Variable exponent Hardy spaces associated with discrete Laplacians on graphs

TL;DR

This paper develops the theory of variable exponent Hardy spaces on infinite graphs with discrete Laplacians, including their atomic decompositions and boundedness of key operators, advancing harmonic analysis on graph structures.

Contribution

It introduces a new framework for variable exponent Hardy spaces on graphs and analyzes the boundedness of classical harmonic analysis operators within this setting.

Findings

Established atomic and molecular decompositions for these Hardy spaces.

Proved boundedness of Littlewood-Paley functions, Riesz transforms, and spectral multipliers.

Extended harmonic analysis tools to variable exponent spaces on graphs.

Abstract

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

Equations

$${\rho }_{p(\cdot )}(f)={\int }_X|f(x){|}^{p(x)}d\mu (x).$$

$$\Vert f{\Vert }_{L^{p(\cdot )}(X)}=\inf \left\{\lambda >0:{\rho }_{p(\cdot )}\left(\frac{f}{\lambda }\right)\le 1\right\},f\in L^{p(\cdot )}(X).$$

$$|p(x)-p(y)|\le \frac{C}{\log (e+1/d(x,y))},x,y\in X,x\ne y.$$

$$|p(x)-a|\le \frac{C}{\log (e+d(x,x_0))},x\in X.$$

$${\mathcal{P}}^{log}(X)=\{p:X\to (0,\infty ):1/p\text{mbox}islog-H\ddot{o}ldercontinuousinX\}.$$

$$p_{+}=esssu{p}_{X}p\text{mbox}and{p}_{-}=essin{f}_{X}p.$$

$$\Vert \mathcal{M}f{\Vert }_{L^{p(\cdot )}(X)}\le C\Vert f{\Vert }_{L^{p(\cdot )}(X)},f\in L^{p(\cdot )}(X).$$

$$\mu (\{x\})=\sum _{x\sim y}\nu (x,y).$$

$$p(x,y)=\frac{\nu (x,y)}{\mu (x)},x,y\in \Gamma .$$

$$P(f)(x)=\sum _{y\in \Gamma }p(x,y)f(y),x\in \Gamma ,$$

$$p_n(x,y)=\sum _{z\in \Gamma }p(x,z)p_{n-1}(z,y)x,y\in \Gamma .$$

$$P^n(f)(x)=\sum _{y\in \Gamma }{p}_n(x,y)f(y),x\in \Gamma ,$$

$$\frac{\mu(B(x,r))}{\mu(B(x,s))}\leq C\Big{(}\frac{r}{s}\Big{)}^{D},\,\,\,x\in\Gamma,\,\,\,0<s\leq r.$$

$$p_n(x,y)\le C\frac{\mu (y)}{\mu (B(x,\sqrt{n}))}{e}^{-c{d}^2(x,y)/n},x,y\in \Gamma \text{mbox}andn\in {\mathbb{N}}_{+}.$$

$$|\widetilde{p_{n,k}}(x,y)|\le C_4\frac{\mu (y)}{n^k\mu (B(x,\sqrt{n}))}{e}^{-c_4{d}^2(x,y)/n},$$

$$S_{L}(f)(x)=\Big{(}\sum_{k=1}^{\infty}\sum_{d(x,y)<k}\frac{|k(I-P)P^{[k/2]}(f)(y)|^{2}}{k\mu(B(y,k))}\mu(y)\Big{)}^{1/2}.$$

$$\Vert f{\Vert }_{H_L^{p(\cdot )}(\Gamma )}=\Vert S_L(f){\Vert }_{p(\cdot )},f\in {\mathbb{H}}_L^{p(\cdot )}(\Gamma ).$$

$$\Upsilon_{\beta}(x)=\Big{\{}(y,k)\in\Gamma\times\mathbb{N}_{+}:d(y,x)<\beta k\Big{\}}.$$

$$T_{\beta }(E)=(\bigcup _{x\in E^c}{{\rm Y}}_{\beta }(x){)}^c.$$

$$T(B)=\Big{\{}(x,k)\in\Gamma\times\mathbb{N}_{+}:d(x_{0},x)\leq r_{0}-k\Big{\}}.$$

$$(\mathcal{A}f)(x)=\Big{(}\sum_{(y,k)\in\Upsilon(x)}\frac{|f(y,k)|^{2}}{k\mu(B(x,k))}\mu(y)\Big{)}^{1/2},\quad x\in\Gamma.$$

$$\Vert f{\Vert }_{T_2^{p(\cdot )}(\Gamma )}=\Vert \mathcal{A}(f){\Vert }_{p(\cdot )},f\in T_2^{p(\cdot )}(\Gamma ).$$

$$\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})=\Big{\|}\Big{(}\sum_{j\in\mathbb{N}}\frac{|\lambda_{j}|^{\mathfrak{p}}\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{p(\cdot)}^{\mathfrak{p}}}\Big{)}^{1/{\mathfrak{p}}}\Big{\|}_{p(\cdot)},$$

$$f(x,k)=\sum _{j\in \mathbb{N}}{\lambda }_j{a}_j(x,k),(x,k)\in \Gamma \times {\mathbb{N}}_{+},$$

$$\mathcal{A}(\{{\lambda }_j\},\{B_j\})\le C\Vert f{\Vert }_{T_2^{p(\cdot )}(\Gamma )}.$$

$$\Vert f{\Vert }_{T_2^{p(\cdot )}(\Gamma )}\le C\mathcal{A}(\{{\lambda }_j\},\{B_j\}).$$

$$f=\sum _{j\in \mathbb{N}}{\lambda }_j{a}_j,\text{mbox}in{L}^2(\Gamma ),$$

$$\Vert f{\Vert }_{H_{L,M,at}^{p(\cdot )}(\Gamma )}=\inf \mathcal{A}(\{{\lambda }_j\},\{B_j\}),$$

$$\Vert f{\Vert }_{H_L^{p(\cdot )}(\Gamma )}\le C\mathcal{A}(\{{\lambda }_j\},\{B_j\}),$$

$$f=\sum _{j\in \mathbb{N}}{\lambda }_j{a}_j,\text{mbox}in{H}_L^{p(\cdot )}(\Gamma ),$$

Full text

The authors are partially supported by Spanish Government grant MTM2016-79436-P. The third author is also supported by Nazarbayev University Social Policy Grant.

1. Introduction

Variable exponent $L^{p(\cdot)}$-spaces were consider for the first time by Orlicz [43]. Later, Nakano [41, p. 284] pointed out that those spaces appear as special cases of the ones studied in [41]. Inspired by Nakano’s results, modular spaces were investigated by several authors. An important step in the study of variable exponent spaces was the paper of Kovác̆ik and Rákosník [35] where the main properties of the Lebesgue and Sobolev spaces with variable exponent were established. In this century the variable exponent spaces have been studied systematically and intensively. Interesting connections between variable exponent spaces and other areas have been developed (see, for instance, [1], [2], [12], [21], [47] and [48]). In the recent monographs [17] and [23] the theory of variable exponent Lebesgue and Sobolev spaces is developed in an exhaustive way.

In this paper we are interested in variable exponent Hardy spaces. The classical theory of Hardy spaces in ${\mathbb{R}}^{n}$ has been extended to a variable exponent setting by Nakai and Sawano [39], Sawano [53], and Cruz-Uribe and Wang [20]. Variable exponent Hardy spaces $H^{p(\cdot)}({\mathbb{R}}^{n})$ were characterized by using Riesz transforms by Yang, Zhuo and Nakai [61], and intrinsic square functions by Zhuo, Yang and Liang [64]. Hardy spaces of variable exponent $H^{p(\cdot)}_{L}({\mathbb{R}}^{n})$ associated with certain classes of operators $L$ have been investigated by Zhuo and Yang ([60] and [63]) and Yang, Zhang and Zhuo [58].

Our objective is to study variable exponent Hardy spaces on graphs. Our graphs will be spaces of homogeneous type in the sense of Coifman and Weiss [14]. Hardy spaces on spaces of homogeneous type were firstly studied by Coifman and Weiss [15] and Macías and Segovia ([36] and [37]).

Let $X$ be a set equipped with a non negative quasidistance $d$. Suppose that $\mu$ is a $\sigma$-finite measure defined on a $\sigma$-algebra on $X$ containing the open balls associated with $d$. We assume that $\mu(B)>0$, for every $d$-ball $B$. We say that the triple $(X,\mu,d)$ is a space of homogeneous type when $\mu$ satisfies the following doubling property: there exists $C>0$ such that, for every $x\in X$ and $r>0$, $\mu(B(x,2r))\leq C\mu(B(x,r))$. Note that this implies that there exist $C,D>0$ such that $\mu(B(x,r))\leq C(r/s)^{D}\mu(B(x,s))$, for every $x\in X$ and $0<s<r$.

Assume that $p:\;X\;\rightarrow\;(0,\infty)$ is a $\mu$-measurable function. If $f$ is a complex $\mu$-measurable function on $X$ we define the modular $\rho_{p(\cdot)}(f)$ of $f$ by

[TABLE]

The variable exponent Lebesgue space $L^{p(\cdot)}(X)$ is the collection of all complex $\mu$-measurable functions $f$ such that, there exists $\lambda>0$ for which $\rho_{p(\cdot)}(\lambda f)<\infty$. The quasinorm $\|\cdot\|_{L^{p(\cdot)}(X)}$ on $L^{p(\cdot)}(X)$ is defined by

[TABLE]

As it was mentioned, the theory of variable exponent Lebesgue spaces in $\mathbb{R}^{n}$ can be found in the monographs [17] and [23]. A very important problem in this field was to find conditions for the exponent $p(\cdot)$ such that the Hardy-Littlewood maximal operator $\mathcal{M}$ is bounded in $L^{p(\cdot)}(\Omega)$ when $\Omega$ is a subset of $\mathbb{R}^{n}$. Diening [22] and Pick and Ruzicka [46] obtained the first results concerning this question. Later, Nekvinda [42] and Cruz-Uribe, Fiorenza and Neugebauer [19] improve them.

Maximal functions are useful tools in our developments. The above results on $\mathbb{R}^{n}$ were extended to spaces of homogeneous type in [31], [32] and [34] by considering bounded spaces or using a ball condition as in [22]. The most general result for the Hardy-Littlewood maximal operator $\mathcal{M}$ on $L^{p(\cdot)}(X)$ when $X$ is unbounded was established by Adamowicz, Harjulehto and Hästö [3]. We say that $p:\;X\;\rightarrow\;(0,\infty)$ is log-Hölder continuous in $X$ when the two following properties are satisfied

1. (i)

There exists $C>0$ such that

[TABLE] 2. (ii)

There exist $x_{0}\in X$, $C>0$ and $a\in\mathbb{R}$ such that

[TABLE]

As it was proved in [3, Lemma 2.1] the property (ii) does not depend on $x_{0}$. We consider the following class of exponent $\mathcal{P}^{log}(X)$ defined by

[TABLE]

Note that if $p\in\mathcal{P}^{log}(X)$, then $rp\in\mathcal{P}^{log}(X)$, for every $r>0$. If $p:\;X\;\rightarrow\;(0,\infty)$ is a function we define

[TABLE]

We always assume that $0<p_{-}\leq p_{+}<\infty$.

The following result, [3, Corollary 1.8], will be very useful in the sequel.

Theorem A**.**

Let $X$ be a space of homogeneous type. Assume that $p\in\mathcal{P}^{log}(X)$ with $p_{-}>1$. Then, there exists $C>0$ such that

[TABLE]

Recently, Zhuo, Sawano and Yang [62] have studied Hardy spaces with variable exponents on RD spaces. RD spaces are special spaces of homogeneous type satisfying an inverse doubling condition (see [30], [40] and [59]). If $\mu$ is a measure defining a RD space $X$, then $\mu(\{x\})=0$, $x\in X$. This is an important point because we are going to define Hardy spaces on graphs that are spaces of homogeneous type but they are not RD spaces.

We now describe the graphs that we consider in this paper (see [8] and [10]). By $\Gamma$ we denote a countably infinite set. The elements of $\Gamma$ are called vertices of the graph. $\nu$ represents a nonnegative symmetric function defined on $\Gamma\times\Gamma$. We say that two vertices $x,y\in\Gamma$ are neighbors when $\nu(x,y)>0$, and in that case we write $x\sim y$. For every $x\in\Gamma$ we denote $deg(x)=card\{y\in\Gamma:\,x\sim y\}$. We assume that $deg(x)>0$, for every $x\in\Gamma$, and $\sup_{x\in\Gamma}deg(x)<\infty$. We now define a measure $\mu$ on $\Gamma$ as follows. For every $x\in\Gamma$, $\mu(\{x\})$ is given by

[TABLE]

To simplify we write $\mu(x)$ instead of $\mu(\{x\})$, for every $x\in\Gamma$. For every $A\subset\Gamma$ we define $\mu(A)=\sum_{x\in A}\mu(x)$. Thus, $\mu$ is a measure on $\Gamma$. We assume that $\mu(\Gamma)=+\infty$.

If $x,y\in\Gamma$, we say that they are connected when $x\sim y$ or there exist $x_{1},x_{2},...,x_{n}\in\Gamma$ such that $x\sim x_{1}\sim x_{2}\sim$ … $\sim x_{n}\sim y$, and in that case we call $[x,x_{1},...,x_{n},y]$ a path of length $n+1$ joining $x$ and $y$. We assume that each $x,y\in\Gamma$ are connected. Note that this fact implies that $\mu(x)\neq 0$, for every $x\in\Gamma$. We now define a distance $d$ on $\Gamma$ as follows

• •

$d(x,x)=0$;

• •

$d(x,y)=1$ if, and only if, $x\neq y$ and $x\sim y$;

• •

$d(x,y)=n\geq 2$ when $x\neq y$, $\nu(x,y)=0$ and $n$ is the minimum length of all possible paths joining $x$ and $y$.

As usual we denote by $B(x,r)=\{y\in\Gamma\;:\;d(x,y)<r\}$, for every $x\in\Gamma$ and $r>0$. Note that, in general, a ball has not a unique center and a unique radius. In particular, $B(x,r)=B(x,[r])$, $x\in\Gamma$ and $r>0$, where $[r]=n$ provided that $n-1<r\leq n$, with $n\in\mathbb{N}_{+}$. Here and in the sequel $\mathbb{N}_{+}=\mathbb{N}\setminus\{0\}$. The distance $d$ defines the discrete topology on $\Gamma$.

We define on $\Gamma$ the Markov kernel $p$ by

[TABLE]

The following two useful properties hold

1. (i)

$\sum_{y\in\Gamma}p(x,y)=1,\;\;x\in\Gamma$, 2. (ii)

$p(x,y)\mu(x)=p(y,x)\mu(y),\;\;x,y\in\Gamma$.

By using the Markov kernel we define the Markov operator $P$ by

[TABLE]

where $f$ is a complex function defined on $\Gamma$. The operator $L=I-P$ is the discrete Laplace operator on $\Gamma$ associated with $\nu$.

For every $n\in\mathbb{N}$, $n\geq 2$, we consider the $n$-th convolution power of the Markov kernel $p_{n}$ defined by

[TABLE]

We consider $p_{1}=p$. Then, for every $n\in\mathbb{N}_{+}$, we have that

[TABLE]

and $P^{0}f=f$, where again $f$ is a complex function defined on $\Gamma$.

In our setting we assume as in [6] the following conditions.

