Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates

Abstract

In this paper we introduce variable exponent local Hardy spaces \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) associated with a non-negative self-adjoint operator L. We assume that, for every \(t>0\), the operator \(e^{-tL}\) has an integral representation whose kernel satisfies a Gaussian upper bound. We define \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) by using an area square integral involving the semigroup \(\{e^{-tL}\}_{t>0}\). A molecular characterization of \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) is established. As an application of the molecular characterization, we prove that \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) coincides with the (global) Hardy space \(H_L^{p(\cdot )}({\mathbb {R}}^n)\) provided that 0 does not belong to the spectrum of L. Also, we show that \(h_L^{p(\cdot )}({\mathbb {R}}^n)=H_{L+I}^{p(\cdot )}({\mathbb {R}}^{n})\).

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The authors thank the referee for his/her corrections and insightful comments, which helped to improve the quality of the manuscript.

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Correspondence to Lourdes Rodríguez-Mesa.

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This paper is partially supported by MTM2016–79436–P. Third author was partially supported by PICT 2012–2568 (ANPCyT) and CAI+D 2015–026 (UNL).

The Space \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\)

The Space \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\)

In this appendix, we introduce the spaces \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) that can be seen as a variable exponent version of \(L_{\mathcal {Q}}^q({\mathbb {R}}^{n})\), \(0<q<\infty \), considered by [11, Section 3] and [12, Section 4]. After proving an atomic characterization of \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), we show that \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) coincides with \(H_I^{p(\cdot )}({\mathbb {R}}^n)\).

A sequence of cubes \({\mathcal {Q}}=\{Q_j\}_{j\in {\mathbb {N}}}\) is said to be a unit cube structure on \({\mathbb {R}}^{n}\) when the following properties are satisfied:

(i):

\(\cup _{j\in {\mathbb {N}}}\ Q_j={\mathbb {R}}^{n}\);

(ii):

\(Q_i^\circ \cap Q_j^\circ =\emptyset \), \(i,j\in {\mathbb {N}}\), \(i\ne j\);

(iii):

there exist \(0<\delta \le 1\) and a sequence \(\{B_j\}_{j\in {\mathbb {N}}}\) of balls in \({\mathbb {R}}^{n}\) with radius 1 such that \(\delta B_j\subset Q_j \subset B_j\), for each \(j\in {\mathbb {N}}\).

Here \(A^\circ \) denotes the interior of the set A.

Suppose that \({\mathcal {Q}}\) is a unit cube structure. The space \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) consists of all those measurable functions f on \({\mathbb {R}}^{n} \) for which the series

$$\begin{aligned} \sum \limits _{j\in {\mathbb {N}}} |Q_j|^{-1/2} \left\| f\chi _{Q_j}\right\| _2\chi _{Q_j} \end{aligned}$$

converges in \(L^{p(\cdot )}({\mathbb {R}}^{n})\). We define, for \(f \in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), the quantity

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}=\left\| \sum \limits _{j\in {\mathbb {N}}} |Q_j|^{-1/2} \left\| f\chi _{Q_j}\right\| _2\chi _{Q_j}\right\| _{p(\cdot )}. \end{aligned}$$

It is clear that \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\subset L^{2}_{{\mathrm{loc}}}({\mathbb {R}}^{n})\).

Note that, according to the property (iii) of \({\mathcal {Q}}\), for every \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\)

$$\begin{aligned} (\delta ^n\omega _n)^{1/2}\Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}\le \left\| \sum \limits _{j\in {\mathbb {N}}} \left\| f\chi _{Q_j}\right\| _2\chi _{Q_j}\right\| _{p(\cdot )}\le \omega _n^{1/2}\Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)} \end{aligned}$$
(A.1)

where \(\omega _n=|B(0,1)|\). Thus, a function \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) if and only if the series \(\sum \limits _{j\in {\mathbb {N}}} \left\| f\chi _{Q_j}\right\| _2 \chi _{Q_j}\) converges in \(L^{p(\cdot )}({\mathbb {R}}^{n})\).

