Boundedness of θ-type Calderón-Zygmund operators on Hardy spaces with non-doubling measures

Abstract

Let μ be a non-negative Radon measure on $R^{d}$ which only satisfies some growth condition. In this paper, we obtain the boundedness of θ-type Calderón-Zygmund operators on the Hardy space $H^{1}(\mu)$.

1 Introduction and preliminaries

During the last few decades, the theory Calderón-Zygmund operators has played a central part of modern harmonic analysis with lots of extensive applications in the other fields of mathematics. One of the most general settings to which Calderón-Zygmund theory extends naturally is the spaces of homogeneous type in the sense of Coifman and Weiss [1]. Many results from real and harmonic analysis on Euclidean spaces have their natural extensions on these spaces (see, for example, [1–3]). A metric space $(X,d)$ equipped with a non-negative Borel measure μ is called a space of homogeneous type if $(X,d,\mu)$ satisfies the measure doubling condition that there exists a positive constant $C_{\mu}$, depending on μ, such that for any ball $B(x,r)=\{y\in X: d(x,y)< r \}$ with $x\in X$ and $r\in(0,\infty)$,

$$\begin{aligned} \mu\bigl(B(x,2r)\bigr)\leq C_{\mu}\mu\bigl(B(x,r)\bigr). \end{aligned}$$

(1.1)

This definition was introduced by Coifman and Weiss in [1]. The doubling condition (1.1) for measures plays a key role in the classical theory of Calderón-Zygmund operators. However, many results on the classical Calderón-Zygmund theory have been proved still valid if the doubling condition is replaced by some weaker conditions. In recent years, many papers focus on the analysis on $R^{d}$ with non-doubling measure; see [4–8] and their references. Throughout this paper, the Euclidean space $R^{d}$ is endowed with a non-negative Radon measure μ which only satisfies the following growth condition, that is, there exists $C>0$ such that

$$\begin{aligned} \mu\bigl(B(x,r)\bigr)\leq Cr^{n} \end{aligned}$$

(1.2)

for all $x\in R^{d}$ and $r>0$, where $B(x,r)=\{y\in R^{d}:|{x-y}|< r\}$, n is a fixed number satisfying $0< n\leq d$. Such a measure need not satisfy the doubling condition (1.1). In [6], Tolsa established Calderón-Zygmund theory for non-doubling measures.

The definition of θ-type Calderón-Zygmund operator was introduced by Yabuta in [9] as follows.

Definition 1.1

Let θ be a non-negative, non-decreasing function on $R^{+}=(0,\infty)$ satisfying

$$\begin{aligned} \int_{0}^{1}\frac{\theta(t)}{t}\,dt< \infty. \end{aligned}$$

(1.3)

A kernel $K(\cdot,\cdot)\in L^{1}_{\mathrm{loc}}(X\times X\backslash\{ (x,y):x=y\})$ is called a θ-type Calderón-Zygmund kernel if the following conditions hold:

$$\begin{aligned} \bigl|{K(x,y)}\bigr|\leq C|{x-y}|^{-n} \end{aligned}$$

(1.4)

and

$$\begin{aligned} \bigl|{K(x,y)-K\bigl(x',y\bigr)}\bigr|+\bigl|{K(y,x)-K \bigl(y,x'\bigr)}\bigr|\leq C\theta \biggl(\frac {|{x-x'}|}{|{x-y}|} \biggr)|{x-y}|^{-n}, \end{aligned}$$

(1.5)

when $|{x-y}|\geq2|{x-x'}|$.

A linear operator T is called the θ-type Calderón-Zygmund operator with kernel $K(\cdot,\cdot)$ satisfying (1.4) and (1.5) if for all $f\in L^{\infty}(\mu)$ with bounded support and $x\notin \operatorname {supp}f$,

$$\begin{aligned} Tf(x)=\int_{R^{d}}K(x,y)f(y)\,d\mu(y). \end{aligned}$$

(1.6)

In [10], the authors proved that the θ-type Calderón-Zygmund operator which is bounded on $L^{2}(\mu)$ is also bounded from $L^{\infty}(\mu)$ into $RBMO(\mu)$ and from $H^{1,\infty }_{atb}(\mu )$ into $L^{1}(\mu)$ on the Euclidean space with non-doubling measures.

