Equivalent Norms of $cmo^{p}(\mathbb {R}^{n})$ and Applications

Abstract

In this paper, we prove that $cmo^{p}(\mathbb {R}^{n})$ and $\Lambda _{n(\frac{1}{p}-1)}$, the dual spaces of local Hardy space $h^{p}(\mathbb {R}^{n})$, are coincide with equivalent norms for $\frac{n}{n+1}<p\le 1$. Moreover, this space can be characterized by another simple norm. As an application, we prove the $h^{p}(\mathbb {R}^{n})$ boundedness of inhomogeneous para-product operators.

1 Introduction

It is well known that Hardy space $H^{p}(\mathbb R^{n})$ has much better functional properties than the space $L^{p}(\mathbb R^{n})$ for $p<1$ [5]. However, as Goldberg in [14] pointed out that $H^{p}(\mathbb {R}^{n})$ space is well suited only to the Fourier analysis, and is not stable under multiplications by the Schwartz test functions. One reason is that $H^{p}(\mathbb {R}^{n})$ does not contain $\mathcal {S}(\mathbb {R}^{n})$, the space of the Schwartz test functions. To circumvent those drawbacks, Goldberg in [14] introduced the local Hardy spaces $h^{p}(\mathbb R^{n}),0<p<\infty$. Let $\Phi \in \mathcal {S}(\mathbb {R}^{n})$ with $\int \Phi \ne 0,\Phi _{t}(x)=t^{-n}\Phi (\frac{x}{t})$ and set

$$\begin{aligned} \mathcal {M}_{\Phi }(f)(x)=\sup _{0<t<1}|\Phi _{t}*f(x)|. \end{aligned}$$

Then $h^{p}(\mathbb {R}^{n})=\{f\in \mathcal {S}'(\mathbb {R}^{n}),\mathcal {M}_{\Phi }(f)\in L^{p}(\mathbb {R}^{n})\},$ where $\mathcal {S}'(\mathbb {R}^{n})$ is the dual of $\mathcal {S}(\mathbb {R}^{n})$.

Goldberg showed that the dual of $h^{1}(\mathbb {R}^{n})$ is $bmo(\mathbb {R}^{n})$, which is defined as the set of $f\in L^{1}_{\mathrm {loc}}(\mathbb {R}^{n})$ such that

$$\begin{aligned} S_{1}(f)=\sup _{|Q|<1}\frac{1}{|Q|}\int _{Q}|f(x)-f_{Q}|\mathrm{d}x<\infty \end{aligned}$$

and

$$\begin{aligned} S_{2}(f)=\sup _{|Q|\ge 1}\frac{1}{|Q|}\int _{Q}|f(x)|\mathrm{d}x<\infty , \end{aligned}$$

equipped with the norm $\Vert f\Vert _{bmo(\mathbb {R}^{n})}=\max \{S_{1}(f),S_{2}(f)\}$, where $f_{Q}$ is the mean of f over Q, i.e., $f_{Q}=\frac{1}{|Q|}\int _{Q}f(x)\mathrm{d}x$. For $0<r<1,$ let $\Lambda _{r}=\{f\in L^{\infty }(\mathbb {R}^{n}): \sup _{x\ne y}\frac{|f(x)-f(y)|}{|x-y|^{r}}<\infty \}$. Then, it is well known that the dual of $h^{p}(\mathbb {R}^{n})$ is $\Lambda _{n(\frac{1}{p}-1)}$ for $\frac{n}{n+1}<p<1$. We refer the reader to [14] for more details of $\Lambda _{r}$ when $r\ge 1$. It is convenient to denote $\Lambda _{0}=bmo(\mathbb {R}^{n})$. For more results about local Hardy spaces, we refer the reader to [1,2,3, 18, 19, 25,26,27,28,29, 31,32,33]. Some recent developments of multi-parameter local Hardy spaces can be seen in [4, 10, 11].

In [31], Rychkov obtained that $h^{p}(\mathbb {R}^{n})$ can be characterized by a continuous Littlewood–Paley–Stein square function. More precisely, let

$$\begin{aligned} \varphi _{0}\in \mathcal {D}({\mathbb {R}^{n}}) \hbox { with nonzero integral, and } \varphi (x)=\varphi _{0}(x)-2^{-n}\varphi _{0}(\frac{x}{2}), \end{aligned}$$

(1.1)

where $\mathcal {D}({\mathbb {R}^{n}})$ is the set of all smooth functions with compact support on $\mathbb {R}^{n}$, then for any $N\ge 0$, there exist two functions $\phi _{0}, \phi \in \mathcal {D}({\mathbb {R}^{n}})$ such that $\phi$ has vanishing moments up to order N (i.e., $\int x^{\alpha }\phi (x)\mathrm{d}x=0$ for all multi-indices with $|\alpha |\le N$) and

$$\begin{aligned} f(x)=\sum \limits _{j\in \mathbb {N}}\phi _{j}*\varphi _{j}*f(x),\ \hbox { in}\ \mathcal {D}'({\mathbb {R}^{n}}), \end{aligned}$$

(1.2)

where $\mathcal {D}'({\mathbb {R}^{n}})$ is dual space of $\mathcal {D}({\mathbb {R}^{n}})$, $\mathbb {N}$ is the set of all natural numbers. Here and the following, we use dyadic dilations defined by $g_{j}(x)=2^{jn} g(2^jx)$ for $j\in \mathbb {N}, j\ge 1$, and $g_{j}(x)$ for $j=0$ is just the value of a function $g_{0}$. For any $j\in \mathbb Z$, denote $\Pi _{j}=\{Q:$ Q are dyadic cubes in $\mathbb R^{n}$ with the side length $l(Q)=2^{-j}$, and the left lower corners of Q are $x_{Q}=2^{-j}\ell$, $\ell \in \mathbb Z^{n}\}$, and $\Pi =\cup _{j\in \mathbb {N}}\Pi _{j}$. At last, denote $\bar{\Pi }=\{Q:$ Q are cubes in $\mathbb R^{n}\}$. Using continuous local Calderón’s identity (1.2), Rychkov proved that $f\in h^{p}(\mathbb {R}^{n})$ if and only if

$$\begin{aligned} \left\| \left( \sum _{j=0}^{\infty }|\varphi _{j}*f|^{2}\right) ^{1/2}\right\| _{L^{p}(\mathbb {R}^{n})}<\infty . \end{aligned}$$

(1.3)

Moreover, the continuous local Calderón’s identity (1.2) also holds in $\mathcal {S}({\mathbb {R}^{n}})$, $\mathcal {S}'({\mathbb {R}^{n}})$. We want to remark that using the fact pointed out in p160 of [31], the vanishing moments of $\varphi$ in (1.3) can be up to any fixed order.

Recently, the authors in [9] obtained the discrete Littlewood–Paley–Stein characterization of $h^{p}(\mathbb {R}^{n})$. More precisely, let $\psi _{0},\psi \in {\mathcal S}(\mathbb R^{n})$ with

$$\begin{aligned} \hbox {supp} \widehat{\psi _{0}}\subseteq \{\xi \in \mathbb R^{n}: |\xi |\le 2\};\ \widehat{\psi _{0}}(\xi )=1,\ \hbox {if}\ |\xi |\le 1, \end{aligned}$$

(1.4)

and

$$\begin{aligned} \hbox {supp} \widehat{\psi }\subseteq \{\xi \in \mathbb R^{n}: \frac{1}{2}\le |\xi |\le 2\}, \end{aligned}$$

(1.5)

and

$$\begin{aligned} |\widehat{\psi _{0}}(\xi )|^{2}+\sum \limits ^{\infty }_{j =1}|\widehat{\psi }(2^{-j}\xi )|^2=1, \ \hbox { for all } \xi \in \mathbb R^{n}, \end{aligned}$$

(1.6)

then one has the following continuous Calderón reproducing formula

$$\begin{aligned} \qquad f(x)= & {} \sum \limits _{j=0}^{+\infty } \psi _{j}*\psi _j*f(x), \end{aligned}$$

(1.7)

and the discrete Calderón reproducing formula is the following

$$\begin{aligned} \qquad f(x)= & {} \sum \limits _{j=0}^{+\infty } \sum \limits _{Q\in \Pi _{j}}|Q|(\psi _{j}*f)(x_{Q})\times \psi _{j}(x-x_{Q}), \end{aligned}$$

(1.8)

where the both series converge in $L^{2}(\mathbb R^{n})$, $\mathcal S (\mathbb R^{{n}})$ and $\mathcal S'(\mathbb R^{{n}})$ [13]. It is proved that in [9] $f\in h^{p}(\mathbb {R}^{n})$ if and only if

$$\begin{aligned} \left\| \Big (\sum _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j}}|\psi _{j}*f(x_{Q})|^2\chi _{Q}(x)\Big )^{\frac{1}{2}}\right\| _{L^p(\mathbb {R}^{n})}<+\infty , \end{aligned}$$

where $\psi _{0}$ and $\psi \in {\mathcal S}(\mathbb R^{n})$ are functions satisfying conditions (1.4)–(1.6). Moreover, they proved that $cmo^{p}(\mathbb {R}^{n})$ defined as following is also the dual space of $h^{p}(\mathbb {R}^{n})$.

