Dual of Hardy-amalgam spaces and norms inequalities

TL;DR

This paper characterizes the dual spaces of generalized Hardy spaces with Wiener amalgam norms and demonstrates the boundedness of classical operators like Calderón-Zygmund, convolution, and Riesz potentials within these spaces.

Contribution

It introduces a new duality framework for Hardy spaces using Wiener amalgam norms and establishes boundedness of key operators in this setting.

Findings

Dual spaces characterized for Hardy spaces with Wiener amalgam norms

Classical operators are bounded in these generalized Hardy spaces

Provides a new perspective on Hardy space duality and operator boundedness

Abstract

We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces, we prove that some classical linear operators such as Calder\'on-Zygmund, convolution and Riesz potential operators are bounded.

Equations

$${\mathcal{M}}_{\phi }(f):=\underset{t>0}{\sup }|f\ast {\phi }_t|$$

$${\Vert f\Vert }_{q,p}:={\Vert {\left\{{\Vert f{\chi }_{{}_{Q_k}}\Vert }_q\right\}}_{k\in {\mathbb{Z}}^d}\Vert }_{{\ell }^p}<\infty ,$$

$$\mathfrak{M}(f)(x)=\underset{r>0}{\sup }|B(x,r){|}^{-1}{\int }_{B(x,r)}|f(y)|dy,x\in {\mathbb{R}}^d.$$

$${\mathcal{N}}_m(\psi ):=\underset{x\in {\mathbb{R}}^d}{\sup }(1+|x|{)}^m\sum _{|\beta |\le m}|{\partial }^{\beta }\psi (x)|$$

$$\sum _{n=0}^m{\Vert f_n\Vert }_{q,p}\le {\Vert \sum _{n=0}^m|f_n|\Vert }_{q,p}.$$

$$\sum _{n=0}^m{\Vert f_n\Vert }_{q,1}\le \sum _{k\in {\mathbb{Z}}^d}{\Vert \sum _{n=0}^m|f_n{\chi }_{{}_{Q_k}}|\Vert }_q={\Vert \sum _{n=0}^m|f_n|\Vert }_{q,1},$$

$$\sum _{n=0}^m{\Vert f_n\Vert }_{q,p}$$

$${\Vert {\left(\underset{n=0}{\sum ^{+\infty }}{\left[\mathfrak{M}(f_n)\right]}^u\right)}^{\frac{1}{u}}\Vert }_{q,p}\approx {\Vert {\left(\underset{n=0}{\sum ^{+\infty }}|f_n{|}^u\right)}^{\frac{1}{u}}\Vert }_{q,p},$$

$${\Vert \sum _{n\ge 0}{\left(\frac{|{\lambda }_n|}{{\Vert {\chi }_{{}_{Q^n}}\Vert }_q}\right)}^{\eta }{\chi }_{{}_{Q^n}}\Vert }_{\frac{q}{\eta },\frac{p}{\eta }}<+\infty ,$$

$${\Vert f\Vert }_{{\mathcal{H}}^{(q,p)}}\approx \inf {\Vert \sum _{n\ge 0}{\left(\frac{|{\lambda }_n|}{{\Vert {\chi }_{{}_{Q^n}}\Vert }_q}\right)}^{\eta }{\chi }_{{}_{Q^n}}\Vert }_{\frac{q}{\eta },\frac{p}{\eta }}^{\frac{1}{\eta }}.$$

$${\Vert f\Vert }_{{\mathcal{H}}_{fin}^{(q,p)}}:=\inf \left\{{\Vert \sum _{n=0}^m{\left(\frac{|{\lambda }_n|}{{\Vert {\chi }_{{}_{Q^n}}\Vert }_q}\right)}^{\eta }{\chi }_{{}_{Q^n}}\Vert }_{\frac{q}{\eta },\frac{p}{\eta }}^{\frac{1}{\eta }}:f=\sum _{n=0}^m{\lambda }_n{\mathfrak{a}}_n,m\in {\mathbb{Z}}_{+}\right\},$$

$${\Vert \sum _{n\ge 0}{\left(\frac{|{\lambda }_n|}{{\Vert {\chi }_{{}_{Q^n}}\Vert }_q}\right)}^{\eta }{\chi }_{{}_{Q^n}}\Vert }_{\frac{q}{\eta },\frac{p}{\eta }}<+\infty ,$$

$${\Vert f\Vert }_{{\mathcal{H}}^{(q,p)}}\stackrel{<}{\sim }{\Vert \sum _{n\ge 0}{\left(\frac{|{\lambda }_n|}{{\Vert {\chi }_{{}_{Q^n}}\Vert }_q}\right)}^{\eta }{\chi }_{{}_{Q^n}}\Vert }_{\frac{q}{\eta },\frac{p}{\eta }}^{\frac{1}{\eta }}.$$

$$L_{\mathrm{comp}}^{r,\delta }({\mathbb{R}}^d):=\left\{f\in L_{\mathrm{comp}}^r({\mathbb{R}}^d):{\int }_{{\mathbb{R}}^d}f(x)x^{\beta }dx=0,|\beta |\le \delta \right\}$$

$$L^{r,\delta }(Q):=\left\{f\in L^r(Q):{\int }_{Q}f(x)x^{\beta }dx=0,|\beta |\le \delta \right\},$$

$${\Vert f\Vert }_{L^r(Q)}:={\left({\int }_Q|f(x){|}^{r}dx\right)}^{\frac{1}{r}}.$$