In this paper, we discuss the boundedness of the θ-type Calderón-Zygmund operator T in the Hardy space $H^{1}(\mu)$. In order to state our main result, we recall some necessary notations and the known results. The following grand maximal operator was introduced by Tolsa in [11].

Definition 1.2

Given $f\in L^{1}_{\mathrm{loc}}(\mu)$, we set

$$\begin{aligned} M_{\Phi} f(x)=\sup_{\varphi\sim x}\biggl\vert {\int _{R^{d}}f\varphi \,d\mu }\biggr\vert , \end{aligned}$$

where the notation $\varphi\sim x$ means that $\varphi\in L^{1}(\mu )\cap C^{1}(R^{d})$ and satisfies

• (i) $\|{\varphi}\|_{L^{1}(\mu)}\leq1$,

• (ii) $0\leq\varphi(y)\leq|{y-x}|^{-n}$ for all $y\in R^{d}$, and

• (iii) $|{\nabla\varphi(y)}|\leq|{y-x}|^{-(n+1)}$ for all $y\in R^{d}$, where $\nabla=(\partial/\partial x_{1},\ldots,\partial /\partial x_{d})$.

In [11], Tolsa obtained the following result.

Theorem 1.1

A function f belongs to $H^{1,\infty}_{atb}(\mu)$ if and only if $f\in L^{1}(\mu)$, $\int f\,d\mu=0$ and $M_{\Phi}f\in L^{1}(\mu)$. Moreover, in this case

$$\begin{aligned} \|{f}\|_{H^{1,\infty}_{atb}(\mu)}\approx\|{f}\|_{L^{1}(\mu)}+\| {M_{\Phi}f} \|_{L^{1}(\mu)}. \end{aligned}$$

In [12], the authors introduced a new atomic characterization of the Hardy space $H^{1}(\mu)$. Given two cubes $Q\subset R$ in $R^{d}$, set

$$\begin{aligned} K_{Q,R}=1+\sum_{k=1}^{N_{Q,R}} \frac{\mu(2^{k}Q)}{l(2^{k}Q)^{n}}, \end{aligned}$$

where $N_{Q,R}$ is the smallest positive integer k such that $l(2^{k}Q)\geq l(R)$; see [6] for some positive of $K_{Q,R}$. The definition of the $(p,\gamma)$-atomic block is given as follows.

Definition 1.3

Let $\rho>1$, $1< p\leq\infty$ and $\gamma\in N$. A function $b\in L^{1}_{\mathrm{loc}}(\mu)$ is called a $(p,\gamma)$-atomic block if

• (1) there exists some cube R such that $\operatorname {supp}(b)\subset R$,

• (2) $\int_{R^{d}}b\,d(\mu)=0$,

• (3) there are functions $a_{1}$, $a_{2}$ supported on cubes $Q_{1},Q_{2}\subset R$ and numbers $\lambda_{1},\lambda_{2}\in\mathbb{R}$ such that $b=\lambda_{1}a_{1}+\lambda_{2}a_{2}$, and

  $$\begin{aligned} \|{a_{j}}\|_{L^{p}(\mu)}\leq \bigl(\mu(\rho Q_{j}) \bigr)^{1/p-1} (K_{Q_{j},R} )^{-\gamma}, \quad j=1,2. \end{aligned}$$

We denote $|{b}|_{H^{1,p}_{atb,\gamma}(\mu)}=|{\lambda_{1}}|+|{\lambda _{2}}|$. We say that $f\in H^{1,p}_{atb,\gamma}(\mu)$ if there are $(p,\gamma)$-atomic blocks $b_{j}$ such that

$$\begin{aligned} f=\sum_{i=1}^{\infty}b_{i}, \end{aligned}$$

(1.7)

with $\sum_{i=1}^{\infty}|{b_{i}}|_{H^{1,p}_{atb,\gamma}(\mu )}<\infty$ (notice that this implies that the sum in (1.7) converges in $L^{1}(\mu)$). The $H^{1,p}_{atb,\gamma}(\mu)$ norm of f is defined by

$$\begin{aligned} \|{f}\|_{H^{1,p}_{atb,\gamma}(\mu)}=\inf\sum_{i=1}^{\infty }|{b_{i}}|_{H^{1,p}_{atb,\gamma}(\mu)}, \end{aligned}$$

where the infimum is taken over all the possible decompositions of f into $(p,\gamma)$-atomic blocks.