1. (a)

The triple $(\Gamma,\mu,d)$ is a space of homogeneous type. Hence, the measure $\mu$ is doubling with respect to the distance $d$ an there exist $C,D>0$ such that

[TABLE] 2. (b)

There exist $C,c>0$ such that

[TABLE]

If $\mu$ is doubling you only need that (1.2) be true when $x=y$ in order to (1.2) hold for any pair of $x,y\in\Gamma$. 3. (c)

For certain $\alpha>0$, $\Gamma$ satisfies property $\Delta(\alpha)$, that is, for every $x\in\Gamma$, $x\sim x$, and $\nu(x,y)\geq\alpha\mu(x)$, for every $x,y\in\Gamma$ such that $x\sim y$.

We recall (see [10, (3)]) that when $\mu$ is doubling, $\Gamma$ satisfies $\Delta(\alpha)$ with $\alpha>0$, and $(p_{n})_{n=1}^{\infty}$ verifies (1.2), then there exist $c_{4},C_{4}>0$ such that, for every $n\in\mathbb{N}_{+}$, $k\in\mathbb{N}$ and $x,y,\in\Gamma$,

[TABLE]

where $\widetilde{p_{n,k}}$ represents the kernel of the operator $(I-P)^{k}P^{n}$, for every $n\in\mathbb{N}_{+}$ and $k\in\mathbb{N}$.

Our first objective is to define Hardy spaces with variable exponents by using Littlewood-Paley square functions. We consider the square function $S_{L}$ defined by

[TABLE]

$S_{L}$ defines a bounded operator in $L^{2}(\Gamma)$ ([10, p. 3460]).

In [8] and [10] Hardy spaces on $\Gamma$ were defined. We extend definitions and results in [10] to variable exponent settings.

We say that $f\in L^{2}(\Gamma)$ is in $\mathbb{H}^{p(\cdot)}_{L}(\Gamma)$ when $S_{L}(f)\in L^{p(\cdot)}(\Gamma)$. We define the Hardy space $H^{p(\cdot)}_{L}(\Gamma)$ as the completion of $\mathbb{H}^{p(\cdot)}_{L}(\Gamma)$ with respect to the quasi-norm $\|\cdot\|_{H^{p(\cdot)}_{L}(\Gamma)}$ where

[TABLE]

Here and in the sequel we write $\|\cdot\|_{p(\cdot)}$ to refer to $\|\cdot\|_{L^{p(\cdot)}(\Gamma)}$.

Before giving a characterization of $H^{p(\cdot)}_{L}(\Gamma)$ by using atoms we introduce tent spaces in our variable exponent setting.

If $\beta>0$ and $x\in\Gamma$, we denote by $\Upsilon_{\beta}(x)$ the cone with vertex $x$ and aperture $\beta$ defined by

[TABLE]

To simplify we write $\Upsilon(x)$ to denote $\Upsilon_{1}(x)$. If $E\subset\Gamma$ and $\beta>0$ we define the tent $T_{\beta}(E)$ over $E$ with overture $\beta$ as follows:

[TABLE]

We also write $T(E)$ to refer to $T_{1}(E)$. If $x\in\Gamma$ and $r>0$ we can see that the tent $T(B)$ over $B=B(x_{0},r_{0})$ is the set given by

[TABLE]

Assume that $f$ is a complex valued function defined on $\Gamma\times\mathbb{N}_{+}$. We consider the following sublinear operator

[TABLE]

We define the tent space $T_{2}^{p(\cdot)}(\Gamma)$ as follows. A complex valued function $f$ defined on $\Gamma\times\mathbb{N}_{+}$ is said to be in $T_{2}^{p(\cdot)}(\Gamma)$ provided that $\mathcal{A}(f)\in L^{p(\cdot)}(\Gamma)$. On $T_{2}^{p(\cdot)}(\Gamma)$ we consider the quasi-norm $\|\cdot\|_{T_{2}^{p(\cdot)}(\Gamma)}$ given by

[TABLE]

If $0<q<\infty$, we say that a complex valued function $a$ defined on $\Gamma\times\mathbb{N}_{+}$ is a $(T_{2}^{p(\cdot)},q)$-atom when there exists a ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, such that

1. (i)

$\operatorname{supp}a\subseteq T(B)$, 2. (ii)

$\|a\|_{T_{2}^{q}(\Gamma)}\leq\mu(B)^{1/q}\|\chi_{B}\|_{p(\cdot)}^{-1}$.

Here and in the sequel if $E\subset\Gamma$, we denote by $\chi_{E}$ the characteristic function supported in $E$. If $0<r<\infty$, $T_{2}^{r}(\Gamma)$ represents the tent space $T_{2}^{p(\cdot)}(\Gamma)$ when $p(x)=r$, $x\in\Gamma$.

For every sequences $(\lambda_{j})_{j\in\mathbb{N}}$ of complex numbers and $(B_{j})_{j\in\mathbb{N}}$ of balls we define

[TABLE]

where $\mathfrak{p}=\min\{1,p_{-}\}$.

We now establish a description of $T_{2}^{p(\cdot)}(\Gamma)$ by using atoms.

Theorem 1.1**.**

Let $p(\cdot)\in\mathcal{P}^{log}(\Gamma)$ and $1<q<\infty$.

(i) For certain $C>0$ the following property is satisfied: if $f\in T_{2}^{p(\cdot)}(\Gamma)$ there exist, for each $j\in\mathbb{N}$, $\lambda_{j}>0$ and a $(T_{2}^{p(\cdot)},q)$-atom $a_{j}$ associated with a ball $B_{j}$ such that

[TABLE]

where the series converges absolutely for every $(x,k)\in\Gamma\times\mathbb{N}_{+}$ and

[TABLE]

Moreover, $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ in the sense of convergence in $T_{2}^{p(\cdot)}(\Gamma)$.

(ii) There exists $C>0$ such that if, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(T_{2}^{p(\cdot)}q)$-atom associated with the ball $B_{j}$ such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$, then the series $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ converges absolutely for every $(x,k)\in\Gamma\times\mathbb{N}_{+}$ and in $T_{2}^{p(\cdot)}(\Gamma)$. Furthermore

[TABLE]

(iii) If $0<r<\infty$ and $f\in T_{2}^{p(\cdot)}(\Gamma)\cap T_{2}^{r}(\Gamma)$, then the series in (1.4) also converges to $f$ in $T_{2}^{r}(\Gamma)$.

By applying Theorem 1.1 we obtain an atomic decomposition for the Hardy space $H^{p(\cdot)}_{L}(\Gamma)$.

Let $M\in\mathbb{N}_{+}$ and $1<q<\infty$. We say that $a\in L^{q}(\Gamma)$ is a $(q,p(\cdot),M)$-atom associated with a ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, when there exists $b\in L^{q}(\Gamma)$ satisfying that:

$(i)$ $a=L^{M}b$;

$(ii)$ $\operatorname{supp}L^{k}b\subset B$, $k=0,...,M$;

$(ii)$ $\|L^{k}b\|_{q}\leq(r_{B})^{M-k}(\mu(B))^{1/q}\|\chi_{B}\|_{p(\cdot)}^{-1}$, $k=0,...,M$.

A function $f\in L^{2}(\Gamma)$ is in $\mathbb{H}^{p(\cdot)}_{L,M,at}(\Gamma)$ when there exist, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(2,p(\cdot),M)$-atom $a_{j}$ associated with the ball $B_{j}$ such that

[TABLE]

and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$. For every $f\in\mathbb{H}^{p(\cdot)}_{L,M,at}(\Gamma)$ we define $\|f\|_{H_{L,M,at}^{p(\cdot)}(\Gamma)}$ by

[TABLE]

where the infimum is taken of over all the pair of sequences $\{\lambda_{j}\}_{j\in\mathbb{N}}$ and $\{B_{j}\}_{j\in\mathbb{N}}$ satisfying that, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and there exists a $(2,p(\cdot),M)$-atom $a_{j}$ associated with the ball $B_{j}$ such that $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j},$ in $L^{2}(\Gamma)$ and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$. By $H^{p(\cdot)}_{L,M,at}(\Gamma)$ we represent the completion of $\mathbb{H}^{p(\cdot)}_{L,M,at}(\Gamma)$ with respect to the quasi-norm $\|\cdot\|_{H_{L,M,at}^{p(\cdot)}(\Gamma)}$.

By using Theorem 1.1 we prove that the Hardy spaces $H^{p(\cdot)}_{L}(\Gamma)$ and $H^{p(\cdot)}_{L,M,at}(\Gamma)$ coincide.

Theorem 1.2**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$, $r\geq 2$, $r>p_{+}$, $M\in\mathbb{N}_{+}$, and $M>2D/p_{-}$. The following assertions hold.

(a) There exists $C>0$ satisfying that: if, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(r,p(\cdot),M)$-atom associated with the ball $B_{j}$ such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$, then the series $\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ converges in $H^{p(\cdot)}_{L}(\Gamma)$ and

[TABLE]

where $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$.

(b) There exists $C>0$ such that, for every $f\in H^{p(\cdot)}_{L}(\Gamma)$, there exist, for each $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(r,p(\cdot),M)$-atom $a_{j}$ associated with the ball $B_{j}$ such that

[TABLE]

and

[TABLE]

The Hardy space $H^{p(\cdot)}_{L}(\Gamma)$ coincides with the space $L^{p(\cdot)}(\Gamma)$ provided that $p\in\mathcal{P}^{\log}(\Gamma)$ and $p_{-}>1$ (see Proposition 4.7).

Next we introduce the molecules in our setting that will be useful to study the boundedness of operators in $H^{p(\cdot)}_{L}(\Gamma)$ (see Section 5).

Let $M\in\mathbb{N}_{+}$, $1<q<\infty$, and $\varepsilon>0$. We say that a function $m:\Gamma\longrightarrow\mathbb{C}$ is a $(q,p(\cdot),M,\varepsilon)$-molecule when there exist a function $b:\Gamma\longrightarrow\mathbb{C}$ and a ball $B=B(x_{B},r_{B})$ with $x_{B}\in\Gamma$ and $r_{B}\geq 1$ such that

(i) $m=L^{M}b$ and,

(ii) For every $k=0,...,M$,

[TABLE]

where, for every $j\in\mathbb{N}_{+}$, $\mathfrak{S}_{j}(B)=B(x_{B},2^{j+1}r_{B})\setminus B(x_{B},2^{j-1}r_{B})$, and $\mathfrak{S}_{0}(B)=B$.

Every $(q,p(\cdot),M)$-atom is also a $(q,p(\cdot),M,\varepsilon)$-molecule, for every $\varepsilon>0$.

Theorem 1.3**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$, $M\in\mathbb{N}_{+}$, $M>2D/p_{-}$ and $\varepsilon>D/p_{-}$, $q\geq 2$, and $q>p_{+}$. There exists $C>0$ satisfying that: if, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $m_{j}$ is a $(q,p(\cdot),M,\varepsilon)$-molecule associated with the ball $B_{j}$, such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$, then the series $\sum_{j\in\mathbb{N}}\lambda_{j}m_{j}\in H^{p(\cdot)}_{L}(\Gamma)$ converges in $H^{p(\cdot)}_{L}(\Gamma)$ and

[TABLE]

where $f=\sum_{j\in\mathbb{N}}\lambda_{j}m_{j}$.

We now consider, for every $f:\Gamma\longrightarrow\mathbb{C}$, the radial maximal function ${\mathcal{M}}_{+}(f)$ given by

[TABLE]

We define

[TABLE]

and we denote by $H^{p(\cdot)}_{L,+}(\Gamma)$ the completion of $\mathbb{H}^{p(\cdot)}_{L,+}(\Gamma)$ with respect to the quasi-norm $\|\cdot\|_{H^{p(\cdot)}_{L,+}(\Gamma)}$ defined by

[TABLE]

We establish (see Proposition 4.8) that $H^{p(\cdot)}_{L}(\Gamma)$ is a subspace of $H^{p(\cdot)}_{L,+}(\Gamma)$.

As it was mentioned Zhuo, Sawano and Yang [62] defined Hardy spaces with variable exponent on homogeneous spaces $(X,\mu,d)$ of $RD$-type. If $(X,\mu,d)$ is a $RD$-space of homogeneous type, $\mu(x)=0$, for every $x\in X$. Hence, our graphs are not $RD$-spaces.

In [15] in the context of homogeneous type spaces atomic Hardy spaces are considered. Macías and Segovia [36] characterized those Hardy spaces by using a grand-maximal function provided that $(X,\mu,d)$ is a normal homogeneous type space. We say that a homogeneous type space is normal when the following property holds: there exist $A_{1},A_{2},K>0$ such that, for every $x\in X$,

1. (i)

$A_{1}r\leq\mu(B(x,r))$, $r>0$, 2. (ii)

$A_{2}r\geq\mu(B(x,r))$, $r\geq K\mu(x)$, 3. (iii)

$B(x,r)=\{x\}$, $0<r<K\mu(x)$.

Also, a normal space $(X,d,\mu)$ is said to have order $\alpha>0$ when

[TABLE]

for every $x,y,z\in X$, $d(x,z)<r$, $d(y,z)<r$. Macías and Segovia [37] proved that if $(X,\mu,d)$ is a space of homogeneous type there exists a quasimetric $d_{1}$ on $X$ defining the same topology as $d$ on $X$ such that $(X,\mu,d_{1})$ is a normal space of order $\alpha$, for some $\alpha>0$. In general, $d$ and $d_{1}$ are not comparable.

Later Uchiyama ([54, Theorem 1 and Corollary 1]) characterized Hardy spaces $H^{p}(X)$ by using radial maximal functions associated to certain nonnegative continuous functions provided that $(X,\mu,d)$ is a space of homogeneous type such that $\mu(B(x,r))\sim r$, for every $x\in X$ and $r>0$. This result was extended to $RD$-spaces by Grafakos, Liu, and Yang [29].

By putting all together the above ideas (see also [50]) we could think on proving a characterization of our $H^{p(\cdot)}_{L}(\Gamma)$ by using the radial maximal function $\mathcal{M}_{+}$. The problem is that we would need to consider a quasimetric $d_{1}$ topologically equivalent to the graph metric $d$ for which the space $(\Gamma,\mu,d_{1})$ is normal and then proceed, for instance, as in the proof of [28, Theorem 1.6]. But when we change the quasimetric we can not be sure that our Markov kernel $p_{n}$ satisfies the sufficient estimates (exponential upper bounds,…). At this moment we do not know how to prove the characterization of the Hardy space $H^{p(\cdot)}_{L}(\Gamma)$ by using the radial maximal function $\mathcal{M}_{+}$.

Yang and Zhuo [60] and [63], and Yang, Zhang and Zhuo [58] studied variable Hardy spaces associated with operators $\mathcal{L}$ on $\mathbb{R}^{n}$ such that the semigroup generated by $\mathcal{L}$ satisfies some kind of Gaussian or off diagonal estimates. However, it is not clear for the semigroup generated by the discrete Laplacian $L$ whether those kind of estimates hold or not (see [16], [44] and [45]). Our study relies on the upper Gaussian estimates for the iterates of the Markov operators (1.2).

Bui, Cao, Ky, Yang and Yang [9] and D. Yang and S. Yang ([56] and [57]) studied Musielak-Orlicz-Hardy spaces $\mathcal{H}_{\varphi,L}(\mathbb{R}^{n})$ associated with operators. Here $\varphi$ is known as a Musielak-Orlicz function. It is an interesting question to define Musielak-Orlicz-Hardy spaces in our discrete settings. When we consider as Musielak-Orlicz function $\varphi(x,t)=t^{p(x)}$, $t>0$ and $x\in\mathbb{R}^{n}$, the Musielak-Orlicz-Hardy space $\mathcal{H}_{\varphi,L}(\mathbb{R}^{n})$ reduces to the variable exponent Hardy space $\mathcal{H}_{L}^{p(\cdot)}(\mathbb{R}^{n})$. However, the Musielak-Orlicz-Hardy spaces $\mathcal{H}_{\varphi,L}(\mathbb{R}^{n})$ are defined requiring certain conditions for $\varphi$ (for instance, $\varphi$ is a uniform Muckenhoupt weight) that are not always satisfied when $\varphi(x,t)=t^{p(x)}$ and $p$ is log-Holder continuous. Hence, the studies about Musielak-Orlicz-Hardy spaces and variable exponent Hardy spaces associated with operators do not cover each other.

