Discrete para-product operators on variable Hardy spaces

TL;DR

This paper establishes the boundedness of para-product operators and Calderón-Zygmund operators on variable Hardy spaces with variable exponents, extending classical results using discrete analysis techniques.

Contribution

It introduces new boundedness results for para-product and Calderón-Zygmund operators on variable Hardy spaces, employing discrete Calderón reproducing formulas and Littlewood-Paley theory.

Findings

Boundedness of para-product operators on variable Hardy spaces.

Characterization of bounded Calderón-Zygmund operators via $T^*1=0$.

Results extend to spaces of homogeneous type.

Abstract

Let $p(\cdot):\mathbb R^n\rightarrow(0,\infty)$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this paper, we obtain the boundedness of para-product operators $\pi_b$ on variable Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$, where $b\in BMO(\mathbb R^n)$. As an application, we show that non-convolution type Calder\'on-Zygmund operators $T$ are bounded on $H^{p(\cdot)}(\mathbb R^n)$ if and only if $T^\ast1=0$, where $\frac{n}{n+\epsilon}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $\epsilon$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calder\'on's reproducing formula, discrete Littlewood-Paley-Stein theory and almost orthogonal estimates. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

Equations

$$\Vert f{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}=\inf \left\{\lambda >0:{\int }_{{\mathbb{R}}^n}{\left(\frac{|f(x)|}{\lambda }\right)}^{p(x)}dx\le 1\right\}.$$

$$|p(x)-p(y)|\le \frac{C}{-\log (|x-y|)},|x-y|\le 1/2$$

$$|p(x)-p(y)|\le \frac{C}{\log |x|+e},|y|\ge |x|.$$

$$\Vert f{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}=\Vert \mathcal{M}f{\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}<\infty .$$

$$M_{\varphi }(f)(x)=\underset{k}{\sup }|{\varphi }_k\ast f(x)|.$$

$$\Vert f{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}\sim \Vert M_{\varphi }(f){\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}.$$

$$\Vert b{\Vert }_{BMO({\mathbb{R}}^n)}=\underset{Q}{\sup }\frac{1}{|Q|}{\int }_Q|f(x)-f_Q|dx<\infty ,$$

$$\sum _{j\in \mathbb{Z}}|\hat{\phi }(2^{-j}\xi ){|}^2=1$$

$${\pi }_b(f)(x)=\sum _k\sum _Q|Q|{\phi }_k(x-x_Q){\tilde{\phi }}_k\ast b(x_Q){\varphi }_k\ast f(x_Q)$$

$${\pi }_b^{\ast }(f)(x)=\sum _k\sum _Q|Q|{\phi }_k\ast f(x_Q){\tilde{\phi }}_k\ast b(x_Q){\varphi }_k(x_Q-x),$$

$$\big{<}Tf,g\big{>}=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K(x,y)f(y)g(x)dydx,$$

$$|K(x,y)|\le \frac{C}{|x-y{|}^n},$$

$$|K(x,y)-K(x^{\prime },y)|\le C\frac{|x-x^{\prime }{|}^{ϵ}}{|x-y{|}^{n+ϵ}},\text{mbox}if|x-x^{\prime }|\le \frac{1}{2}|x-y|,$$

$$|K(x,y)-K(x,y^{\prime })|\le C\frac{|y-y^{\prime }{|}^{ϵ}}{|x-y{|}^{n+ϵ}},\text{mbox}if|y-y^{\prime }|\le \frac{1}{2}|x-y|.$$

$$\big{<}T1,\eta\big{>}=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K(x,y)\eta(x)dydx,$$

$$\big{<}T^{\ast}1,\eta\big{>}=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}K(x,y)\eta(y)dydx.$$

$$|{\partial }_x^{\alpha }K(x,y)|+|{\partial }_y^{\alpha }K(x,y)|\le \frac{C}{|x-y{|}^{n+|\alpha |}},\text{mbox}if0\le |\alpha |\le N,$$

$$|{\partial }_x^{\alpha }K(x,y)-{\partial }_y^{\alpha }K(x^{\prime },y)|\le C\frac{|x-x^{\prime }{|}^{ϵ}}{|x-y{|}^{n+N+ϵ}},\text{mbox}if|\alpha |=N,|x-x^{\prime }|\le \frac{1}{2}|x-y|,$$

$$|{\partial }_x^{\alpha }K(x,y)-{\partial }_y^{\alpha }K(x,y^{\prime })|\le C\frac{|y-y^{\prime }{|}^{ϵ}}{|x-y{|}^{n+N+ϵ}},\text{mbox}if|\alpha |=N,|y-y^{\prime }|\le \frac{1}{2}|x-y|,$$