Definition 1.1

Let $0<p\le 1$. Suppose that $\psi _{0}$ and $\psi \in {\mathcal S}(\mathbb R^{n})$ are functions satisfying conditions (1.4)–(1.6). $cmo^{p}(\mathbb {R}^{n})$ is defined by

$$\begin{aligned} cmo^{p}(\mathbb {R}^{n})=\{f\in \mathcal S'(\mathbb {R}^{n}):\ \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}<\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}= \sup \limits _{P\in \bar{\Pi }}\Big (\frac{1}{|P|^{\frac{2}{p}-1}} \int _{P}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}(|\psi _{j}*f(x_{Q})|^2 \chi _{Q}(x)) \mathrm{d}x\Big )^{1/2}.\nonumber \\ \end{aligned}$$

(1.9)

Obviously, $cmo^{p}(\mathbb {R}^{n})$ is coincident with $\Lambda _{n(\frac{1}{p}-1)}$ for $\frac{n}{n+1}<p\le 1$ since they are all the dual spaces of $h^{p}(\mathbb {R}^{n})$. A natural question is: can $cmo^{p}(\mathbb {R}^{n})$ be coincident with $\Lambda _{n(\frac{1}{p}-1)}$ for $\frac{n}{n+1}<p\le 1$ by norms? In this paper, we give a positive answer.

Theorem 1.1

Suppose that $\psi _{0}$ and $\psi \in {\mathcal S}(\mathbb R^{n})$ are functions satisfying conditions (1.4)–(1.6). Then for $\frac{n}{n+1}<p\le 1$, $f\in \Lambda _{n(\frac{1}{p}-1)}$ if and only if $f\in cmo^{p}(\mathbb {R}^{n})$. Moreover, $\Vert f\Vert _{\Lambda _{n(\frac{1}{p}-1)}}\approx \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}.$

Furthermore, we give another equivalent norm of $cmo^{p}(\mathbb {R}^{n})$ which has a very simple form.

Definition 1.2

Let $0<p\le 1$. Suppose that $\psi _{0}$ and $\psi \in {\mathcal S}(\mathbb R^{n})$ are functions satisfying conditions (1.4)–(1.6). $Lip^{p}(\mathbb {R}^{n})$ is defined by

$$\begin{aligned} Lip^{p}(\mathbb {R}^{n})=\{f\in \mathcal S'(\mathbb {R}^{n}):\ \Vert f\Vert _{Lip^{p}(\mathbb {R}^{n})}<\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{Lip^{p}(\mathbb {R}^{n})}= \sup \limits _{j\ge 0,x\in \mathbb {R}^n}2^{jn(\frac{1}{p}-1)}|\psi _{j}*f(x)|. \end{aligned}$$

(1.10)

We remark that using continuous local Calderón’s identity (1.2), the test functions in (1.10) can be replaced by those given in (1.1).

Theorem 1.2

Suppose that $\psi _{0}$ and $\psi \in {\mathcal S}(\mathbb R^{n})$ are functions satisfying conditions (1.4)–(1.6). Then for $0<p\le 1$, $f\in Lip^{p}(\mathbb {R}^{n})$ if and only if $f\in cmo^{p}(\mathbb {R}^{n})$. Moreover, $\Vert f\Vert _{Lip^{p}(\mathbb {R}^{n})}\approx \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}.$

Since $Lip^{p}(\mathbb {R}^{n})$ is decreasing as p increases, we have the following corollary.

Corollary 1.3

For $0<p\le 1$, $cmo^{p}(\mathbb {R}^{n})$ is increasing as p increases. Precisely, for $0<p_{1}<p_{2}\le 1$, we have

$$\begin{aligned} \Vert f\Vert _{cmo^{p_{2}}(\mathbb {R}^{n})}\le C\Vert f\Vert _{cmo^{p_{1}}(\mathbb {R}^{n})}, \end{aligned}$$

where C is a constant independent of f.

The organization of this paper is as follows. In Sect. 2 we establish the equivalence between $cmo^{p}(\mathbb {R}^{n})$ and $Lip^{p}(\mathbb {R}^{n})$ for $0<p\le 1$. The proof of Theorem 1.1 is presented in Sect. 3. For this, we prove that $Lip^{p}(\mathbb {R}^{n})$ identifies $\Lambda _{n(\frac{1}{p}-1)}$ with equivalent norms for $\frac{n}{n+1}<p<1$ and $\Lambda _{0}$ equals to $cmo^{1}(\mathbb {R}^{n})$ with equivalent norms, that is, Theorems 3.1 and 3.2, respectively. As an application of equivalent theorems, we discuss the boundedness of inhomogeneous para-product operators on $h^{p}(\mathbb {R}^{n})$ in last section.

Finally, we make some conventions. Throughout the paper, C denotes a positive constant that is independent of the main parameters involved, but whose value may vary from line to line. Constants with subscript, such as $C_{1}$, do not change in different occurrences. We denote $f\le Cg$ by $f\lesssim g$. If $f\lesssim g \lesssim f$, we write $f\approx g$.

2 Equivalence Between $cmo^{p}(\mathbb {R}^{n})$ and $Lip^{p}(\mathbb {R}^{n})$

In this section, we prove the equivalence between $cmo^{p}(\mathbb {R}^{n})$ and $Lip^{p}(\mathbb {R}^{n})$ for $0<p\le 1$, that is Theorem 1.2.

We first prove

$$\begin{aligned} \Vert f \Vert _{cmo^{p}(\mathbb {R}^{n})}\lesssim \Vert f \Vert _{Lip^{p}(\mathbb {R}^{n})}. \end{aligned}$$

(2.1)

Firstly for any fixed $P\in \bar{\Pi }$,

$$\begin{aligned}&\frac{1}{|P|^{\frac{2}{p}-1}} \int _{P}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}(|\psi _{j}*f(x_{Q})|^2 \chi _{Q}(x)) \mathrm{d}x \\&\quad =\frac{1}{|P|^{\frac{2}{p}-1}}\sum _{j\in N}\sum _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f(x_{Q})|^{2}. \end{aligned}$$

We consider two cases $|P|<1$ and $|P|\ge 1$. When $|P|<1$ and $P\in \bar{\Pi }$, there exists $j_{0}\in \mathbb {N}$ such that $2^{-j_{0}}\le \ell (P)<2^{-j_{0}+1}$. Then

$$\begin{aligned}&\frac{1}{|P|^{\frac{2}{p}-1}}\sum _{j\in N}\sum _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f(x_{Q})|^{2}\\&\quad \le \Vert f\Vert ^{2}_{Lip^{p}(\mathbb {R}^{n})}|P|^{1-\frac{2}{p}}\sum _{j\ge j_{0}}\sum _{Q\in \Pi _{j},Q\subseteq P} |Q|2^{-2jn(\frac{1}{p}-1)}:=A. \end{aligned}$$

Fix $\tau \in (0,1)$, then for any $p\in (0,1]$,

$$\begin{aligned} A\le & {} \Vert f\Vert _{Lip^{p}(\mathbb {R}^{n})}|P|^{2-\tau -\frac{2}{p}}\sum _{j\ge j_{0}}2^{-jn\tau }2^{-2jn(\frac{1}{p}-1)}\\\lesssim & {} \Vert f\Vert _{Lip^{p}(\mathbb {R}^{n})}|P|^{2-\tau -\frac{2}{p}}2^{-j_{0}n\tau }2^{-2j_{0}n(\frac{1}{p}-1)}\lesssim \Vert f\Vert ^{2}_{Lip^{p}(\mathbb {R}^{n})}, \end{aligned}$$

If $|P|\ge 1$,

$$\begin{aligned}&\frac{1}{|P|^{\frac{2}{p}-1}}\sum _{j\in \mathbb {N}}\sum _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f(x_{Q})|^{2}\\&\quad =\frac{1}{|P|^{\frac{2}{P}-1}}\sum _{j\in \mathbb {N}}\sum _{Q\in \Pi _{j},Q\subseteq P} |Q|(2^{jn(\frac{1}{p}-1)}|\psi _{j}*f(x_{Q})|)^{2}2^{-2jn(\frac{1}{p}-1)}\\&\quad \le \Vert f\Vert ^{2}_{Lip^{p}(\mathbb {R}^{n})}|P|^{1-\tau -\frac{2}{p}}\sum _{j\in \mathbb {N}}2^{-2jn\tau }2^{-2jn(\frac{1}{p}-1)} \lesssim \Vert f\Vert ^{2}_{Lip^{p}(\mathbb {R}^{n})}. \end{aligned}$$

Thus, we obtain (2.1).