This paper is organized in the following way. Section 2 is dedicated to establish some results that will be very useful throughout this work. Theorem 1.1 is proved in Section 3 while the proofs of Theorems 1.2 and 1.3 are established in Section 4. Theorem 1.2 is separated in Propositions 4.3 and 4.4. In Section 5 we use atomic and molecular characterizations of $H^{p(\cdot)}_{L}(\Gamma)$ to study $H^{p(\cdot)}_{L}(\Gamma)$-boundedness properties of certain Littlewood-Paley square functions, Riesz transforms, and spectral multipliers for the discrete Laplacian $L$.

From now on $C$ and $c$ represent positive constants that can change in each occurrence.

$\mathbf{Acknowledgements}$. The authors would strongly like to give thanks to Professor Dachun Yang for sending us his paper [62] (jointly with C. Zhuo and Y. Sawano).

2. Auxiliary results

In this section we present some results that will be very useful in the sequel.

By using Rubio de Francia extrapolation theorem (see, for instance, [18]) we can obtain the following Fefferman-Stein vector valued inequality in our variable exponent setting. This property also can be seen as a special case of [62, Theorem 2.7].

Lemma 2.1**.**

Assume that $p\in\mathcal{P}^{log}(\Gamma)$, $p_{-}>1$, and $1<q<\infty$. Then, there exists $C>0$ such that, for every sequence $\{f_{j}\}_{j\in\mathbb{N}}\subset L^{p(\cdot)}(\Gamma)$, we have that

[TABLE]

The following result is a special case of [62, Proposition 2.11].

Lemma 2.2**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$ and $q\in[1,\infty]\cap(p_{+},\infty]$. There exists $C>0$ such that if, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$, $a_{j}\in L^{q}(\Gamma)$ and $B_{j}$ is a ball in $\Gamma$ satisfying that

(i) $supp(a_{j})\subset B_{j}$,

(ii) $\|a_{j}\|_{q}\leq\mu(B_{j})^{1/q}\|\chi_{B_{j}}\|_{p(\cdot)}^{-1}$,

then

[TABLE]

The results proved in the next lemma are consequence of Theorem A and Lemma 2.2.

Lemma 2.3**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$.

(i) If $0<w<p_{-}$, there exists $C>0$ such that, for every $x\in\Gamma$, $\beta>1$ and $r>0$, we have that

[TABLE]

(ii) If $q\in[1,\infty)\cap(p_{+},\infty)$, there exists $C>0$ such that, for every $x\in\Gamma$, $\beta>1$ and $r>0$, we have that

[TABLE]

Proof.

We consider $x_{0}\in\Gamma$, $r_{0}>0$, and $\beta>1$.

$(i)$ We choose $0<w<p_{-}$. We have that, for every $x\in B(x_{0},\beta x_{0})$,

[TABLE]

By Theorem A we deduce that

[TABLE]

$(ii)$ Let $q\in[1,\infty)\cap(p_{+},\infty)$. We define the function $a_{0}=\lambda_{0}\chi_{B(x_{0},r_{0})}$, where $\lambda_{0}=(\mu(B(x_{0},\beta r_{0}))/\mu(B(x_{0},r_{0})))^{1/q}$. It is clear that $\operatorname{supp}(a_{0})\subset B(x_{0},\beta r_{0})$ and $\|a_{0}\|_{q}\leq(\mu(B(x_{0},\beta r_{0})))^{1/q}$. Then, according to Lemma 2.2, we get

[TABLE]

and the proof is completed. ∎

By proceeding as in the last proof but using Lemma 2.1 instead of Theorem A we can obtain the following lemma.

Lemma 2.4**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$ and $0<w<p_{-}$.Then, there exists $C>0$ such that, for every $\beta\geq 1$ and every sequences $\{x_{j}\}_{j\in\mathbb{N}}\subset\Gamma$, $\{r_{j}\}_{j\in\mathbb{N}}\subset(0,\infty)$, and $\{\lambda_{j}\}_{j\in\mathbb{N}}\subset\mathbb{C}$ we have that

[TABLE]

3. Tent spaces of variable exponents on graphs. (Proof of Theorem 1.1)

Tent spaces were introduced by Coifman, Meyer and Stein in [13]. These spaces play an important role in the development of the theory of Hardy spaces in different settings. Discrete tent spaces were considered in [10] (see also [8]) to define Hardy spaces associated with operators on graphs. In this section we study tent spaces of variable exponents on graphs. Discrete tent spaces in [10] are particular cases of our variable exponent tent spaces on graphs.

Proof of Theorem 1.1.

In order to prove this result we follow the ideas developed in [13, Proof of Theorem 1, (c)] (see also [52, Theorem 1.1]). We need to make some modifications because we have variable exponents (see [64, Theorem 2.16] for a proof in the continuous case). We need to introduce the concept of $\gamma$-density. Suppose that $F$ is a subset of $\Gamma$ such that the complement $F^{c}$ of $F$ has finite $\mu$-measure. Let $0<\gamma<1$. We say that $x\in\Gamma$ has global $\gamma$-density with respect to $F$ when, for every $r>0$,

[TABLE]

We denote by $F_{\gamma}^{*}$ the set of all those elements of $\Gamma$ with global $\gamma$-density with respect to $F$. It is clear that $F_{\gamma}^{*}\subset F$. Also we have that

[TABLE]

Here $\mathcal{M}$ denotes the centered Hardy-Littlewood maximal function. Since $\mathcal{M}$ is of weak type (1,1) and we are dealing with a space of homogeneous type, there exists $C>0$ such that

[TABLE]

Here $C$ does not depend on $F$.

$(i)$ Suppose that $f\in T^{p(\cdot)}_{2}(\Gamma)\cap T^{2}_{2}(\Gamma)$. Let $k\in\mathbb{Z}$. We define

[TABLE]

and $F_{k}=O_{k}^{c}$. Since $\mathcal{A}(f)\in L^{p(\cdot)}(\Gamma)$ and $p_{+}<\infty$, we have that $\sum_{x\in\Gamma}|\mathcal{A}(f)(x)|^{p(x)}\mu(x)<\infty$ and then $\mu(O_{k})<\infty$.

Fix $\eta,\,\gamma\in(0,1)$. We could take for instance $\eta=\gamma=1/2$ but to simplify we prefer keep writing $\eta$ and $\gamma$. A careful reading of [52, Lemma 2.1] allows us to ensure that there exists $C>0$ for which

[TABLE]

Since $f\in T_{2}^{2}(\Gamma)$ by using dominated convergence theorem we get that

[TABLE]

We have that

[TABLE]

It follows that $f(y,t)=0$, $(y,t)\in\cap_{k\in\mathbb{Z}}(\cup_{x\in(F_{k})_{\gamma}^{*}}\Upsilon_{1-\eta}(x))$. Hence,

[TABLE]

We apply [52, Lemma 2.2] to $\Omega_{k}=((F_{k})_{\gamma}^{*})^{c}$, $k\in\mathbb{Z}$. Note that $\mu(O_{k})\leq\mu(\Omega_{k})\leq\frac{C}{1-\gamma}\mu(O_{k})<\infty$ and $\Omega_{k}\not=\Gamma$, $k\in\mathbb{Z}$. There exists $\mathcal{C}>0$ such that for every $k\in\mathbb{Z}$ there exists a set $I_{k}\subset\mathbb{N}$, and, for every $n\in I_{k}$, $x_{n}^{k}\in\Gamma$ and $\varphi_{n}^{k}:\Gamma\longrightarrow[0,\infty)$ satisfying, by taking $r_{n}^{k}=d(x_{n}^{k},\Omega_{k}^{c})/10$, that:

• •

$\Omega_{k}=\cup_{n\in I_{k}}B(x_{n}^{k},r_{n}^{k})$;

• •

$B(x_{i}^{k},r_{i}^{k}/4)\cap B(x_{j}^{k},r_{j}^{k}/4)=\emptyset$, provided that $i,j\in I_{k}$, $i\not=j$;

• •

$card\{m\in I_{k}:B(x_{n}^{k},5r_{n}^{k})\cap B(x_{m}^{k},5r_{m}^{k})\not=\emptyset\}\leq\mathcal{C},\quad n\in I_{k};$

• •

$\mbox{supp}\;\varphi_{n}^{k}\subset B(x_{n}^{k},2r_{n}^{k})$, $n\in I_{k}$;

• •

$\varphi_{n}^{k}(x)\geq\mathcal{C}^{-1}$, $x\in B(x_{n}^{k},r_{n}^{k})$, $n\in I_{k}$;

• •

$\sum_{n\in I_{k}}\varphi_{n}^{k}=\chi_{\Omega_{k}}$.

Since $\mbox{supp}\;f\subset\cup_{k\in\mathbb{Z}}T_{1-\eta}(\Omega_{k})$ as in [52, p. 131] we have that, for every $(x,t)\in\Gamma\times\mathbb{N}_{+}$,

[TABLE]

We now write the equality (3.2) as follows

[TABLE]

where

[TABLE]

with $C_{\eta}=(2+12/(1-\eta))$ and, for each $(y,t)\in\Gamma\times\mathbb{N}_{+}$,

[TABLE]

for every $k\in\mathbb{Z}$ and $j\in I_{k}$.

Note that, for every $k\in\mathbb{Z}$ and $j\in I_{k}$, $\mbox{supp }(a_{j}^{k})\subset T(B(x_{j}^{k},C_{\eta}r_{j}^{k}))\cap T_{1-\eta}(\Omega_{k})\cap(T_{1-\eta}(\Omega_{k+1}))^{c}$ [52, p. 132].

We are going to see that there exists $C>0$ such that, for every $k\in\mathbb{Z}$ and $j\in I_{k}$, $Ca_{j}^{k}$ is a $(T_{2}^{p(\cdot)},2)$-atom.

Let $k\in\mathbb{Z}$ and $j\in I_{k}$. Suppose that $h\in T_{2}^{2}(\Gamma)$ with $\|h\|_{T_{2}^{2}(\Gamma)}\leq 1$. By using [52, Lemma 2.1] and Hölder’s inequality we get

[TABLE]

because $x\in B(x_{j}^{k},C_{\eta}r_{j}^{k})$ provided that $(y,t)\in\Upsilon(x)\cap T(B(x_{j}^{k},C_{\eta}r_{j}^{k})))$.

Then Hölder’s inequality leads to

[TABLE]

Since $(T_{2}^{2}(\Gamma))^{\prime}=T_{2}^{2}(\Gamma)$ we conclude that

[TABLE]

Hence $a_{j}^{k}/C$ is a $(T_{2}^{p(\cdot)},2)$-atom. Note that $C$ does not depend on $k$ nor on $j$.

In a similar way we can see that, for every $r\in(1,\infty)$, since $(T_{2}^{r}(\Gamma))^{\prime}=T_{2}^{r^{\prime}}(\Gamma)$, there exists $C>0$ such that, for every $k\in\mathbb{Z}$ and $j\in I_{k}$,

[TABLE]

and then $a_{j}^{k}/C$ is a $(T_{2}^{p(\cdot)},r)$-atom.

Our next objective is to see that $\mathcal{A}(\{\lambda_{j}^{k}\},\{B(x_{j}^{k},C_{\eta}r_{j}^{k})\})\leq C\|f\|_{T_{2}^{p(\cdot)}(\Gamma)}$. In order to do this we proceed as in [64, p. 1569]. By (2), for every $j,k\in\mathbb{N}$, we have that

[TABLE]

Then, we get

[TABLE]

Here $0<r<\mathfrak{p}$. According to Lemma 2.1 it follows that

[TABLE]

If $x\in\Omega_{k}$, then $\mathcal{M}(\chi_{O_{k}})(x)>1-\gamma$. Hence, $\chi_{\Omega_{k}}\leq\frac{1}{1-\gamma}\mathcal{M}(\chi_{O_{k}})$. By proceeding as above we obtain

[TABLE]

Since $\{x\in\Gamma:\mathcal{A}(f)(x)=+\infty\}=\varnothing$, we can write

[TABLE]

We get

[TABLE]

We are going to see that the equality (3.3) also holds in $T_{2}^{p(\cdot)}(\Gamma)$, that is, the series converges to $f$ in $T_{2}^{p(\cdot)}(\Gamma)$. Assume that $\{(k_{\ell},j_{\ell})\}_{\ell\in\mathbb{N}}$ represents an ordenation in the set $\{(k,j):k\in\mathbb{Z},\,\,j\in I_{k}\}$. Note that the series in (3.3) is absolutely pointwisely convergent in $\Gamma\times\mathbb{N}_{+}$. Then, we can write

[TABLE]

Since $f\in T_{2}^{p(\cdot)}(\Gamma)$, according to (3.5), the series

[TABLE]

converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$.

For every $\ell\in\mathbb{N}$, since $\operatorname{supp}a_{j_{\ell}}^{k_{\ell}}\subset T(B(x_{j_{\ell}}^{k_{\ell}},C_{\eta}r_{j_{\ell}}^{k_{\ell}}))$, we have that

[TABLE]

Then, by using (3.4), Lemma 2.2 leads to

[TABLE]

provided that $\alpha,\beta\in\mathbb{N}$, $\alpha<\beta$.

Hence the series $\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}$ converges in $T_{2}^{p(\cdot)}(\Gamma)$ to a certain $g\in T_{2}^{p(\cdot)}(\Gamma)$. Then (see, for instance, [20, Lemma 2.4]),

[TABLE]

and it follows that, for every $x\in\Gamma$,

[TABLE]

We deduce that

[TABLE]

for every $(y,t)\in\Gamma\times\mathbb{N}_{+}$. Hence $g=f$. Thus, we prove that the series in (3.3) converges to $f$ in $T_{2}^{p(\cdot)}(\Gamma)$.

Suppose now that $1<r<\infty$ and $f\in T_{2}^{p(\cdot)}(\Gamma)$. Since $T_{2}^{p(\cdot)}(\Gamma)\cap T_{2}^{2}(\Gamma)$ is dense in $T_{2}^{p(\cdot)}(\Gamma)$, for every $k\in\mathbb{N}_{+}$, there exists $f_{k}\in T_{2}^{p(\cdot)}(\Gamma)\cap T_{2}^{2}(\Gamma)$ such that $\|f_{k}-f\|_{T_{2}^{p(\cdot)}(\Gamma)}\leq 2^{-k}\|f\|_{T_{2}^{p(\cdot)}(\Gamma)}$. Also, we take $f_{0}=0$. We have that $f=\sum_{k=1}^{\infty}(f_{k}-f_{k-1})$, where the series converges in $T_{2}^{p(\cdot)}(\Gamma)$.

According to the first part of this proof, for every $k\in\mathbb{N}_{+}$, we can write

[TABLE]

where the series is absolutely pointwisely convergent and it converges in $T_{2}^{p(\cdot)}(\Gamma)$ where, for each $j\in\mathbb{N}_{+}$, $a_{j}^{k}$ is a $(T_{2}^{p(\cdot)},r)$-atom associated with the ball $B_{j}^{k}$ and $\lambda_{j}^{k}>0$ satisfying that $\sum_{j=1}^{\infty}(\lambda_{j}^{k})^{\mathfrak{p}}\chi_{B_{j}^{k}}\|\chi_{B_{j}^{k}}\|^{-\mathfrak{p}}_{p(\cdot)}$ converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$ and

[TABLE]

Here $C>0$ does not depend on $k$.