$$\mathcal{A}(\{{\lambda }_j{\}}_{j=1}^{\infty },\{Q_j{\}}_{j=1}^{\infty })={\Vert {\left\{\sum _j{\left(\frac{|{\lambda }_j|{\chi }_{Q_j}}{\Vert {\chi }_{Q_j}{\Vert }_{L^{p(\cdot )}}}\right)}^{p^{-}}\right\}}^{\frac{1}{p^{-}}}\Vert }_{L^{p(\cdot )}}.$$

$$\mathcal{A}(\{{\lambda }_j{\}}_{j=1}^{\infty },\{Q_j{\}}_{j=1}^{\infty })\le C\Vert f{\Vert }_{H^{p(\cdot )}},$$

$$\mathcal{A}(\{{\lambda }_j{\}}_{j=1}^{\infty },\{Q_j{\}}_{j=1}^{\infty })<\infty ,$$

$$\Vert f{\Vert }_{H^{p(\cdot )}}\le C\mathcal{A}(\{{\lambda }_j{\}}_{j=1}^{\infty },\{Q_j{\}}_{j=1}^{\infty }).$$

$$\Vert \Vert \mathbb{M}(f){\Vert }_{l^q}{\Vert }_{L^{p(\cdot )}}\le C\Vert \Vert f|{|}_{l^q}{\Vert }_{L^{p(\cdot )}},$$

$$\sum _{j\in \mathbb{Z}}|\hat{\phi }(2^{-j}\xi ){|}^2=1$$

$$\Vert f{\Vert }_{L^q({\mathbb{R}}^n)}\sim \Vert h{\Vert }_{L^q({\mathbb{R}}^n)}\text{mbox}and\Vert f{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}\sim \Vert h{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}$$

$$f(x)$$

$$=\sum _{j\in \mathbb{Z}}\sum _Q|Q|{\phi }_j\ast h(x_Q){\phi }_j(x-x_Q),$$

$$\Vert f{\Vert }_{H^{p(\cdot )}({\mathbb{R}}^n)}\sim {\Vert {\left(\sum _{j\in \mathbb{Z}}\sum _Q|{\phi }_j\ast h(x_Q){|}^2{\chi }_Q\right)}^{1/2}\Vert }_{L^{p(\cdot )}({\mathbb{R}}^n)}.$$

$${\Omega }_{\ell }=\{x\in {\mathbb{R}}^n:M_{\varphi }(f)(x)>2^{\ell }\}$$

$$B_{\ell }=\left\{Q\text{mbox}isadyadiccube:|Q\cap {\Omega }_{\ell }|>\frac{1}{2}|Q|,|Q\cap {\Omega }_{\ell +1}|\le \frac{1}{2}|Q|\right\}.$$

$${\tilde{\Omega }}_{\ell }=\{x\in {\mathbb{R}}^n:M({\chi }_{{\Omega }_{\ell }})(x)>\frac{1}{1000}\}.$$

$${\pi }_b(f)(x)$$

Full text

1. Introduction and statements of results

The real-variable theory of Hardy spaces $H^{p}(\mathbb{R}^{n})$ was initiated by Stein and Weiss [2] and systematically developed by Fefferman and Stein in [11]. The Hardy space $H^{p}(\mathbb{R}^{n})$ with $0<p\leq 1$, which is a suitable substitute of the Lebesgue space $L^{p}(\mathbb{R}^{n})$, plays an important role in the study of operators and their application to partial differential equations.

In this paper, we consider the variable Hardy space estimates for non-convolution type Calderón-Zygmund singular integral operators. Our proofs mainly use the $H^{p(\cdot)}(\mathbb{R}^{n})-$boundedness of the discrete para-product operators, almost orthogonality estimates and the discrete version of Calderón’s reproducing formula. These techniques and estimates used in our proofs are different from the approaches which have been used in the existing proofs of many other variable Hardy spaces inequalities. For example, the boundedness of operators on variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^{n})$ was usually established via using atomic decompositions together with maximal operator estimate for the action of $T$ on atoms in recent years. We refer to the work of Cruz-Uribe and his collaborators [5, 6, 7], Nakai and Sawano [22, 24] and the author [26, 27, 28, 29]. The method to variable Hardy spaces in this paper can be applied to more general cases even when the maximal function characterizations are absent. On the other hand, the para-product operators can be used to nonlinear analysis. See Bony [1] or Meyer [21] for more details. There are many different forms of para-products operators, such as continuous para-product operator [8], frame para-product operator [9] and wavelet para-product operator [19]. Motivated by these, we introduce a new discrete para-product operator $\pi_{b}$ and obtain the $H^{p(\cdot)}(\mathbb{R}^{n})-$boundedness of $\pi_{b}$, which also has its own interests.