Conversely, first we claim that

$$\begin{aligned} |\psi _{j'}*f(x)|\lesssim & {} \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}\Vert \psi _{j'}(x-\cdot )\Vert _{h^{p}(\mathbb {R}^{n})}. \end{aligned}$$

(2.2)

Then, if we obtain that

$$\begin{aligned} \Vert \psi _{j'}(x-\cdot )\Vert _{h^{p}(\mathbb {R}^{n})}\lesssim 2^{j'n(1-\frac{1}{p})}, \end{aligned}$$

(2.3)

one can complete the proof. Now we prove (2.2). For any fixed $x\in \mathbb {R}^{n}$, set $h(t)=\psi _{j'}(x-t)$. Define

$$\begin{aligned} S(h)(\cdot )=\big (\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j}}|(\psi _{j}*h)(x_{Q})|^{2}\chi _{Q}(\cdot )\big )^{1/2}. \end{aligned}$$

For any $i\in \mathbb {Z}$, let

$$\begin{aligned} \Omega _i =\left\{ x \in \mathbb R^{n}: S(h)(x)> 2^i\right\} ,\tilde{\Omega }_i=\left\{ x \in \mathbb R^{n}: M(\chi _{\Omega _i}) (x)> \frac{1}{(10)^{n}}\right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {B}_i=\left\{ Q: Q\in \cup _{j\ge 0}\Pi _{j}, \ |Q \cap \Omega _i| > \frac{1}{2} |Q|, \ \ |Q \cap \Omega _{i+1}| \le \frac{1}{2}|Q|\right\} . \end{aligned}$$

Then, $\cup _{Q\in \mathcal {B}_i}Q\subseteq \tilde{\Omega }_i$. By local reproducing formula (1.8), one has

$$\begin{aligned} \psi _{j'}*f(x)= & {} \sum _{j\in \mathbb {N}}\sum _{Q\in \Pi _{j}}\psi _{j}*f(x_{Q})\cdot \psi _{j}*\psi _{j'}(x-x_{Q})\\= & {} \sum _{Q\in \Pi }\psi _{Q}*f(x_{Q}) \psi _{Q}*\psi _{j'}(x-x_{Q})\\= & {} \sum ^{+\infty }_{i=-\infty }\sum _{Q^{*}\in B_{i}}\sum _{Q\subseteq Q^{*},Q\in B_{i}}|Q||\psi _{Q}*f(x_{Q})||\psi _{Q}*\psi _{j'}(x-x_{Q})|. \end{aligned}$$

in $L^{2}(\mathbb {R}^{n})$, where $Q^{*}$ are the maximal dyadic cubes in $\mathcal {B}_i$, and $\varphi _{j}$ are denoted by $\varphi _{Q}$.

Hence,

$$\begin{aligned} |\psi _{j'}*f(x)|\le & {} \big \{\sum ^{+\infty }_{i=-\infty }\sum _{Q^{*}\in B_{i}}|Q^{*}|^{1-\frac{p}{2}}\big (\sum _{Q\subseteq Q^{*},Q\in B_{i}} |Q|(\psi _{Q}*\psi _{j'}(x-x_{Q}))^{2}\big )^{\frac{p}{2}}\\&\cdot \big (\frac{1}{|Q^{*}|^{\frac{2}{p}-1}} \sum _{Q\subseteq Q^{*},Q\in B_{i}}|Q||\psi _{Q}*f(x_{Q})|^{2}\big )^{\frac{p}{2}}\big \}^{\frac{1}{p}}\\\le & {} \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}\big \{\sum ^{+\infty }_{i=-\infty }\sum _{Q^{*}\in B_{i}}|Q^{*}|^{1-\frac{p}{2}}\nonumber \\&\big (\sum _{Q\subseteq Q^{*},Q\in B_{i}}|Q||\psi _{Q}*\psi _{j'}(x-x_{Q})|^{2}\big )^{\frac{p}{2}}\big \}^{\frac{1}{p}}. \end{aligned}$$

Since $Q\in \mathcal {B}_i$, one has $|Q|\le 2|Q\cap \widetilde{\Omega }_{i}\backslash \Omega _{i+1}|$. Hence,

$$\begin{aligned} \sum _{Q\subseteq Q^{*},Q\in \mathcal {B}_i}|Q||(\varphi _{Q}*h)(x_{Q})|^{2}\le & {} 2\sum _{Q\in \mathcal {B}_i}|Q\cap \widetilde{\Omega }_{i}\backslash \Omega _{i+1}||(\varphi _{Q}*h)(x_{Q})|^{2}\\= & {} 2\int _{\widetilde{\Omega }_{i}\backslash \Omega _{i+1}}\sum _{Q\in \mathcal {B}_i}|(\varphi _{Q}*h)(x_{Q})|^{2}\chi _{Q}(x)dx\\\le & {} 2\int _{\widetilde{\Omega }_{i}\backslash \Omega _{i+1}}(S(h)(x))^{2}dx\lesssim 2^{2i}|\widetilde{\Omega }_{i}|, \end{aligned}$$

which implies that

$$\begin{aligned}&|\psi _{j'}*f(x)|\quad \lesssim \quad \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}\left\{ \sum \limits _{i=-\infty }^{+\infty }|\tilde{\Omega } _i|^{1-\frac{p}{2}}(2^{2i} |\tilde{\Omega }_i|)^{p/2}\right\} ^{\frac{1}{p}}\nonumber \\&\quad \lesssim \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})}\{\sum \limits _{i=-\infty }^{+\infty }2^{ip}|\Omega _{i}|\}^{\frac{1}{p}}\\&\quad \lesssim \quad \Vert f\Vert _{cmo^{p}(\mathbb {R}^{n})} \Vert h\Vert _{h^{p}(\mathbb {R}^{n})}. \end{aligned}$$

It is well known that

$$\begin{aligned} \Vert S(h)\Vert _{L^p(\mathbb {R}^{n})}\approx \Vert \big (\sum \limits _{j\in \mathbb {N}} |\psi _{j}*h(\cdot )|^{2}\big )^{1/2}\Vert _{L^{p}(\mathbb {R}^{n})}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert \psi _{j'}(x-\cdot )\Vert ^{p}_{h^{p}(\mathbb {R}^{n})}\approx & {} \int \big (\sum \limits _{j\in \mathbb {N}} |\psi _{j}*[\psi _{j'}(x-\cdot )](y)|^{2}\big )^{p/2}\mathrm{d}y\\= & {} \int \big (\sum \limits _{j\in \mathbb {N}} |\psi _{j}*\psi _{j'}(y)|^{2}\big )^{p/2}\mathrm{d}y\\\lesssim & {} \int \sum _{j\in \mathbb {N}}\frac{2^{(j\wedge j')np}2^{-|j-j'| Np}}{(1+2^{(j\wedge j')}|y|)^{Lp}}dy\\\lesssim & {} \sum _{j\in \mathbb {N}}2^{(j\wedge j')(np-n)}\cdot 2^{-|j-j'| Np}\\= & {} 2^{j'n(p-1)}\sum _{j\in \mathbb {N}}2^{(j\wedge j')n(p-1)}\cdot 2^{j'n(1-p)}\cdot 2^{-|j-j'| Np}\\= & {} 2^{j'n(p-1)}\sum _{j\in \mathbb {N}}2^{[(j'-j\wedge j')]n(1-p)}\cdot 2^{-|j-j'| Np}. \end{aligned}$$

Since$|j'-j\wedge j'|\le | j-j'|$, one has

$$\begin{aligned} \Vert \psi _{j'}(x-\cdot )\Vert ^{p}_{h^{p}(\mathbb {R}^{n})}\lesssim & {} 2^{j'n(p-1)}\sum _{j\in \mathbb {N}}2^{|j-j'|n(1-p)}\cdot 2^{-|j-j'| Np}\\= & {} 2^{j'n(p-1)}\sum _{j\in \mathbb {N}}2^{-|j-j'|(Np-n(1-p))}\\= & {} 2^{j'n(p-1)}\sum _{j\in \mathbb {N}}2^{-|j-j'|p(N-n(\frac{1}{p}-1))}. \end{aligned}$$

Choosing N big enough such that $N-n(\frac{1}{p}-1)>0$, one obtain (2.3).