Suppose now that \(\alpha :{\mathbb {N}}\rightarrow {\mathbb {N}}\) is a bijective mapping and \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). We define, for every \(m\in {\mathbb {N}}\), \(F_m=\sum \limits _{j=1}^m \left\| f\chi _{Q_{\alpha (j)}}\right\| _2\chi _{Q_{\alpha (j)}}\). It is clear that the sequence \(\{F_m\}_{m\in {\mathbb {N}}}\) is increasing and verifies \(F_m\le \sum \limits _{j\in {\mathbb {N}}} \left\| f\chi _{Q_j}\right\| _2\chi _{Q_j}\). Then, since \(p^+<\infty \), according to [23, Lemma 3.2.8 (c)], \(F_m\rightarrow \sum _{j\in {\mathbb {N}}} \left\| f\chi _{Q_{\alpha (j)}}\right\| _2\chi _{Q_{\alpha (j)}}\), when \(m \rightarrow \infty ,\) in \(L^{p(\cdot )}({\mathbb {R}}^{n})\), and

$$\begin{aligned} \sum _{j\in {\mathbb {N}}} \left\| f\chi _{Q_{\alpha (j)}}\right\| _2\chi _{Q_{\alpha (j)}}\le \sum _{j\in {\mathbb {N}}} \left\| f\chi _{Q_j}\right\| _2\chi _{Q_j}. \end{aligned}$$

Similarly, we can see that

$$\begin{aligned} \sum _{j\in {\mathbb {N}}}\left\| f\chi _{Q_j}\right\| _2\chi _{Q_j}\le \sum _{j\in {\mathbb {N}}} \left\| f\chi _{Q_{\alpha (j)}}\right\| _2\chi _{Q_{\alpha (j)}} . \end{aligned}$$

Hence, the series \(\sum \limits _{j\in {\mathbb {N}}}\left\| f\chi _{Q_{\alpha (j)}}\right\| _2\chi _{Q_{\alpha (j)}}\) converges unconditionally in \(L^{p(\cdot )}({\mathbb {R}}^{n})\). By [31, Theorem 2.4], for every \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), we know that

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}\sim \left( \sum _{j\in {\mathbb {N}}} \left( \left\| f\chi _{Q_j}\right\| _2\left\| \chi _{Q_j}\right\| _{p(\cdot )}\right) ^{p_\infty }\right) ^{1/p_\infty } \end{aligned}$$

where \(p_\infty \) was defined in (LH2).

The space \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) is endowed with the topology associated with the quasinorm \(\Vert \cdot \Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}\). Thus, \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) is a quasi-Banach space. Indeed, let \(\{f_n\}_{n\in {\mathbb {N}}}\) be a Cauchy sequence in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). Then, for every \(j\in {\mathbb {N}}\), \(\{f_n\}_{n\in {\mathbb {N}}}\) is a Cauchy sequence in \(L^2(Q_j)\), and there exists \(g_j\in L^2(Q_j)\) such that \(f_n\rightarrow g_j\), as \(n\rightarrow \infty \), in \(L^2(Q_j)\). We define a measurable function g as follows

$$\begin{aligned}g(x)=\left\{ \begin{array}{ll} g_j(x), &{}\quad x\in Q_j^\circ ,j\in {\mathbb {N}},\\ 0, &{}\quad \text {otherwise}. \end{array}\right. \end{aligned}$$

We are going to see that \(g\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and \(f_n\rightarrow g\), as \(n\rightarrow \infty \), in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\).