We first recall some notations and known results on variable function spaces that will be used in this paper. See [3, 7, 10, 20, 22, 30] for more information. For a measurable subset $E\subset\mathbb{R}^{n}$, we denote $p^{-}(E)=\inf_{x\in E}p(x)$ and $p^{+}(E)=\sup_{x\in E}p(x).$ Especially, we denote $p^{-}=p^{-}(\mathbb{R}^{n})$ and $p^{+}=p^{+}(\mathbb{R}^{n})$. Let $p(\cdot)$: $\mathbb{R}^{n}\rightarrow(0,\infty)$ be a measurable function with $0<p^{-}\leq p^{+}<\infty$ and $\mathcal{P}^{0}(\mathbb{R}^{n})$ be the set of all these $p(\cdot)$. Let $\mathcal{P}(\mathbb{R}^{n})$ be the set of all measurable functions $p(\cdot):\mathbb{R}^{n}\rightarrow[1,\infty)$ such that $1<p^{-}\leq p^{+}<\infty.$ Let $\mathcal{B}(\mathbb{R}^{n})$ be the set of $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})$ such that the Hardy-littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}$. For any Schwartz functions $\psi$ on $\mathbb{R}^{n}$, we denote that $\psi_{t}(x)=t^{-n}\psi(t^{-1}x),\;t>0$. Hereafter, for simplicity we use $\psi_{k}(x)$ to denote $\psi_{2^{-k}}(x)=2^{kn}\psi(2^{k}x),\;k\in\mathbb{N}$.

The variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^{n})$ is defined as the set of all measurable function $f$ for which the quantity $\int_{\mathbb{R}^{n}}|\varepsilon f(x)|^{p(x)}dx$ is finite for some $\varepsilon>0$ and

[TABLE]

As a special case of the theory of Nakano and Luxemberg, we see that $L^{p(\cdot)}(\mathbb{R}^{n})$ is a quasi-normed space. Especially, when $p^{-}\geq 1$, $L^{p(\cdot)}(\mathbb{R}^{n})$ is a Banach space. In the study of variable exponent function spaces it is common to assume that the exponent function $p(\cdot)$ satisfies $LH(\mathbb{R}^{n})$ conditions. We say that $p(\cdot)\in LH(\mathbb{R}^{n})$, if $p(\cdot)$ satisfies

[TABLE]

and

[TABLE]

It is well known that $p(\cdot)\in\mathcal{B}(\mathbb{R}^{n})$ if $p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\cap LH(\mathbb{R}^{n}).$ Moreover, examples shows that the above $LH(\mathbb{R}^{n})$ conditions are necessary in certain sense, see Pick and R${\mathring{\mbox{u}}}$žička ([23]) for more details.

Denote by $\mathcal{M}$ the grand maximal operator given by $\mathcal{M}f(x)=\sup\{|\psi_{t}\ast f(x)|:t>0,\psi\in\mathcal{F}_{N}\}$ for any fixed large integer $N$, where $\mathcal{F}_{N}=\{\psi\in\mathcal{S}:\int\psi(x)dx=1,\sum_{|\alpha|\leq N}\sup(1+|x|)^{N}|\partial^{\alpha}\psi(x)|\leq 1\}$. The variable Hardy space ${H}^{p(\cdot)}(\mathbb{R}^{n})$ is the set of all $f\in\mathcal{S}^{\prime}$ for which the quantity

[TABLE]

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ with $\int\phi(x)dx=1$. Denote that

[TABLE]

If $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}^{0}(\mathbb{R}^{n})$,

[TABLE]

Throughout this paper, $C$ or $c$ denotes a positive constant that may vary at each occurrence but is independent to the main parameter, and $A\sim B$ means that there are constants $C_{1}>0$ and $C_{2}>0$ independent of the the main parameter such that $C_{1}B\leq A\leq C_{2}B$. Given a measurable set $S\subset\mathbb{R}^{n}$, $|S|$ denotes the Lebesgue measure and $\chi_{S}$ means the characteristic function. For a cube $Q$, let $Q^{\ast}$ denote with the same center and $100$ its side length, i.e. $l(Q^{\ast})=100l(Q)$. We define $\mathcal{D}(\mathbb{R}^{n})$ to be the space of all smooth functions with compact support and $\mathcal{D}_{0}(\mathbb{R}^{n})$ to be the space of all smooth functions with compact support and integral zero. We also use the notations $j\wedge j^{\prime}=\min\{j,j^{\prime}\}$.