Thus we complete the proof. $\square$

3 Equivalence Between $Lip^{p}(\mathbb {R}^{n})$ and $\Lambda _{r}$

Theorem 3.1

$f\in \Lambda _r, 0<r<1$ if and only if $f\in \mathcal S'(\mathbb {R}^{n})$ and $\sup _{j\in \mathbb N, x\in \mathbb R^n}2^{jr}|\psi _j*f(x)|<+\infty .$ Moreover, $\Vert f\Vert _{\Lambda _r}\approx \sup _{j\in \mathbb N, x\in \mathbb R^n}2^{jr}|\psi _j*f(x)|.$

Proof

Suppose that $f\in \Lambda _r, 0<r<1,$ that is, f is a bounded and continuous function. Then it is easy to check that $f\in \mathcal S'(\mathbb {R}^{n}), |\psi _0*f(x)|\le C \Vert f\Vert _{L^{\infty }(\mathbb {R}^{n})}$, and for $j\ge 1,|\psi _j*f(x)|=|\int \psi _j(x-y)f(y)\mathrm{d}y|=|\int \psi _j(x-y)[f(y)-f(x)]\mathrm{d}y|$ since $\int \psi _j(x)\mathrm{d}x=0.$ Thus, $|\psi _j*f(x)|\le C 2^{-rj}\Vert f\Vert _{\Lambda _r}.$ These estimates imply that $\sup _{j\in \mathbb N, x\in \mathbb R^n}2^{jr}|\psi _j*f(x)|\le C\Vert f\Vert _{\Lambda _r}.$ Conversely, if $f\in \mathcal S'(\mathbb R^{n}),$ by the continuous Calderón reproducing formula (1.7), $f(x)=\sum \limits _{j=0}^{\infty } \psi _{j}*\psi _j*f(x)$ in $\mathcal S'(\mathbb R^{n}).$ Note that if $\sup _{j\in \mathbb N, x\in \mathbb R^n}2^{jr}|\psi _j*f(x)|\le C,$ then $|\psi _{j}*\psi _j*f(x)|\le C2^{-rj}$ and hence, the series $\sum \limits _{j=0}^{\infty } \psi _{j}*\psi _j*f(x)$ converges uniformly. This implies that $\sum \limits _{j=0}^{\infty } \psi _{j}*\psi _j*f(x)$ is a bounded and continuous function, thus $f(x)=\sum \limits _{j=0}^{\infty } \psi _{j}*\psi _j*f(x)$ for all $x\in \mathbb R^n.$ To see that $f\in \Lambda _r, 0<r<1,$ we only need to show that for any given x and y with $|x-y|\le 1, |f(x)-f(y)|\le C|x-y|^r\sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)|.$ To do this, let $j_0\in \mathbb N$ such that $2^{-j_0}\le |x-y|<2^{1-j_0}.$ Split the series into two parts: $I(x)=\sum \limits _{j=0}^{j_0}\psi _{j}*\psi _j*f(x)$ and $II(x)=\sum \limits _{j=j_0+1}^{\infty }\psi _{j}*\psi _j*f(x).$ Write $I(x)-I(y)=\sum \limits _{j=0}^{j_0}\int [\psi _{j}(x-z)-\psi _j(y-z)]\psi _j*f(z)\mathrm{d}z.$ Applying the estimate that $\int |[\psi _{j}(x-z)-\psi _j(y-z)]\psi _j*f(z)\mathrm{d}z|\le C(2^j|x-y|)2^{-rj}\sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)|,$ we obtain that

$$\begin{aligned} |I(x)-I(y)|\le & {} C2^{j_0(1-r)}|x-y|\sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)|\\\le & {} C |x-y|^r\sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)| . \end{aligned}$$

Applying the size condition, it follows that

$$\begin{aligned} |II(x)-II(y)|\le & {} C2^{-rj_0}\sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)\\\le & {} C|x-y|^r \sup _{j\in \mathbb N, u\in \mathbb R^n}2^{jr}|\psi _j*f(u)|. \end{aligned}$$

Thus, we complete the proof. $\square$

Theorem 3.2

$f\in bmo(\mathbb R^n)$ if and only if $f\in cmo(\mathbb R^n).$ Moreover, $\Vert f\Vert _{bmo(\mathbb R^n)}\approx \Vert f\Vert _{cmo(\mathbb R^n)}.$

Note that, for convenience, here we denote $cmo(\mathbb {R}^{n})=cmo^{1}(\mathbb {R}^{n})$ .

Proof

Given any $f\in cmo(\mathbb {R}^{n})$, we now prove

$$\begin{aligned} \Vert f\Vert _{bmo(\mathbb {R}^{n})}\lesssim \Vert f\Vert _{cmo(\mathbb {R}^{n})}. \end{aligned}$$

To do this, define a linear functional on $h^{1}(\mathbb {R}^{n})$ by

$$\begin{aligned} \mathcal {L}_{f}(g)=\langle f, g\rangle \end{aligned}$$

for $g \in h^{1}(\mathbb {R}^{n}).$ By the above the duality argument, namely $(h^{1}(\mathbb {R}^{n}))^{*}=cmo(\mathbb {R}^{n}),$ we have

$$\begin{aligned} |\mathcal {L}_{f}(g)|\le \Vert f\Vert _{cmo(\mathbb {R}^{n})}\Vert g\Vert _{h^{1}(\mathbb {R}^{n})}. \end{aligned}$$

Now fix a $Q\in \bar{\Pi }$, and let $L^{2}_{Q}$ denote the space of all square integrable functions supported in Q. If $\ell (Q)\ge 1$, it is easy to see that each $g\in L^{2}_{Q}$ is a multiple of an (1, 2) atom of $h^{1}(\mathbb {R}^{n})$ with $\Vert g\Vert _{h^{1}(\mathbb {R}^{n})}\le C|Q|^{\frac{1}{2}}\Vert g\Vert _{L^{2}(\mathbb {R}^{n})}$. Hence, $\mathcal {L}_{f}$ is a linear functional on $L^{2}_{Q}$ with norm at most $C|Q|^{\frac{1}{2}}\Vert f\Vert _{cmo(\mathbb {R}^{n})}$. Then, the Riesz representation theorem for Hilbert spaces $L^{2}_{Q}$ tells us that there exists $F^{Q}\in L^{2}_{Q}$ such that

$$\begin{aligned} \mathcal {L}_{f}(g)=\langle f, g\rangle =\int _{Q}F^{Q}(x)g(x)\mathrm{d}x, \forall g\in L^{2}_{Q} \end{aligned}$$

with $\Vert F^{Q}\Vert _{L^{2}_{Q}}=\Vert \mathcal {L}_{f}\Vert \le C|Q|^{\frac{1}{2}}\Vert f\Vert _{cmo(\mathbb {R}^{n})},$ which yields that f must be a square integrable function on Q and $f=F^{Q}$ on Q. Therefore, for any cubes Q satisfying $\ell (Q)\ge 1$,

$$\begin{aligned} \left\{ \frac{1}{|Q|}\int _{Q}|f(x)|^{2}\mathrm{d}x\right\} ^{1/2} =\left\{ \frac{1}{|Q|}\int _{Q}|F^{Q}(x)|^{2}\mathrm{d}x\right\} ^{1/2}\le C\Vert f\Vert _{cmo(\mathbb {R}^{n})}, \end{aligned}$$

which implies that

$$\begin{aligned} \sup _{|Q|\ge 1}\frac{1}{|Q|}\int _{Q}|f(x)|\mathrm{d}x\le C\Vert f\Vert _{cmo(\mathbb {R}^{n})}. \end{aligned}$$

On the other hand, if $\ell (Q)< 1$, let $L^{2}_{Q,0}=\{f\in L_Q^2, \int f=0\}$. Recall that a function a(x) supported in a cube Q is said to be a (p, 2) atom of $h^{p}(\mathbb {R}^{n}), 0<p\le 1,$ if it satisfies a size condition $\Vert a\Vert _{L^{2}(\mathbb {R}^{n})}\le |Q|^{\frac{1}{2}-\frac{1}{p}}$ and a cancellation condition $\int x^{\alpha }a(x)\mathrm{d}x=0, \alpha \in \mathbb {N}^{n}, |\alpha |\le n(\frac{1}{p}-1)$, when $\ell (Q)<1$ [14]. Then, each $g\in L^{2}_{Q,0}$ is a multiple of an (1, 2) atom of $h^{1}(\mathbb {R}^{n})$ with $\Vert g\Vert _{h^{1}(\mathbb {R}^{n})}\le C|Q|^{\frac{1}{2}}\Vert g\Vert _{L^{2}(\mathbb {R}^{n})}$, and $\mathcal {L}_{f}$ is a linear functional on $L^{2}_{Q,0}$ with norm at most $C|Q|^{\frac{1}{2}}\Vert f\Vert _{cmo(\mathbb {R}^{n})}$. By the Riesz representation theorem, there exists some $F^{Q}\in L^{2}_{Q,0}$ such that