Assume that $\{(k_{\ell},j_{\ell}):\,\ell\in\mathbb{N}\}$ is an ordenation of $\mathbb{N}_{+}\times\mathbb{N}_{+}$. We are going to see that $\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}=f$ in the sense of convergence in $T_{2}^{p(\cdot)}(\Gamma)$. Indeed, let $\epsilon>0$. There exists $\delta\in\mathbb{N}_{+}$ such that $\|\sum_{k=1}^{\delta}(f_{k}-f_{k-1})-f\|_{T_{2}^{p(\cdot)}(\Gamma)}<\epsilon$. By Lemma 2.2 we can write, for every $m\in\mathbb{N}_{+}$,

[TABLE]

Since, for every $k\in\mathbb{N}_{+}$, the series $\sum_{j=1}^{\infty}\frac{|\lambda_{j}^{k}|^{\mathfrak{p}}}{\|\chi_{B_{j}^{k}}\|_{p(\cdot)}^{\mathfrak{p}}}\chi_{B_{j}^{k}}$ converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$, there exists $m_{0}\in\mathbb{N}_{+}$ such that

[TABLE]

for every $k\in\mathbb{N}_{+}$, $k=1,...,\delta$.

We now choose $L_{0}\in\mathbb{N}$ such that

[TABLE]

We have that, for every $L>L_{0}$,

[TABLE]

Thus, we proved that $\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}=f$ in $T_{2}^{p(\cdot)}(\Gamma)$.

$(ii)$ Suppose now that $1<q<\infty$ and that, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(T^{p(\cdot)}_{2},q)$-atom associated with the ball $B_{j}$, satisfying that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty.$ By proceeding as above we get that $\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ converges in $T_{2}^{p(\cdot)}(\Gamma)$ and if $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ in $T_{2}^{p(\cdot)}(\Gamma)$, we have that

[TABLE]

On the other hand, for every $j\in\mathbb{N}$, $|a_{j}|$ is $(T^{p(\cdot)}_{2},q)$-atom associated with the ball $B_{j}$, and $\mathcal{A}(\{|\lambda_{j}|\},\{B_{j}\})<\infty$. Then, the series $\sum_{j\in\mathbb{N}}|\lambda_{j}||a_{j}|$ converges in $T_{2}^{p(\cdot)}(\Gamma)$. By taking in mind [20, Lemma 2.4] that also holds in our setting, we deduce that $\sum_{j\in\mathbb{N}}|\lambda_{j}||a_{j}(x,k)|<\infty$, for every $(x,k)\in\Gamma\times\mathbb{N}_{+}$.

$(iii)$ Let $0<r<\infty$. Assume that $f\in T_{2}^{p(\cdot)}(\Gamma)\cap T_{2}^{r}(\Gamma)\cap T_{2}^{2}(\Gamma)$. We are going to see that the equality in (3.3) also holds in $T_{2}^{r}(\Gamma)$. As in (3.6) we write

[TABLE]

where $\{(k_{\ell},j_{\ell}):\ell\in\mathbb{N}\}$ represents an ordenation in the set $\{(k,j):k\in\mathbb{Z},j\in I_{k}\}$.

Suppose firstly that $0<r\leq 1$. We define, for every $k\in\mathbb{Z}$ and $j\in I_{k}$,

[TABLE]

and

[TABLE]

It is clear that $\widetilde{a_{j}^{k}}\widetilde{\lambda_{j}^{k}}=a_{j}^{k}\lambda_{j}^{k}$, $k\in\mathbb{Z}$, $j\in I_{k}$. By proceeding as in the proof of (3.4) we can see that there exists $C>0$ such that for every $k\in\mathbb{Z}$ and $j\in I_{k}$, $C\widetilde{a_{j}^{k}}$ is a $(T_{2}^{q}(\Gamma),r)$-atom associated with the ball $B(x_{j}^{k},C_{\eta}r_{j}^{k})$. Also, we have that

[TABLE]

From Lemma 2.2 being $p(x)=r$, $x\in\Gamma$, we deduce that

[TABLE]

Hence, the series $\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}$ converges in $T_{2}^{r}(\Gamma)$ and $f=\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}$ in $T_{2}^{r}(\Gamma)$.

Suppose now $r\in(1,\infty)$. We have that

[TABLE]

Then, $F_{k-1}\subset F_{k}$, $k\in\mathbb{Z}$, and $\cap_{k\in\mathbb{Z}}F_{k}=\{x\in\Gamma:\mathcal{A}(f)(x)=0\}$. Hence, since $f\in T_{2}^{r}(\Gamma)$, the monotone convergence theorem leads to

[TABLE]

On the other hand, we recall that, for every $k\in\mathbb{Z}$, $\Omega_{k}=((F_{k})^{*}_{\gamma})^{c}$, and, by (3.1),

[TABLE]

Also, $O_{k+1}\subset O_{k}$, $k\in\mathbb{Z}$, and $\cap_{k\in\mathbb{Z}}O_{k}=\{x\in\Gamma:\mathcal{A}(f)(x)=+\infty\}=\emptyset$. Hence, $\lim_{k\rightarrow+\infty}\mu(O_{k})=0$, and then $\lim_{k\rightarrow+\infty}\mu(\Omega_{k})=0$. Since $f\in T_{2}^{r}(\Gamma)$, the dominated convergence theorem implies that

[TABLE]

Let $\varepsilon>0$. By (3.7) and (3.8), there exists $m_{0}\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

For $k\in\mathbb{Z}$ and $j\in I_{k}$, we define $A_{k,j}=\operatorname{supp}(a_{k}^{j})$. Then,

[TABLE]

For every $k\in\mathbb{Z}$, by taking into account the Whitney covering $\{B(x_{j}^{k},r_{j}^{k})\}_{j\in I_{k}}$ it follows that

[TABLE]

We define

[TABLE]

We have that, for $k\in\mathbb{Z}$, $k<-m_{0}$ and $j\in I_{k}$,

[TABLE]

By using [52, Lemma 2.1] we get, for every $h\in T_{2}^{r^{\prime}}(\Gamma)$ such that $\|h\|_{T_{2}^{r^{\prime}}(\Gamma)}\leq 1$,

[TABLE]

We obtain that

[TABLE]

We define

[TABLE]

Note that

[TABLE]

Then, $\operatorname{supp}\mathcal{A}(\mathcal{H}_{0})\subset\Omega_{m_{0}}$. We can write

[TABLE]

By (3.9) and (3.10) we obtain, for every $m_{1}\in\mathbb{N}$,

[TABLE]

Let $m_{1}\in\mathbb{N}$. We define

[TABLE]

Since $\operatorname{supp}A_{k,j}\subset T_{1}(B(x_{j}^{k},C_{\eta}r_{j}^{k}))$, we have that

[TABLE]

where

[TABLE]

and we get

[TABLE]

By using (3.1), the doubling property of $\mu$ and the Whitney covering properties we obtain, for each $k\in\mathbb{Z}$,

[TABLE]

Then, we have that

[TABLE]

and since $f\in T_{2}^{r}(\Gamma)$ it follows that

[TABLE]

Then, there exists $m_{1}\in\mathbb{N}$ such that

[TABLE]

We conclude that

[TABLE]

There exists $\ell_{0}\in\mathbb{N}$ such that $|k_{\ell}|+j_{\ell}>m_{0}+m_{1}$ provided that $\ell>\ell_{0}$. We obtain that

[TABLE]

Thus we establish that the series $\sum_{\ell\in\mathbb{N}}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}$ converges in $T_{2}^{r}(\Gamma)$. Also, we have that $f=\sum_{\ell\in\mathbb{N}}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}$ in $T_{2}^{r}(\Gamma)$.

We define $T_{c}(\Gamma)$ the space of complex functions $f$ defined on $\Gamma\times\mathbb{N}_{+}$ such that $\operatorname{supp}f$ is finite. Note that in $\Gamma\times\mathbb{N}_{+}$ the compact sets are the finite sets. $T_{c}(\Gamma)$ is a dense subspace of $T_{2}^{p(\cdot)}(\Gamma)$ and in $T_{2}^{r}(\Gamma)$. Suppose now that $f\in T_{2}^{p(\cdot)}(\Gamma)\cap T_{2}^{q}(\Gamma)$. There exists a sequence $\{f_{k}\}_{k\in\mathbb{N}}\subset T_{c}(\Gamma)$ such that $f_{0}=0$, and $\|f_{k}-f\|_{T_{2}^{p(\cdot)}(\Gamma)}\leq 2^{-k}\|f\|_{T_{2}^{p(\cdot)}(\Gamma)}$ and $\|f_{k}-f\|_{T_{2}^{r}(\Gamma)}\leq 2^{-k}\|f\|_{T_{2}^{r}(\Gamma)}$. By the above arguments, for every $k\in\mathbb{N}_{+}$, we can write

[TABLE]

where the series is absolutely pointwisely convergent, and also in both $T_{2}^{p(\cdot)}(\Gamma)$ and in $T^{r}_{2}(\Gamma)$, and, for each $j\in\mathbb{N}_{+}$, $a_{j}^{k}$ is a $(T_{2}^{p(\cdot)},r)$-atom associated with the ball $B_{j}^{k}$ and $\lambda_{j}^{k}>0$ satisfying that $\sum_{j=1}^{\infty}(\lambda_{j}^{k})^{\mathfrak{p}}\chi_{B_{j}^{k}}\|\chi_{B_{j}^{k}}\|^{-\mathfrak{p}}_{p(\cdot)}$ converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$ and

[TABLE]

Here $C>0$ does not depend on $k$.

By proceeding as in the end of part (i) we can prove that if $\{(k_{\ell},j_{\ell}):\,\ell\in\mathbb{N}\}$ is an ordenation of $\mathbb{Z}\times\mathbb{N}_{+}$, then $\sum_{\ell=0}^{\infty}\lambda_{j_{\ell}}^{k_{\ell}}a_{j_{\ell}}^{k_{\ell}}=f$ in the sense of convergence in $T_{2}^{p(\cdot)}(\Gamma)$ and in $T^{q}_{2}(\Gamma)$.

∎

4. Variable exponents Hardy spaces on graphs

We recall the definition of our Hardy spaces. We define, for every $f\in L^{2}(\Gamma)$,

[TABLE]

The Hardy space $H^{p(\cdot)}_{L}(\Gamma)$ is defined as the completion of

[TABLE]

with respect to the quasinorm $\|\cdot\|_{H^{p(\cdot)}_{L}(\Gamma)}$ given by

[TABLE]

The following result was established in [10, Theorem 3.7] (see also [27, Proposition 2.1]).

Proposition 4.1**.**

Let $M\in\mathbb{N}_{+}$. Then, for every $f\in L^{2}(\Gamma)$ we have

[TABLE]

on $L^{2}(\Gamma)$, where the coefficients $c_{k,N}$, $N\in\mathbb{N}_{+}$, are defined as follows:

(i) $c_{k,1}=1$, $k\in\mathbb{N}$,

(ii) $c_{k,N+1}=\sum_{j=0}^{k}c_{j,N}$, $k\in\mathbb{N}$.

As above we denote by $T_{c}(\Gamma)$ the space of complex functions $f$ defined on $\Gamma\times\mathbb{N}_{+}$ such that $\operatorname{supp}f$ is finite.

Motivated by the representation (4.1) and the definition of Hardy spaces we introduce the operator $\Pi_{M}$, $M\in\mathbb{N}_{+}$, as follows

[TABLE]

for every $f\in T_{c}(\Gamma)$. These operators were considered in [10]. Next we generalize [10, Proposition 3.14].

Proposition 4.2**.**

Let $M\in\mathbb{N}_{+}$. The operator $\Pi_{M}$ can be extended from $T_{c}(\Gamma)$ to:

(i) $T_{2}^{q}(\Gamma)$ as a bounded operator from $T_{2}^{q}(\Gamma)$ into $L^{q}(\Gamma)$, for every $1<q<\infty$. Moreover, if we also continue denoting $\Pi_{M}$ to the extension operator, for every $f\in T_{2}^{q}(\Gamma)$, with $1<q<\infty$, we have that

[TABLE]

(ii) $T_{2}^{p(\cdot)}(\Gamma)$ as a bounded operator from $T_{2}^{p(\cdot)}(\Gamma)$ into $H^{p(\cdot)}_{L}(\Gamma)$, provided that $p\in\mathcal{P}^{\log}(\Gamma)$ and $M>2D/p_{-}$.

Proof.

We recall that $T_{c}(\Gamma)$ is a dense subspace of $T_{2}^{p(\cdot)}(\Gamma)$, and in particular of $T_{2}^{q}(\Gamma)$, $0<q<\infty$.

$(i)$ Let $1<q<\infty$. Assume that $f\in T_{c}(\Gamma)$ and $h\in L^{q^{\prime}}(\Gamma)$. We have that

[TABLE]

Note that the series appearing above are actually finite sums because $f\in T_{c}(\Gamma)$.

We can write

[TABLE]

because $p_{n}(x,y)\mu(x)=p_{n}(y,x)\mu(y)$, $x,y\in\Gamma$ and $n\in\mathbb{N}$.

Since $0\leq c_{k,M+1}\leq(k+1)^{M}$, $k\in\mathbb{N}$, by using [10, Proposition 3.2, (a)], we get

[TABLE]

where

[TABLE]

Note that, since

[TABLE]

according to [8, Proposition 4.6] we obtain that $\mathcal{A}(g)\in L^{q^{\prime}}(\Gamma)$ and by using Hölder’s inequality we obtain

[TABLE]

Hence,

[TABLE]

Thus we prove that $\Pi_{M}$ can be extended from $T_{c}(\Gamma)$ to $T_{2}^{q}(\Gamma)$ as a bounded operator from $T_{2}^{q}(\Gamma)$ into $L^{q}(\Gamma)$. We continue denoting by $\Pi_{M}$ to a such extension.

Let now $f\in T_{2}^{q}(\Gamma)$. For every $\ell,m\in\mathbb{N}_{+}$,

[TABLE]

and

[TABLE]

Here $x_{0}\in\Gamma$ is fixed.

Let $\ell\in\mathbb{N}_{+}$. By using dominated convergence theorem we deduce that $f_{\ell,m}\to f_{\ell}$, as $m\to\infty$, in $T_{2}^{q}(\Gamma)$. Then,

[TABLE]

in the sense of convergence in $L^{q}(\Gamma)$. For every $t\in\mathbb{N}_{+}$, $f_{\ell,m}(.,t)\to f_{\ell}(.,t)$, as $m\to\infty$, in $L^{q}(\Gamma)$. Indeed, let $t\in\mathbb{N}_{+}$. By taking into account that $\mu$ is doubling (see (1.1)) we have that, for every $m,\ell\in\mathbb{N}_{+}$,

[TABLE]

Since $P$ is a bounded operator in $L^{q}(\Gamma)$, it follows that

[TABLE]

Also, $f_{\ell}\to f$, as $\ell\to\infty$, in $T_{2}^{q}(\Gamma)$. Then, $\Pi_{M}(f_{\ell})\to\Pi_{M}(f)$, as $\ell\to\infty$, in $L^{q}(\Gamma)$. Hence, we obtain (4.2).

$(ii)$ Let $f\in T_{c}(\Gamma)$. We take $q=2\max\{1,p_{+}\}$. According to Theorem 1.1, for every $j\in\mathbb{N}$, there exist $\lambda_{j}\in(0,\infty)$ and a $(T_{2}^{p(\cdot)},q)$-atom $a_{j}$ associated to a ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that

[TABLE]

where the series converges in both $T_{2}^{p(\cdot)}(\Gamma)$ and $T_{2}^{2}(\Gamma)$, and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})\leq C\|f\|_{T_{2}^{p(\cdot)}(\Gamma)}$, where $C$ does not depend on $f$. Note that as it was shown in (3.4), for every $j\in\mathbb{N}$, $a_{j}$ is a $(T_{2}^{p(\cdot)},r)$-atom for every $1<r<\infty$.