A local integrable function $b$ is in $BMO(\mathbb{R}^{n})$, if

[TABLE]

where the supremum ranges over all finite cubes $Q\in\mathbb{R}^{n}$. Let $\varphi$ be Schwartz functions with support on the unit ball satisfying the conditions: for all $\xi\in\mathbb{R}^{n}$,

[TABLE]

and $\int_{\mathbb{R}^{n}}\varphi(x)x^{\alpha}dx=0$ for all $0\leq|\alpha|\leq M$. For any $b\in BMO(\mathbb{R}^{n})$, the discrete para-product operator $\pi_{b}$ is defined by

[TABLE]

and

[TABLE]

where $x_{Q}$ is any point in $Q$ and the summation of $Q$ is taken over all dyadic cubes $Q$ with side length $2^{-k-N}$ in $\mathbb{R}^{n}$ for each $k\in\mathbb{Z}$ and a fixed large integer $N$.

First we obtain the $H^{p(\cdot)}(\mathbb{R}^{n})$ boundedness of $\pi_{b}$.

Theorem 1.1**.**

Suppose that $b\in BMO(\mathbb{R}^{n})$ and $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}^{0}(\mathbb{R}^{n})$. Then $\pi_{b}$ is bounded on $H^{p(\cdot)}(\mathbb{R}^{n})$ and bounded from $H^{p(\cdot)}(\mathbb{R}^{n})$ to $L^{p(\cdot)}(\mathbb{R}^{n})$.

Let $\epsilon$ be the regularity exponent with $0<\epsilon\leq 1$. An operator $T$ is a non-convolution Calderón-Zygmund operators, denoted by $T\in CZO(\epsilon)$, if it is bounded operator on $L^{2}(\mathbb{R}^{n})$, and if for all $f,g\in\mathcal{D}(\mathbb{R}^{n})$ with $\mbox{supp}(f)\cap\mbox{supp}(g)=\emptyset,$

[TABLE]

where the distributional kernel coincides with a function $K$ defined away the diagonal on $\mathbb{R}^{2n}$ and satisfies the following conditions

[TABLE]

Obviously, for any $\eta\in\mathcal{D}_{0}$,

[TABLE]

and

[TABLE]

Then we state the following result.

Theorem 1.2**.**

Suppose that $p(\cdot)\in LH(\mathbb{R}^{n})$ and $T\in CZO(\epsilon),$ where $0<\epsilon\leq 1$. Then $T$ extends to a continuous operator on $H^{p(\cdot)}(\mathbb{R}^{n})$,$\frac{n}{n+\epsilon}<p^{-}\leq p^{+}\leq 1$ if and only if $T^{\ast}1=0$.

Remark 1.3*.*

Theorem 1.1 and Theorem 1.2 still hold for $H^{p(\cdot)}(\mathcal{X})$ under some suitable conditions, where $\mathcal{X}$ is a space of homogeneous type introduced by R. Coifman and G. Weiss in [2]. We will see that, in the next sections, our proofs of these theorems only depends on the atomic decomposition characterization of variable Hardy spaces, the discrete Calderón reproducing formula, the discrete Littlewood-Paley-Stein theory and almost orthogonality estimates. The atomic decomposition theory of $H^{p(\cdot)}(\mathcal{X})$ was established in [31]. The formula, the discrete Littlewood-Paley-Stein theory and the estimates still hold for the spaces of homogeneous type. For more details, see [9, 15, 16]. Therefore, the theorems still hold for this setting.

To study the boundedness of the Calderón-Zygmund operators $T$ on other variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^{n})$, we need to assume more additional conditions on $T$. We say that an operator $T\in CZO(N,\epsilon)$, if the distributional kernel coincides with a function $K$ defined away the diagonal on $\mathbb{R}^{2n}$ and satisfies the following conditions

[TABLE]

where $0<\epsilon\leq 1$ and $N>0$ is an integer.

Finally, we obtain the following theorem.

Theorem 1.4**.**

Suppose that $p(\cdot)\in LH(\mathbb{R}^{n})$ and $T\in CZO(N,\epsilon)\cap CZO,$ where $0<\epsilon\leq 1$. Then $T$ extends to a continuous operator on $H^{p(\cdot)}(\mathbb{R}^{n})$,$\frac{n}{n+N+\epsilon}<p^{-}\leq p^{+}\leq 1$ if and only if $T^{\ast}x^{\beta}=0$ with $|\beta|\leq N$.