Let \(\varepsilon >0\). Since \(\{f_n\}_{n\in {\mathbb {N}}}\) is a Cauchy sequence in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), there exists \(n_0\in {\mathbb {N}}\) such that \(\Vert f_n-f_m\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}<\varepsilon \), for every \(n,m\in {\mathbb {N}}\) with \(n,m\ge n_0\). Also, for every \(\ell \in {\mathbb {N}}\), there exists \(n_\ell \in {\mathbb {N}}, n_\ell \ge n_0\) verifying that

$$\begin{aligned} |Q_j|^{-1/2}\Vert f_{n_\ell }-g\Vert _{L^2(Q_j)}\Vert \chi _{Q_j}\Vert _{p(\cdot )}<\varepsilon \ell ^{-1/{\underline{p}}},\quad j=1,\dots ,\ell . \end{aligned}$$

Then, for every \(\ell \in {\mathbb {N}}\), we can write

$$\begin{aligned}&\left\| \sum \limits _{j=1}^\ell |Q_j|^{-1/2}\Vert f_n-g\Vert _{L^2(Q_j)} \chi _{Q_j}\right\| _{p(\cdot )}^{{\underline{p}}}\\&\;\; \le \left\| \sum \limits _{j=1}^\ell |Q_j|^{-1/2}\Vert f_n-f_{n_\ell }\Vert _{L^2(Q_j)} \chi _{Q_j}\right\| _{p(\cdot )}^{{\underline{p}}}+\left\| \sum \limits _{j=1}^\ell |Q_j|^{-1/2}\Vert f_{n_\ell }-g\Vert _{L^2(Q_j)} \chi _{Q_j}\right\| _{p(\cdot )}^{{\underline{p}}}\\&\;\; \le \Vert f_n-f_{n_\ell }\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}^{{\underline{p}}}+\sum _{j=1}^\ell \left( |Q_j|^{-1/2}\Vert f_{n_\ell }-g\Vert _{L^2(Q_j)}\Vert \chi _{Q_j}\Vert _{p(\cdot )}\right) ^{\underline{p}}\\&\;\;\le 2\varepsilon ^{{\underline{p}}}, \quad n\in {\mathbb {N}}, \,n\ge n_0. \end{aligned}$$

By using the Monotone Convergence Theorem, we obtain that

$$\begin{aligned} \Vert f_n-g\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}=\left\| \sum _{j\in {\mathbb {N}}}|Q_j|^{-1/2}\Vert f_n-g\Vert _{L^2(Q_j)} \chi _{Q_j}\right\| _{p(\cdot )}\le 2^{1/{\underline{p}}}\varepsilon ,\quad n\in {\mathbb {N}}, \,n\ge n_0. \end{aligned}$$

Hence, \(g\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and \(f_n\rightarrow g\), as \(n\rightarrow \infty \), in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\).

The space \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) does not depend on the unit cube structure on \({\mathbb {R}}^{n}\) as we show in the following proposition.

Proposition A.1

Let \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) and suppose that \({\mathcal {Q}}\) and \({\mathcal {R}}\) are two unit cube structures on \({\mathbb {R}}^{n}\). Then, \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)=L^{p(\cdot )}_{{\mathcal {R}}}({\mathbb {R}}^{n}),\) algebraically and topologically.

Proof

We write \({\mathcal {Q}}=\{Q_j\}_{j\in {\mathbb {N}}}\) and \({\mathcal {R}}=\{R_j\}_{j\in {\mathbb {N}}}\). There exists \(J_0\in {\mathbb {N}}\) such that, for every \(j\in {\mathbb {N}}\), the set

$$\begin{aligned} A_j=\{i\in {\mathbb {N}}: R_i\cap Q_j\ne \emptyset \} \end{aligned}$$

has at most \(J_0\) elements. Indeed, let \(j\in {\mathbb {N}}\). We define \(S_j\) as the cube in \({\mathbb {R}}^{n}\) whose side-length is \(\ell (S_j)=6\) and which has \(Q_j\) in its center. Then, by using property (iii) of the unit cube structures, if \(k\in {\mathbb {N}}\) and \(R_k\cap ({\mathbb {R}}^{n}{\setminus } S_j)\ne \emptyset \), we have that \(R_k\cap Q_j= \emptyset \). If \(\delta _{{\mathcal {R}}}\) denotes the constant associated with \({\mathcal {R}}\) in property (iii), we deduce that

$$\begin{aligned} \mathop {\mathrm {card}}(A_j)\le (3/\delta _{{\mathcal {R}}})^n. \end{aligned}$$