We remark that the definition when $\gamma=1$ was introduced by Tolsa in [6]. It was proved in [6, 12] that the definition of $H^{1,p}_{atb,\gamma}(\mu)$ is independent of the chosen constant $\rho >1$, and for any integer $\gamma\geq1$ and $1< p\leq\infty$, all the atomic Hardy spaces $H^{1,p}_{atb,\gamma}(\mu)$ are just the Hardy space $H^{1,\infty}_{at}(\mu)$ with equivalent norms.

Let $T^{*}$ be the transpose of T. As mentioned in [13], we have to assume that $T^{*}1=0$. Here, by $T^{*}1=0$, we mean that for any bounded function b with compact support and $\int_{R^{d}}b\mu=0$,

$$\begin{aligned} \int_{R^{d}}Tb(x)\,d\mu(x)=0. \end{aligned}$$

(1.8)

The main result of our paper is given as follows.

Theorem 1.2

Let T be a θ-type Calderón-Zygmund operator defined by (1.6) as above, which is bounded on $L^{2}(\mu)$ and $T^{*}1=0$ as in (1.8). Then T is bounded on $H^{1}(\mu)$.

Throughout this paper, C always means a positive constant independent of the main parameters involved, but it may be different in different contents.

2 Proof of our main result

The following lemma will be used in the proof of Theorem 1.2.

Lemma 2.1

Let $M_{\Phi}$ be as in Definition  1.2 and $1< p<\infty$. Then $M_{\Phi}$ is bounded on $L^{p}(\mu)$.

In fact, Tolsa proved that $M_{\Phi}$ is bounded from $H^{1}(\mu)$ into $L^{1}(\mu)$; see Lemma 3.1 in [11]. On the other hand, it is obvious that $M_{\Phi}$ is bounded on $L^{\infty}(\mu)$ for $1< p<\infty$. By Theorem 7.2 in [6], we obtain that $M_{\Phi}$ is bounded on $L^{p}(\mu)$ for $1< p<\infty$.

Now we will prove Theorem 1.2.

Proof of Theorem 1.2

By the standard argument, it suffices to verify that for any atomic block b as in Definition 1.3 with $\rho=4$, $p=\infty$ and $\gamma=2$, Tb is in $H^{1}(\mu)$ with norm $C|{b}|_{H^{1,\infty}_{atb,2}}$. By Definition 1.3, it follows

$$\begin{aligned} \|{a_{j}}\|_{L^{\infty}(\mu)}\leq \bigl( \mu(4Q_{j})K^{2}_{Q_{j},R} \bigr)^{-1}, \end{aligned}$$

(2.1)

where $j=1,2$. The assumption that $T^{*}1=0$ tells us that $\int_{R^{d}}Tb\,d(\mu)=0$. Recalling that T is bounded from $H^{1}(\mu)$ into $L^{1}(\mu)$ (see [6]), we obtain

$$\begin{aligned} \|{Tb}\|_{L^{1}(\mu)}\leq C|{b}|_{H^{1,\infty}_{atb}(\mu)}. \end{aligned}$$

By this and Theorem 1.1, we deduce that the proof of Theorem 1.2 can be reduced to proving that

$$\begin{aligned} \bigl\| {M_{\Phi}(Tb)}\bigr\| _{L^{1}(\mu)}\leq C|{b}|_{H^{1,\infty}_{atb}(\mu)}. \end{aligned}$$

(2.2)

We can write

$$\begin{aligned} \int_{R^{d}}M_{\Phi}(Tb) (x)\,d\mu(x)=\int _{R^{d}\backslash4R}M_{\Phi}(Tb) (x)\,d\mu (x)+\int _{4R}M_{\Phi}(Tb) (x)\,d\mu(x)=I_{1}+I_{2}. \end{aligned}$$