Remark 1.5*.*

After we were completing this paper, we learned that Cruz-Uribe, Moen and Nguyen also obtained the boundedness of non-convolution Calderón-Zygmund operators on variable Hardy spaces via the different method, which mainly contains finite atomic decompositions and the extrapolation method (See [6, Theorem 1.11]).

2. Proof of Theorem 1.1

In this section, we will establish the $H^{p(\cdot)}(\mathbb{R}^{n})$ boundedness of $\pi_{b}$. To do this, we need the atomic decomposition theory for $H^{p(\cdot)}(\mathbb{R}^{n})$. We recall the atoms of variable Hardy spaces ${H}^{p(\cdot)}(\mathbb{R}^{n})$ in [22]. Let $p(\cdot):\mathbb{R}^{n}\rightarrow(0,\infty)$, $0<p^{-}\leq p^{+}\leq 1<q\leq\infty$. Fix an integer $d\geq d_{p(\cdot)}\equiv\min\{d\in\mathbb{N}\bigcup\{0\}:p^{-}(n+d+1)>n\}.$ A function $a$ on $\mathbb{R}^{n}$ is called a $(p(\cdot),q)$-atom, if there exists a cube $Q$ such that ${\rm supp}\,a\subset Q$; $\|a\|_{L^{q}(\mathbb{R}^{n}}\leq\frac{|Q|^{1/q}}{\|\chi_{Q}\|_{L^{p(\cdot)}(\mathbb{R}^{n})}}$; $\int_{\mathbb{R}^{n}}a(x)x^{\alpha}dx=0\;{\rm for}\;|\alpha|\leq d$.

Theorem 2.1**.**

[7, 22, 25]** Let $p(\cdot)\in LH(\mathbb{R}^{n}),0<p^{-}\leq p^{+}\leq 1<q<\infty$ and

[TABLE]

If $f\in L^{q}(\mathbb{R}^{n})\cap H^{p(\cdot)}(\mathbb{R}^{n})$, there is a sequence of $(p(\cdot),q)-$atoms $\{a_{j}\}$ and a sequence of scalars $\{\lambda_{j}\}$ with

[TABLE]

such that $f=\sum_{j}\lambda_{j}a_{j}$, where the series converges to $f$ in both $H^{p(\cdot)}(\mathbb{R}^{n})$ and $L^{q}(\mathbb{R}^{n})$ norms. Conversely, if

[TABLE]

then $f=\sum_{j}\lambda_{j}a_{j}$ converges in $\mathcal{S}_{\infty}^{\prime}$, belongs to $H^{p(\cdot)}$ and satisfies

[TABLE]

We also need the following Fefferman-Stein vector valued inequality.

Proposition 2.2**.**

[4]** Let $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}^{0}(\mathbb{R}^{n})$. Then for any $q>1$, $f=\{f_{i}\}_{i\in\mathbb{Z}}$, $f_{i}\in L_{loc}(\mathbb{R}^{n})$, $i\in\mathbb{Z}$

[TABLE]

where $\mathbb{M}(f)=\{M(f_{i})\}_{i\in\mathbb{Z}}$.

Before we prove the theorem, we recall the discrete Calderón-type identity. The well-known discrete Calderón identity was first introduced by Frazier and Jawerth [12]. We will need the following discrete Calderón-type identity, which can be found in [25] for $\mathbb{R}^{n}$ and was first used in [14, 18] for spaces of homogeneous type.

Lemma 2.3**.**

Suppose that $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}^{0}(\mathbb{R}^{n})$, $L,N\in\mathbb{N}$ are large enough. Let $\varphi$ be Schwartz functions with support on the unit ball satisfying the conditions: for all $\xi\in\mathbb{R}^{n}$,

[TABLE]

and $\int_{\mathbb{R}^{n}}\varphi(x)x^{\alpha}dx=0$ for all $0\leq|\alpha|\leq N$. Then for all $f\in H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$, $1<q<\infty$, there exists a function $h\in H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$ with

[TABLE]

such that

[TABLE]

where $\tilde{\varphi}\in\mathcal{S}(\mathbb{R}^{n})$ with $\int_{\mathbb{R}^{n}}\tilde{\varphi}(x)x^{\alpha}dx=0$, $|\alpha|\leq L$ , and the series converges in both norms of $L^{q}(\mathbb{R}^{n})$ and $H^{p(\cdot)}(\mathbb{R}^{n})$, and where $x_{Q}$ is any point in $Q$ and the summation of $Q$ is taken over all dyadic cubes $Q$ with side length $2^{-j-N}$ in $\mathbb{R}^{n}$ for each $k\in\mathbb{Z}$ and a fixed large integer $N$.