Also, \(\cup _{i\in A_j} R_i\subset S_j\). According to [70, Lemma 2.6], we know that there exists \(C>0\) such that, for every \(j\in {\mathbb {N}}\),

$$\begin{aligned} \frac{\Vert \chi _{S_j}\Vert _{p(\cdot )}}{\Vert \chi _{R_i}\Vert _{p(\cdot )}}\le C\left( \frac{|S_j|}{|R_i|}\right) ^{1/p^-}\le C,\quad i\in A_j. \end{aligned}$$

Hence, given \(f\in L^{p(\cdot )}_{{\mathcal {R}}}({\mathbb {R}}^{n})\),

$$\begin{aligned} \sum \limits _{j\in {\mathbb {N}}}\left( \left\| f\chi _{Q_j}\right\| _2\left\| \chi _{Q_j}\right\| _{p(\cdot )}\right) ^{p_\infty }&\le \sum \limits _{j\in {\mathbb {N}}}\left( \left\| f\chi _{\cup _{i\in A_j} R_i}\right\| _2\left\| \chi _{S_j}\right\| _{p(\cdot )}\right) ^{p_\infty }\\&\le C \sum \limits _{j\in {\mathbb {N}}}\left( \sum \limits _{i\in A_j}\left\| f\chi _{R_i}\right\| _2\left\| \chi _{R_i}\right\| _{p(\cdot )}\right) ^{p_\infty }\\&\le C \sum \limits _{i\in {\mathbb {N}}}\left( \left\| f\chi _{R_i}\right\| _2\left\| \chi _{R_i}\right\| _{p(\cdot )}\right) ^{p_\infty }, \end{aligned}$$

where we have used that there exists \(J_1\in {\mathbb {N}}\) such that for every \(i\in {\mathbb {N}}\), the set \(\{j\in {\mathbb {N}}: R_i\cap Q_j\ne \emptyset \}\) has at most \(J_1\) elements. Thus, \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). By repeating the argument used above, interchanging the roles of \({\mathcal {Q}}\) and \({\mathcal {R}}\), we obtain the desired result.\(\square \)

Let B be a ball in \({\mathbb {R}}^{n}\) with radius greater or equal than 1. A measurable function a on \({\mathbb {R}}^{n}\) is called a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom associated with B when a is supported on B and \(\left\| a\right\| _2\le |B|^{1/2}\Vert \chi _B\Vert _{p(\cdot )}^{-1}\).

Next, we establish a characterization of \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) by using atoms.

Theorem A.2

Assume that \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) with \(p^+<2\), \({\mathcal {Q}}\) is a unit cube structure on \({\mathbb {R}}^{n}\), and f a measurable function on \({\mathbb {R}}^{n}\). The following assertions are equivalent.

(i):

\(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\);

(ii):

for every \(j\in {\mathbb {N}}\), there exist \(\lambda _j>0\) and a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom \(a_j\) associated with a ball \(B_j\) with radius greater or equal than 1 such that \(f=\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and almost everywhere in \({\mathbb {R}}^{n}\), and

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}} \right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}<\infty . \end{aligned}$$
Moreover, the quantities \(\Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}\) and \(\Vert f\Vert _{L^{p(\cdot )}_{{\mathcal {Q}},at}({\mathbb {R}}^{n})}\) are comparable, where

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}_{{\mathcal {Q}},at}({\mathbb {R}}^{n})}=\inf \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}} \right) ^{1/{\underline{p}}}\right\| _{p(\cdot )} \end{aligned}$$

and the infimum is taken over all the sequences \(\{(\lambda _j,B_j)\}_{j\in {\mathbb {N}}}\) verifying that \(\lambda _j>0\) and there exists a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom \(a_j\) associated with the ball \(B_j\) with radius greater or equal than 1 such that \(f=\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\).

Remark A.3

The statement (i) \(\Rightarrow \) (ii) is still true when \(p\in C^{\mathrm{log}}({\mathbb {R}}^n)\) with \(p_+<\infty \).

