite measure, we mean a possibly non-unique, real number $m_f(E)$ such that

\[ \max(|\{x\in E: f(x)>m_f(E)\}|,\quad |\{x\in E: f(x)< m_f(E)\}|)\leq|E|/2. \]

In order to introduce the equivalent definition of $BMO_\nu (\mathbb {R}^{n})$, we recall the definition of local mean oscillation.

Definition 4.2 (cf. [Reference Lerner, Ombrosi and Rivera-Ríos29]) For a complex-valued measurable function $f$, we define the local mean oscillation of $f$ over a cube $Q$ by

\[ a_{\tau}(f;Q):=\inf_{c\in \mathbb{C}}((f-c)\chi_Q)^{*}(\tau|Q|)\quad (0<\tau<1), \]

where $f^{\ast }$ denotes the non-increasing rearrangement of $f$.

For $\tau \in (0,\frac {1}{2^{n+2}}]$, the following equivalent relation is valid:

(4.1)\begin{equation} \|f\|_{BMO_\nu(\mathbb{R}^{n})}\sim \sup_Q \frac{|Q|}{\nu(Q)}a_{\tau}(f;Q). \end{equation}

We refer readers to [Reference Lerner, Ombrosi and Rivera-Ríos29, lemma 2.1] for more details.

Now we prove theorem 1.2.

Proof of theorem 1.2. We first show that $(1)\Rightarrow (2)$. Without loss of generality, we assume that $b$ and $\phi$ are real-valued, and $\phi (z)\geq 1$ for $z\in B(z_0,\delta )$, where $|z_0|=1$ and $\delta >0$ is a small constant. For any cube $Q$, denote by

\[ P:=Q-10\sqrt{n}\delta^{{-}1}l_Qz_0 \]

the cube associated with $Q$. For $0<\tau <1$, by the definition of $a_{\tau }(f;Q)$, there exists a subset $\tilde Q$ of $Q$, such that $|\tilde Q|=\tau |Q|$ and for any $x\in \widetilde Q$,

\[ a_{\tau}(b;Q)\leq |b(x)-m_b(P)|. \]

Then, by the definition of $m_b(P)$, there exist subsets $E\subset \tilde Q$ and $F\subset P$ such that

\[ |E|=|\tilde Q|/2=\tau|Q|/2,\ |F|=|P|/2=|Q|/2, \]

and

\[ a_{\tau}(b;Q)\leq |b(x)-b(y)|,\quad\forall\, x\in E,\, y\in F, \]

and $b(x)-b(y)$ does not change sign in $E\times F$. Let

\[ f(x):=\left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}\chi_F(x). \]

Then,

\begin{align*} \mathcal{V}_{\rho}((\Phi\star f)_b)(x)& =\sup_{t_{k}\downarrow 0}\left(\sum_{k=1}^{\infty}\left|\int_{F}(b(x)-b(y)) (\phi_{t_k}(x-y)-\phi_{t_{k+1}}(x-y))\,\textrm{d}y\right|^{\rho}\right)^{1/\rho}\\ & \quad\times\left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}\\ & \geq\varliminf_{t\rightarrow 0}\left|\int_{F}(b(x)-b(y))(\phi_{10\sqrt{n}\delta^{{-}1}l_Q}(x-y)-\phi_{t}(x-y))\,\textrm{d}y\right|\\ & \quad\times\left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}. \end{align*}

For $x\in E\subset Q, y\in F\subset P$, we have

\begin{align*} & x-y\in 2l_QQ_0+10\sqrt{n}\delta^{{-}1}l_Qz_0\subset (l_QQ_0)^{c},\\ & \frac{x-y}{10\sqrt{n}\delta^{{-}1}l_Q}\in \frac{\delta}{5\sqrt{n}}Q_0+z_0\subset B(z_0,\delta), \end{align*}

where $Q_0$ is the cube centred at origin with side length 1. From this, for $x\in E$, $y\in F$, we have the following estimates