Then repeating the same argument in [25, Proposition 2.3] and applying Lemma 2.3 yield that

[TABLE]

We are ready to prove Theorem 1.1.

Proof of Theorem 1.1. For $f\in L^{2}(\mathbb{R}^{n})\cap H^{p(\cdot)}(\mathbb{R}^{n})$ and $\ell\in\mathbb{N}$, define

[TABLE]

and

[TABLE]

Let

[TABLE]

Then $\Omega_{\ell}\subset\widetilde{\Omega_{\ell}}$. By the $L^{2}$ boundedness of $M$, $|\widetilde{\Omega_{\ell}}|\leq C|\Omega_{\ell}|.$

We write $\varphi_{Q}:=\phi_{k},$ if $l(Q)=2^{-k-N}$ and $x_{Q}$ is any point in $Q$. Denote $\tilde{Q}\in B_{\ell}$ are maximal dyadic cubes in $B_{\ell}$, we rewrite

[TABLE]

where

[TABLE]

and

[TABLE]

By the definition of $b_{\tilde{Q}}$ and the support of $\varphi$, we have that $b_{\tilde{Q}}$ is supported in $\tilde{Q}^{\ast}$. Given $h\in L^{2}(\mathbb{R}^{n})$ with $\|h\|_{L^{2}(\mathbb{R}^{n})\leq 1}$. By duality and the Hölder’s inequalities, we have

[TABLE]

where the last inequality follows form the $L^{2}$ estimates of the discrete Littlewood-Paley square function estimates.

Then the estimate implies that

[TABLE]

Hence, together with the cancellation conditions of $\varphi$, we have obtain that $b_{\tilde{Q}}$ is a $(p(\cdot),2)-$atom.

Therefore, applying the atomic decomposition of $H^{p(\cdot)}(\mathbb{R}^{n})$ in Theorem 2.1,

[TABLE]

On the other hand, for any $x\in Q\in B_{\ell}$, we have $M\chi_{Q\cap{\tilde{\Omega}_{\ell}\setminus\Omega_{\ell+1}}}(x)>\frac{1}{2}$. Then we have

[TABLE]

For any $x_{Q}\in Q\cap\tilde{\Omega}_{\ell}\setminus\Omega_{\ell+1}$, we have $M_{\phi}(f)(x_{Q})\leq 2^{\ell+1}$. By the Fefferman-Stein vector valued inequality, we have the following estimate:

[TABLE]

where the last inequality follows from [9, page 110, Theorem 4.13].

Therefore,

[TABLE]

Since

[TABLE]

and the equivalent characterization of variable Hardy spaces

[TABLE]

then by using the Fefferman-Stein vector valued inequality in Proposition 2.2, we yields that

[TABLE]

If we combine these estimates we get that

[TABLE]

for $f\in L^{2}(\mathbb{R}^{n})\cap H^{p(\cdot)}(\mathbb{R}^{n})$. Since that $L^{2}(\mathbb{R}^{n})\cap H^{p(\cdot)}(\mathbb{R}^{n})$ is dense in ${H}^{p(\cdot)}(\mathbb{R}^{n})$, then by the density argument $\pi_{b}$ can be extended to a bounded operator on $H^{p(\cdot)}(\mathbb{R}^{n})$.

Now we prove that $\pi_{b}$ is bounded on $L^{2}(\mathbb{R}^{n})$. By applying Hölder’s inequality and Carleson’s condition,

[TABLE]

for any $f\in L^{2}(\mathbb{R}^{n})$. Thus, $\pi_{b}$ is bounded on $L^{2}(\mathbb{R}^{n})$ and $H^{p(\cdot)}(\mathbb{R}^{n}).$ Note that $\sup\limits_{k}|\phi_{k}\ast f(x)|\leq CM(f)(x)$. Then for $q>1$, $\sup\limits_{k}|\phi_{k}\ast f(x)|\in L^{q}(\mathbb{R}^{n})$. Since

[TABLE]

there is a sequence of $t_{j}\rightarrow 0$ such that $\lim_{t_{j}\rightarrow 0}\phi_{t_{j}}\ast f(x)=f(x)$ for a.e. $x\in\mathbb{R}^{n}$. Then for $p(\cdot)\in\mathcal{P}^{0}$,

[TABLE]

and thus

[TABLE]

Therefore, we get that

[TABLE]

for $f\in L^{2}(\mathbb{R}^{n})\cap H^{p(\cdot)}(\mathbb{R}^{n})$.