By Proposition 4.2, $(i)$ $\Pi_{M}$ can be extended to $T_{2}^{2}(\Gamma)$ as a bounded operator from $T_{2}^{2}(\Gamma)$ to $L^{2}(\Gamma)$. We have that

[TABLE]

We are going to see that $S_{L}(\Pi_{M}(f))\in L^{p(\cdot)}(\Gamma)$ and that

[TABLE]

for a certain $C>0$ that does not depend on $f$.

For every $j\in\mathbb{N}$, we write $\alpha_{j}=\Pi_{M}(a_{j})$. The operator $P$ is bounded in $L^{2}(\Gamma)$. Then, for every $k\in\mathbb{N}_{+}$,

[TABLE]

Hence, for every $k\in\mathbb{N}$,

[TABLE]

By using Fatou’s lemma we get

[TABLE]

We recall that, according to [8, Proposition 4.6], $S_{L}$ is a bounded operator from $L^{q}(\Gamma)$ into itself.

If $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, we define $S_{0}(B)=B(x_{B},8Mr_{B})$, and, for every $i\in\mathbb{N}_{+}$, $S_{i}(B)=B(x_{B},2^{i+3}Mr_{B})\setminus B(x_{B},2^{i+2}Mr_{B})$. As above we denote $\mathfrak{p}=\min\{1,p_{-}\}$. We can write

[TABLE]

By using $(i)$ we get

[TABLE]

Since $q\in[1,\infty)\cap(p_{+},\infty)$, by Lemmas 2.2, 2.3, (i), and 2.4, and Theorem 1.1 it follows that

[TABLE]

Let $j\in\mathbb{N}_{+}$. In order to study $S_{L}(\alpha_{j})$ in $\Gamma\setminus S_{0}(B_{j})$ we write

[TABLE]

where

[TABLE]

Note that $\operatorname{supp}(\beta_{j})\subset B(x_{B_{j}},2r_{B_{j}})$. We have that, for every $x\in\Gamma$,

[TABLE]

Here $S_{M,k}$ denotes the kernel of the operator $(I-P)^{M+1}P^{[k/2]}$, for every $k\in\mathbb{N}$, $k\geq 2$.

Since $\operatorname{supp}(\beta_{j})\subset B(x_{B_{j}},2r_{B_{j}})$, we obtain

[TABLE]

Also, $d(x,x_{B_{j}})\geq 8Mr_{B_{j}}\geq(M+4)r_{B_{j}}$, provided that $x\in\Gamma\setminus S_{0}(B_{j})$. Then, $(I-P)^{M+1}(\beta_{j})(x)=0$, for every $x\in\Gamma\setminus S_{0}(B_{j})$. We can write

[TABLE]

If $z\in B(x_{B_{j}},2r_{B_{j}})$, $y\in B(x,k)$, $k\leq d(x,x_{B_{j}})/2$ and $x\in\Gamma\setminus S_{0}(B_{j})$, then $d(y,z)\geq d(x,x_{B_{j}})/4$. Also, $B(x,d(x,x_{B_{j}}))\subset B(y,2d(x,x_{B_{j}}))$, provided that $y\in B(x,k)$, $k\leq d(x,x_{B_{j}})/2$. By using (1.1) and (1.3) we deduce that

[TABLE]

On the other hand, by invoking again (1.3), we get

[TABLE]

We have that

[TABLE]

and

[TABLE]

Estimates (4), (4) and Jensen’s inequality lead to

[TABLE]

If $x\in\Gamma\setminus S_{0}(B_{j})$, $B_{j}\subset B(x_{B_{j}},d(x,x_{B_{j}}))$, and then

[TABLE]

By combining the above estimates we obtain

[TABLE]

Hence,

[TABLE]

We now study $\|\beta_{j}\|_{2}$. We recall that

[TABLE]

Assume that $h\in L^{2}(\Gamma)$. We have that

[TABLE]

Since $P$ is a selfadjoint and contractive operator in $L^{2}(\Gamma)$ and $c_{k,M+1}\leq(k+1)^{M}$, $k\in\mathbb{N}$, we can write

[TABLE]

Since $a_{j}$ is a $(T_{2}^{p(\cdot)},2)$-atom associated with $B_{j}$ it follows that

[TABLE]

We get

[TABLE]

We choose $0<w<p_{-}$. Since $q\in[1,\infty)\cap(p_{+},\infty)$, according to Lemmas 2.2, 2.3, (i), and 2.4, we obtain

[TABLE]

Since $M>2D/p_{-}$, by choosing above $0<w<p_{-}$ such that $M>2D/w$, it follows from (4), (4) and (4) that

[TABLE]

∎

We recall the definitions of atoms in $H_{L}^{p(\cdot)}(\Gamma)$. Let $M\in\mathbb{N}_{+}$ and $1<r<\infty$. We say that $a\in L^{r}(\Gamma)$ is a $(r,p(\cdot),M)$-atom associated with a ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, when there exists $b\in L^{r}(\Gamma)$ satisfying that:

$(i)$ $a=L^{M}b$;

$(ii)$ $\operatorname{supp}L^{k}b\subset B$, $k=0,...,M$;

$(iii)$ $\|L^{k}b\|_{r}\leq(r_{B})^{M-k}(\mu(B))^{1/r}\|\chi_{B}\|_{p(\cdot)}^{-1}$, $k=0,...,M$.

If $f\in L^{2}(\Gamma)$, we say that $f$ has a $(2,p(\cdot),M)$-atomic representation when

[TABLE]

where, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(2,p(\cdot),M)$-atom associated with the ball $B_{j}$, satisfying that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$.

We define the atomic Hardy space $H_{L,M,at}^{p(\cdot)}(\Gamma)$ as follows: $f\in L^{2}(\Gamma)$ is in $\mathbb{H}_{L,M,at}^{p(\cdot)}(\Gamma)$ if and only if $f$ has a $(2,p(\cdot),M)$-atomic representation. On $\mathbb{H}_{L,M,at}^{p(\cdot)}(\Gamma)$ as usual we consider the quasinorm $\|\cdot\|_{H_{L,M,at}^{p(\cdot)}(\Gamma)}$ given by

[TABLE]

where

[TABLE]

for every sequence $\{\lambda_{j}\}_{j\in\mathbb{N}}$ of complex numbers and $\{B_{j}\}_{j\in\mathbb{N}}$ of balls, and where the infimum is taken over all the sequences $\{\lambda_{j}\}_{j\in\mathbb{N}}$ and $\{B_{j}\}_{j\in\mathbb{N}}$ such that $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$, being, for every $j\in\mathbb{N}$, $a_{j}$ a $(2,p(\cdot),M)$-atom associated with $B_{j}$.

The atomic Hardy space $H_{L,M,at}^{p(\cdot)}(\Gamma)$ is defined as the completion of $\mathbb{H}_{L,M,at}^{p(\cdot)}(\Gamma)$ with respect to the quasinorm $\|\cdot\|_{H_{L,M,at}^{p(\cdot)}(\Gamma)}$.

We now prove Theorem 1.2 by establishing the next two propositions.

Proposition 4.3**.**

Assume that $p\in\mathcal{P}^{\log}(\Gamma)$, $r\geq 2$, $r>p_{+}$ and $M\in\mathbb{N}_{+}$, $M>2D/p_{-}$. There exists $C>0$ satisfying that: if, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(r,p(\cdot),M)$-atom associated with the ball $B_{j}$ such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$, then the series $\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ converges in $H^{p(\cdot)}_{L}(\Gamma)$ and

[TABLE]

where $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$.

Proof.

In order to proof this fact we proceed as in the proof of Proposition 4.2, (ii). If $B=B(x_{B},r_{B})$ is a ball we recall that $S_{0}(B)=B(x_{B},8Mr_{B})$, and, for every $i\in\mathbb{N}_{+}$, $S_{i}(B)=B(x_{B},2^{i+3}Mr_{B})\setminus B(x_{B},2^{i+2}Mr_{B})$. Assume that, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $a_{j}$ is a $(r,p(\cdot),M)$-atom associated with the function $b_{j}$ and the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$. Let $\ell_{1},\ell_{2}\in\mathbb{N}$, $\ell_{1}<\ell_{2}$. We have that

[TABLE]

Since $S_{L}$ is a bounded operator in $L^{r}(\Gamma)$, the properties of the $(r,p(\cdot),M)$-atoms lead to

[TABLE]

Since $r>\max\{1,p_{+}\}$, by using Lemmas 2.2, 2.3, (i), and 2.4 we get

[TABLE]

Let $j\in\mathbb{N}_{+}$. We can write

[TABLE]

Here $S_{M,k}$ denotes as above the kernel of the operator $(I-P)^{M+1}P^{[k/2]}$, for every $k\in\mathbb{N}$, $k\geq 2$.

Since $\operatorname{supp}(b_{j})\subset B_{j}$, then $\operatorname{supp}((I-P)^{M+1}(b_{j}))\subset B(x_{B_{j}},(M+2)r_{B_{j}})$, so $(I-P)^{M+1}(b_{j})(x)=0$, $x\in\Gamma\setminus S_{0}(B_{j})$. Hence,

[TABLE]

We have that (see (4))

[TABLE]

Also, since $r\geq 2$, by using Jensen’s inequality we get

[TABLE]

The properties of $b_{j}$ and Lemma 2.3, (i), lead to

[TABLE]

where $0<R<p_{-}$.

According to Lemmas 2.2 and 2.4, since $r>\max\{p_{+},1\}$, we can write

[TABLE]

Then, we deduce that

[TABLE]

because $M>2D/p_{-}$.

Since the series \sum_{j\in\mathbb{N}}\Big{(}\frac{|\lambda_{j}|\chi_{B_{j}}}{\|\chi_{B_{j}}\|_{p(\cdot)}}\Big{)}^{\mathfrak{p}} converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$, we conclude that the series $\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$ converges in $H^{p(\cdot)}_{L}(\Gamma)$, and

[TABLE]

where $f=\sum_{j\in\mathbb{N}}\lambda_{j}a_{j}$.

∎

Our next objective is to see that each $f\in H^{p(\cdot)}_{L}(\Gamma)$ admits $(r,p(\cdot),M)$-representations.

Proposition 4.4**.**

Assume that $p\in\mathcal{P}^{\log}(\Gamma)$, $r\geq 2$, $r>p_{+}$, $M\in\mathbb{N}_{+}$ and $M>2D/p_{-}$. There exists $C>0$ such that, for every $f\in H^{p(\cdot)}_{L}(\Gamma)$ there exist, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(r,p(\cdot),M)$-atom $a_{j}$ associated with $B_{j}$ such that

[TABLE]

and

[TABLE]

Proof.

Let $f\in L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$. According to Proposition 4.1, we have that

[TABLE]

where $c_{k,1}=1$ and $c_{k,N+1}=\sum_{j=0}^{k}c_{j,N}$, $N\in\mathbb{N}_{+}$ and $k\in\mathbb{N}$.

We can write

[TABLE]

Since $f\in L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$, $k(I-P)P^{[k/2]}(f)\in T_{2}^{2}(\Gamma)\cap T_{2}^{p(\cdot)}(\Gamma)$, $k\in\mathbb{N}_{+}$, and according to Theorem 1.1 there exist, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(T^{p(\cdot)}_{2},r)$-atom $a_{j}$ associated to $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, satisfying that

[TABLE]

and

[TABLE]

where the series converges in both $T_{2}^{p(\cdot)}(\Gamma)$ and $T_{2}^{2}(\Gamma)$.

Note that, for every $m\in\mathbb{N}$, $\zeta_{m}=\sum_{j=0}^{m}\lambda_{j}a_{j}\in T_{c}(\Gamma)$. By Proposition 4.2 we have that

[TABLE]

in $L^{2}(\Gamma)$, and in $H^{p(\cdot)}_{L}(\Gamma)$. Also, for every $x\in\Gamma$,

[TABLE]

We are going to see that there exists $C>0$ such that, for every $j\in\mathbb{N}$, $C\Pi_{M}(a_{j})$ is a $(r,p(\cdot),M)$-atom.

Let $j\in\mathbb{N}$. We write $\Pi_{M}(a_{j})=(I-P)^{M}b_{j}$, where

[TABLE]

Note that, for every $\ell\in\mathbb{N}_{+}$, $a_{j}(.,\ell)\in L^{r}(\Gamma)$, because $\operatorname{supp}(a_{j}(.,\ell))$ is finite.

Hence, since $P$ is a contraction in $L^{r}(\Gamma)$, $b_{j}\in L^{r}(\Gamma)$. As it was proved in [10, p. 3463], $\operatorname{supp}L^{k}b_{j}\subset B(x_{B_{j}},M+2r_{B_{j}})\subset B(x_{B_{j}},(M+2)r_{B_{j}})=\mathbb{B}_{j}$, for every $k=0,...,M$.

In order to estimate $\|L^{k}b_{j}\|_{r}$, $k=0,...,M$, we proceed as in [8, p. 828-829]. Let $k\in\mathbb{N}$, $0\leq k\leq M$. Assume that $h\in L^{r^{\prime}}(\Gamma)$. By taking into account that $c_{R,M+1}\leq(R+1)^{M}$, $R\in\mathbb{N}$, the operator $P$ is selfadjoint, [10, Proposition 3.2, (a)], and Hölder’s inequality, we obtain

[TABLE]

where, as in (4.3) ,

[TABLE]

Since $\mathcal{S}_{k,L}$ defines a bounded operator from $L^{r^{\prime}}(\Gamma)$ into itself ([8, Proposition 4.6]), by using Lemma 2.3, (i), we get

[TABLE]

where $r_{\mathbb{B}_{j}}=(M+2)r_{B_{j}}$. We deduce that

[TABLE]

Thus our aim is proved.

On the other hand, by using Lemma 2.4, we get

[TABLE]

Suppose now that $f\in H^{p(\cdot)}_{L}(\Gamma)$. Since $H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$ is a dense subspace of $H^{p(\cdot)}_{L}(\Gamma)$, there exists a sequence $(f_{k})_{k\in\mathbb{N}}\subset H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$ such that $f_{0}=0$, $f_{k}\longrightarrow f$, as $k\rightarrow\infty$, in $H^{p(\cdot)}_{L}(\Gamma)$, and $\|f_{k}-f\|_{H^{p(\cdot)}_{L}(\Gamma)}\leq 2^{-k}\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}$, $k\in\mathbb{N}$. According to the first part of this proof, for every $k\in\mathbb{N}$,

[TABLE]

where, for every $j\in\mathbb{N}$, $\lambda_{j,k}\in\mathbb{C}$ and $a_{j,k}$ is a $(r,p(\cdot),M)$-atom associated with the ball $\mathbb{B}_{j,k}$, such that

[TABLE]

We have that

[TABLE]

Suppose that $\{(j_{\ell},k_{\ell})\}_{\ell\in\mathbb{N}}$ is an ordenation of $\mathbb{N}\times\mathbb{N}$. By using Minkowski’s inequality we can write

[TABLE]

According to Proposition 4.3 we deduce that the series $\sum_{\ell\in\mathbb{N}}\lambda_{j_{\ell},k_{\ell}}a_{j_{\ell},k_{\ell}}$ converges in $H^{p(\cdot)}_{L}(\Gamma)$.

We are going to see that $f=\sum_{\ell\in\mathbb{N}}\lambda_{j_{\ell},k_{\ell}}a_{j_{\ell},k_{\ell}}$.