Proof

We write \({\mathcal {Q}}=\{Q_j\}_{j\in {\mathbb {N}}}\). Since \({\mathcal {Q}}\) is a unit cube structure, there exists \(0<\delta \le 1\) such that, for every \(j\in {\mathbb {N}}\), \(\delta B_j \subset Q_j \subset B_j\) for certain ball \(B_j\) with radius 1.

Suppose first that \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). For every \(j\in {\mathbb {N}}\) for which \(\left\| f\chi _{Q_j}\right\| _2\ne 0\), we define

$$\begin{aligned} \lambda _j=\frac{\left\| f\chi _{Q_j}\right\| _2 \Vert \chi _{B_j}\Vert _{p(\cdot )}}{|B_j|^{1/2}}\quad \text {and}\quad a_j=\frac{1}{\lambda _j}f\chi _{Q_j}, \end{aligned}$$

and in other case, \(\lambda _j=0\) and \(a_j=0\). Clearly, \(f=\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) a.e. in \({\mathbb {R}}^{n}\).

Let \(j_1,j_2\in {\mathbb {N}}\), \(j_1\le j_2\). We have that

$$\begin{aligned} \sum \limits _{j=j_1}^{j_2} \lambda _j a_j = \sum \limits _{j=j_1}^{j_2} f\chi _{Q_j}\quad \text {a.e. in }{\mathbb {R}}^n, \end{aligned}$$

and, by (A.1), and since \(Q_j^\circ \cap Q_k^\circ =\emptyset \), \(j,k\in {\mathbb {N}}\), \(j\ne k\),

$$\begin{aligned} \left\| \sum \limits _{j=j_1}^{j_2} \lambda _j a_j\right\| _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}=\left\| \sum \limits _{j=j_1}^{j_2} f \chi _{Q_j}\right\| _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}\le C\left\| \sum \limits _{j=j_1}^{j_2} \left\| f\chi _{Q_j}\right\| _2 \chi _{Q_j}\right\| _{p(\cdot )}. \end{aligned}$$

This means that \(\left\{ \sum _{j=1}^m \lambda _j a_j\right\} _{m\in {\mathbb {N}}}\) is a Cauchy sequence in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and the completeness of \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) implies that \(\lim _{m\rightarrow \infty } \sum _{j=1}^m \lambda _j a_j=g\) in the sense of \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), for certain \(g\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). Then, there exists an increasing function \(h:{\mathbb {N}}\rightarrow {\mathbb {N}}\) such that \(\lim _{m\rightarrow \infty } \sum _{j=1}^{h(m)} \lambda _j a_j=g\), a.e. in \({\mathbb {R}}^{n}\). Hence, \(f=g,\) a.e. in \({\mathbb {R}}^n\).

It is clear that, for each \(j\in {\mathbb {N}}\), \(a_j\) is a \(L_{{\mathcal {Q}}}^{p(\cdot )}\)-atom associated with \(B_j\).

Also, by [70, Lemma 2.6] and since \(Q_j\subset B_j\subset \delta ^{-1}Q_j\), we have that \(\Vert \chi _{B_j}\Vert _{p(\cdot )}\sim \Vert \chi _{Q_j}\Vert _{p(\cdot )}\sim \Vert \chi _{\delta ^{-1} Q_j}\Vert _{p(\cdot )}\), for each \(j\in {\mathbb {N}}\). Here, if \(\alpha >0\) and Q is a cube with sides of length \(\ell _Q\), \(\alpha Q\) represents the cube with the same center as Q and with sides of length \(\alpha \ell _Q\). By using [64, Remark 3.16] and that \(Q_j^\circ \cap Q_k^\circ =\emptyset \), \(j,k\in {\mathbb {N}}\), \(j\ne k\), we deduce that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}} \right\| _{p(\cdot )}&\le C \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{\delta ^{-1}Q_j}}{\Vert \chi _{\delta ^{-1}Q_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}} \right\| _{p(\cdot )}\\&\le C \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{Q_j}}{\Vert \chi _{Q_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}} \right\| _{p(\cdot )}\\&\le C \left\| \sum \limits _{j\in {\mathbb {N}}} \frac{\lambda _j \chi _{Q_j}}{\Vert \chi _{Q_j}\Vert _{p(\cdot )}}\right\| _{p(\cdot )}\le C\Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}<\infty . \end{aligned}$$