Let us now estimate $I_{1}$. Let $x_{R}$ be the center of the cube R. From the fact $T^{*}1=0$, we obtain

$$\begin{aligned} I_{1}={}&\int_{R^{d}\backslash4R}\sup_{\varphi\sim x} \biggl\vert {\int_{R^{d}}Tb(y)\bigl[\varphi(y)- \varphi(x_{R})\bigr]\,d\mu(y)}\biggr\vert \,d\mu(x) \\ \leq{}&\int_{R^{d}\backslash4R}\sup_{\varphi\sim x}\biggl\vert { \int_{2R}Tb(y)\bigl[\varphi(y)-\varphi(x_{R}) \bigr]\,d\mu(y)}\biggr\vert \,d\mu(x) \\ &{}+\int_{R^{d}\backslash4R}\sup_{\varphi\sim x}\biggl\vert {\int _{R^{d}\backslash 2R}Tb(y)\bigl[\varphi(y)-\varphi(x_{R})\bigr]\,d\mu(y)}\biggr\vert \,d\mu(x) \\ ={}&I_{11}+I_{12}. \end{aligned}$$

Note that for any $z\in2R$, $x\in2^{k+1}R\backslash2^{k}R$, and $k\geq 2$, we have $|{x-z}|\geq l(2^{k-2}R)$.

This together with Definition 1.2 and the mean value theorem leads to

$$\begin{aligned} \bigl|{\varphi(y)-\varphi(x_{R})}\bigr|\leq C\frac{l(R)}{l(2^{k-2}R)^{n+1}}. \end{aligned}$$

(2.3)

For $j=1,2$, denote $N_{Q_{j},2R}$ simply by $N_{j}$ for $y\in2R$. By (2.3), (1.4), Hölder’s inequality, the boundedness of T in $L^{2}(\mu)$ and (2.1), we have

$$\begin{aligned} I_{11}={}&\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sup_{\varphi\sim x} \biggl[\int _{2R\backslash 2Q_{j}}\bigl|{Ta_{j}(y)}\bigr|\bigl|{\varphi(y)- \varphi(x_{R})}\bigr|\,d\mu(y) \biggr]\,d\mu(x) \\ &{}+\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{\infty}\int_{2^{k+1}R\backslash 2^{k}R} \sup_{\varphi\sim x} \biggl[\int_{2Q_{j}}\bigl|{Ta_{j}(y)}\bigr|\bigl|{ \varphi (y)-\varphi(x_{R})}\bigr|\,d\mu(y) \biggr]\,d\mu(x) \\ \leq{}& C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\frac{l(R)}{l(2^{k-2}R)^{n+1}}\sum_{l=1}^{N_{j}-1} \int_{2^{l+1}Q_{j}\backslash2^{l}Q_{j}}\int_{Q_{j}}\frac {|{a_{j}(z)}|}{|{y-z}|^{n}}\,d\mu(z)\,d\mu(y)\,d\mu(x) \\ &{}+C\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{\infty}\int_{2^{k+1}R\backslash 2^{k}R} \frac{l(R)}{l(2^{k-2}R)^{n+1}}\bigl\| {(Ta_{j})\chi_{2Q_{j}}}\bigr\| _{L^{1}(\mu )}\,d\mu(x) \\ \leq{}& C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{\infty}2^{-k} \sum_{l=1}^{N_{j}-1}\frac{\mu(2^{l+1}Q_{j})}{l(2^{l+1}Q_{j})^{n}} \|{a_{j}}\|_{L^{\infty}(\mu)}\mu(Q_{j}) \\ &{}+C\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{\infty}2^{-k} \bigl\| {(Ta_{j})\chi _{2Q_{j}}}\bigr\| _{L^{2}(\mu)}\mu(2Q_{j})^{1/2} \\ \leq{}& C\sum_{j=1}^{2}|{ \lambda_{j}}|K_{Q_{j},R}\|{a_{j}}\|_{L^{\infty}(\mu )}\mu (Q_{j})+C\sum_{j=1}^{2}|{ \lambda_{j}}| \|{a_{j}}\|_{L^{2}(\mu)} \mu(2Q_{j})^{1/2} \\ \leq{}& C\sum_{j=1}^{2}|{ \lambda_{j}}|, \end{aligned}$$