$$\begin{aligned} \mathcal {L}_{f}(g)=\langle f, g\rangle =\int _{Q}F^{Q}(x)g(x)\mathrm{d}x, \forall g\in L^{2}_{Q,0} \end{aligned}$$

with $\Vert F^{Q}\Vert _{L^{2}_{Q}}=\Vert \mathcal {L}_{f}\Vert \le C|Q|^{\frac{1}{2}}\Vert f\Vert _{cmo(\mathbb {R}^{n})},$ which yields that f must be a square integrable function on Q and $f=F^{Q}+c_{Q}$ on Q for some constant $c_{Q}$ since $\int g(x)\mathrm{d}x=0$. Therefore, for any cube Q satisfying $\ell (Q)< 1$,

$$\begin{aligned} \left\{ \frac{1}{|Q|}\int _{Q}|f(x)-c_{Q}|^{2}\mathrm{d}x\right\} ^{1/2} =\left\{ \frac{1}{|Q|}\int _{Q}|F^{Q}(x)|^{2}\mathrm{d}x\right\} ^{1/2}\le C\Vert f\Vert _{cmo(\mathbb {R}^{n})}, \end{aligned}$$

which implies that

$$\begin{aligned} \sup _{|Q|<1}\big \{\frac{1}{|Q|}\int _{Q}|f(x)-c_{Q}|^{2}\mathrm{d}x\big \}^{1/2}\le C\Vert f\Vert _{cmo(\mathbb {R}^{n})}. \end{aligned}$$

Hence

$$\begin{aligned} \sup _{|Q|<1}\big \{\frac{1}{|Q|}\int _{Q}|f(x)-f_{Q}|^{2}\mathrm{d}x\big \}^{1/2}\le C\Vert f\Vert _{cmo(\mathbb {R}^{n})}. \end{aligned}$$

The estimates above imply that $\Vert f\Vert _{bmo(\mathbb {R}^{n})}\lesssim \Vert f\Vert _{cmo(\mathbb {R}^{n})}.$

We now prove

$$\begin{aligned} \Vert f\Vert _{cmo(\mathbb {R}^{n})}\lesssim \Vert f\Vert _{bmo(\mathbb {R}^{n})}. \end{aligned}$$

By the definition of $bmo(\mathbb {R}^{n})$, it is easy to see that,

$$\begin{aligned} \sup _{Q}\frac{1}{|Q|}\int _{Q}|f(x)-f_{Q}|\mathrm{d}x\le 2\Vert f\Vert _{bmo(\mathbb {R}^{n})}, \end{aligned}$$

where $\sup$ is taken over all cubes with sides parallel to the coordinate axes in $\mathbb {R}^{n}$.

For any $P\in \bar{\Pi }$, let $P^{*}=3\sqrt{n}P$, the cube with the same center of P and side length $3\sqrt{n}\ell (P)$. Split f as $f=f_{1}+f_{2}+f_{3}$ with $f_{1}=(f-f_{P^{*}})\chi _{P^{*}}$, $f_{2}=(f-f_{P^{*}})\chi _{(P^{*})^{c}}$ and $f_{3}=f_{P^{*}}$. It is easy to see that

$$\begin{aligned}&\frac{1}{|P|} \int _{P}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}(|\psi _{j}*f(x_{Q})|^2 \chi _{Q}(x))\\&\quad \lesssim \sum _{i=1}^{3}\int _{P}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}(|\psi _{j}*f_{i}(x_{Q})|^2 \chi _{Q}(x)). \end{aligned}$$

For $f_{1}$, by Littlewood–Paley theory,

$$\begin{aligned}&\frac{1}{|P|} \int _{P}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}(|\psi _{j}*f_{1}(x_{Q})|^2 \chi _{Q}(x)) \mathrm{d}x\\&\quad =\frac{1}{|P|}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f_{1}(x_{Q})|^2\\&\quad \le \frac{1}{|P|}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j}}|Q||\psi _{j}*f_{1}(x_{Q})|^2\\&\quad \lesssim \frac{1}{|P|}\Vert f_{1}\Vert _{L^{2}(\mathbb {R}^{n})}\lesssim \frac{1}{|P|}\int _{P^{*}}|f(y)-f_{P^{*}}|^{2}\mathrm{d}y\lesssim \Vert f\Vert ^{2}_{bmo(\mathbb {R}^{n})}. \end{aligned}$$

For $f_{2}$, first of all, it is easy to see that $|\psi _{j}*f_{2}(x_{Q})| =|\int \psi _{j}(x_{Q}-y)f_{2}(y)dy|$ is dominated by

$$\begin{aligned} \int _{(P^{*})^{c}}\frac{2^{-j}}{(2^{-j}+|x_{Q}-y|)^{n+1}}|f_{2}(y)|\mathrm{d}y \approx \int _{(P^{*})^{c}}\frac{2^{-j}}{(2^{-j}+|x_{P}-y|)^{n+1}}|f_{2}(y)|\mathrm{d}y, \end{aligned}$$

where $x_{P}$ is the center of P. Hence

$$\begin{aligned}&|\psi _{j}*f_{2}(x_{Q})|\\&\quad \lesssim 2^{-j}\sum _{k=1}^{+\infty }\int _{2^{k-1}\ell (P)\le |x_{P}-y|<2^{k}\ell (P)} \frac{1}{(2^{-j}+|x_{P}-y|)^{n+1}}|f_{2}(y)|\mathrm{d}y\\&\quad \lesssim 2^{-j}\sum _{k=1}^{+\infty }\frac{1}{(2^{k}\ell (P))^{n+1}}\int _{|x_{P}-y|<2^{k}\ell (P)}|f_{2}(y)|\mathrm{d}y\\&\quad \lesssim 2^{-j}\sum _{k=1}^{+\infty }\frac{1}{(2^{k}\ell (P))^{n+1}}\left\{ \int _{|x_{P}-y|<2^{k}\ell (P)}|f(y)\right. \\&\left. \qquad -f_{2^{k}P^{*}}|\mathrm{d}y+(2^{k}\ell (P))^{n} |f_{2^{k}P^{*}}- f_{P^{*}}| \right\} , \end{aligned}$$

combining with classical result about $BMO(\mathbb {R}^{n})$ functions, gives

$$\begin{aligned} |\psi _{j}*f_{2}(x_{Q})|\lesssim 2^{-j}\ell (P)^{-1}\Vert f\Vert _{bmo(\mathbb {R}^{n})}. \end{aligned}$$

Note that there exists $j_{0}\in \mathbb {Z}$ such that $2^{-j_{0}}<\ell (P)\le 2^{-j_{0}+1}$. Then,

$$\begin{aligned}&\frac{1}{|P|}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f_{2}(x_{Q})|^2\\&\qquad \lesssim \frac{1}{|P|}\sum \limits _{j=j_{0}}^{\infty }\sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q|2^{-2(j-j_{0})}\Vert f\Vert ^{2}_{bmo(\mathbb {R}^{n})}\lesssim \Vert f\Vert ^{2}_{bmo(\mathbb {R}^{n})}. \end{aligned}$$

For the constant item $f_{3}$, note that $\int \psi _{0}=1$ and $\int \psi _{j}=0, j\ge 1$. Hence if $\ell (P)<1$,

$$\begin{aligned} \frac{1}{|P|}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f_{3}(x_{Q})|^2=\frac{1}{|P|}\sum \limits _{j\ge 1} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f_{3}(x_{Q})|^2=0. \end{aligned}$$

Otherwise, if $\ell (P)\ge 1$

$$\begin{aligned} \frac{1}{|P|}\sum \limits _{j\in \mathbb {N}} \sum \limits _{Q\in \Pi _{j},Q\subseteq P}|Q||\psi _{j}*f_{3}(x_{Q})|^2\le |f_{3}|^{2}\le (S_{2}(f))^{2}\le \Vert f\Vert ^{2}_{bmo(\mathbb {R}^{n})}, \end{aligned}$$

since $f_{3}=f_{P^{*}}$ with $\ell (P^{*})>1$.

Thus, we complete the proof. $\square$

We want to point out that the results of this section can also be seen in [8]. For completeness, we give their proofs.

4 Applications

In this section, we discuss the boundedness of inhomogeneous para-product operators on $h^{p}(\mathbb {R}^{n})$. Firstly, we recall some about non-convolution singular integral operators.