$${\int }_Q\left[f(x)-P_Q^{\delta }(f)(x)\right]\mathfrak{q}(x)dx=0.$$

$$\underset{x\in Q}{\sup }|P_Q^{\delta }(f)(x)|\le \frac{C}{|Q|}{\int }_Q|f(x)|dx,$$

$${\left({\int }_Q|P_Q^{\delta }(f)(x){|}^{r}dx\right)}^{\frac{1}{r}}\le C{\left({\int }_Q|f(x){|}^{r}dx\right)}^{\frac{1}{r}},$$

$${\Vert f\Vert }_{{\mathcal{L}}_{r,\varphi ,\delta }}:=\underset{Q\in \mathcal{Q}}{\sup }\frac{1}{\varphi (Q)}{\left(\frac{1}{|Q|}{\int }_Q{\left|f(x)-P_Q^{\delta }(f)(x)\right|}^{r}dx\right)}^{\frac{1}{r}},$$

$${\Vert f\Vert }_{{\mathcal{L}}_{r,\varphi ,\delta }}:=\underset{Q\in \mathcal{Q}}{\sup }\frac{1}{\varphi (Q)}{\Vert f-P_Q^{\delta }(f)\Vert }_{L^{\infty }(Q)},$$

$${\varphi }_1(Q)=\frac{{\Vert {\chi }_{{}_Q}\Vert }_{q,p}}{|Q|},$$

$$\Vert T\Vert :={\Vert T\Vert }_{{\left({\mathcal{H}}^{(q,p)}\right)}^{\ast }}=\underset{\underset{{\Vert f\Vert }_{{\mathcal{H}}^{(q,p)}}\le 1}{f\in {\mathcal{H}}^{(q,p)}}}{\sup }|T(f)|.$$

$$T_g:f\in {\mathcal{H}}_{fin}^{(q,p)}⟼{\int }_{{\mathbb{R}}^d}g(x)f(x)dx,$$

$$\Vert T_g\Vert \le C{\Vert g\Vert }_{{\mathcal{L}}_{r^{\prime },{\varphi }_1,\delta }},$$

$$T(f)={\int }_{{\mathbb{R}}^d}g(x)f(x)dx,\text{ for all }f\in {\mathcal{H}}_{fin}^{(q,p)},$$

$${\Vert g\Vert }_{{\mathcal{L}}_{r^{\prime },{\varphi }_1,\delta }}\le C\Vert T\Vert ,$$

$$T_g(f)={\int }_{{\mathbb{R}}^d}g(x)f(x)dx,\forall f\in {\mathcal{H}}_{fin}^{(q,p)}.$$

$$|T_g(f)|$$

Full text

1. Introduction

Let $\varphi\in\mathcal{C}^{\infty}(\mathbb{R}^{d})$ with support on $B(0,1)$ such that $\int_{\mathbb{R}^{d}}\varphi dx=1$, where $B(0,1)$ is the unit open ball centered at [math] and $\mathcal{C}^{\infty}(\mathbb{R}^{d})$ denotes the space of infinitely differentiable complex valued functions on $\mathbb{R}^{d}$. For $t>0$, we denote by $\varphi_{t}$ the dilated function $\varphi_{t}(x)=t^{-d}\varphi(x/t)$, $x\in\mathbb{R}^{d}$. The Hardy-amalgam spaces $\mathcal{H}^{(q,p)}$ ( $0<q,p<\infty$) introduced in [1], are a generalization of the classical Hardy spaces $\mathcal{H}^{q}$ in the sense that they are the spaces of tempered distributions $f$ such that the maximal function

[TABLE]

belongs to the Wiener amalgam spaces $(L^{q},\ell^{p})(\mathbb{R}^{d})$.

We recall that for $0<p,q\leq\infty$, a locally integrable function $f$ belongs to the amalgam space $(L^{q},\ell^{p})$ if

[TABLE]

where $Q_{k}=k+\left[0,1\right)^{d}$ for $k\in\mathbb{Z}^{d}$. Endowed with the (quasi)-norm $\left\|\cdot\right\|_{q,p}$, the amalgam space $(L^{q},\ell^{p})$ is a complete space, and a Banach space for $1\leq q,p\leq+\infty$. These spaces coincide with the Lebesgue spaces whenever $p=q$, the class of spaces decreases with respect to the local component and increases with respect to the global component. Hardy-amalgam spaces are equal to $(L^{q},\ell^{p})$ spaces for $1<p,q<\infty$ and the dual space of $(L^{q},\ell^{p})$ is known to be $(L^{q^{\prime}},\ell^{p^{\prime}})$ under the assumption that $1\leq q,p<\infty$ where $\frac{1}{q}+\frac{1}{q^{\prime}}=1$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ with the usual conventions (see [3], [5], [8], [13] and [21]).

We also know that the Hardy-Littlewood maximal operator $\mathfrak{M}$ defined for a locally integrable function $f$ by

[TABLE]

is bounded in $(L^{q},\ell^{p})$ ($1<q,p<\infty$) (see [6] Theorems 4.2 and 4.5).