where we have used the fact that

$$\begin{aligned} K_{Q_{j},2R}\leq CK_{Q_{j},R}. \end{aligned}$$

For $I_{12}$, we get

$$\begin{aligned} I_{12}={}&\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sup_{\varphi \sim x}\biggl\vert {\int _{R^{d}\backslash2R}Tb(y)\bigl[\varphi(y)-\varphi (x_{R})\bigr]\,d\mu (y)}\biggr\vert \,d\mu(x) \\ \leq{}&\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}M_{\Phi}\bigl[|{Tb}|\chi_{2^{k+2}R\backslash2^{k-1}R} \bigr](x)\,d\mu(x) \\ &{}+\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sup_{\varphi\sim x} \biggl[\int _{2^{k+2}R\backslash2^{k-1}R}\bigl|{Tb(y)}\bigr|\varphi(x_{R})\,d\mu (y) \biggr]\,d\mu(x) \\ &{}+\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sup_{\varphi\sim x} \biggl[\int _{R^{d}\backslash2^{k+2}R}\bigl|{Tb(y)}\bigr|\bigl(\varphi(y)+\varphi (x_{R}) \bigr)\,d\mu(y) \biggr]\,d\mu(x) \\ &{}+\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sup_{\varphi\sim x} \biggl[\int _{2^{k-1}R\backslash2R}\bigl|{Tb(y)}\bigr|\bigl(\varphi(y)+\varphi (x_{R}) \bigr)\,d\mu(y) \biggr]\,d\mu(x) \\ ={}&I_{121}+I_{122}+I_{123}+I_{124}. \end{aligned}$$

From Lemma 2.1, the fact that $\int_{R^{d}}b\,d(\mu)=0$ and (1.5), we can deduce that

$$\begin{aligned} I_{121}\leq{}&\sum_{k=2}^{\infty}\mu \bigl(2^{k+1}R\bigr)^{1/2}\bigl\Vert {M_{\Phi}\bigl[|{Tb}|\chi_{2^{k+2}R\backslash2^{k-1}R} \bigr]}\bigr\Vert _{L^{2}(\mu)} \\ \leq{}& C\sum_{k=2}^{\infty}\mu \bigl(2^{k+1}R\bigr)^{1/2} \biggl(\int_{2^{k+2}R\backslash2^{k-1}R} \biggl\vert {\int_{R}\bigl(K(y,z)-K(y,x_{R}) \bigr)b(z)\,d\mu (z)}\biggr\vert ^{2}\,d\mu(y) \biggr)^{1/2} \\ \leq{}& C\sum_{k=2}^{\infty}\mu \bigl(2^{k+1}R\bigr)^{1/2} \\ &{}\times\biggl(\int_{2^{k+2}R\backslash2^{k-1}R} \biggl[\int_{R}\theta \biggl(\frac {|{z-x_{R}}|}{|{y-x_{R}}|} \biggr) |{y-x_{R}}|^{-n}\bigl|{b(z)}\bigr|\,d\mu(z) \biggr]^{2}\,d\mu(y) \biggr)^{1/2} \\ \leq{}& C\sum_{k=2}^{\infty}\frac{\mu(2^{k+1}R)}{l(2^{k}R)^{n}} \theta \bigl(2^{-k}\bigr)\|{b}\|_{L^{1}(\mu)} \leq C\int _{0}^{1}\frac{\theta(t)}{t}\,dt\|{b}\|_{L^{1}(\mu)} \leq C\sum_{j=1}^{2}|{\lambda_{j}}|, \end{aligned}$$

where we have used the following inequality:

$$\begin{aligned} \int_{0}^{1}\frac{\theta(t)}{t}\geq\sum \int_{2^{k}}^{2^{1-k}}\frac {\theta(2^{-k})}{2^{1-k}}\geq C \sum _{k=1}^{\infty}\theta\bigl(2^{-k}\bigr), \end{aligned}$$

and the fact

$$\begin{aligned} \|{b}\|_{L^{1}(\mu)}\leq\sum_{j=1}^{2}|{ \lambda_{j}}|\|{a_{j}}\|_{L^{1}(\mu )}\leq C\sum _{j=1}^{2}|{\lambda_{j}}|. \end{aligned}$$