A locally integral function $\mathcal {K}(x,y)$ defined away from the diagonal $x=y$ in $\mathbb {R}^{n}\times \mathbb {R}^{n}$ is called a Calderón–Zygmund kernel with regularity exponent $\varepsilon >0$ if there exists a constant $C >0$ such that

$$\begin{aligned} |\mathcal {K}(x,y)|\le C\frac{1}{|x-y|^{n}},\quad \ \hbox {for}\quad \ x\ne y, \end{aligned}$$

(4.1)

and

$$\begin{aligned} |\mathcal {K}(x,y)-\mathcal {K}(x,y')|\le C \frac{|y-y'|^{\varepsilon }}{|x-y|^{n+\varepsilon }}, \end{aligned}$$

(4.2)

whenever $|y-y'|\le \frac{1}{2}|x-y|$, and

$$\begin{aligned} |\mathcal {K}(x,y)-\mathcal {K}(x',y)|\le C\frac{|x-x'|^{\varepsilon }}{|x-y|^{n+\varepsilon }}, \end{aligned}$$

(4.3)

whenever $|x-x'|\le \frac{1}{2}|x-y|$. The operator T is said to be non-convolution Calderón–Zygmund singular integral if T is a continuous linear operator from $\mathcal {D}(\mathbb {R}^{n})$ to $\mathcal {D}'(\mathbb {R}^{n})$ defined by

$$\begin{aligned} \langle Tf,g\rangle =\int \mathcal {K}(x,y)f(y)g(x)\mathrm{d}x\mathrm{d}y \end{aligned}$$

for all $f,g\in \mathcal {D}(\mathbb {R}^{n})$ with disjoint supports, where $\mathcal {K}$ is a Calderón–Zygmund kernel. The fundamental result for the third generation Calderón–Zygmund singular integrals is the Theorem T1 contained in [6]. See [7, 16, 17, 20,21,22, 30] for other versions of the T1 theorems for Hardy, Besov and Triebel–Lizorkin spaces. If the Calderón–Zygmund kernel $\mathcal {K}$ satisfies a restrictve size condition, namely, the condition (4.1) is replaced by

$$\begin{aligned} |\mathcal {K}(x,y)|\le C\min \{\frac{1}{|x-y|^{n}},\frac{1}{|x-y|^{n+\delta }}\},\ \hbox {for some }\delta >0 \ \hbox {and}\ \ x\ne y, \end{aligned}$$

(4.4)

we then obtain an inhomogeneous Calderón–Zygmund kernel associate with regularity exponent $\varepsilon , \delta > 0$, and inhomogeneous Calderón–Zygmund singular integral associate with regularity exponent $\varepsilon , \delta > 0$, respectively. It is well known that each pseudo-differential operator $T_{\sigma }f(x)=\int \sigma (x,\xi )e^{2\pi ix\xi }\hat{f}(\xi )d\xi$ with $\sigma \in S_{1,0}^{0}$ is an inhomogeneous Calderón–Zygmund singular integral. For the boundedness of operators on local Hardy spaces, we refer the readers to the work in [14, 26]

Using atomic decomposition, one has the following result.

Theorem 4.1

Suppose that T is an inhomogeneous Calderón–Zygmund singular integral associate with regularity exponent $\varepsilon , \delta > 0$. Then, if T is bounded on $L^{2}(\mathbb {R}^{n})$, T is bounded from $h^{p}(\mathbb {R}^{n})$ to $L^{p}(\mathbb {R}^{n})$ if $\max \{\frac{n}{n+\varepsilon },\frac{n}{n+\delta }\}<p\le 1$.

The proof of Theorem 4.1 is standard, and we refer the readers to [15] for non-convolution Calderón–Zygmund singular integral on $H^{p}(\mathbb {R}^{n})$, to [12] for Journé’s type of multi-parameter singular integral operators on multi-parameter Hardy spaces $H^{p}(\mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}})$, to [10] for inhomogeneous Journé’s type of multi-parameter singular integral operators on multi-parameter Hardy spaces $h^{p}(\mathbb {R}^{n_{1}} \times \mathbb {R}^{n_{2}})$, so we omit its proof. We want to remark that it is enough to prove Theorem 4.1 under conditions (4.4) and (4.2).

Now we begin to define inhomogeneous para-product operators. In this section, we always suppose that $\psi _{0},\psi , \varphi$ belong to $\mathcal {D}(\mathbb {R}^{n})$ supported on unit ball centered at origin satisfying that $\psi$ has vanishing moments up to some proper order N and $\int \psi _{0}(x)\mathrm{d}x=\int \varphi (x)\mathrm{d}x=1$. Set $\psi _{j}(x)=2^{jn} \psi (2^jx)$ for $j\in \mathbb {N}, j\ge 1$, and $\varphi _{j}(x)=2^{jn} \varphi (2^jx)$ for $j\in \mathbb {N}$.

By Corollary 1.3, $bmo(\mathbb {R}^{n})\supseteq cmo^{p}(\mathbb {R}^{n})$ for all $0<p\le 1$. Fixed any $b\in bmo(\mathbb {R}^{n})$, inhomogeneous para-product operators are defined as following:

$$\begin{aligned} \pi _{b}(f)(x)=\sum _{j\in N}\sum _{Q\in \Pi _{j}}|Q|\psi _{j}*b(x_{Q})\cdot \psi _{j}(x-x_{Q})\varphi _{j}*f (x_{Q}), \end{aligned}$$

and its adjoint operator

$$\begin{aligned} \pi _{b}^{*}(f)(x)= \sum \limits _{k\in N} \sum \limits _{Q\in \Pi _{k}}|Q|\varphi _{k}(x-x_{Q})\psi _{k}*b(x_{Q})\cdot \psi _{k}*f(x_{Q}). \end{aligned}$$

One can check that $\pi _{b},\pi _{b}^{*}$ are inhomogeneous Calderón–Zygmund singular integrals associate with regularity exponent $\varepsilon =\delta =1$ and bounded on $L^{2}(\mathbb {R}^{n})$. Hence $\pi _{b},\pi _{b}^{*}$ are bounded from $h^{p}(\mathbb {R}^{n})$ to $L^{p}(\mathbb {R}^{n})$ if $\frac{n}{n+1}<p\le 1$. Now we give the main result of this section.

Theorem 4.2

Suppose that $b\in Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})$. Then, $\pi _{b},\pi _{b}^{*}$ are bounded on $h^{p}(\mathbb {R}^{n})$ if $\frac{n}{n+1}<p\le 1$.

To prove Theorem 4.2, we will follow the approach in [23, 24] by reducing the $h^{p}(\mathbb {R}^{n})$ boundedness of $\pi _{b}$ and $\pi _{b}^{*}$ to $h^{p}(\mathbb {R}^{n})\rightarrow L^{p}(\mathbb {R}^{n})$ boundedness.

Since the proofs of the $h^{p}(\mathbb {R}^{n})$ boundedness of $\pi _{b}^{*}$ is similar as $\pi _{b}'s$ but more difficult, we only give the proof of $\pi _{b}^{*}$. Obviously, $L^{2}(\mathbb {R}^{n})\cap h^{p}(\mathbb {R}^{n})$ is dense in $h^{p}(\mathbb {R}^{n})$ since $\mathcal {S}(\mathbb {R}^{n})$ is dense in $h^{p}(\mathbb {R}^{n})$. For $f\in L^{2}(\mathbb {R}^{n})\cap h^{p}(\mathbb {R}^{n})$, using (1.3), one has

$$\begin{aligned} \Vert \pi _{b}^{*}(f)\Vert _{h^{p}(\mathbb R^{n})}=\left\| \left\{ \sum \limits _{k\in \mathbb {N }}|\psi _{k}*\pi _{b}^{*}(f)(x)|^{2}\right\} ^{\frac{1}{2}}\right\| _{L^{p}(\mathbb R^{n})}. \end{aligned}$$

It is easy to see that $\{\sum \limits _{k\in \mathbb {N }}|\psi _{k}*\pi _{b}^{*}(f)(x)|^{2}\}^{\frac{1}{2}}$ is bounded on $L^{2}(\mathbb {R}^{n})$, since $b\in Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})\subseteq bmo(\mathbb {R}^{n})$. Moreover,

$$\begin{aligned}&\psi _{k}*\pi _{b}^{*}(f)(x)=\int \psi _{k}(x-z)\pi _{b}^{*}(f)(z)\mathrm{d}z\\&\quad =\int \psi _{k}(x-z)\sum \limits _{k'\in \mathbb {N}} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}*f(x_{Q'})\mathrm{d}z\\&\quad =\iint \sum \limits _{k'\in \mathbb {N}} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\psi _{k}(x-z)\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)f(y)\mathrm{d}z\mathrm{d}y. \end{aligned}$$

Set

$$\begin{aligned} S_{k}(x,y)=\int \sum \limits _{k'\in \mathbb {N}} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\psi _{k}(x-z)\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)\mathrm{d}z. \end{aligned}$$