Conversely, assume that \(f=\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and almost everywhere in \({\mathbb {R}}^{n}\), where for every \(j\in {\mathbb {N}}\), \(\lambda _j>0\) and \(a_j\) is a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom associated with the ball \(B_j\) with radius greater or equal than 1 satisfying that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}} \right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}<\infty . \end{aligned}$$

For every \(j\in {\mathbb {N}}\), we define the set \(J_j=\{i\in {\mathbb {N}}: B_j\cap Q_i\ne \emptyset \}\). Then, we can write

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}&\le \left\| \sum \limits _{i\in {\mathbb {N}}} \sum \limits _{j\in {\mathbb {N}}} \lambda _j \left\| a_j\chi _{Q_i}\right\| _{2} \chi _{Q_i}\right\| _{p(\cdot )}=\left\| \sum \limits _{j\in {\mathbb {N}}} \lambda _j \sum \limits _{i\in J_j} \left\| a_j\chi _{Q_i}\right\| _{2} \chi _{Q_i}\right\| _{p(\cdot )}\\&=\left\| \sum \limits _{j\in {\mathbb {N}}} \lambda _j b_j\right\| _{p(\cdot )}\le \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \lambda _j b_j\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}, \end{aligned}$$

where \(b_j=\sum _{i\in J_j} \left\| a_j\chi _{Q_i}\right\| _{2} \chi _{Q_i}\), \(j\in {\mathbb {N}}\). It is clear that, since the radius of \(B_j\) is at least 1 and \(Q_j\) is contained in a ball of radius 1, then \(\mathop {\mathrm {supp}}b_j\subset 3B_j\). Thus,

$$\begin{aligned} \Vert b_j\Vert _2=\left( \sum \limits _{i\in J_j} \left\| a_j \chi _{Q_i}\right\| _2^2\right) ^{1/2}\le \left\| a_j\right\| _2\le |B_j|^{1/2}\Vert \chi _{B_j}\Vert _{p(\cdot )}^{-1}\le C|3B_j|^{1/2}\Vert \chi _{3B_j}\Vert _{p(\cdot )}^{-1}. \end{aligned}$$

Moreover, since \(p^+<2\), according to [53, Lemma 4.1] and [64, Remark 3.16] we have that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \lambda _j b_j\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}&\le C\left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{3B_j}}{\Vert \chi _{3B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}\\&\le C\left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}. \end{aligned}$$

The proof is thus finished. \(\square \)

Remark A.4

In the proof of Theorem A.2, it is established that, if \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) then, for every \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), there exist, for every \(j\in {\mathbb {N}}\), \(\lambda _j>0\) and a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom \(a_j\) associated with a ball \(B_j\) with radius 1 such that \(f=\sum \limits _{j\in {\mathbb {N}}} \lambda _j a_j\) in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) and

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}} \right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}\sim \Vert f\Vert _{L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)}. \end{aligned}$$

As a consequence of Theorem A.2, we obtain the following result.

Corollary A.5

Assume that \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) with \(p^+<2\), and \({\mathcal {Q}}\) is a unit cube structure on \({\mathbb {R}}^{n}\). Then, \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\) is contained in \(L^{p(\cdot )}({\mathbb {R}}^{n})\).