An argument similar to the estimate for $I_{121}$ tells us that

$$\begin{aligned} I_{122}\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

Finally, we estimate $I_{123}$. By the fact that $\int_{R^{d}}b\,d\mu=0$, Definition 1.2 and (1.5), we obtain

$$\begin{aligned} I_{123}\leq{}&\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sum_{l=k+2}^{\infty}\int _{2^{l+1}R\backslash2^{l}R}\int_{R}\bigl|{K(y,z)-K(y,x_{R})}\bigr|\bigl|{b(z)}\bigr|\,d\mu(z) \\ &{}\times \biggl[\frac{1}{|{y-x}|^{n}}+\frac{1}{|{x_{R}-x}|^{n}} \biggr]\,d\mu (y)\,d\mu(x) \\ \leq{}& C\sum_{k=2}^{\infty}\int _{2^{k+1}R\backslash2^{k}R}\sum_{l=k+2}^{\infty}\int _{2^{l+1}R\backslash2^{l}R} \int_{R}\theta \biggl( \frac{|{z-x_{R}}|}{|{y-x_{R}}|} \biggr) |{y-x_{R}}|^{-n}\bigl|{b(z)}\bigr|\,d\mu(z) \\ &{}\times \biggl[\frac{1}{|{y-x}|^{n}}+\frac{1}{|{x_{R}-x}|^{n}} \biggr]\,d\mu (y)\,d\mu(x) \\ \leq{}& C\sum_{k=2}^{\infty}\sum _{l=k+2}^{\infty}\theta\bigl(2^{-l}\bigr) \frac {\mu (2^{l+1}R)}{l(2^{l+1}R)^{n}} \frac{\mu(2^{k+1}R)}{l(2^{k+1}R)^{n}}\|{b}\|_{L^{1}(\mu)} \\ \leq{}& C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

An argument similar to the estimate for $I_{123}$ indicates that

$$\begin{aligned} I_{124}\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

Combining the estimate for $I_{121}$, $I_{122}$, $I_{123}$ and $I_{124}$, we obtain the desired estimate for $I_{12}$. The estimates for $I_{11}$ and $I_{12}$ tell us that

$$\begin{aligned} I_{1}=\int_{R^{d}\backslash4R}M_{\Phi}(Tb) (x)\,d\mu(x)\leq C|{b}|_{H^{1,\infty}_{atb,2}}(\mu). \end{aligned}$$

(2.4)

For $I_{2}$, by the sublinearity of $M_{\Phi}$, it follows

$$\begin{aligned} I_{2}\leq\int_{4R}M_{\Phi}\bigl[(Tb) \chi_{8R} \bigr](x)\,d\mu(x)+ \int_{4R}M_{\Phi}\bigl[(Tb)\chi_{R^{d}\backslash8R} \bigr](x)\,d\mu (x)=I_{21}+I_{22}. \end{aligned}$$

From $Q_{j}\subset R$, Definition 1.2 and (2.1), we obtain

$$\begin{aligned} I_{22}&\leq\int_{4R}\sup_{\varphi\sim x} \biggl[\int_{R^{d}\backslash 8R}\bigl|{Tb(y)}\bigr|\varphi(y)\,d\mu(y) \biggr]\,d\mu(x) \\ &\leq\sum_{j=1}^{2}|{\lambda_{j}}| \int_{4R}\sum_{k=2}^{\infty}\int_{2^{k+1}R\backslash2^{k}R}\biggl\vert {\int_{Q_{j}}K(y,z)a_{j}(z)\,d\mu (z)}\biggr\vert \frac {1}{|{x-y}|^{n}}\,d\mu(y)\,d\mu(x) \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=3}^{\infty}\|{a_{j}}\| _{L^{\infty}(\mu)}\mu(Q_{j}) \frac{\mu(2^{k+1}R)}{l(2^{k-2}R)^{n}} \frac{\mu(4R)}{l(2^{k-2}R)^{n}} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