\[ \phi_{10\sqrt{n}\delta^{{-}1}l_Q}(x-y)\gtrsim \frac{1}{|Q|}\phi\left(\frac{x-y}{10\sqrt{n}\delta^{{-}1}l_Q}\right)\geq\frac{1}{|Q|}, \]

and

\begin{align*} \lim_{t\rightarrow 0}\left|\phi_{t}(x-y))\right|& =\lim_{t\rightarrow 0}\frac{1}{t^{n}}\left|\phi\left(\frac{x-y}{t}\right)\right|\lesssim \lim_{t\rightarrow 0}\frac{1}{t^{n}}\left(\frac{|x-y|}{t}\right)^{{-}n-1} \\ & \lesssim \lim_{t\rightarrow 0}t(|x-y|)^{{-}n-1}=0. \end{align*}

Hence, for $x\in E$,

\begin{align*} \mathcal{V}_{\rho}((\Phi\star f)_b)(x)& \ge\varliminf_{t\rightarrow 0}\int_{F}|b(x)-b(y)||\phi_{10\sqrt{n}\delta^{{-}1}l_Q}(x-y)-\phi_{t}(x-y)|\,\textrm{d}y \\ & \quad \times \left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}\\ & \geq \int_{F}|b(x)-b(y)|\varliminf_{t\rightarrow 0}|\phi_{10\sqrt{n}\delta^{{-}1}l_Q}(x-y)-\phi_{t}(x-y)|\,\textrm{d}y\\ & \quad \times \left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}\\ & =\int_{F}|b(x)-b(y)||\phi_{10\sqrt{n}\delta^{{-}1}l_Q}(x-y)|\,\textrm{d}y\left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}\\ & \gtrsim a_{\tau}(b;Q)\left(\int_F\mu(x)\,\textrm{d}x\right)^{{-}1/p}, \end{align*}

which yields that

(4.2)\begin{equation} \int_E\mathcal{V}_{\rho}((\Phi\ast f)_b)(x)\,\textrm{d}x\gtrsim\tau|Q|a_{\tau}(b;Q)\left(\int_P\mu(x)\,\textrm{d}x\right)^{{-}1/p}. \end{equation}

On the other hand, by the Hölder inequality and $(1)$ of theorem 1.2, we have

\begin{align*} \int_E\mathcal{V}_{\rho}((\Phi\star f)_b)(x)\,\textrm{d}x& \leq\left(\int_E\mathcal{V}_{\rho}((\Phi\star f)_b)(x)^{p}\lambda(x)\,\textrm{d}x\right)^{1/p}\left(\int_Q\lambda(x)^{{-}p'/p}\,\textrm{d}x\right)^{1/p'}\\ & \lesssim\left(\int_Q\lambda(x)^{{-}p'/p}\,\textrm{d}x\right)^{1/p'}. \end{align*}

This, together with (4.2) and $P\subset KQ$ for some $K>0$, gives that

(4.3)\begin{equation} a_{\tau}(b;Q)\lesssim\left(\frac{1}{|Q|}\int_Q\mu(x)\,\textrm{d}x\right)^{1/p}\left(\frac{1}{|Q|}\int_Q\lambda(x)^{{-}p'/p}\,\textrm{d}x\right)^{1/p'}. \end{equation}

Noting that

\[ \frac{1}{|Q|}\int_Q\mu(x)\,\textrm{d}x\lesssim\left(\frac{1}{|Q|}\int_Q\mu(x)^{1/(p+1)}\,\textrm{d}x\right)^{p+1} \]

(see [Reference Lerner, Ombrosi and Rivera-Ríos29]), using Hölder's inequality and $\mu =\nu ^{p}\lambda$, we obtain

\[ \left(\frac{1}{|Q|}\int_Q\mu(x)^{1/(p+1)}\,\textrm{d}x\right)^{p+1}\leq\left(\frac{1}{|Q|}\int_Q\nu(x)\,\textrm{d}x\right)^{p} \left(\frac{1}{|Q|}\int_Q\lambda(x)\,\textrm{d}x\right). \]