Similarly, by the density argument $\pi_{b}$ can be extended to a bounded operator from $H^{p(\cdot)}(\mathbb{R}^{n})$ to $H^{p(\cdot)}(\mathbb{R}^{n})$. Thus we complete the proof of Theorem 1.1. $\hfill\Box$

3. Proof of Theorem 1.2 and Theorem 1.4

In this section, we will prove the $H^{p(\cdot)}$ boundedness of Calderón-Zygmund operators. For the proof we first need the discrete Littlewood-Paley characterizations for $H^{p(\cdot)}$ in [25, Proposition 2.3]. Let $\psi\in\mathcal{S}(\mathbb{R}^{n})$ satisfy

[TABLE]

and

[TABLE]

Denote by $\mathcal{S}_{\infty}(\mathbb{R}^{n})$ the functions $f\in\mathcal{S}(\mathbb{R}^{n})$ satisfying $\int_{\mathbb{R}^{n}}f(x)x^{\alpha}dx=0$ for all muti-indices $\alpha\in\mathbb{Z}_{+}^{n}:=(\{0,1,2,\cdots\})^{n}$ and $\mathcal{S}^{\prime}_{\infty}(\mathbb{R}^{n})$ its topological dual space. For $f\in\mathcal{S}^{\prime}_{\infty}(\mathbb{R}^{n})$, we recall the definition of the Littlewood-Paley-Stein square function

[TABLE]

and the discrete Littlewood-Paley-Stein square function

[TABLE]

where $Q$ denote dyadic cubes in $\mathbb{R}^{n}$ with side-lengths $2^{-j}$ and the lower left-corners of $Q$ are $2^{-j}\mathbf{k}$. If $p(\cdot)\in LH(\mathbb{R}^{n})\cap\mathcal{P}^{0}(\mathbb{R}^{n})$,

[TABLE]

Then by discrete Calderón-type identity, the discrete Littlewood-Paley characterizations for $H^{p(\cdot)}$ and almost orthogonal estimates, we show the following theorem.

Theorem 3.1**.**

Suppose that $p(\cdot)\in LH(\mathbb{R}^{n})$ and $T\in CZO(\epsilon),$ where $0<\epsilon\leq 1$. If $T1=T^{\ast}1=0$, then $T$ extends to a continuous operator on $H^{p(\cdot)}(\mathbb{R}^{n})$,$\frac{n}{n+\epsilon}<p^{-}\leq p^{+}\leq 1$.

Now we are ready to prove Theorem 3.1.

Proof of Theorem 3.1 Given $q>1$, since that the subspace $H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$ is dense in $H^{p(\cdot)}(\mathbb{R}^{n})$, we only need to prove that $T$ is bounded from $H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$ to $H^{p(\cdot)}(\mathbb{R}^{n})$. For any $f\in H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$ and $T\in CZO(\epsilon)$, we have $Tf\in L^{q}(\mathbb{R}^{n})$. Then by Lemma 2.3,

[TABLE]

and

[TABLE]

where

[TABLE]

and $h\in H^{p(\cdot)}(\mathbb{R}^{n})\cap L^{q}(\mathbb{R}^{n})$ with

[TABLE]

We claim that

[TABLE]

To prove (3.1), we consider the four cases: (1) $j>j^{\prime}$ and $|x_{Q}-x_{Q^{\prime}}|\leq 5\;2^{-j^{\prime}}$; (2) $j>j^{\prime}$ and $|x_{Q}-x_{Q^{\prime}}|\geq 5\;2^{-j^{\prime}}$; (3) $j\leq j^{\prime}$ and $|x_{Q}-x_{Q^{\prime}}|\leq 5\;2^{-j^{\prime}}$; (4) $j\leq j^{\prime}$ and $|x_{Q}-x_{Q^{\prime}}|\geq 5\;2^{-j^{\prime}}$. The idea we used here comes from [17].