In order to estimate $I_{21}$, we write

$$\begin{aligned} I_{21}\leq{}&\sum_{j=1}^{2}|{ \lambda_{j}}|\int_{4Q_{j}}M_{\Phi}\bigl[(Ta_{j})\chi _{8R} \bigr](x)\,d\mu(x) +\sum_{j=1}^{2}|{\lambda_{j}}| \int_{4R\backslash4Q_{j}}M_{\Phi}\bigl[(Ta_{j}) \chi_{2Q_{j}} \bigr](x)\,d\mu(x) \\ &{}+\sum_{j=1}^{2}|{\lambda_{j}}| \int_{4R\backslash4Q_{j}}M_{\Phi}\bigl[(Ta_{j}) \chi_{8R\backslash2Q_{j}} \bigr](x)\,d\mu(x) \\ &=I_{211}+I_{212}+I_{213}. \end{aligned}$$

Hölder’s inequality, Lemma 2.1, the boundedness of T in $L^{2}(\mu )$ and (2.1) lead to

$$\begin{aligned} I_{211}&\leq\sum_{j=1}^{2}|{ \lambda_{j}}|\mu(4Q_{j})^{1/2}\bigl\| {M_{\Phi}\bigl[(Ta_{j})\chi_{8R}\bigr]}\bigr\| _{L^{2}(\mu)} \\ &\leq C\sum _{j=1}^{2}|{\lambda_{j}}| \mu(4Q_{j})^{1/2}\|{Ta_{j}}\|_{L^{2}(\mu )} \leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\mu(4Q_{j})^{1/2}\|{a_{j}} \|_{L^{2}(\mu)} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\mu(4Q_{j})\|{a_{j}} \|_{L^{\infty}(\mu)} \leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

By Definition 1.2, Hölder’s inequality, the boundedness of T in $L^{2}(\mu)$ and (2.1), we get

$$\begin{aligned} I_{212}&\leq\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}\sup_{\varphi\sim x} \biggl\vert {\int _{2Q_{j}}Ta_{j}(y)\varphi(y)\,d\mu(y)}\biggr\vert d \mu(x) \\ &\leq\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{N_{Q_{j},4R}}\int_{2^{k+1}Q_{j}\backslash2^{k}Q_{j}} \frac{1}{l(2^{k-2}Q_{j})^{n}}\,d\mu(x) \int_{2Q_{j}}\bigl|{Ta_{j}(y)}\bigr|\,d\mu(y) \\ &\leq\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{N_{Q_{j},4R}}\frac{\mu (2^{k+1}Q_{j})}{l(2^{k-2}Q_{j})^{n}} \|{Ta_{j}}\| _{L^{2}(\mu)}\mu(2Q_{j})^{1/2} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|K_{Q_{j},R}\mu(2Q_{j})^{1/2} \|{a_{j}}\| _{L^{2}(\mu)} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|, \end{aligned}$$

where we have used the fact that

$$\begin{aligned} K_{Q_{j},4R}\leq CK_{Q_{j},R}. \end{aligned}$$

(2.5)

For $I_{213}$, we can write

$$\begin{aligned} I_{213}={}&\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[ (Ta_{j}) \chi_{8R\backslash2Q_{j}} \bigr](x)\,d\mu(x) \\ \leq{}&\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[|{Ta_{j}}|\chi _{2^{k+2}Q_{j}\backslash2^{k-1}Q_{j}} \bigr](x)\,d\mu(x) \\ &{}+\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{N_{Q_{j},4R}}\int_{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[|{Ta_{j}}|\chi_{\max\{ 2^{k+2}Q_{j},8R\}\backslash2^{k+2}Q_{j}} \bigr](x)\,d\mu (x) \\ &{}+\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{N_{Q_{j},4R}}\int_{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[|{Ta_{j}}|\chi _{2^{k-1}Q_{j}\backslash2Q_{j}} \bigr](x)\,d\mu(x) \\ ={}&J_{1}+J_{2}+J_{3}. \end{aligned}$$