As for the classical Hardy spaces, not only the definition does not depend on the particular function $\varphi$, but the Hardy-amalgam spaces can be characterized in terms of grand maximal function, and we can also replace the regular function $\varphi$ by the Poisson kernel. The Hardy-Lorentz spaces $H^{q,\alpha}$ and $H^{p,s}$ defined by Wael Abu-Shammala and Alberto Torchinsky in [2], are continuously embedded in $\mathcal{H}^{(q,p)}$ provided $q\leq p$, $\alpha\leq q$ and $s\leq p$. In the case $0<q\leq\min(1,p)$ and $0<p<\infty$, we proved in [1] that $\mathcal{H}^{(q,p)}$ has an atomic characterization, with atoms which are exactly those of classical Hardy space.

It is also well known (see [19]) that in the classical Hardy spaces Riesz transforms $R_{j}$ ($j=1,\ldots,d$), convolution operators such as Riesz potential and Calderón-Zygmund type operators are bounded in $\mathcal{H}^{p}$ under appropriate conditions.

The aim of this paper is to characterize the dual of $\mathcal{H}^{(q,p)}$ spaces when $0<q\leq p\leq 1$, and to extend some of these results and others, known in the case of classical $\mathcal{H}^{q}$ spaces to Hardy-amalgam spaces $\mathcal{H}^{(q,p)}$, when $0<q\leq 1$ and $q\leq p<+\infty$, just as it is done for other generalizations of $\mathcal{H}^{q}$ (see [4], [9], [16], [17] and [18]). For this purpose, we organize this paper as follows.

In Section 2, we recall some properties of Wiener amalgam spaces $(L^{q},\ell^{p})$ and Hardy-amalgam spaces $\mathcal{H}^{(q,p)}$ obtained in [1] that we will need. We characterize the dual spaces of $\mathcal{H}^{(q,p)}$ ($0<q\leq p\leq 1$) in Section 3. The last section, consisting of three subsections, is devoted to some applications of the results of Sections 2 and 3. In the first two subsections, we respectively study the boundedness of Calderón-Zygmund and convolution operators in $\mathcal{H}^{(q,p)}$ spaces when $0<q\leq 1$ and $q\leq p<+\infty$. In the last subsection, we prove that the Riesz potential operators $I_{\alpha}$ extend to bounded operators from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q_{1},p_{1})}$ when $0<\alpha<d$, $0<q\leq 1$; $q\leq p<+\infty$; $0<q_{1}\leq p_{1}<+\infty$; $1/{q_{1}}=1/q-\alpha/d$ and $1/{p_{1}}=1/p-\alpha/d$.

Throughout the paper, we always let $\mathbb{N}=\left\{1,2,\ldots\right\}$ and $\mathbb{Z}_{+}=\left\{0,1,2,\ldots\right\}$. We use $\mathcal{S}:=\mathcal{S}(\mathbb{R}^{d})$ to denote the Schwartz class of rapidly decreasing smooth functions equipped with the topology defined by the family of norms $\left\{\mathcal{N}_{m}\right\}_{m\in\mathbb{Z}_{+}}$, where for all $m\in\mathbb{Z}_{+}$ and $\psi\in\mathcal{S}$,

[TABLE]

with $|\beta|=\beta_{1}+\ldots+\beta_{d}$, ${\partial}^{\beta}=\left(\partial/{\partial x_{1}}\right)^{\beta_{1}}\ldots\left(\partial/{\partial x_{d}}\right)^{\beta_{d}}$ for all $\beta=(\beta_{1},\ldots,\beta_{d})\in\mathbb{Z}_{+}^{d}$ and $|x|:=(x_{1}^{2}+\ldots+x_{d}^{2})^{1/2}$. The dual space of $\mathcal{S}$ is the space of tempered distributions denoted by $\mathcal{S}^{\prime}:=\mathcal{S}^{\prime}(\mathbb{R}^{d})$ equipped with the weak-$\ast$ topology. If $f\in\mathcal{S^{\prime}}$ and $\theta\in\mathcal{S}$, we denote the evaluation of $f$ on $\theta$ by $\left\langle f,\theta\right\rangle$. The letter $C$ will be used for non-negative constants independent of the relevant variables that may change from one occurrence to another. When a constant depends on some important parameters $\alpha,\gamma,\ldots$, we denote it by $C(\alpha,\gamma,\ldots)$. Constants with subscript, such as $C_{\alpha,\gamma,\ldots}$, do not change in different occurrences but depend on the parameters mentioned in it. We propose the following abbreviation \mathrm{\bf A}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\mathrm{\bf B} for the inequalities $\mathrm{\bf A}\leq C\mathrm{\bf B}$, where $C$ is a positive constant independent of the main parameters. If \mathrm{\bf A}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\mathrm{\bf B} and \mathrm{\bf B}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\mathrm{\bf A}, then we write $\mathrm{\bf A}\approx\mathrm{\bf B}$. For any given (quasi-) normed spaces $\mathcal{A}$ and $\mathcal{B}$ with the corresponding (quasi-) norms $\left\|\cdot\right\|_{\mathcal{A}}$ and $\left\|\cdot\right\|_{\mathcal{B}}$, the symbol $\mathcal{A}\hookrightarrow\mathcal{B}$ means that if $f\in\mathcal{A}$, then $f\in\mathcal{B}$ and \left\|f\right\|_{\mathcal{B}}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\left\|f\right\|_{\mathcal{A}}.