Thus, by (4.3) and $\lambda \in A_p$, we conclude that

\begin{align*} a_{\tau}(b;Q)& \lesssim\left(\frac{1}{|Q|}\int_Q\nu(x)\,\textrm{d}x\right)\left(\frac{1}{|Q|}\int_Q\lambda(x)\,\textrm{d}x\right)^{1/p} \left(\frac{1}{|Q|}\int_Q\lambda(x)^{{-}p'/p}\,\textrm{d}x\right)^{1/p'}\\ & \lesssim\frac{1}{|Q|}\int_Q\nu(x)\,\textrm{d}x. \end{align*}

This implies that $b\in BMO_\nu ({{\mathbb {R}^{n}}})$ by choosing $\tau =1/2^{n+2}$ and invoking (4.1).

Next, we show that $(2)\Rightarrow (1)$. Indeed, using lemma 2.4, following the standard steps of [Reference Lerner, Ombrosi and Rivera-Ríos28], there exist $3^{n}$ sparse families $\mathcal {S}_j$ such that

(4.4)\begin{equation} \mathcal{V}_\rho((\Phi\star f)_b)(x)\lesssim \sum_{j=1}^{3^{n}}(\mathcal{T}_{\mathcal{S}_j,b}(f)(x)+\mathcal{T}_{\mathcal{S}_j,b}^{{\ast}} (f)(x)), \end{equation}

where

\begin{align*} \mathcal{T}_{\mathcal{S},b}f(x)& :=\sum_{Q\in\mathcal{S}}|b(x)-\langle b\rangle_Q|\langle |f|\rangle_Q\chi_Q(x),\\ \mathcal{T}_{\mathcal{S},b}^{{\ast}} f(x)& :=\sum_{Q\in\mathcal{S}}\langle |(b-\langle b\rangle_Q)f|\rangle_Q\chi_Q(x). \end{align*}

In [Reference Lerner, Ombrosi and Rivera-Ríos28], the authors proved that

\[ \|\mathcal{T}_{\mathcal{S},b}f+\mathcal{T}_{\mathcal{S},b}^{{\ast}} f\|_{L^{p}(\lambda)}\lesssim ([\mu]_{A_p}[\lambda]_{A_p})^{\max\{1,\frac{1}{p-1}\}}\|b\|_{BMO_\nu({{\mathbb{R}^{n}}})}\|f\|_{L^{p}(\mu)}, \]

where $\mu,\lambda \in A_p~(1< p<\infty )$, $\nu =(\mu \lambda ^{-1})^{1/p}$ and $b\in BMO_\nu ({{\mathbb {R}^{n}}})$. This, together with (4.4), shows that $(2)$ implies $(1)$. Theorem 1.2 is proved.

5. The characterization of $\mathcal {BMO}_\omega ({{\mathbb {R}^{n}}})$ spaces

This section is devoted to the proof of theorem 1.3. We first recall and establish two lemmas.

Lemma 5.1 (cf. [Reference Liang, Ky and Yang30]) Let $\tilde {\phi }\in \mathcal {S}(\mathbb {R}^{n})$ such that $\tilde {\phi }(x)=1$ for all $x\in B(0,1)$, and $M_{\tilde {\phi }}$ be defined as in (1.5). Suppose that $f$ is a measurable function such that $\operatorname {supp} f\subset B:=B(x_B,r)$ with some $x_B\in \mathbb {R}^{n}$ and $r\in (0,\infty )$. Then for all $x\not \in B$,

\[ \frac{1}{|x-x_B|^{n}}\left|\int_{B}f(y)\,\textrm{d}y\right|\lesssim M_{\tilde{\phi}}(f)(x). \]

