Open Access

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

Journal of Inequalities and Applications20152015:323

https://doi.org/10.1186/s13660-015-0847-5

Received: 28 April 2015

Accepted: 28 September 2015

Published: 9 October 2015

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 \(H^{1}(\mu)\).

Keywords

non-doubling measure space θ-type Calderón-Zygmund operatorHardy space

MSC

42B20

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, [13]). 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 [48] 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
  1. (i)

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

     
  2. (ii)

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

     
  3. (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. (1)

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

     
  2. (2)

    \(\int_{R^{d}}b\,d(\mu)=0\),

     
  3. (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 \(RBMO(\mu)\).

Declarations

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 11271091).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kim Hyong Jik Normal University
(2)
School of Mathematical Sciences and LPMC, Nankai University

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