Then, the $h^{p}(\mathbb {R}^{n})$ boundedness of $\pi _{b}^{*}$ can be reduced to $h^{p}(\mathbb {R}^{n})\rightarrow L^{p}(\mathbb {R}^{n})$ boundedness of vector value inhomogeneous Calderón–Zygmund singular integral with kernel $\{S_{k}(x,y)\}$. We will finish the proof of Theorem 4.2 by proving that $\{S_{k}(x,y)\}$ satisfies vector value inhomogeneous Calderón–Zygmund kernel conditions, namely,

Lemma 4.3

Let $S_{k}(x,y)=\int \sum \limits _{k'\in \mathbb {N}} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\psi _{k}(x-z)\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)dz,$ and suppose that $b\in Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})$. Then for every $\varepsilon \in (0,1)$,

• (i) $\{\sum \limits _{k\in \mathbb {N}}|S_{k}(x,y)|^{2}\}^{1/2}\le C\min \{\frac{1}{|x-y|^{n}},\frac{1}{|x-y|^{n+1}}\}$ if $|x-y|>0;$

• (ii) $\{\sum \limits _{k\in \mathbb {N}}|S_{k}(x,y')-S_{k}(x,y)|^{2}\}^{1/2}\le C \frac{|y-y'|^{\varepsilon }}{|x-y|^{n+\varepsilon }}$, if $|y-y'|\le \frac{1}{2}|x-y|.$

Proof

Split $S_{k}(x,y)$ as following

$$\begin{aligned}S_{k}(x,y)= & {} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\int \psi _{k}(x-z)\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)\mathrm{d}z\\&+ \sum \limits _{k'<k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\int \psi _{k}(x-z)\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)\mathrm{d}z\\&=S^{1}_{k}(x,y)+S^{2}_{k}(x,y). \end{aligned}$$

We first prove size condition (i). For $S^{1}_{k}$, one has

$$\begin{aligned} S^{1}_{k}= & {} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\int [\psi _{k}(x-z-x_{Q'})\\&-\psi _{k}(x-x_{Q'})]\varphi _{k'}(z)\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)\mathrm{d}z\\&+\sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\psi _{k}(x-x_{Q'})\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)=A+B. \end{aligned}$$

We first show that

$$\begin{aligned} |B|\lesssim \frac{2^{-k}}{(2^{-k}+|x-y|)^{n+1}}. \end{aligned}$$

(4.5)

If $|x-y|\le 2\cdot 2^{-k}$,

$$\begin{aligned} |B|\le & {} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'||\psi _{k}(x-x_{Q'})||\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|\nonumber \\\le & {} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\frac{2^{-k\varepsilon '}}{(2^{-k}+|x-x_{Q'}|)^{n+\varepsilon '}}|\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|\nonumber \\\le & {} \frac{2^{-k}}{2^{-k(n+1)}}\sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'||\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|\nonumber \\\lesssim & {} \frac{2^{-k}}{(2^{-k}+|x-y|)^{n+1}}\sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'||\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|. \end{aligned}$$

(4.6)

When $|x-y|> 2\cdot 2^{-k}$. Note that $|x_{Q'}-y|\le 2^{-k'}\le 2^{-k}$. Hence, $|x-x_{Q'}|\ge |x-y|-|y-x_{Q'}|\ge \frac{1}{2}|x-y|,$ which also yields (4.6). Thus, (4.5) is obtained if we can prove

$$\begin{aligned} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'||\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|\le C. \end{aligned}$$

Indeed, using $b\in Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})$, one has

$$\begin{aligned}&\sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'||\psi _{k'}*b(x_{Q'})||\psi _{k'}(x_{Q'}-y)|\\&\qquad \le \Vert b\Vert _{Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})}\sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}} |Q'|2^{k'(1-\frac{n+1}{n})}|\psi _{k'}(x_{Q'}-y)|\\&\qquad \lesssim \sum \limits _{k'\ge 0}2^{k'(1-\frac{n+1}{n})}\sum \limits _{Q'\in \Pi _{k'}}\int _{Q'}\frac{2^{-k'}}{(2^{-k'}+|x_{Q'}-y|)^{n_1}}dz\\&\qquad \lesssim \int _{Q'}\frac{2^{-k'}}{(2^{-k'}+|z-y|)^{n_1}}dz\le C, \end{aligned}$$

since $2^{-k'}+|x_{Q'}-y|\approx 2^{-k'}+|z-y|$ for any $z\in Q'$.

For A, firstly, by the fact that $\psi _{k'}*b(x_{Q'})$ is bounded uniformly, one has

$$\begin{aligned} |A|\lesssim & {} \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\int |\psi _{k}(x-z-x_{Q'})-\psi _{k}(x-x_{Q'})||\varphi _{k'}(z)||\psi _{k'}(x_{Q'}-y)|\mathrm{d}z. \end{aligned}$$

Hence, by the following classical result, for any $\varepsilon \in (0,1)$,

$$\begin{aligned}&|\psi _{k}(x-z-x_{Q'})-\psi _{k}(x-x_{Q'})|\\&\qquad \lesssim \left( \frac{|z|}{2^{-k}}\right) ^{\varepsilon }\left( \frac{2^{-k}}{(2^{-k}+|x-z-x_{Q'}|)^{n+1}}+\frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}}\right) , \end{aligned}$$

and using the fact that the support of $\psi$ is unit ball at origin, A is dominated by

$$\begin{aligned}&\sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}}|Q'|\\&\qquad \int \left( \frac{2^{-k'}}{2^{-k}}\right) ^{\varepsilon } \left( \frac{2^{-k}}{(2^{-k}+|x-z-x_{Q'}|)^{n+1}}+\frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}}\right) \\&\qquad \frac{2^{-k'}}{(2^{-k'}+|z|)^{n+1}}|\psi _{k'}(x_{Q'}-y)|\mathrm{d}z\\&\quad =\sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}}|Q'|2^{-(k'-k)\varepsilon }|\psi _{k'}(x_{Q'}-y)|\\&\qquad \int \frac{2^{-k}}{(2^{-k}+|x-z-x_{Q'}|)^{n+1}} \frac{2^{-k'}}{(2^{-k'}+|z|)^{n+1}}\mathrm{d}z\\&\qquad +\sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}} |Q'|2^{-(k'-k)\varepsilon }\frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}}| \psi _{k'}(x_{Q'}-y)|\\&\qquad \int \frac{2^{-k'}}{(2^{-k'}+|z|)^{n+1}}\mathrm{d}z\\&\quad \lesssim \sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}}|Q'|2^{-(k'-k)\varepsilon }|\psi _{k'}(x_{Q'}-y)| \frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}}\\&\quad \lesssim \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|2^{-(k'-k)\varepsilon } \frac{2^{-k'}}{(2^{-k'}+|y-x_{Q'}|)^{n+1}}\frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}}\\&\quad =\sum \limits _{k'\ge k}\sum \limits _{Q'\in \Pi _{k'}} 2^{-(k'-k)\varepsilon '}\int _{Q'}\frac{2^{-k'(1-\varepsilon ')}}{(2^{-k'}+|y-x_{Q'}|)^{n+1-\varepsilon '}}\frac{2^{-k(1-\varepsilon ')}}{(2^{-k}+|x-x_{Q'}|)^{n+1-\varepsilon '}}\mathrm{d}z\\&\thickapprox \sum \limits _{k'\ge k} \sum \limits _{Q'\in \Pi _{k'}}2^{-(k'-k)\varepsilon '}\int _{Q'}\frac{2^{-k'(1-\varepsilon ')}}{(2^{-k'}+|y-z|)^{n+1-\varepsilon '}}\frac{2^{-k(1-\varepsilon ')}}{(2^{-k}+|x-z|)^{n+1-\varepsilon '}}\mathrm{d}z, \end{aligned}$$

since $2^{-k'}+|y-x_{Q'}|\thickapprox 2^{-k'}+|y-z|,2^{-k}+|x-x_{Q'}|\thickapprox 2^{-k}+|x-z|$ if $z\in Q'$. Then

$$\begin{aligned} |A|\lesssim & {} \sum \limits _{k'\ge k} 2^{-(k'-k)\varepsilon '}\frac{2^{-k}}{(2^{-k}+|x-y|)^{n+1}}\lesssim \frac{2^{-k}}{(2^{-k}+|x-y|)^{n+1}}. \end{aligned}$$

(4.7)

By estimates (4.5) and (4.7), one has

$$\begin{aligned} |S^{1}_{k}(x,y)| \lesssim \frac{2^{-k}}{(2^{-k}+|x-y|)^{n+1}}. \end{aligned}$$

(4.8)

For $S^{2}_{k}$, using the cancelation condition of $\psi$, one has

$$\begin{aligned}&S^{2}_{k}(x,y)\\&\quad =\sum \limits _{k'<k} \sum \limits _{Q'\in \Pi _{k'}}|Q'|\int \psi _{k}(x-z)[\varphi _{k'}(z-x_{Q'})\\&\qquad -\varphi _{k'}(x-x_{Q'})]\psi _{k'}*b(x_{Q'})\psi _{k'}(x_{Q'}-y)\mathrm{d}z. \end{aligned}$$

Then with a similar process to estimate A, one can see that (4.8) also holds for $S^{2}_{k}$. Thus, we can complete the proof of size condition (i).