For $\lambda>0$ and a cube $Q\subset\mathbb{R}^{d}$ (by a cube we mean a cube whose edges are parallel to the coordinate axes), we write $\lambda Q$ for the cube with same center as $Q$ and side-length $\lambda$ times side-length of $Q$, while $\left\lfloor\lambda\right\rfloor$ stands for the greatest integer less or equal to $\lambda$. Also, for $x\in\mathbb{R}^{d}$ and $\ell>0$, $Q(x,\ell)$ will denote the cube centered at $x$ and side-length $\ell$. We use the same notations for balls. For a measurable set $E\subset\mathbb{R}^{d}$, we denote by $\chi_{{}_{E}}$ the characteristic function of $E$ and $\left|E\right|$ for its Lebesgue measure.

Throughout this paper, without loss of generality and unless otherwise specified, we assume that cubes are closed and denote by $\mathcal{Q}$ the set of all cubes.

Acknowledgement. The authors are very grateful to Aline Bonami for her support and to the referee for his meticulous reading of the manuscript.

2. Prerequisites on Wiener amalgam and Hardy-amalgam spaces

2.1. Wiener amalgam spaces $(L^{q},\ell^{p})$

Let $0<q<1$ and $0<p\leq 1$. We have the following well known reverse Minkowski’s inequality for $(L^{q},\ell^{p})$. We give a proof for the reader’s convenience.

Proposition 2.1**.**

Let $0<q<1$ and $0<p\leq 1$. For all finite sequence $\left\{f_{n}\right\}_{n=0}^{m}$ of elements of $(L^{q},\ell^{p})$, we have

[TABLE]

Proof.

Let $0<q<1$, $0<p\leq 1$ and a finite sequence $\left\{f_{n}\right\}_{n=0}^{m}$ of elements of $(L^{q},\ell^{p})$.

For $p=1$, it is immediate that

[TABLE]

by the reverse Minkowski’s inequality in $L^{q}$ (see [11], 1.1.5 (b), pp. 11-12).

Let $0<p<1$ and set $a_{n}:=\left\{\left\|f_{n}\chi_{{}_{Q_{k}}}\right\|_{q}\right\}_{k\in\mathbb{Z}^{d}}$, for $n=0,1,\ldots,m$. We have

[TABLE]

by the reverse Minkowski’s inequality in $\ell^{p}$ and $L^{q}$. This completes the proof. ∎

Notice that by monotone convergence theorem, (2.1) holds for any sequence in $(L^{q},\ell^{p})$, for $0<q<1$ and $0<p\leq 1$.

Another very useful result is the following.

Proposition 2.2** ([14], Proposition 11.12).**

Let $1<p,u\leq+\infty$ and $1<q<+\infty$. Then, for all sequence of measurable functions $\left\{f_{n}\right\}_{n\geq 0}$,

[TABLE]

with the implicit equivalent positive constants independent of $\left\{f_{n}\right\}_{n\geq 0}$ .

2.2. Hardy-amalgam spaces $\mathcal{H}^{(q,p)}$

Let $0<q,p<+\infty$ and $\varphi\in\mathcal{S}$ with $\text{supp}(\varphi)\subset B(0,1)$ and $\int_{\mathbb{R}^{d}}\varphi(x)dx=1$. The Hardy-amalgam spaces $\mathcal{H}^{(q,p)}$ are Banach spaces whenever $1\leq q,p<+\infty$ and quasi-Banach spaces otherwise (see [1], Proposition 3.8, p. 1912).

From now on, we assume that $0<q\leq 1$ and $q\leq p<+\infty$. Also, for simplicity, we just denote $\mathcal{M}_{\varphi}$ by $\mathcal{M}_{0}$. We recall the following definition which is also the one of an atom for $\mathcal{H}^{q}$.

Definition 2.3**.**

Let $1<r\leq+\infty$ and $s\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer. A function a is a $(q,r,s)$-atom on $\mathbb{R}^{d}$ for $\mathcal{H}^{(q,p)}$ if there exists a cube $Q$ such that

1. (1)

$\text{supp}(\textbf{a})\subset Q$* ,* 2. (2)

$\left\|\textbf{a}\right\|_{r}\leq|Q|^{\frac{1}{r}-\frac{1}{q}}$* ;* 3. (3)

$\int_{\mathbb{R}^{d}}x^{\beta}\textbf{a}(x)dx=0$, for all multi-indexes $\beta$ with $|\beta|\leq s$ .

We denote by $\mathcal{A}(q,r,s)$ the set of all $(\textbf{a},Q)$ such that a is a $(q,r,s)$-atom and $Q$ is the associated cube (with respect to Definition 2.3). As it was proved in ([1], Theorems 4.3 and 4.4), we have the following atomic decomposition theorem.

Theorem 2.4**.**

Let $0<\eta\leq 1$, $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer and $f\in\mathcal{S}^{\prime}$. Then, $f\in\mathcal{H}^{(q,p)}$ if and only if there exist a sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n\geq 0}$ in $\mathcal{A}(q,\infty,\delta)$ and a sequence of scalars $\left\{{\lambda}_{n}\right\}_{n\geq 0}$ such that

[TABLE]

and $f=\sum_{n\geq 0}{\lambda}_{n}\textbf{a}_{n}$ in $\mathcal{S^{\prime}}$ and $\mathcal{H}^{(q,p)}$.

Moreover, we have

[TABLE]

where the infimum is taken over all decompositions of $f$ in sum of multiple of elements of $\mathcal{A}(q;\infty,\delta)$.

Notice that in this theorem, one can replace $(q,\infty,\delta)$-atoms by $(q,r,\delta)$-atoms, provided that $\max(p,1)<r<\infty$ and $0<\eta<q$ (see[1], Theorem 4.6).