Proof

Let \(f\in L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). By virtue of Theorem A.2, we can write \(f=\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) in \(L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\), where for every \(j\in {\mathbb {N}}\), \(\lambda _j>0\) and \(a_j\) is a \(L^{p(\cdot )}_{\mathcal {Q}}\)-atom associated with a ball \(B_j\) with radius at least 1 satisfying that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}} \right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}<\infty . \end{aligned}$$

Let \(j_1,j_2\in {\mathbb {N}}\), \(j_1\le j_2\). Since \(p^+<2\), according to [69, Proposition 2.11], we deduce that

$$\begin{aligned} \left\| \sum \limits _{j=j_1}^{j_2} \lambda _j a_j\right\| _{p(\cdot )}\le \left\| \left( \sum \limits _{j=j_1}^{j_2} (\lambda _j a_j)^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}\le C\left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}. \end{aligned}$$

Then, the series \(\sum _{j\in {\mathbb {N}}} \lambda _j a_j\) defines a function g in \(L^{p(\cdot )}({\mathbb {R}}^{n})\). Moreover, we can find an increasing function \(h:{\mathbb {N}}\rightarrow {\mathbb {N}}\) such that \(\lim _{m\rightarrow \infty } \sum \limits _{j=1}^{h(m)} \lambda _j a_j=g,\) a.e. in \({\mathbb {R}}^{n}\). Hence, \(f=g,\) a.e. in \({\mathbb {R}}^n\), so \(f\in L^{p(\cdot )}({\mathbb {R}}^{n})\). \(\square \)

Remark A.6

The inclusion established in Corollary A.5 is proper. In order to prove this fact, we modify the function considered in [10, Remark 3.10]. Suppose that \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) with \(p^+<2\), so \(p^-<2\). We define

$$\begin{aligned} f(x)=|x|^{\sigma -\frac{n}{p^-}}\chi _{B(0,1)(x)},\quad x\in {\mathbb {R}}^{n}, \end{aligned}$$

where \(n(1/p^--1/p^+)<\sigma <n(1/p^--1/2)\). It is not hard to see that \(f\notin L^2_{{\mathrm{loc}}}({\mathbb {R}}^{n})\). Then, \(f\notin L^{p(\cdot )}_{\mathcal {Q}}({\mathbb {R}}^n)\). On the other hand, we have that

$$\begin{aligned} \int _{{\mathbb {R}}^{n}} |f(x)|^{p(x)} \mathrm{d}x=\int _{B(0,1)} |x|^{\big (\sigma -\frac{n}{p^-}\big )p(x)} \mathrm{d}x\le \int _{B(0,1)} |x|^{\big (\sigma -\frac{n}{p^-}\big )p^+} \mathrm{d}x<\infty . \end{aligned}$$

Hence, \(f\in L^{p(\cdot )}({\mathbb {R}}^{n})\).

As a consequence of Propositions 3.2 and 3.4, we can prove the following result.

Theorem A.7

Let \(p\in C^{\mathrm{log}}({\mathbb {R}}^{n})\) such that \(p^+<2\) and \({\mathcal {Q}}\) a unit cube structure. Then, \(L_{{\mathcal {Q}}}^{p(\cdot )}({\mathbb {R}}^n)=H_I^{p(\cdot )}({\mathbb {R}}^n),\) algebraically and topologically.

Proof

Note firstly that if a is a \(L_{\mathcal {Q}}^{p(\cdot )}\)-atom associated with a ball B with radius greater or equal than 1, then a is also a \((p(\cdot ),2,M)\)-atom, and then a \((p(\cdot ), 2, M,\varepsilon )\)-molecule associated with B, for every \(M\in {\mathbb {N}}_0\) and \(\varepsilon >0\).

Let \(f\in L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)\cap L^2({\mathbb {R}}^n)\). According to Remark A.4 we can write \(f=\sum _{j\in {\mathbb {N}}}\lambda _ja_j\) in \(L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)\), where for every \(j\in {\mathbb {N}}\), \(\lambda _j>0\), and \(a_j\) is a \(L_{\mathcal {Q}}^{p(\cdot )}\)-atom associated with the ball \(B_j\) having radius equal to 1, and satisfying that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}\le C\Vert f\Vert _{L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)}. \end{aligned}$$

Here C does not depend on f.