Lemma 5.2 Let $\omega \in A_q$ with $q\in (1,1+1/n)$. Then for any $b\in BMO(\mathbb {R}^{n})$ and $(1,q,s)_\omega$-atom $a$ with $s\ge 0$ and $\operatorname {supp} a\subset B:=B(x_B,r)$, there holds that

\[ \|(b-\langle b\rangle_B)\mathcal{V}_\rho(\Phi\star a)\|_{L^{1}(\omega)}\lesssim\|b\|_{BMO(\mathbb{R}^{n})}. \]

Proof. We prove this lemma by considering the following two terms:

\[ I_1:=\int_{4B}|b(x)-\langle b\rangle_{B}|\mathcal{V}_\rho(\Phi\star a)\omega(x)\,\textrm{d}x, \]

and

\[ I_2:=\int_{(4B)^{c}}|b(x)-\langle b\rangle_{B}|\mathcal{V}_\rho(\Phi\star a)\omega(x)\,\textrm{d}x. \]

Note that for any $\omega \in A_\infty$, $q\in [1,\infty )$ and $B\subset \mathbb {R}^{n}$,

(5.1)\begin{equation} \left[\frac{1}{\omega(B)}\int_B|b(x)-\langle b\rangle_B|^{q}\omega(x)\,\textrm{d}x\right]^{1/q}\lesssim\|b\|_{BMO({{\mathbb{R}^{n}}})}. \end{equation}

By Hölder's inequality, theorem 2.5 and definition 3.1, we have

\begin{align*} I_1& \leq\left(\int_{4B}|b(x)-\langle b\rangle_{B}|^{q'}\omega(x)\,\textrm{d}x\right)^{1/q'}\left(\int_{\mathbb{R}^{n}}\mathcal{V}_\rho(\Phi\star a)(x)^{q}\omega(x)\,\textrm{d}x\right)^{1/q}\\ & \lesssim\left[\left(\int_{4B}|b(x)-\langle b\rangle_{4B}|^{q'}\omega(x)\,\textrm{d}x\right)^{1/q'}+\omega(4B)^{1/q'}\|b\|_{BMO(\mathbb{R}^{n})}\right]\|a\|_{L^{q}(\omega)}\\ & \lesssim\omega(4B)^{1/q'}\|b\|_{BMO(\mathbb{R}^{n})}\omega(B)^{{-}1/q'}\sim\|b\|_{BMO(\mathbb{R}^{n})}. \end{align*}

For $I_2$, noting that $\omega \in A_q$, $\omega (2^{j+1}B)\lesssim 2^{(j+1)nq}\omega (B)$ and invoking the vanishing property of $a$, it follows from (3.5) and (5.1) that

\begin{align*} I_2& \lesssim\int_{(4B)^{c}}|b(x)-\langle b\rangle_B|\int_B|a(y)|\frac{|y-x_B|}{|x-x_B|^{n+1}}\,\textrm{d}y\omega(x)\,\textrm{d}x\\ & \lesssim\int_B|a(y)|\sum_{j=2}^{\infty}\int_{2^{j+1}B\backslash 2^{j}B}|b(x)-\langle b\rangle_B|\frac{|y-x_B|}{|x-x_B|^{n+1}}\omega(x)\,\textrm{d}x\,\textrm{d}y\\ & \leq\left(\int_B|a(y)|^{q}\omega(y)\,\textrm{d}y\right)^{1/q}\left(\int_B\omega(y)^{{-}q'/q}\,\textrm{d}y\right)^{1/q'}\\ & \quad\times\sum_{j=2}^{\infty}\int_{2^{j+1}B}\frac{r}{(2^{j}r)^{n+1}}(|b(x)-\langle b\rangle_{2^{j+1}B}|+|\langle b\rangle_{2^{j+1}B}-\langle b\rangle_B|)\omega(x)\,\textrm{d}x\\ & \lesssim\frac{|B|}{\omega(B)}\sum_{j=2}^{\infty}2^{{-}j(n+1)}j\frac{\omega(2^{j+1}B)}{|B|}\|b\|_{BMO(\mathbb{R}^{n})}\\ & \lesssim\|b\|_{BMO(\mathbb{R}^{n})}\sum_{j=2}^{\infty}2^{{-}j(n+1-qn)}j\lesssim\|b\|_{BMO(\mathbb{R}^{n})}. \end{align*}

Combining the estimates of $I_1$ and $I_2$, we finish the proof of lemma 5.2.