Let $\epsilon>0$. There exists $L_{0}\in\mathbb{N}$ such that if $\ell_{0}\in\mathbb{N}$, $\ell_{0}\geq L_{0}$, then

[TABLE]

Let $\ell_{0}\in\mathbb{N}$, $\ell_{0}\geq L_{0}$. We define $\hat{k}_{0}=\max\{k_{\ell},\,\,\ell=0,...,\ell_{0}\}$ and $J_{0}=\max\{j_{\ell},\,\,\ell=0,...,\ell_{0}\}$. There exists $\hat{k}_{1}\in\mathbb{N}$ such that $\hat{k}_{1}>\hat{k}_{0}$ and

[TABLE]

We choose $J_{1}\in\mathbb{N}$, $J_{1}>J_{0}$, and such that

[TABLE]

for every $k=0,...,\hat{k}_{1}$.

Then, by defining $U_{0}=\{(j,k):j=1,...,J_{1},\,k=0,...,\hat{k}_{1}\}$ and $W_{0}=\{(j_{\ell},k_{\ell}):\,\,\ell\leq\ell_{0}\}$ we have that $W_{0}\subset U_{0}$ and

[TABLE]

By combining the above estimates we conclude that

[TABLE]

Note that the constant $C>0$ does not depend on $\ell_{0}$.

∎

As a consequence of Propositions 4.3 and 4.4 we deduce the following result.

Corollary 4.5**.**

Assume that $p\in\mathcal{P}^{\log}(\Gamma)$, $p_{+}<2$ and $M\in\mathbb{N}_{+}$, $M>2D/p_{-}$. Then, $H^{p(\cdot)}_{L}(\Gamma)=H^{p(\cdot)}_{L,M,at}(\Gamma)$ and the quasinorms are equivalent.

We now establish that $H^{p(\cdot)}_{L}(\Gamma)=L^{p(\cdot)}(\Gamma)$ provided that $p_{-}>1$. In order to prove this property we need to see that $S_{L}$ defines a bounded sublinear operator from $L^{p(\cdot)}(\Gamma)$ into itself when $p_{-}>1$.

We recall the definitions of Muckenhoupt classes of weights $A_{r}(\Gamma)$, $1\leq r\leq\infty$, in our setting. If $1<r<\infty$ we say that a function $w:\,\Gamma\longrightarrow(0,\infty)$ is in $A_{r}(\Gamma)$ when there exists $C>0$ such that, for every ball $B\subset\Gamma$,

[TABLE]

where $r^{\prime}=\frac{r}{r-1}$ is the exponent conjugated of $r$.

A function $w:\Gamma\longrightarrow(0,\infty)$ is said to be in $A_{1}(\Gamma)$ when there is a constant $C>0$ such that, for every ball $B\subset\Gamma$,

[TABLE]

By $A_{\infty}(\Gamma)$ is denoted, as usual, the union $\cup_{1\leq p<\infty}A_{p}(\Gamma)$.

If $1\leq r<\infty$ and $w:\Gamma\longrightarrow(0,\infty)$, $L^{r}(\Gamma,w)$ denotes the weighted $L^{p}$-space on $(\Gamma,\mu,d)$.

Proposition 4.6**.**

(i) Let $1<q<\infty$ and $w\in A_{q}(\Gamma)$. Then, $S_{L}$ defines a bounded operator from $L^{q}(\Gamma,w)$ into itself.

(ii) Let $p\in\mathcal{P}^{\log}(\Gamma)$ such that $p_{-}>1$. Then, $S_{L}$ defines a bounded operator from $L^{p(\cdot)}(\Gamma)$ into itself.

Proof.

(i) We adapt to our context some arguments developed in [38, Sections 3.1, 3.2, and 4.1.1]. We divide the proof in four steps and we sketch the proof of each of them.

$\mathbf{Step\,\,1}$. For every $\beta\geq 1$ we define the operator $\mathcal{A}^{\beta}$ as follows

[TABLE]

where $F:\Gamma\times\mathbb{N}_{+}\longrightarrow\mathbb{C}$.

Let $\beta\geq 1$, $1<s<\infty$ and $w\in A_{s}(\Gamma)$. There exists $C>0$ such that

[TABLE]

Note that (4.11) is non trivial when $\mathcal{A}(F)\in L^{s}(\Gamma,w)$.

To prove (4.11) we can show firstly that (4.11) holds when $s=2$ and $w\in A_{r}(\Gamma)$, $1<r<\infty$, and then to argue by using extrapolation (see the proof of [38, Proposition 3.2]).

$\mathbf{Step\,\,2}$. We define, for every $r\in(0,\infty)$, the maximal operator $C_{r}$ by

[TABLE]

For every $0<r<\infty$, $1<s<\infty$ and $w\in A_{s}(\Gamma)$, we have that there exists $C>0$ for which

[TABLE]

when $F:\Gamma\times\mathbb{N}_{+}\longrightarrow\mathbb{C}$. The proof of (4.12) can be completed by following some ideas developed in the proof of [38, Proposition 3.34]. We need to use a Whitney covering theorem in our setting that can be find in [36, Lemma 2.9] to establish (4.12) for every complex function $F$ with compact (finite) support in $\Gamma\times\mathbb{N}_{+}$ by employing a good $\lambda$ argument involving $\mathcal{A}^{\beta}$ for certain $\beta>1$ and using (4.11). Then, (4.12) can be proved for general complex valued functions $F$ on $\Gamma\times\mathbb{N}_{+}$ by approximation.

$\mathbf{Step\,\,3}$. We consider, for every $r\in(0,\infty)$, the maximal operator $\mathfrak{C}_{r}$ given by

[TABLE]

for every $x\in\Gamma$, where $f:\Gamma\longrightarrow\mathbb{C}$.

For every $1<r<\infty$,

[TABLE]

We can prove (4.13) arguing as in the proof of [38, Proposition 4.1]. We need to make some modifications in our setting. Let $1<r<\infty$, $x\in\Gamma$ and $f:\Gamma\longrightarrow\mathbb{C}$. We take a ball $B=B(x_{B},r_{B})$ such that $x\in B$. We decompose $f$ in the following way $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{B(x_{B},4r_{B})}$.

According to [8, Proposition 4.6, (ii)] $S_{L}$ is bounded from $L^{r}(\Gamma)$ into itself. Then, we can write

[TABLE]

We have taken into account that $\mu$ is doubling.

We define, for every $j\in\mathbb{N}$, $\mathcal{Z}_{j}(B)=B(x_{B},2^{j+1}r_{B})\setminus B(x_{B},2^{j}r_{B})$. By taking into account that $(I-P)f_{2}(y)=0$ when $y\in B(x_{B},2r_{B})$, and using (1.3) we get

[TABLE]

Since $B(x_{B},r_{B})\subset B(y,3r_{B})$, when $y\in B(x_{B},2r_{B})$, and $\mu$ is doubling it follows that

[TABLE]

for every $y\in B(x_{B},2r_{B})$, $j,k\in\mathbb{N}_{+}$, $k\leq r_{B}$ and $j\geq 2$.

Hence, we obtain

[TABLE]

By taking the supremum over all the balls $B$ such that $x\in B$ we establish (4.13).

$\mathbf{Step\,\,4}$. We now combine (4.12) and (4.13) to prove that $S_{L}$ is bounded from $L^{q}(\Gamma,w)$ into itself.

We choose $1<r<q$ such that $w\in A_{q/r}(\Gamma)$. Then we can write

[TABLE]

We have applied that $\mathcal{M}$ is bounded from $L^{s}(\Gamma,v)$ into itself, for every $s\in(1,\infty)$ and $v\in A_{s}(\Gamma)$.

(ii) This assertion can be proved by using (i) and extrapolation arguments as in [18, Theorem 1.3]. ∎

Note that Proposition 4.6 is an extension of [8, Proposition 4.6, (ii)].

Proposition 4.7**.**

Let $p\in\mathcal{P}^{\log}(\Gamma)$ such that $p_{-}>1$. Then, $H^{p(\cdot)}_{L}(\Gamma)=L^{p(\cdot)}(\Gamma)$, algebraic and topologically.

Proof.

Suppose that $f\in L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$. According to Proposition 4.4 by taking $r\geq 2$, $r>p_{+}$, $M\in\mathbb{N}_{+}$, and $>2D/p_{-}$, there exist, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(r,p(\cdot),M)$-atom $a_{j}$ associated with the ball $B_{j}$ such that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$, where the series converges in $H^{p(\cdot)}_{L}(\Gamma)$ and pointwisely, and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})\leq C\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}$. By using Lemma 2.2 we obtain, for every $j_{1},j_{2}\in\mathbb{N}$ and $j_{1}<j_{2}$,

[TABLE]

Note that in this case $\mathfrak{p}=1$. Since the series

[TABLE]

is convergent in $L^{p(\cdot)}(\Gamma)$, the series $\sum_{j=0}^{\infty}\lambda_{j}a_{j}$ also converges in $L^{p(\cdot)}(\Gamma)$ and

[TABLE]

So we have that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$ in the sense of convergence in $L^{p(\cdot)}(\Gamma)$.

On the other hand, if $f\in L^{2}(\Gamma)\cap L^{p(\cdot)}(\Gamma)$, from Proposition 4.6 we deduce that $S_{L}(f)\in L^{p(\cdot)}(\Gamma)$ and $\|S_{L}(f)\|_{p(\cdot)}\leq C\|f\|_{p(\cdot)}$, that is, $f\in L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$ and $\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}\leq C\|f\|_{p(\cdot)}$.

By taking closures we conclude that $H^{p(\cdot)}_{L}(\Gamma)=L^{p(\cdot)}(\Gamma)$ and the two norms are equivalent. ∎

We recall the definition of molecules. Let $p\in\mathcal{P}^{\log}(\Gamma)$ and $M\in\mathbb{N}_{+}$, $1<q<\infty$, and $\varepsilon>0$. We say that a function $m\in L^{q}(\Gamma)$ is a $(q,p(\cdot),M,\varepsilon)$-molecule when there exists a function $b\in L^{q}(\Gamma)$ and a ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, such that $m=L^{M}b$ and, for every $k=0,...,M$,

[TABLE]

where $\mathfrak{S}_{j}(B)=B(x_{B},2^{j+1}r_{B})\setminus B(x_{B},2^{j-1}r_{B})$, $j\in\mathbb{N}_{+}$ and $\mathfrak{S}_{0}(B)=B$.

Suppose that $M\in\mathbb{N}$, $1<q<\infty$ and $a$ is a $(q,p(\cdot),M)$-atom associated with the ball $B$ and the function $b\in L^{q}(\Gamma)$. Let $k\in\mathbb{N}$, $k=0,...,M$. Since $\operatorname{supp}L^{k}b\subset B$, $\|L^{k}b\|_{L^{q}(\mathfrak{S}_{j}(B))}=0$, $j\in\mathbb{N}_{+}$. Also, $\|L^{k}b\|_{L^{q}(\mathfrak{S}_{0}(B))}=\|L^{k}b\|_{q}\leq r_{B}^{M-k}(\mu(B))^{1/q}\|\chi_{B}\|_{p(\cdot)}^{-1}$. Hence, $a$ is a $(q,p(\cdot),M,\varepsilon)$-molecule, for every $\varepsilon>0$.

Also, there exists $C>0$ such that $C\Pi_{M}(a)$ is a $(q,p(\cdot),M,\varepsilon)$-molecule provided that $a$ is a $(T_{2}^{p(\cdot)},q)$-atom, $2\leq q<\infty$, $\varepsilon>0$, and $M\in\mathbb{N}_{+}$. Indeed, assume that $2\leq q<\infty$, $\varepsilon>0$, $M\in\mathbb{N}_{+}$, and $a$ is a $(T_{2}^{p(\cdot)},q)$-atom associated with the ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$. We define $\alpha=\Pi_{M}(a)$. We can write $\alpha=L^{M}b$, where

[TABLE]

Assume that $j_{0}\in\mathbb{N}_{+}$ such that $2^{j_{0}-1}>M+1$. As it was established in (4.10) we have that, for every $\ell\in\mathbb{N}$, $0\leq\ell\leq M$,

[TABLE]

Hence, according to Lemma 2.3, (i), there exists $C>0$ such that

[TABLE]

for every $\ell\in\mathbb{N}$, $0\leq\ell\leq M$, $j\in\mathbb{N}$ and $j\leq j_{0}$.

Suppose now $\ell\in\mathbb{N}$, $0\leq\ell\leq M$, and $j\in\mathbb{N}$, $j>j_{0}$. Let $h\in L^{q^{\prime}}(\Gamma)\cap L^{2}(\Gamma)$ such that $\operatorname{supp}(h)\subset\mathfrak{S}_{j}(B)$. By proceeding as in [8, Proposition 3.2, (a)] we obtain

[TABLE]

where, for every $x\in\Gamma$,

[TABLE]

Then,

[TABLE]

being $q^{\prime}=q/(q-1)$.

Note that $d(x,x_{B})<r_{B}$ provided that $d(x,y)<k$ and $(y,k)\in T(B)$. Also, since $\operatorname{supp}(h)\subset B(x_{B},2^{j+1}r_{B})\setminus B(x_{B},2^{j-1}r_{B})$, we have that

[TABLE]

Hence $L^{\ell}(h)(y)=0$, $y\in B$. Since $q\geq 2$, Jensen’s inequality leads to

[TABLE]

We deduce that

[TABLE]

By Lemma 2.3, (i), it follows that

[TABLE]

Note that $C$ does not depend on $b$ and $j$. Thus, we have shown that $\alpha$ is a $(q,p(\cdot),M,\varepsilon)$-molecule.

Assume now that $q\in[1,\infty)\cap(p_{+},\infty)$ and $m$ is a $(q,p(\cdot),M,\varepsilon)$-molecule associated with the ball $B=B(x_{B},r_{B})$ and the function $b$ as in the definition. By using Lemma 2.3, (ii), we deduce, for every $k=0,...,M$, that

[TABLE]

We are going to prove Theorem 1.3.

Proof of Theorem 1.3..

In order to establish this result we proceed as in the proof of Proposition 4.3. Suppose that, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and $m_{j}$ is a $(q,p(\cdot),M,\varepsilon)$-molecule associated to the function $b_{j}$ and the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$.

We are going to see that, for every $\epsilon>0$, there exists $j_{0}\in\mathbb{N}$ such that, for every $j_{1},j_{2}\in\mathbb{N}$, $j_{0}\leq j_{1}<j_{2}$,

[TABLE]

Let $j_{1},j_{2}\in\mathbb{N}$, $j_{1}<j_{2}$. We have that

[TABLE]

If $B=B(x_{B},r_{B})$ is a ball, we recall that $\mathcal{S}_{0}(B)=B(x_{B},8Mr_{B})$ and, for every $i\in\mathbb{N}_{+}$, $\mathcal{S}_{i}(B)=B(x_{B},2^{i+3}Mr_{B})\setminus B(x_{B},2^{i+2}Mr_{B})$.

Then, we get

[TABLE]

Since $S_{L}$ is bounded from $L^{q}(\Gamma)$ into itself ([8, p. Proposition 4.6, (ii)]), by (4) we get,

[TABLE]

By using Lemmas 2.2, 2.3, (i), and 2.4 we obtain

[TABLE]

because $q\geq 2$ and $q>p_{+}$.

We are going to estimate

[TABLE]

for every $i\in\mathbb{N}_{+}$.