Since \(f\in L^2({\mathbb {R}}^n)=L_{\mathcal {Q}}^2({\mathbb {R}}^n)\), \(f=\sum _{j\in {\mathbb {N}}}\lambda _ja_j\) also in \(L^2({\mathbb {R}}^n)\). Then, according to Proposition 3.4, we obtain that \(f\in H_I^{p(\cdot )}({\mathbb {R}}^n)\) and \(\Vert f\Vert _{H_I^{p(\cdot )}({\mathbb {R}}^n)}\le C\Vert f\Vert _{L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)}\).

Assume now that \(f\in L^2({\mathbb {R}}^n)\cap H_I^{p(\cdot )}({\mathbb {R}}^n)\). We choose \(M\in {\mathbb {N}}_0\) such that \(2M>n(1/p^--1/2)\). By Proposition 3.2, we have that \(f=\sum _{j\in {\mathbb {N}}}\lambda _ja_j\) in \(L^2({\mathbb {R}}^n)\) and in \(H_{I,{\mathrm{at}}, M}^{p(\cdot )}({\mathbb {R}}^n)\), where for every \(j\in {\mathbb {N}}\), \(\lambda _j>0\) and \(a_j\) is a \((p(\cdot ), 2,M)\)-atom associated with the ball \(B_j\), satisfying that

$$\begin{aligned} \left\| \left( \sum \limits _{j\in {\mathbb {N}}} \left( \frac{\lambda _j \chi _{B_j}}{\Vert \chi _{B_j}\Vert _{p(\cdot )}}\right) ^{{\underline{p}}}\right) ^{1/{\underline{p}}}\right\| _{p(\cdot )}\le C\Vert f\Vert _{H_I^{p(\cdot )}({\mathbb {R}}^n)}. \end{aligned}$$

Here \(C>0\) does not depend on f.

If the radius \(r_{B_j}\) of \(B_j\) is greater or equal than 1, then \(a_j\) is also a \(L_{\mathcal {Q}}^{p(\cdot )}\)-atom associated with \(B_j\). Suppose now that a is a \((p(\cdot ), 2,M)\)-atom associated with a ball \(B=B(x_B,r_B)\) with \(x_B\in {\mathbb {R}}^n\) and \(0<r_B<1\). Our objective is to see that there exists \(C_0>0\) such that \(C_0a\) is a \(L_{\mathcal {Q}}^{p(\cdot )}\)-atom associated with \(B(x_B,1)\).

According to [66, Lemma 3.9], we have that

$$\begin{aligned} \Vert a\Vert _2&\le r_B^{2M}|B|^{1/2}\Vert \chi _B\Vert _{p(\cdot )}^{-1}\le Cr_B^{2M+n/2-n/p^-}|B(x_B,1)|^{1/2}\Vert \chi _{B(x_B,1)}\Vert _{p(\cdot )}^{-1}\\&\le C|B(x_B,1)|^{1/2}\Vert \chi _{B(x_B,1)}\Vert _{p(\cdot )}^{-1}. \end{aligned}$$

Hence, \(C_0a\) is a \(L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)\)-atom associated with \(B(x_B,1)\) for a certain \(C_0>0\) which does not depend on a.

We deduce from Theorem A.2 that \(f=\sum _{j\in {\mathbb {N}}}\lambda _ja_j\) in \(L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)\) and

$$\begin{aligned} \Vert f\Vert _{L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)}\le C\Vert f\Vert _{H_I^{p(\cdot )}({\mathbb {R}}^n)}. \end{aligned}$$

Since \(L^2({\mathbb {R}}^n)\cap {\mathbb {H}}\) is dense in \({\mathbb {H}}\), whenever \({\mathbb {H}}=L_{\mathcal {Q}}^{p(\cdot )}({\mathbb {R}}^n)\) or \({\mathbb {H}}=H_I^{p(\cdot )}({\mathbb {R}}^n)\), the proof is finished. \(\square \)

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Almeida, V., Betancor, J.J., Dalmasso, E. et al. Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates. J Geom Anal 30, 3275–3330 (2020). https://doi.org/10.1007/s12220-019-00199-y

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Keywords

  • Hardy spaces
  • Molecules
  • Local
  • Variable exponent

Mathematics Subject Classification

  • 42B35
  • 42B30
  • 42B25