Let $1<r\leq+\infty$ and $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer. For simplicity, we denote by $\mathcal{H}_{fin}^{(q,p)}$ the subspace of $\mathcal{H}^{(q,p)}$ consisting of finite linear combinations of $(q,r,\delta)$-atoms. Then, $\mathcal{H}_{fin}^{(q,p)}$ is a dense subspace of $\mathcal{H}^{(q,p)}$ under the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$ (see [1], Remark 4.7, pp. 1922-1923).

For given $(q,r,\delta)$-atoms and $f$ belonging to the corresponding $\mathcal{H}_{fin}^{(q,p)}$ space, we put:

[TABLE]

if $r=+\infty$ and $0<\eta\leq 1$ is a fix real number, and

[TABLE]

if $\max\left\{p,1\right\}<r<+\infty$ and $0<\eta<q$ a fix real number. The infimum in both definitions, are taken over all finite decompositions of $f$ using $(q,r,\delta)$-atom $\mathfrak{a}_{n}$ supported on the cube $Q^{n}$ and $\left\|\cdot\right\|_{\mathcal{H}_{fin}^{(q,p)}}$ defines a quasi-norm on $\mathcal{H}_{fin}^{(q,p)}$.

With these definitions, we obtained the following result.

Theorem 2.5** ([1], Theorem 4.9).**

Let $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer and $\max\left\{p,1\right\}<r\leq+\infty$.

1. (1)

If $r<+\infty$ and $0<\eta<q$, then $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$ and $\left\|\cdot\right\|_{\mathcal{H}_{fin}^{(q,p)}}$ are equivalent on $\mathcal{H}_{fin}^{(q,p)}$. 2. (2)

If $r=+\infty$ and $0<\eta\leq 1$, then, $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$ and $\left\|\cdot\right\|_{\mathcal{H}_{fin}^{(q,p)}}$ are equivalent on $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$, where $\mathcal{C}(\mathbb{R}^{d})$ denotes the space of continuous complex functions on $\mathbb{R}^{d}$.

Furthermore, by [1], Lemma 4.10, p. 1929, if $\mathcal{H}_{fin}^{(q,p)}$ is the subspace of $\mathcal{H}^{(q,p)}$ consisting of finite linear combinations of $(q,\infty,\delta)$-atoms, then $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$ is dense in $\mathcal{H}^{(q,p)}$ with respect to the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$, since $\mathcal{H}_{fin}^{(q,p)}$ is dense in $\mathcal{H}^{(q,p)}$ in the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$.

We recall the following definition of a molecule.

Definition 2.6** ([1], Definition 5.1).**

Let $1<r\leq+\infty$ and $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer. A measurable function m is a $(q,r,\delta)$-molecule centered at a cube $Q$ if it satisfies the following conditions:

1. (1)

$\left\|\textbf{m}\chi_{{}_{\widetilde{Q}}}\right\|_{r}\leq|Q|^{\frac{1}{r}-\frac{1}{q}}$* , where $\widetilde{Q}:=2\sqrt{d}Q$,* 2. (2)

$|\textbf{m}(x)|\leq|Q|^{-\frac{1}{q}}\left(1+\frac{|x-x_{Q}|}{\ell(Q)}\right)^{-2d-2\delta-3}$, for all $x\notin\widetilde{Q}$, where $x_{Q}$ and $\ell(Q)$ respectively denote the center and the sidelength of $Q$. 3. (3)

$\int_{\mathbb{R}^{d}}x^{\beta}\textbf{m}(x)dx=0$, for all multi-indexes $\beta$ with $|\beta|\leq\delta$.

We denote by $\mathcal{M}\ell(q,r,\delta)$ the set of all $(\textbf{m},Q)$ such that m is a $(q,r,\delta)$-molecule centered at $Q$. We obtained in [1] the following theorem.

Theorem 2.7** ([1], Theorem 5.3).**

Let $\max\left\{p,1\right\}<r<+\infty$, $0<\eta<q$ and $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer. Then, for all sequences $\left\{(\textbf{m}_{n},Q^{n})\right\}_{n\geq 0}$ in $\mathcal{M}\ell(q,r,\delta)$ and for all sequences of scalars $\left\{{\lambda}_{n}\right\}_{n\geq 0}$ such that

[TABLE]

the series $f:=\sum_{n\geq 0}{\lambda}_{n}\textbf{m}_{n}$ converges in $\mathcal{S^{\prime}}$ and $\mathcal{H}^{(q,p)}$, with

[TABLE]

3. A characterization of the dual space of $\mathcal{H}^{(q,p)}$

3.1. Some results about Campanato’s spaces

In this section, we consider an integer $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$. Let $1<r\leq+\infty$. We denote by $L_{\mathrm{comp}}^{r}(\mathbb{R}^{d})$, the subspace of $L^{r}$- functions with compact support. We set

[TABLE]

and, for all cubes $Q$,

[TABLE]

where $L^{r}(Q)$ stands for the subspace of $L^{r}$-functions supported in $Q$. Also, $L^{r}(Q)$ endowed with the norm $\left\|\cdot\right\|_{L^{r}(Q)}$ defined by

[TABLE]