In Case (1), since $T1=0$, we have

[TABLE]

Choose a smooth function $\eta_{0}$ such that $\mbox{supp}\;\eta_{0}\subset\{x:|x|\leq 6\}$ and let $\eta_{0}=1$ when $|x|\leq 2$. Set $\eta_{1}=1-\eta_{0}$. Then we get that

[TABLE]

For $I$, we denote $\tilde{\varphi}(v)=(\varphi_{j^{\prime}}(v-x_{Q^{\prime}})-\varphi_{j^{\prime}}(x_{Q}-x_{Q^{\prime}}))\eta_{0}(2^{j}(v-x_{Q}))$ and $\omega(u)=\varphi_{j}(x_{Q}-u)$. Applying Hölder’s inequality and the $L^{2}(\mathbb{R}^{n})$ boundedness of $T$ yield that

[TABLE]

We now deal with the term $II$. By the cancellation condition of $\varphi$, we get that

[TABLE]

In Case (2), observe that $|x_{Q}-x_{Q^{\prime}}|\sim|u-v|$. The smoothness condition on the kernel $K(u,v)$ implies

[TABLE]

The other cases are similar to Case (1) and Case (2). So we prove the claim.

Then using the claim and the Fefferman-Stein vector-valued maximal inequality, we have

[TABLE]

where the second inequality follows from the lemma in [13, pages 147-148] and $\frac{n}{n+\epsilon}<\delta<p^{-}$.

Thus, we have completed the proof of Theorem 3.1. $\hfill\Box$

We now turn to the

Proof of Theorem 1.2: Let $p(\cdot)\in LH(\mathbb{R}^{n})$ and $T\in CZO(\epsilon),$ where $0<\epsilon\leq 1$. First, by using para-product operator we prove that $T$ is bounded on $H^{p(\cdot)}(\mathbb{R}^{n})$,$\frac{n}{n+\epsilon}<p^{-}\leq p^{+}\leq 1$ if $T^{\ast}1=0$. From Lemma 2.3, we obtain

[TABLE]

and

[TABLE]

In the proof of Theorem 1.1, we have showed that $\pi_{b}$ is bounded on $L^{2}(\mathbb{R}^{n})$. Note that the kernel of $\pi_{b}$ is the function

[TABLE]

fulfilling the all conditions (1.1), (1.2) and (1.3). In fact, applying the size condition of $\phi,\varphi$ and $\tilde{\varphi}\in H^{1}(\mathbb{R}^{n})$ yield

[TABLE]

Repeating the similar argument, the kernel $K_{b}(x,y)$ also satisfies the conditions (1.2) and (1.3) and then $\pi_{b}$ is a Calderón-Zygmund operator. We define the new operator $\tilde{T}$ by

[TABLE]

By Theorem 3.1, $\tilde{T}$ is bounded on $H^{p(\cdot)}(\mathbb{R}^{n})$ since $\tilde{T}1=T1-\pi_{T1}1=0$ and $\tilde{T}^{\ast}1=T^{\ast}1-\pi^{\ast}_{T1}1=0$. By Theorem 1.1, $\pi_{b}$ is bounded on $H^{p(\cdot)}(\mathbb{R}^{n})$ and then $T$ is bounded on $H^{p(\cdot)}(\mathbb{R}^{n})$. Therefore, the condition $T^{\ast}1=0$ is sufficient. On the other hand, this condition is obviously necessary. Indeed, for any $\varphi\in\mathcal{D}_{0}$, then $\varphi\in H^{1}(\mathbb{R}^{n})$ and $T\varphi\in H^{1}(\mathbb{R}^{n})$. Thus, $\int T\varphi(x)dx=0$. Therefore, we conclude the proof of Theorem 1.2. $\hfill\Box$

Proof of Theorem 1.4 The proof is similar to the ones of Theorem 3.1 and Theorem 1.2. We only need to observe that $\varphi$ has the zero vanishing moment up to order $N$ and $T^{\ast}(x^{\beta})=0$ with $|\beta|\leq N$. Then by repeating the similar argument together with using Taylor expansion of the kernel of $T$ and the high order moment condition of $\varphi$, then we can establish the desired almost orthogonal estimate. On the other hand, for any bounded, compactly supported function $\varphi$ fulfilling the moment condition $\int x^{\alpha}\varphi(x)dx=0$ for all $|\alpha|\leq N$, then we also have $\int x^{\alpha}T\varphi(x)dx=0$. The integral is well defined since $T\varphi(x)=\mathcal{O}(|x|^{-n-N})$ at infinity. This means $T^{\ast}x^{\alpha}=0.$ $\hfill\Box$

Acknowledgments. The project is sponsored by Natural Science Foundation of Jiangsu Province of China (grant no. BK20180734), Natural Science Research of Jiangsu Higher Education Institutions of China (grant no. 18KJB110022) and Nanjing University of Posts and Telecommunications Science Foundation (grant no. NY217151).