Now, we are in the position to prove theorem 1.3.

Proof of theorem 1.3. First, we show that $(2)$ implies $(1)$. In view of lemma 3.2, we only need to prove that for any $(1,\infty,s)$-atom $a$ with $s\ge 0$ and $\operatorname {supp} a\subset B:=B(x_B,r)$, there holds that

\[ \|\mathcal{V}_\rho((\Phi\star a)_b)\|_{L^{1}(\omega)}\lesssim\|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}. \]

Write

\[ \|\mathcal{V}_\rho((\Phi\star a)_b)\|_{L^{1}(\omega)}\leq\|\mathcal{V}_\rho(\Phi\star((b-\langle b\rangle_B)a))\|_{L^{1}(\omega)}+ \|(b-\langle b\rangle_B)\mathcal{V}_\rho(\Phi\star a)\|_{L^{1}(\omega)}. \]

Since $(1,\infty,s)$-atom is $(1,q,s)$-atom and $\|b\|_{BMO(\mathbb {R}^{n})}\lesssim \|b\|_{\mathcal {BMO}_\omega (\mathbb {R}^{n})}$, by lemma 5.2 and theorem 1.1, it suffices to show that $(b-\langle b\rangle _B)a\in H^{1}(\omega )$ with

(5.2)\begin{equation} \|(b-\langle b\rangle_B)a\|_{H^{1}(\omega)}\lesssim\|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}. \end{equation}

We now show (5.2). For $x\not \in 2B$, note that

\begin{align*} M_\phi((b-\langle b\rangle_B)a)(x)& \leq\sup_{t>0}t^{{-}n}\int_B|b(y)-\langle b\rangle_B||a(y)|\left|\phi\left(\frac{x-y}{t}\right)\right|\textrm{d}y\\ & \lesssim\sup_{t>0}t^{{-}n}\int_B|b(y)-\langle b\rangle_B||a(y)|(1+|x-y|/t)^{{-}n}\,\textrm{d}y\\ & \lesssim\frac{1}{|x-x_B|^{n}}\int_B|b(y)-\langle b\rangle_B||a(y)|\,\textrm{d}y. \end{align*}

Hence, by the definition of $\mathcal {BMO}_\omega (\mathbb {R}^{n})$ and $\|a\|_{L^{\infty }({{\mathbb {R}^{n}}})}\leq \omega (B)^{-1}$, we have

(5.3)\begin{align} & \int_{(2B)^{c}}M_\phi((b-\langle b\rangle_B)a)(x)\omega(x)\,\textrm{d}x\nonumber\\ & \quad\lesssim\frac{1}{\omega(B)}\left(\int_{(2B)^{c}}\frac{\omega(x)}{|x-x_B|^{n}}\,\textrm{d}x\right)\left(\int_B|b(y)-\langle b\rangle_B|\,\textrm{d}y\right)\leq\|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}. \end{align}

Meanwhile, by the $L^{q}(\omega )$-boundedness of $M_\phi$, $\|a\|_{L^{\infty }({{\mathbb {R}^{n}}})}\leq \omega (B)^{-1}$ and (5.1), we obtain

\begin{align*} \int_{2B}M_\phi((b-\langle b\rangle_B)a)(x)\omega(x)\,\textrm{d}x& \leq\left[\int_{\mathbb{R}^{n}}M_\phi((b-\langle b\rangle_B)a)(x)^{q}\omega(x)\,\textrm{d}x\right]^{1/q} \\ & \quad \times \left(\int_{2B}\omega(x)\,\textrm{d}x\right)^{1/q'}\\ & \lesssim\|(b-\langle b\rangle_B)a\|_{L^{q}(\omega)}\omega(2B)^{1/q'}\\ & \lesssim\omega(B)^{{-}1/q}\left(\int_B|b(x)-\langle b\rangle_B|^{q}\omega(x)\,\textrm{d}x\right)^{1/q}\\ & \lesssim\|b\|_{BMO(\mathbb{R}^{n})}\lesssim\|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}. \end{align*}

This, together with (5.3), implies (5.2), and proves that $(2) \Rightarrow (1)$.

