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 \(H^{1}(\mu)\).
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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 \(r\in(0,\infty)\),
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
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
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:
and
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 \(x\notin \operatorname {supp}f\),
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
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
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
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
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
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 \(\int_{R^{d}}b\mu=0\),
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
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
By this and Theorem 1.1, we deduce that the proof of Theorem 1.2 can be reduced to proving that
We can write
Let us now estimate \(I_{1}\). Let \(x_{R}\) be the center of the cube R. From the fact \(T^{*}1=0\), we obtain
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
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
where we have used the fact that
For \(I_{12}\), we get
From Lemma 2.1, the fact that \(\int_{R^{d}}b\,d(\mu)=0\) and (1.5), we can deduce that
where we have used the following inequality:
and the fact
An argument similar to the estimate for \(I_{121}\) tells us that
Finally, we estimate \(I_{123}\). By the fact that \(\int_{R^{d}}b\,d\mu=0\), Definition 1.2 and (1.5), we obtain
An argument similar to the estimate for \(I_{123}\) indicates that
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
For \(I_{2}\), by the sublinearity of \(M_{\Phi}\), it follows
From \(Q_{j}\subset R\), Definition 1.2 and (2.1), we obtain
In order to estimate \(I_{21}\), we write
Hölder’s inequality, Lemma 2.1, the boundedness of T in \(L^{2}(\mu )\) and (2.1) lead to
By Definition 1.2, Hölder’s inequality, the boundedness of T in \(L^{2}(\mu)\) and (2.1), we get
where we have used the fact that
For \(I_{213}\), we can write
Lemma 2.1, (1.4) and (2.1) imply that
By (ii) of Definition 1.2, (1.4), (2.5) and (2.1), we have
With the argument similar to the estimate for \(J_{2}\) it follows that
Thus
From the estimation of \(I_{21}\) and \(I_{22}\), we obtain
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 \(RBMO(\mu)\).
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (Grant No. 11271091).
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Ri, C., Zhang, Z. Boundedness of θ-type Calderón-Zygmund operators on Hardy spaces with non-doubling measures. J Inequal Appl 2015, 323 (2015). https://doi.org/10.1186/s13660-015-0847-5
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DOI: https://doi.org/10.1186/s13660-015-0847-5