To prove (ii), we only estimate the smoothness of $S^{1}_{k}(x,y)$ with the second variable y since the proof to obtain the smoothness of $S^{2}_{k}(x,y)$ is similar and easier. Set

$$\begin{aligned}&S_{k}^{1}(x,y)-S_{k}^{1}(x,y')\\&\quad =\int \sum _{k'\ge k}\sum _{Q'\in \Pi _{k'}}|Q'|\psi _{k}(x-z)\psi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'}) [\psi _{k'}(x_{Q'}-y)]\\&\qquad -\psi _{k'}(x_{Q'}-y')]\mathrm{d}z\\&\quad =\int \sum _{k'\ge k}\sum _{Q'\in \Pi _{k'}}|Q'|[\psi _{k}(x-z)-\psi _{k}(x-x_{Q'})]\varphi _{k'}(z-x_{Q'})\psi _{k'}*b(x_{Q'})\\&\qquad \cdot [\psi _{k'}(x_{Q'}-y)-\psi _{k'}(x_{Q'}-y')]dz\\&\qquad +\sum _{k'\ge k}\sum _{Q'\in \Pi _{k'}}|Q'| \psi _{k}(x-x_{Q'})\psi _{k'}*b(x_{Q'})\left[ \psi _{k'}(x_{Q'}-y)-\psi _{k'}(x_{Q'}-y')\right] \\&\quad =A^{1}_{k}(x,y)+A^{2}_{k}(x,y). \end{aligned}$$

For any $\varepsilon \in (0,1)$, using the support condition, it is easy to have

$$\begin{aligned}&|A^{1}_{k}(x,y)|\\&\quad \lesssim \int _{| z-x_{Q'}|\le 2^{-k'}}\sum _{k'\ge k}\sum _{Q'\in \Pi _{k'}}\mid Q'\mid \left( \frac{| z-x_{Q'}|}{2^{-k}}\right) ^{\varepsilon }\left[ \frac{2^{-k}}{(2^{-k}+\mid x-z\mid )^{n+1}}\right. \\&\left. \qquad +\frac{2^{-k}}{(2^{-k}+\mid x-x_{Q'}\mid )^{n+1}}\right] \frac{2^{-k'}}{(2^{-k'}+\mid z-x_{Q'}\mid )^{n+1}}\mid \left[ \psi _{k'}(x_{Q'}-y)\right. \\&\left. \qquad -\psi _{k'}(x_{Q'}-y')\right] \mid \mathrm{d}z, \end{aligned}$$

since $\mid \psi _{k'}*b(x_{Q'})\mid \le C$, uniformly for $k'$ and $x_{Q'}$. Then,

$$\begin{aligned} |A^{1}_{k}(x,y)|\lesssim & {} \sum _{k'\ge k}2^{(k-k')\varepsilon }\sum _{Q'\in \Pi _{k'}}|Q'|\frac{2^{-k}}{(2^{-k}+| x-x_{Q'}|)^{n+1}}\left( \frac{| y-y'|}{2^{-k'}}\right) ^{\varepsilon '}\\&\cdot \left[ \frac{2^{-k'}}{(2^{-k'}+|x_{Q'}-y|)^{n+1}}+\frac{2^{-k'}}{(2^{-k'}+| x_{Q'}-y'|)^{n+1}}\right] \\\lesssim & {} \sum _{k'\ge k}2^{(k-k')\varepsilon }\int \frac{2^{-k}}{(2^{-k}+| x-u|)^{n+1}}\left( \frac{| y-y'|}{2^{-k'}}\right) ^{\varepsilon '}\\&\cdot \left[ \frac{2^{-k'}}{(2^{-k'}+| u-y|)^{n+1}}+\frac{2^{-k'}}{(2^{-k'}+| u-y'|)^{n+1}}\right] \mathrm{d}u\\\lesssim & {} \sum _{k'\ge k}2^{(k-k')\varepsilon }2^{(k'-k)\varepsilon '}(\frac{| y-y'|}{2^{-k}})^{\varepsilon '}\left[ \frac{2^{-k}}{(2^{-k}+| x-y|)^{n+1}}\right. \\&\left. +\frac{2^{-k}}{(2^{-k}+| x-y'|)^{n+1}}\right] \\\lesssim & {} \left( \frac{| y-y'|}{2^{-k}}\right) ^{\varepsilon '}\left[ \frac{2^{-k}}{(2^{-k}+| x-y|)^{n+1}}+\frac{2^{-k}}{(2^{-k}+| x-y'|)^{n+1}}\right] \end{aligned}$$

provided $\varepsilon >\varepsilon '$. Hence, for any $\varepsilon '\in (0,1)$, when $|y-y'|\le \frac{1}{2}|y-x|$,

$$\begin{aligned} \sum _{k\in \mathbb {N}}|A^{1}_{k}(x,y)|\lesssim & {} \sum _{k\in \mathbb {N}}\left( \frac{\mid y-y'\mid }{2^{-k}}\right) ^{\varepsilon '}\left[ \frac{2^{-k}}{(2^{-k}+\mid x-y\mid )^{n+1}}+\frac{2^{-k}}{(2^{-k}+| x-y'|)^{n+1}}\right] \\\lesssim & {} \frac{|y-y'|^{\varepsilon '}}{|x-y|^{n+\varepsilon '}}. \end{aligned}$$

At last, for any $\varepsilon \in (0,1)$,

$$\begin{aligned} |A^{2}_{k}(x,y)|&\lesssim \sum _{k'\ge k}\sum _{Q'\in \Pi _{k'}}|Q'|\frac{2^{-k}}{(2^{-k}+|x-x_{Q'}|)^{n+1}} |\psi _{k'}*b(x_{Q'})|\left( \frac{| y-y'|}{2^{-k'}}\right) ^{\varepsilon }\\&\qquad \cdot \left[ \frac{2^{-k'}}{(2^{-k'}+| x_{Q'}-y|)^{n+1}}+\frac{2^{-k'}}{\left( 2^{-k'}+| x_{Q'}-y'|\right) ^{n+1}}\right] \\&\quad \lesssim \sum _{k'> k}\sum _{Q'\in \Pi _{k'}}\int _{Q'}\frac{2^{-k}}{(2^{-k}+|x-u|)^{n+1}} |\psi _{k'}*b(x_{Q'})|(\frac{| y-y'|}{2^{-k'}})^{\varepsilon }\\&\qquad \cdot \left[ \frac{2^{-k'}}{(2^{-k'}+| u-y|)^{n+1}}+\frac{2^{-k'}}{(2^{-k'}+| u-y'|)^{n+1}}\right] du\\&\quad \lesssim \Vert b\Vert _{Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})}\sum _{k'> k}2^{-k'}\sum _{Q'\in \Pi _{k'}}\int _{Q'}\frac{2^{-k}}{(2^{-k}+|x-u|)^{n+1}} \left( \frac{| y-y'|}{2^{-k'}}\right) ^{\varepsilon }\\&\qquad \cdot \left[ \frac{2^{-k'}}{(2^{-k'}+| u-y|)^{n+1}}+\frac{2^{-k'}}{(2^{-k'}+| u-y'|)^{n+1}}\right] \mathrm{d}u\\&\quad \lesssim \Vert b\Vert _{Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})}\sum _{k'> k}2^{-k'}(\frac{| y-y'|}{2^{-k'}})^{\varepsilon }\frac{2^{-k}}{(2^{-k}+| x-y|)^{n+1}}\\&\quad \lesssim \Vert b\Vert _{Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})}\frac{2^{-k}}{(2^{-k}+| x-y|)^{n+1}}|y-y'|^{\varepsilon }\sum _{k'> k}2^{-k'(1-\varepsilon )}\\&\quad \lesssim \Vert b\Vert _{Lip^{\frac{n}{n+1}}(\mathbb {R}^{n})}\frac{2^{-k}}{(2^{-k}+| x-y|)^{n+1}}|y-y'|^{\varepsilon }. \end{aligned}$$

Thus, for any $\varepsilon \in (0,1)$,

$$\begin{aligned} \sum _{k\in \mathbb {N}}|A^{2}_{k}(x,y)|\lesssim & {} \frac{|y-y'|^{\varepsilon }}{|x-y|^{n+\varepsilon }}. \end{aligned}$$

Then, we complete the proof. $\square$