Lemma 2.1, (1.4) and (2.1) imply that

$$\begin{aligned} J_{1}&=\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\mu \bigl(2^{k+1}Q_{j}\bigr)^{1/2}\bigl\Vert {M_{\Phi}\bigl[f |{Ta_{j}}| \chi _{2^{k+2}Q_{j}\backslash2^{k-1}Q_{j}} \bigr]}\bigr\Vert _{L^{2}(\mu)} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\mu \bigl(2^{k+1}Q_{j}\bigr)^{1/2} \times\biggl(\int _{2^{k+2}Q_{j}\backslash 2^{k-1}Q_{j}}\biggl\vert {\int_{Q_{j}}K(y,z)a_{j}(z)\,d\mu(z)}\biggr\vert ^{2}\,d\mu (y) \biggr)^{1/2} \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}} \frac{\mu (2^{k+2}Q_{j})}{l(2^{k-3}Q_{j})^{n}} \|{a_{j}}\|_{L^{\infty}(\mu)}\mu(Q_{j}) \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

By (ii) of Definition 1.2, (1.4), (2.5) and (2.1), we have

$$\begin{aligned} J_{2}&=\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}\sup_{\varphi\sim x} \biggl[ \int _{2^{k-1}Q_{j}\backslash2Q_{j}}\bigl|{Ta_{j}(y)}\bigr|\varphi(y)\,d\mu(y) \biggr]\,d\mu(x) \\ &\leq\sum_{j=1}^{2}|{\lambda_{j}}| \sum_{k=2}^{N_{Q_{j},4R}}\int_{2^{k+1}Q_{j}\backslash2^{k}Q_{j}} \sum_{l=1}^{k-2}\int_{2^{l+1}Q_{j}\backslash2^{l}Q_{j}} \biggl\vert {\int_{Q_{j}}K(y,z)a_{j}(z)\,d\mu (z)} \biggr\vert \frac {1}{|{y-x}|^{n}}\,d\mu(y)\,d\mu(x) \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}} \frac{\mu (2^{k+1}Q)}{l(2^{k+1}Q_{j})^{n}} \sum_{l=1}^{k-2} \frac{\mu(2^{l+1}Q)}{l(2^{l+1}Q_{j})^{n}}\|{a_{j}}\| _{L^{\infty}(\mu)}\mu(Q_{j}) \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|(K_{Q_{j},R})^{2}\|{a_{j}} \|_{L^{\infty}(\mu )}\mu(Q_{j}) \\ &\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

With the argument similar to the estimate for $J_{2}$ it follows that

$$\begin{aligned} J_{3}=\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[|{Ta_{j}}|\chi _{2^{k-1}Q_{j}\backslash2Q_{j}} \bigr](x)\,d\mu(x)\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

Thus

$$\begin{aligned} I_{213}=\sum_{j=1}^{2}|{ \lambda_{j}}|\sum_{k=2}^{N_{Q_{j},4R}}\int _{2^{k+1}Q_{j}\backslash2^{k}Q_{j}}M_{\Phi}\bigl[ (Ta_{j}) \chi_{8R\backslash2Q_{j}} \bigr](x)\,d\mu(x)\leq C\sum_{j=1}^{2}|{ \lambda_{j}}|. \end{aligned}$$

From the estimation of $I_{21}$ and $I_{22}$, we obtain

$$\begin{aligned} I_{2}=\int_{4R}M_{\Phi}(Tb) (x)\,d\mu(x)\leq C\sum_{j=1}^{2}|{\lambda _{j}}|=C|{b}|_{H^{1,\infty}_{atb,2}}. \end{aligned}$$

(2.6)

The estimates (2.4) and (2.6) lead to (2.2), and this completes the proof of our theorem. □

Remark 2.2

It is known that the dual space of $H^{1}(\mu)$ is the space $RBMO(\mu)$, which is introduced in [12]. From Theorem 1.2, the fact that $RBMO(\mu)={ (H^{1}(\mu) )}^{*}$ and a standard dual argument, it is easy to deduce the boundedness of the transpose operator of T on the $RBMO(\mu)$ space as below.

Corollary 2.3

Let T be the same as in Theorem  1.2. Then $T^{*}$, the transpose operator of T, is bounded on $RBMO(\mu)$.