Next, we show that $(1)\Rightarrow (2)$. For any ball $B:=B(x_B,r)$, take $h:=\textrm {sgn}(b-\langle b\rangle _B)$ and

\[ a:=\frac{1}{2\omega(B)}(h-\langle h\rangle_B)\chi_B. \]

Then $\operatorname {supp} a\subset B$, $\|a\|_{L^{\infty }({{\mathbb {R}^{n}}})}\leq \omega (B)^{-1}$ and $\int _Ba(y)dy=0.$ By lemma 5.2 and assumption $(1)$ in theorem 1.3, we obtain

\begin{align*} \|\mathcal{V}_\rho(\Phi\star((b-\langle b\rangle_B)a))\|_{L^{1}(\omega)}& \leq\|\mathcal{V}_\rho((\Phi\star a)_b)\|_{L^{1}(\omega)}+ \|(b-\langle b\rangle_B)\mathcal{V}_\rho(\Phi\star a)\|_{L^{1}(\omega)}\\ & \lesssim\|a\|_{H^{1}(\omega)}+\|b\|_{BMO(\mathbb{R}^{n})}. \end{align*}

Hence, by (i) of theorem 1.1 with $\lim \limits _{t\rightarrow 0}\phi _t\ast f=f$ on $L^{1}(\omega )$ for $\omega \in A_1$ (see [Reference Muckenhoupt and Wheeden36]) and (5.1),

\begin{align*} \|(b-\langle b\rangle_B)a\|_{H^{1}(\omega)}& \leq\|(b-\langle b\rangle_B)a\|_{L^{1}(\omega)}+\|\mathcal{V}_\rho(\Phi\star((b-\langle b\rangle_B)a))\|_{L^{1}(\omega)}\\ & \lesssim\frac{1}{\omega(B)}\int_B|b(x)-\langle b\rangle_B|\omega(x)\,\textrm{d}x+\|a\|_{H^{1}(\omega)}+\|b\|_{BMO(\mathbb{R}^{n})}\\ & \lesssim\|b\|_{BMO(\mathbb{R}^{n})}+\|a\|_{H^{1}(\omega)}+\|b\|_{BMO(\mathbb{R}^{n})}\lesssim 1. \end{align*}

Also, invoking lemma 5.1, for any $x\not \in B$, we have

\begin{align*} & \frac{1}{2\omega(B)|x-x_B|^{n}}\int_B|b(y)-\langle b\rangle_B|\,\textrm{d}y\\ & \quad=\frac{1}{|x-x_B|^{n}}\int_B(b(y)-\langle b\rangle_B)a(y)\,\textrm{d}y\lesssim M_{\tilde{\phi}}((b-\langle b\rangle_B)a)(x). \end{align*}

Consequently,

\begin{align*} \left(\frac{1}{\omega(B)}\int_{B^{c}}\frac{\omega(x)}{|x-x_B|^{n}}\,\textrm{d}x\right)\left(\int_B|b(y)-\langle b\rangle_B|\,\textrm{d}y\right) & \lesssim\|M_{\tilde{\phi}}((b-\langle b\rangle_B)a)\|_{L^{1}(\omega)}\\ & \lesssim\|(b-\langle b\rangle_B)a\|_{H^{1}(\omega)}, \end{align*}

which implies that

\[ \|b\|_{\mathcal{BMO}_\omega(\mathbb{R}^{n})}\lesssim\sup_B\|(b-\langle b\rangle_B)a\|_{H^{1}(\omega)} \lesssim 1. \]

This finishes the proof of the implication $(1) \Rightarrow (2)$. Theorem 1.3 is proved.