We decompose, for every $j\in\mathbb{N}$, $S_{L}(m_{j})$ as follows

[TABLE]

Let $j\in\mathbb{N}$ and $i\in\mathbb{N}_{+}$. Since $q\geq 2$ and $q>p_{+}$, by (4) and (1.3), we obtain

[TABLE]

In order to study $\sum_{x\in S_{i}(B_{j})}I_{1,j}(x)^{q}\mu(x)$ we split $m_{j}$ in the following way

[TABLE]

Since $S_{L}$ is bounded in $L^{q}(\Gamma)$ we obtain

[TABLE]

where $\ell_{0}\in\mathbb{N}$, $2^{\ell_{0}}\leq M<2^{\ell_{0}+1}$. Molecular properties of $m_{j}$ and Lemma 2.3, (i), imply that

[TABLE]

According to (1.3), (4) and by taking into account that $\operatorname{supp}(L(m_{j,1}))\cap S_{i}(B_{j})=\varnothing$ and that $d(y,z)\geq c2^{i}r_{B_{j}}$ provided that $z\in B(x_{B_{j}},2^{i-3}Mr_{B_{j}})$, $x\in S_{i}(B_{j})$, $d(y,x)<k\leq d(x,x_{B_{j}})/2$, we can write

[TABLE]

On the other hand, $d(y,z)\geq c2^{i}r_{B_{j}}$ provided that $z\notin B(x_{B_{j}},2^{i+4}Mr_{B_{j}})$, $x\in S_{i}(B_{j})$, $d(y,x)<k\leq d(x,x_{B_{j}})/2$. By proceeding as above we deduce that

[TABLE]

By combining the above estimates we get

[TABLE]

By using Lemma 2.3, (i), we obtain

[TABLE]

where $2D/M<w<p_{-}$ and $D/\varepsilon<w$.

According to Lemmas 2.2 and 2.4 we get

[TABLE]

By (4.15) we conclude that

[TABLE]

Let $\epsilon>0$. Since the series

[TABLE]

converges in $L^{p(\cdot)/\mathfrak{p}}(\Gamma)$, there exists $j_{0}\in\mathbb{N}$ such that, for every $j_{1},j_{2}\in\mathbb{N}$, $j_{0}\leq j_{1}<j_{2}$,

[TABLE]

The completeness of $H^{p(\cdot)}_{L}(\Gamma)$ implies that the series $\sum_{j=1}^{\infty}\lambda_{j}m_{j}$ converges in $H^{p(\cdot)}_{L}(\Gamma)$. By writing $f=\sum_{j=1}^{\infty}\lambda_{j}m_{j}$ we have that

[TABLE]

∎

For every complex function $f$ defined on $\Gamma$ we consider the maximal function $\mathcal{M}_{+}(f)$ given by

[TABLE]

According to [26, (2.3)] we have that $|P^{k}(f)|\leq C\mathcal{M}(f)$, $k\in\mathbb{N}_{+}$. Then, $\mathcal{M}_{+}(f)\leq C\mathcal{M}(f)$ and the maximal function $\mathcal{M}_{+}$ defines a bounded (sublinear) operator from $L^{q}(\Gamma)$ into itself, for every $1<q\leq\infty$.

We say that $f\in L^{2}(\Gamma)$ is in $\mathbb{H}^{p(\cdot)}_{L,+}(\Gamma)$ when $\mathcal{M}_{+}(f)\in L^{p(\cdot)}(\Gamma)$. The maximal Hardy space $H^{p(\cdot)}_{L,+}(\Gamma)$ is the completion of $\mathbb{H}^{p(\cdot)}_{L,+}(\Gamma)$ with respect to the norm $\|\cdot\|_{H^{p(\cdot)}_{L,+}(\Gamma)}$ defined by

[TABLE]

Next we establish that $H^{p(\cdot)}_{L}(\Gamma)$ is a subspace of $H^{p(\cdot)}_{L,+}(\Gamma)$.

Proposition 4.8**.**

Let $p\in\mathcal{P}^{\log}(\Gamma)$. There exists $C>0$ such that, for every $f\in H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$ we have that

[TABLE]

Hence, $H^{p(\cdot)}_{L}(\Gamma)$ is contained in $H^{p(\cdot)}_{L,+}(\Gamma)$.

Proof.

Let $f\in H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$. We take $q=2\max\{2,p_{+}\}$ and $M\in\mathbb{N}$, $M>4D/p_{-}$. According to the proof of Proposition 4.4, there exists, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(q,p(\cdot),M)$-atom $a_{j}$ associated to the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that

[TABLE]

and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})\leq C\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}$. Since, for every $n\in\mathbb{N}$, $P^{n}$ is bounded in $L^{2}(\Gamma)$ we have that, for every $n\in\mathbb{N}$,

[TABLE]

and then

[TABLE]

Hence, we can write

[TABLE]

We deduce that

[TABLE]

Since $\mathcal{M}_{+}$ is bounded in $L^{q}(\Gamma)$ we obtain

[TABLE]

By Lemmas 2.2, 2.3, (i), and 2.4 we get

[TABLE]

Let $j\in\mathbb{N}$ and $i\in\mathbb{N}_{+}$. Since $P^{n}(a_{j})(x)=0$, when $x\in S_{i}(B_{j})$ and $n\in\mathbb{N}$, $n<2^{i}$, we can write

[TABLE]

Since $a_{j}$ is a $(q,p(\cdot),M)$-atom we can write $a_{j}=L^{M}b_{j}$, where $\mbox{supp }b_{j}\subset B_{j}$ and $\|b_{j}\|_{q}\leq r_{B_{j}}^{M}\mu(B_{j})^{1/q}\|\chi_{B_{j}}\|_{p(\cdot)}^{-1}$. According to (1.3) and (4), for every $n\in\mathbb{N}_{+}$, the kernel $\widetilde{p}_{n,M}$ of the operator $L^{M}P^{n}$ satisfies that

[TABLE]

We define $\mathbb{P}_{n,M}(x)=\sum_{y\in\Gamma}|\widetilde{p}_{n,M}(x,y)|$, $x\in\Gamma$. By using Jensen’s inequality we obtain, for every complex function $g$ defined on $\Gamma$,

[TABLE]

By (4) it follows that, for every $n\in\mathbb{N}_{+}$,

[TABLE]

We have taken into account that $d(x,y)\geq d(x,x_{B_{j}})/2\geq 4r_{B_{j}}$, when $x\in\Gamma\setminus S_{0}(B_{j})$ and $y\in B_{j}$.

By putting $D_{n}(x)=D$, when $d(x,x_{B_{j}})>\sqrt{n}$, and $D_{n}(x)=0$, when $d(x,x_{B_{j}})\leq\sqrt{n}$, we obtain

[TABLE]

Then,

[TABLE]

We take $2D/M<w<p_{-}$. We recall that $2D/M<p_{-}$. Lemmas 2.2 and 2.4 lead to

[TABLE]

Then,

[TABLE]

∎

We now consider other type of atoms (see [62, Definition 4.1]). We say that a complex valued function $a$ defined in $\Gamma$ is a $(2,p(\cdot))$-atom associated with the ball $B$ when the following properties are satisfied

(i) $\operatorname{supp}(a)\subset B$,

(ii) $\|a\|_{2}\leq(\mu(B))^{1/2}\|\chi_{B}\|_{p(\cdot)}^{-1}$,

(iii) $\sum_{x\in B}a(x)\mu(x)=0$.

Note that if $M\in\mathbb{N}_{+}$ and $a$ is a $(2,p(\cdot),M)$-atom associated with the ball $B$, then $a$ is a $(2,p(\cdot))$-atom ([10, Remark 3.4, (i)]).

A function $f\in L^{2}(\Gamma)$ is in the Hardy space $\mathbb{H}^{p(\cdot)}_{atom}(\Gamma)$ when, for every $j\in\mathbb{N}$, there exist $\lambda_{j}\in\mathbb{C}$ and a $(2,p(\cdot))$-atom $a_{j}$ associated to the ball $B_{j}$ such that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$, in $L^{2}(\Gamma)$, and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$. For every $f\in\mathbb{H}^{p(\cdot)}_{atom}(\Gamma)$ we define $\|f\|_{H^{p(\cdot)}_{atom}(\Gamma)}$ by

[TABLE]

where the infimum is taken over all the sequences $\{\lambda_{j}\}_{j=0}^{\infty}$ of complex numbers and $\{B_{j}\}_{j=0}^{\infty}$ of balls in $\Gamma$ such that, for every $j\in\mathbb{N}$, there exists a $(2,p(\cdot))$-atom associated with the ball $B_{j}$ and $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$, in $L^{2}(\Gamma)$. Thus, $\|.\|_{H^{p(\cdot)}_{atom}(\Gamma)}$ is a (quasi)norm in $\mathbb{H}^{p(\cdot)}_{atom}(\Gamma)$.

We define the space $H^{p(\cdot)}_{atom}(\Gamma)$ as the completion of $\mathbb{H}^{p(\cdot)}_{atom}(\Gamma)$ with respect to $\|.\|_{H^{p(\cdot)}_{atom}(\Gamma)}$. We have that $H^{p(\cdot)}_{L,M,at}(\Gamma)$ is a subspace of $H^{p(\cdot)}_{atom}(\Gamma)$.

We say that a graph $(\Gamma,\mu,d)$ has the Poincaré property when there exists $C>0$ such that for every $f:\;\Gamma\;\rightarrow\;\mathbb{R}$, $x_{0}\in\Gamma$ and $r_{0}>0$,

[TABLE]

where

[TABLE]

In ([50, Lemma 4]) it was established that if $\mu$ is doubling, $\Gamma$ satisfies the Poincaré property and property $\Gamma$ satisfies $\Delta(\alpha)$, with $\alpha>0$, then there exist $c_{3},C_{3}>0$ and $h\in(0,1)$ such that, for every $n\in\mathbb{N}$ and $x,y,y_{0}\in\Gamma$ being $d(y_{0},y)\leq\sqrt{n}$,

[TABLE]

We now prove that, under certain conditions, $H^{p(\cdot)}_{atom}(\Gamma)$ is a subspace of $H^{p(\cdot)}_{L,+}(\Gamma)$.

Proposition 4.9**.**

Let $p\in\mathcal{P}^{log}(\Gamma)$. Assume that $(\Gamma,\mu,d)$ has, in addition to the properties we have adopted from the beginning, the Poincaré property, $p_{+}<2$, and $\frac{D}{D+h}<\mathfrak{p}$, where $h$ is the one in (4.17). Then, $H^{p(\cdot)}_{atom}(\Gamma)$ is continuously contained in $H^{p(\cdot)}_{L,+}(\Gamma)$.

Proof.

Let $f\in H^{p(\cdot)}_{atom}(\Gamma)\cap L^{2}(\Gamma)$. Then, $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$, in $L^{2}(\Gamma)$, where for every $j\in\mathbb{N}$, there exist $\lambda_{j}\in\mathbb{C}$ and a $(2,p(\cdot))$-atom $a_{j}$ associated to the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$ such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})<\infty$. For every $k\in\mathbb{N}$,

[TABLE]

and

[TABLE]

Assume that $a$ is a $(2,p(\cdot))$-atom associated with the ball $B=B(x_{B},r_{B})$.

In the following we are inspired by some ideas developed in [49, p. 60-61].

We now assume that $x\not\in B(x_{B},2r_{B})$. We take firstly $k\in\mathbb{N}$ such that $k\leq r_{B}^{2}$. According to (1.2) we have that

[TABLE]

If $y\in B$, then $d(x,y)\geq\frac{1}{2}d(x,x_{B})$ so, we have

[TABLE]

We now assume that $k>r_{B}^{2}$. Since $\sum_{\Gamma}a(y)\mu(y)=0$, we can write

[TABLE]

By taking into account that $p(x,y)\mu(x)=p(y,x)\mu(y)$, $x,y\in\Gamma$, we also obtain, for every $k\in\mathbb{N}$, $p_{k}(x,y)\mu(x)=p_{k}(y,x)\mu(y)$, $x,y\in\Gamma$. From (4.17) and the relation we have just shown we deduce that

[TABLE]

Since $k>r_{B}^{2}$, we get

[TABLE]

We define $D_{k}(x)=0$, when $r_{B}+d(x,x_{B})<\sqrt{k}$, and $D_{k}(x)=D$, when $r_{B}+d(x,x_{B})\geq\sqrt{k}$. It follows that, for every $k\in\mathbb{N}_{+}$,

[TABLE]

We have used in the last line that $B\subset B(x,r_{B}+d(x,x_{B}))$.

Since $|P^{k}(a)|\leq C\mathcal{M}(a)$, $k\in\mathbb{N}$, the above estimations lead to

[TABLE]

We can write

[TABLE]

Let $j\in\mathbb{N}$. It is clear that $\mbox{supp}\;(\mathcal{M}(a_{j})\chi_{B(x_{B_{j}},2r_{B_{j}})})\subset B(x_{B_{j}},2r_{B_{j}})$. Also, according to Lemma 2.3, $(i)$, there exists $C>0$ such that,

[TABLE]

By Lemmas 2.2, 2.3, $(i)$, and 2.4 we deduce that

[TABLE]

By using Lemma 2.1 we obtain

[TABLE]

because $\mathfrak{p}>\frac{D}{D+h}$.

We conclude that

[TABLE]

∎

Remark 4.10**.**

In [55] (see also [25]) Hardy spaces associated with operators are compared with the classical ones. In [55, Theorem 6.1] it is proved that imposing certain conditions on the heat kernel is sufficient in order that Hardy operators, associated with the operator they are dealing with, coincide with the classical ones. In Yang and Zhou obtained in [59, Theorem 5.3] a version of [55, Theorem 6.1] in the variable exponent setting. As far as we know this question has not been investigated for Hardy spaces associated with operators on graphs even for constant exponents. Our purpose is to study this question in a foregoing paper.

5. Applications

In this section we present some applications of the atomic and molecular characterizations proved in the previous section. We establish $H^{p(\cdot)}_{L}(\Gamma)$-boundedness properties of some square functions, Riesz transforms, and L-spectral multipliers.

5.1. Square functions

Assume that $N\in\mathbb{N}_{+}$. The square function $G_{L,N}$ is defined as follows:

[TABLE]

where $f$ is complex function defined on $\Gamma$.

The sublinear operator $G_{L,N}$ is bounded from $L^{r}(\Gamma)$ into itself, for every $1<r<\infty$ [8, Proposition 4.6] (see also [6], [7], and [26]), and from $H^{r}_{L}(\Gamma)$ into $L^{r}(\Gamma)$, for every $0<r\leq 1$ ([8, Theorem 4.1] and [10, Theorem 6.1]). We now extend [10, Theorem 6.1] to our variable exponent setting.

Proposition 5.1**.**

Let $p\in\mathcal{P}^{\log}(\Gamma)$ and $N\in\mathbb{N}_{+}$. Then, the square function $G_{L,N}$ can be extended from $H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$ to $H^{p(\cdot)}_{L}(\Gamma)$ as a bounded operator from $H^{p(\cdot)}_{L}(\Gamma)$ to $L^{p(\cdot)}(\Gamma)$.

Proof.

We are going to see that there exists $C>0$ such that

[TABLE]

Assume that $f\in H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$, $M\in\mathbb{N}$, $M>D/p_{-}$, and $q=2\max\{2,p_{+}\}$. According to Proposition 4.4, there exist, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(q,p(\cdot),M)$-atom $a_{j}$ associated to the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$, where the series converges in $L^{2}(\Gamma)$ and in $H^{p(\cdot)}_{L}(\Gamma)$, and such that $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})\leq C\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}$. Here $C>0$ does not depend on $f$.

Our objective is to see that

[TABLE]

for a certain $C>0$ independent of $f$. In order to do this we adapt the arguments in the proof of Proposition 4.2.