We point out that $L_{\mathrm{comp}}^{r,\delta}(\mathbb{R}^{d})$ is a dense subspace of $\mathcal{H}^{(q,p)}$. Indeed, $L_{\mathrm{comp}}^{r,\delta}(\mathbb{R}^{d})=\mathcal{H}_{fin}^{(q,p)}$ (the subspace of $\mathcal{H}^{(q,p)}$ consisting of finite linear combinations of $(q,r,\delta)$-atoms) and $\mathcal{H}_{fin}^{(q,p)}$ is dense in $\mathcal{H}^{(q,p)}$. Let $f\in L_{\mathrm{loc}}^{1}$ and $Q$ be a cube. Then, there exists an unique polynomial of $\mathcal{P_{\delta}}$ ($\mathcal{P_{\delta}}:=\mathcal{P_{\delta}}(\mathbb{R}^{d})$ is the space of polynomial functions of degree at most $\delta$) that we denote by $P_{Q}^{\delta}(f)$ such that, for all $\mathfrak{q}\in\mathcal{P_{\delta}}$,

[TABLE]

Relation (3.1) follows from Riesz’s representation theorem. From (3.1), we have $P_{Q}^{\delta}(g)=g$, if $g\in\mathcal{P_{\delta}}$.

Remark 3.1**.**

Let $1\leq r\leq+\infty$ and $f\in L_{\mathrm{loc}}^{r}$. We have, for all $Q\in\mathcal{Q}$,

[TABLE]

and

[TABLE]

where the constant $C>0$ does not depend on $Q$ and $f$.

For (3.2), see [15], Lemma 4.1. Estimate (3.3) follows from (3.2).

Definition 3.2** ([16], Definition 6.1 $\left(\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})\right)$).**

Let $1\leq r\leq+\infty$, a function $\phi:\mathcal{Q}\rightarrow(0,+\infty)$ and $f\in L_{\mathrm{loc}}^{r}$. One denotes

[TABLE]

when $r<+\infty$, and

[TABLE]

when $r=+\infty$. Then, the Campanato space $\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})$ is defined to be the set of all $f\in L_{\mathrm{loc}}^{r}$ such that $\left\|f\right\|_{\mathcal{L}_{r,\phi,\delta}}<+\infty$. One considers elements in $\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})$ modulo polynomials of degree $\delta$ so that $\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})$ is a Banach space. When one writes $f\in\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})$, then $f$ stands for the representative of $\left\{f+\mathfrak{q}:\ \mathfrak{q}\ \text{ is polynomial of degree}\ \delta\right\}$.

3.2. A dual of some Hardy-amalgam spaces

For simplicity, we just denote $\mathcal{L}_{r,\phi,\delta}(\mathbb{R}^{d})$ by $\mathcal{L}_{r,\phi,\delta}$ and define the function $\phi_{1}:\mathcal{Q}\rightarrow(0,+\infty)$ by

[TABLE]

for all $Q\in\mathcal{Q}$. For any $T\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$, we set

[TABLE]

Our duality result for $\mathcal{H}^{(q,p)}$, with $0<q\leq p\leq 1$, can be stated as follows.

Theorem 3.3**.**

Suppose that $0<q\leq p\leq 1$. Let $1<r\leq+\infty$. Then, $\left(\mathcal{H}^{(q,p)}\right)^{\ast}$ is isomorphic to $\mathcal{L}_{r^{\prime},\phi_{1},\delta}$ with equivalent norms, where $\frac{1}{r}+\frac{1}{r^{\prime}}=1$. More precisely, we have the following assertions:

1. (1)

Let $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$. Then, the mapping

[TABLE]

where $\mathcal{H}_{fin}^{(q,p)}$ is the subspace of $\mathcal{H}^{(q,p)}$ consisting of finite linear combinations of $(q,r,\delta)$-atoms, extends to a unique continuous linear functional on $\mathcal{H}^{(q,p)}$ such that

[TABLE]

where $C>0$ is a constant independent of $g$. 2. (2)

Conversely, for any $T\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$, there exists $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$ such that

[TABLE]

and

[TABLE]

where $C>0$ is a constant independent of $T$.

Proof.

For our proof, we borrow some ideas from [4], Theorem 8.3, [10], pp. 289-292; [15], Theorem 4.1 and [16], Theorem 7.5. We distinguish two cases: $1<r<+\infty$ and $r=+\infty$.

Suppose that $1<r<+\infty$. We first prove Assertion (1). Fix $0<\eta<q$. Let $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$ , where $\frac{1}{r}+\frac{1}{r^{\prime}}=1$. Consider the mapping $T_{g}$ defined on $\mathcal{H}_{fin}^{(q,p)}$ by

[TABLE]

It’s easy to verify that $T_{g}$ is well defined and linear. Let $f\in\mathcal{H}_{fin}^{(q,p)}$. Then, there exist a finite sequence $\left\{\left(a_{n},Q^{n}\right)\right\}_{n=0}^{m}$ in $\mathcal{A}(q,r,\delta)$ and a finite sequence of scalars $\left\{\lambda_{n}\right\}_{n=0}^{m}$ such that $f=\sum_{n=0}^{m}\lambda_{n}a_{n}$. Thus, by the vanishing condition of the atoms $a_{n}$, Hölder’s inequality and Proposition 2.1, we have

[TABLE]

Thus,

[TABLE]

by Theorem 2.5. This shows that $g\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$ and

[TABLE]

since $\mathcal{H}_{fin}^{(q,p)}$ is a dense subspace of $\mathcal{H}^{(q,p)}$ with respect to the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$.

Now, we prove the second assertion. Let $T\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$. Fix $0<\eta<q$. Let $Q$ be a cube. We first prove that

[TABLE]

For any $f\in L^{r,\delta}(Q)\backslash\left\{0\right\}$, set

[TABLE]

for all $x\in\mathbb{R}^{d}$. Clearly, $(a,Q)\in\mathcal{A}(q,r,\delta)$. Thus, $f\in\mathcal{H}_{fin}^{(q,p)}\subset\mathcal{H}^{(q,p)}$ and

[TABLE]

Hence for all $f\in L^{r,\delta}(Q)$,

[TABLE]

Thus, $T$ is a continuous linear functional on $L^{r,\delta}(Q)$, with

[TABLE]

This proves (3.6).

From (3.6), since $L^{r,\delta}(Q)$ is a subspace of $L^{r}(Q)$, by Hahn-Banach Theorem, there exists a continuous linear functional $T_{Q}$ which extends $T$ on $L^{r}(Q)$; namely $T_{Q}\in\left(L^{r}(Q)\right)^{\ast}$ and $T(f)=T_{Q}(f)$, for all $f\in L^{r,\delta}(Q)$. Also, by the duality $\left(L^{r}(Q)\right)^{\ast}=L^{r^{\prime}}(Q)$, that is there exists $g^{Q}\in L^{r^{\prime}}(Q)$ such that

[TABLE]

for all $f\in L^{r}(Q)$. Hence

[TABLE]

for all $f\in L^{r,\delta}(Q)$. Let $\left\{Q_{n}\right\}_{n\geq 1}$ be an increasing sequence of cubes which converges to $\mathbb{R}^{d}$; namely $Q_{n}\subset Q_{n+1}$, for all $n\geq 1$, and $\underset{n\geq 1}{\bigcup}Q_{n}=\mathbb{R}^{d}$. From the above result, it follows that for each cube $Q_{n}$, there exists $g^{Q_{n}}\in L^{r^{\prime}}(Q_{n})$, such that

[TABLE]

for all $f\in L^{r,\delta}(Q_{n})$.

Now, we construct a function $g\in L_{\mathrm{loc}}^{r^{\prime}}$ such that

[TABLE]

for all $f\in L^{r,\delta}(Q_{n})$ and all $n\geq 1$. Let us first assume that $f\in L^{r,\delta}(Q_{1})$. Then,

[TABLE]

by (3.9). Moreover, $L^{r,\delta}(Q_{1})\subset L^{r,\delta}(Q_{2})$. Hence $f\in L^{r,\delta}(Q_{2})$ and by (3.9),

[TABLE]

Thus, for all $f\in L^{r,\delta}(Q_{1})$,

[TABLE]

For any $h\in L^{r}(Q_{1})$, we have $h-P_{Q_{1}}^{\delta}(h)\chi_{{}_{Q_{1}}}\in L^{r,\delta}(Q_{1})$, by (3.1). Hence

[TABLE]

for all $h\in L^{r}(Q_{1})$, by (3.11). But

[TABLE]

(see [10], pp. 290-291). Therefore,

[TABLE]

for all $h\in L^{r}(Q_{1})$. It follows from (3.12) that

[TABLE]

for almost all $x\in Q_{1}$. Thus, after changing values of $g^{Q_{1}}$ (or $g^{Q_{2}}$) on a set of measure zero, we have

[TABLE]

Arguing as above, we obtain

[TABLE]

for all $x\in Q_{n}$ and all $n\geq 1$.

Set

[TABLE]

and

[TABLE]

for all $n\geq 1$. Then, we have

[TABLE]

for all $x\in Q_{n+1}$ and all $n\geq 1$, by (3.13). With (3.14) and (3.15), we define the function $g$ on $\mathbb{R}^{d}$ by

[TABLE]

for all $n\geq 1$. We have that $g\in L_{\mathrm{loc}}^{r^{\prime}}$, since $g_{n}\in L_{\mathrm{loc}}^{r^{\prime}}$, for all $n\geq 1$, by definition. Also, it’s easy to see that

[TABLE]

for all $f\in L^{r,\delta}(Q_{n})$ and all $n\geq 1$, by (3.9). Thus, the function $g$ satisfies (3.10).

To finish, we show that $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$ and

[TABLE]

for all $f\in\mathcal{H}_{fin}^{(q,p)}$. To prove (3.16), consider $f\in\mathcal{H}_{fin}^{(q,p)}$. We have $f\in L_{\mathrm{comp}}^{r,\delta}(\mathbb{R}^{d})$, since $L_{\mathrm{comp}}^{r,\delta}(\mathbb{R}^{d})=\mathcal{H}_{fin}^{(q,p)}$. Thus, by the definition of the sequence $\left\{Q_{n}\right\}_{n\geq 1}$, there exists an integer $n\geq 1$ such that $f\in L^{r,\delta}(Q_{n})$. Hence (3.16) holds, by (3.10). It remains to show that $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$. Let $Q$ be a cube and $f\in L^{r}(Q)$ such that $\left\|f\right\|_{L^{r}(Q)}\leq 1$. Set

[TABLE]

for all $x\in\mathbb{R}^{d}$, with $C_{f,Q,\delta,r}:=\left(1+\left\|\left(f-P_{Q}^{\delta}(f)\right)\chi_{{}_{Q}}\right\|_{r}\right)^{-1}$. Notice that $C_{f,Q,\delta,r}^{-1}\leq 2+C$, where $C>0$ is a constant independent of $f$ and $Q$, by (3.3) and the fact that $\left\|f\right\|_{L^{r}(Q)}\leq 1$. It is straightforward that $(a,Q)\in\mathcal{A}(q,r,\delta)$. Hence

[TABLE]

by (3.16). Since $T\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$, by using the vanishing condition of the atom $a$, Theorem 2.5, it follows that

[TABLE]

Combining this result with (3.17) and (3.1), we obtain

[TABLE]

Thus,

[TABLE]

From this inequality, it follows that

[TABLE]

Therefore, \left\|g\right\|_{\mathcal{L}_{r^{\prime},\phi_{1},\delta}}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\left\|T\right\| and $g\in\mathcal{L}_{r^{\prime},\phi_{1},\delta}$. This finishes the proof for the case where $1<r<+\infty$.