As above if $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, we denote $S_{0}(B)=B(x_{B},8Mr_{B})$ and, for every $i\in\mathbb{N}_{+}$, $S_{i}(B)=B(x_{B},2^{i+3}Mr_{B})\setminus B(x_{B},2^{i+2}Mr_{B})$. Since $P$ is bounded in $L^{2}(\Gamma)$ we can write

[TABLE]

and then

[TABLE]

for every $k\in\mathbb{N}_{+}$. It follows that

[TABLE]

Since $G_{L,N}$ is bounded in $L^{q}(\Gamma)$ we deduce that

[TABLE]

By taking into account Lemmas 2.2, 2.3, (i), and 2.4 we get

[TABLE]

Let $j\in\mathbb{N}$ and $i\in\mathbb{N}_{+}$. We have that $a_{j}=L^{M}b_{j}$, where $b_{j}\in L^{q}(\Gamma)$ satisfying that, for every $\ell=0,...,M$,

(i) $\operatorname{supp}(L^{\ell}b_{j})\subset B_{j}$,

(ii) $\|L^{\ell}b_{j}\|_{q}\leq r_{B_{j}}^{M-\ell}\mu(B_{j})^{1/q}\|\chi_{B_{j}}\|_{p(\cdot)}^{-1}$.

Since $d(x,z)\geq d(x,x_{B_{j}})/2$, when $z\in B_{j}$ and $x\in S_{i}(B_{j})$, (1.3) leads to

[TABLE]

where $D_{k}=2D$, when $\sqrt{k}\leq 2^{i+1}Mr_{B_{j}}$, and $D_{k}=0$, when $\sqrt{k}>2^{i+1}Mr_{B_{j}}$.

Then,

[TABLE]

We can write

[TABLE]

According to Lemmas 2.2, 2.3, (i), and 2.4, we have that

[TABLE]

where $0<w<p_{-}$.

By choosing $2D/M<w<p_{-}$, we conclude that

[TABLE]

∎

5.2. Spectral multipliers for the discrete Laplacian $L$

The discrete Laplacian $L$ is a positive operator in $L^{2}(\Gamma)$ and $\|Lf\|_{2}\leq 2\|f\|_{2}$, $f\in L^{2}(\Gamma)$. Then, the spectrum of $L$ is contained in $[0,2]$ and there exists an spectral measure $E_{L}$ such that $L=\int_{[0,2]}\lambda dE_{L}(\lambda)$. If $F$ is a complex bounded measurable function defined in $[0,2]$ the operator $F(L)$ defined by

[TABLE]

is bounded from $L^{2}(\Gamma)$ into itself. The operators $F(L)$ in (5.1) are called spectral multipliers for the operator $L$. If the graph $\Gamma$ satisfies the property $\Delta(\alpha)$, for some $\alpha>0$, $-1$ is not in the spectrum of $P$ and then the exists $a_{0}\in(0,2)$ such that the spectral measure $E_{L}$ is supported in $[0,a_{0}]$ (see [24, Lemma 1.3]).

Our objective is to give conditions on $F$ that allow us to extend the operator $F(L)$ from $L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$ to $H^{p(\cdot)}_{L}(\Gamma)$ as a bounded operator from $H^{p(\cdot)}_{L}(\Gamma)$ into itself.

Let $s>0$. If $f:\mathbb{R}\longrightarrow\mathbb{C}$ is a $C^{[s]}(\mathbb{R})$-function we define $\|f\|_{\mathcal{C}^{s}}$ by

[TABLE]

where

[TABLE]

The space $\mathcal{C}^{s}(\mathbb{R})$ is defined by

[TABLE]

In order to get boundedness properties for spectral multipliers it is usual to consider the following property. We say that a complex bounded measurable function $F$ defined on $[0,\infty)$ satisfies the property $R_{s}$ with $s>0$ when

[TABLE]

Here, $\eta$ is a fixed $C^{\infty}((0,\infty))$ with compact support and it is not identically zero.

It is known that $F(L)$ can be extended from

(i) $L^{p}(\Gamma)\cap L^{2}(\Gamma)$ to $L^{p}(\Gamma)$ as a bounded operator from $L^{p}(\Gamma)$ into itself when $F$ satisfies $R_{s}$ with $s>D/2$, for every $1<p<\infty$ ([11, p. 266, (i)]).

(ii) $H^{p}_{L}(\Gamma)\cap L^{2}(\Gamma)$ to $H^{p}_{L}(\Gamma)$ as a bounded operator from $H^{p}_{L}(\Gamma)$ into itself when $F$ satisfies $R_{s}$ with $s>D(1/p-1/2)$, for every $0<p\leq 1$ ([10, Theorem 6.2] and [33]).

Next we generalize [10, Theorem 6.2] to our variable exponent setting.

Proposition 5.2**.**

Let $p\in\mathcal{P}^{\log}(\Gamma)$, $p_{+}<2$, and $s>2D/p_{-}$. Assume that $F$ is a complex bounded measurable function on $(0,\infty)$ satisfying $R_{s}$. Then, $F(L)$ is bounded from $H^{p(\cdot)}_{L}(\Gamma)$ into itself.

Proof.

We use the procedure in the proof of [10, Theorem 6.2] (see also [4]) with the necessary modifications due to the variable exponent context. We consider $M\in\mathbb{N}$, $M>2D/p_{-}$. Suppose that $f\in L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$. According to Proposition 4.4 there exists, for every $j\in\mathbb{N}$, $\lambda_{j}\in\mathbb{C}$ and a $(2,p(\cdot),2M)$-atom associated to the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, with $x_{B_{j}}\in\Gamma$ and $r_{B_{j}}\geq 1$, such that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$ in the sense of convergence in $L^{2}(\Gamma)$ and in $H^{p(\cdot)}_{L}(\Gamma)$, and

[TABLE]

Since $F(L)$ is bounded in $L^{2}(\Gamma)$ we have that

[TABLE]

We are going to see that there exists $C>0$ such that, for every $j\in\mathbb{N}$, $CF(L)(a_{j})$ is a $(2,p(\cdot),M,\varepsilon)$-molecule for some $\varepsilon>D/p_{-}$. When this fact is proved the proof finishes because from Theorem 1.3 it follows that $F(L)(f)\in H^{p(\cdot)}_{L}(\Gamma)$ and

[TABLE]

Suppose that $a$ is a $(2,p(\cdot),2M)$-atom associated to the ball $B=B(x_{B},r_{B})$, with $x_{B}\in\Gamma$ and $r_{B}\geq 1$, and the function $b\in L^{2}(\Gamma)$. It is clear that $F(L)(a)=L^{M}F(L)(L^{M}(b))$. Since $L$ and $F(L)$ are bounded operators in $L^{2}(\Gamma)$, the function $w=F(L)(L^{M}(b))\in L^{2}(\Gamma)$. We have to prove that, for some $\varepsilon>D/p_{-}$,

[TABLE]

for every $j\in\mathbb{N}$ and $k=0,...,M$, where $\mathfrak{S}_{j}(B)=B(x_{B},2^{j+1}r_{B})\setminus B(x_{B},2^{j-1}r_{B})$, $j\in\mathbb{N}_{+}$ and $\mathfrak{S}_{0}(B)=B$.

Let $k=0,...,M$. By using the properties of the function $b$ and once more that $L$ and $F(L)$ are bounded operators in $L^{2}(\Gamma)$ we deduce that

[TABLE]

Assume that $j\in\mathbb{N}_{+}$. We choose $0\leq\theta\in C^{\infty}(\mathbb{R})$ such that

[TABLE]

and, for every $\ell\in\mathbb{N}_{+}$, we define $\psi_{\ell}\in C^{\infty}(\mathbb{R})$ satisfying that

[TABLE]

and

[TABLE]

We can write the following decomposition [8, p. 840] (see also [4])

[TABLE]

It follows that

[TABLE]

where the convergence is understood in the space of operators in $L^{2}(\Gamma)$. In order to prove this we consider firstly a complex valued function $f$ defined on $\Gamma$ with support in a finite set $\Omega$. According to (1.2) we have that, for every $n\in\mathbb{N}_{+}$,

[TABLE]

Then, since $\mu(\Gamma)=\infty$, $\lim_{n\to\infty}P^{n}(f)(x)=0$, for every $x\in\Gamma$. By using monotone convergence theorem we deduce that

[TABLE]

Hence $\lim_{n\to\infty}\|P^{n}(f)\|_{2}=0$. A density argument allows us to conclude that $\lim_{n\to\infty}\|P^{n}(f)\|_{2}=0$, for every $f\in L^{2}(\Gamma)$.

As in [5, p. 839] we can see that

[TABLE]

Let $\eta>0$. There exists $\ell_{0}\in\mathbb{N}$ such that

[TABLE]

We have that

[TABLE]

Then,

[TABLE]

being $W=\sum_{\ell=0}^{\infty}F_{\ell}(L)(L^{M}b)$.

As in [10, (19) and (20)] we get

[TABLE]

Then, since $s>2D/p_{-}$, $p_{-}<2$ and according to Lemma 2.3, (i), if $0<w<p_{-}$ we obtain

[TABLE]

Let $\ell\in\mathbb{N}_{+}$. We take $0<w<p_{-}$ such that $\delta=s-D/w>0$. We get

[TABLE]

where $0<\varepsilon<\min\{M-D/w,s-D/w\}$ and $\varepsilon>D/p_{-}$. Note that $\varepsilon$ can be chosen in this way.

By combining (5.2) and (5.2) we obtain that

[TABLE]

Thus the proof is finished. ∎

5.3. Riesz transforms associated with $L$ on $\Gamma$

The gradient $\nabla f$ of a complex function $f$ defined on $\Gamma$ is defined by

[TABLE]

It is clear that $\nabla$ is a sublinear operator.

By using spectral theory we can define, for every $\alpha\in\mathbb{R}$, the operator $L^{\alpha}$ as follows

[TABLE]

If $\alpha\geq 0$, then $L^{\alpha}$ is a bounded operator in $L^{2}(\Gamma)$. However, if $\alpha<0$, the operator $L^{\alpha}$ is unbounded in $L^{2}(\Gamma)$ and the domain $D(L^{\alpha})$ of $L^{\alpha}$ is given by

[TABLE]

The square root $L^{1/2}$ of $L$ is bounded in $L^{2}(\Gamma)$ and $\|L^{1/2}f\|_{2}=\|\nabla f\|_{2}$, $f\in L^{2}(\Gamma)$. Thus the equality can be rewritten $\|\nabla L^{-1/2}f\|_{2}=\|f\|_{2}$, $f\in R_{2}(L^{1/2})$, where $R_{2}(L^{1/2})$ denotes the range of $L^{1/2}$ in $L^{2}(\Gamma)$. $R_{2}(L^{1/2})$ is dense in $L^{2}(\Gamma)$.

We define the Riesz transform $R_{L}$ associated with $L$ on $\Gamma$ by

[TABLE]

$R_{L}$ can be extended from $R_{2}(L^{1/2})$ to $L^{2}(\Gamma)$ as an isometry in $L^{2}(\Gamma)$.

In [7, Lemma 1.13] it was proved that, if $f\in R_{2}(L^{1/2})$ then

[TABLE]

in $L^{2}(\Gamma)$, where the sequence $\{\beta_{k}\}_{k\in\mathbb{N}}$ is defined by the expansion

[TABLE]

Hence, we have that

[TABLE]

If the properties (a), (c), and (1.2) for $x=y$ hold for $(\Gamma,\mu,d)$, $R_{L}$ can be extended from $L^{2}(\Gamma)\cap L^{q}(\Gamma)$ to $L^{q}(\Gamma)$ as a bounded operator from $L^{q}(\Gamma)$ into itself, for every $1<q\leq 2$, and from $L^{1}(\Gamma)$ into $L^{1,\infty}(\Gamma)$, the weak $L^{1}(\Gamma)$ (see [51]). In order that $R_{L}$ can be extended from $L^{2}(\Gamma)\cap L^{q}(\Gamma)$ to $L^{q}(\Gamma)$ as a bounded operator from $L^{q}(\Gamma)$ into itself when $q>2$ it is not sufficient with the properties (a), (c), and (1.2) for $x=y$, as the example of two copies of $\mathbb{Z}^{2}$ linked between with an edge shows [51, Section 4]. In [7, Theorem 1.3] it is established that if $(\Gamma,\mu,d)$ satisfies (a), (c), Poincaré inequality and there exists $C>0$ such that, for every $f\in L^{q_{0}}(\Gamma)$ and $k\in\mathbb{N}_{+}$,

[TABLE]

being $q_{0}>2$, then $R_{L}$ can be extended from $L^{2}(\Gamma)\cap L^{q}(\Gamma)$ to $L^{q}(\Gamma)$ as a bounded operator from $L^{q}(\Gamma)$ into itself when $2\leq q<q_{0}$. Weighted $L^{q}$-inequalities for the Riesz transform $R_{L}$ were established by Badr and Martell [6].

Boundedness properties for $R_{L}$ in Hardy spaces $H^{q}(\Gamma)$, $0<q\leq 1$, were proved in [8, Theorem 4.2] and [10, Theorem 6.5]. In the next proposition we extend those results in [8] and [10] to our variable exponent Hardy spaces $H^{p(\cdot)}_{L}(\Gamma)$.

Proposition 5.3**.**

Let $p\in\mathcal{P}^{\log}(\Gamma)$ such that $p_{+}<2$. Then, $R_{L}$ can be extended from $L^{2}(\Gamma)\cap H^{p(\cdot)}_{L}(\Gamma)$ to $H^{p(\cdot)}_{L}(\Gamma)$ as a bounded operator from $H^{p(\cdot)}_{L}(\Gamma)$ into $L^{p(\cdot)}(\Gamma)$.

Proof.

Suppose that $f\in H^{p(\cdot)}_{L}(\Gamma)\cap L^{2}(\Gamma)$. Let $M\in\mathbb{N}$ and $M>2D/p_{-}$. According to Proposition 4.4, for every $j\in\mathbb{N}$, there exist $\lambda_{j}\in\mathbb{C}$ and a $(2,p(\cdot),M)$-atom $a_{j}$ associated with the ball $B_{j}=B(x_{B_{j}},r_{B_{j}})$, such that $f=\sum_{j=0}^{\infty}\lambda_{j}a_{j}$ in $L^{2}(\Gamma)$ and in $H^{p(\cdot)}_{L}(\Gamma)$, and $\mathcal{A}(\{\lambda_{j}\},\{B_{j}\})\leq C\|f\|_{H^{p(\cdot)}_{L}(\Gamma)}$. Since $R_{L}$ is bounded in $L^{2}(\Gamma)$ we have that

[TABLE]

Also, we can write

[TABLE]

Since $R_{L}$ is bounded in $L^{2}(\Gamma)$ it follows that

[TABLE]

According to Lemmas 2.2, 2.3, (i), and 2.4 we deduce that

[TABLE]

Assume now that $a$ is a $(2,p(\cdot),M)$-atom associated with the ball $B=B(x_{B},r_{B})$ and the function $b\in L^{2}(\Gamma)$. Since $a=L^{M}b=L^{1/2}L^{M-1/2}b\in R_{2}(L^{1/2})$ we can write

[TABLE]

where $\widetilde{p_{k,M}}$ represents the kernel of the operator $L^{M}P^{k}$, for every $k\in\mathbb{N}$. Note that, for every $k\in\mathbb{N}$, $\operatorname{supp}(P^{k}a)\subset B(x_{B},r_{B}+k)$. Then, $\operatorname{supp}(\nabla P^{k}(a))\subset B(x_{B},r_{B}+k+1)$. We have that

[TABLE]

By [10, Lemma 6.6], Lemma 2.3, (i), and by proceeding as in [8, p. 838] we get, for every $i\in\mathbb{N}_{+}$,

[TABLE]

Here $0<w<p_{-}$. From Lemma 2.4, for every $i\in\mathbb{N}_{+}$, we deduce that

[TABLE]

Then, since $M>2D/p_{-}$ we conclude that

[TABLE]

and the proof is finished. ∎