We consider now the case where $r=+\infty$. Let $g\in\mathcal{L}_{1,\phi_{1},\delta}$. Consider the mapping $T_{g}$ defined on $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$ by

[TABLE]

Fix $0<\eta<q$. Arguing as in the first case, we obtain

[TABLE]

by Theorem 2.5. This shows that $g\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$ and

[TABLE]

since $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$ is dense in $\mathcal{H}^{(q,p)}$ with respect to the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$. This proves the first assertion.

For the converse, let $T\in\left(\mathcal{H}^{(q,p)}\right)^{\ast}$. Consider $1<s<+\infty$. According to the first case, there exists a function $g\in L_{\mathrm{loc}}^{s^{\prime}}$ such that

[TABLE]

for all $f\in\mathcal{H}_{fin}^{(q,p)}$, where $\mathcal{H}_{fin}^{(q,p)}$ consists of finite linear combinations of $(q,s,\delta)$-atoms, $g\in\mathcal{L}_{s^{\prime},\phi_{1},\delta}$ and \left\|g\right\|_{\mathcal{L}_{s^{\prime},\phi_{1},\delta}}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\left\|T\right\|. Hence

[TABLE]

for all $f\in\mathcal{H}_{fin}^{(q,p)}$, where $\mathcal{H}_{fin}^{(q,p)}$ consists of finite linear combinations of $(q,\infty,\delta)$-atoms, by (3.18) and the fact that $(q,\infty,\delta)$-atoms are $(q,s,\delta)$-atoms. Also, \left\|g\right\|_{\mathcal{L}_{1,\phi_{1},\delta}}\raisebox{-4.30554pt}{{}\stackrel{{\scriptstyle\textstyle<}}{{\sim}}{}}\left\|T\right\|, since $\left\|g\right\|_{\mathcal{L}_{1,\phi_{1},\delta}}\leq\left\|g\right\|_{\mathcal{L}_{s^{\prime},\phi_{1},\delta}}$. This ends the proof in Case $r=+\infty$ and hence of Theorem 3.3. ∎

We mention that we are not able at the moment to characterize the dual of $\mathcal{H}^{(q,p)}$ when $0<q\leq 1<p<+\infty$.

4. Boundedness of some classical linear operators

In this section, unless otherwise specified, we assume that $0<q\leq 1$ and $q\leq p<+\infty$.

4.1. Calderón-Zygmund operator

Let $\triangle:=\left\{(x,x):\ x\in\mathbb{R}^{d}\right\}$ be the diagonal of $\mathbb{R}^{d}\times\mathbb{R}^{d}$. We say that a function $K:\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle\rightarrow\mathbb{C}$ is a standard kernel if there exist a constant $A>0$ and an exponent $\mu>0$ such that:

[TABLE]

[TABLE]

and

[TABLE]

We denote by $\mathcal{SK}(\mu,A)$ the class of all standard kernels $K$ associated with $\mu$ and $A$.

A classical example of standard kernel is the function $K$ defined on $\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle$ by

[TABLE]

where $k$ is a $\mathcal{C}^{\infty}$-function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ such that

[TABLE]

and $|\nabla^{m}k(x)|:=\left(\underset{|\beta|=m}{\sum}|({\partial}^{\beta}k)(x)|^{2}\right)^{\frac{1}{2}}$.

Definition 4.1** ([7], Definition 5.11).**

An operator $T$ is a (generalized) Calderón-Zygmund operator if

1. (1)

$T$* is bounded on $L^{2}$;* 2. (2)

There exists a standard kernel $K$ such that for $f\in L^{2}$ with compact support,

[TABLE]

It is well known that the Calderón-Zygmund operator $T$ is bounded on $L^{r}$, $1<r<+\infty$ (see [7], Theorem 5.10 and [12], Theorem 8.2.1). Our second main result is the following.

Theorem 4.2**.**

Let $T$ be a Calderón-Zygmund operator with kernel $K\in\mathcal{SK}(\mu,A)$. If $\frac{d}{d+\mu}<q\leq 1$, then $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$.

Proof.

Let $r>\max\left\{2;p\right\}$ be a real and $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer. Let $\mathcal{H}^{(q,p)}_{fin}$ be the space of finite linear combinations of $(q,r,\delta)$-atoms, and $f$ an element of this space. There exist a finite sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n=0}^{j}$ in $\mathcal{A}(q,r,\delta)$ and a finite sequence of scalars $\left\{{\lambda}_{n}\right\}_{n=0}^{j}$ such $f=\sum_{n=0}^{j}\lambda_{n}\textbf{a}_{n}$. Set $\widetilde{Q^{n}}:=2\sqrt{d}Q^{n}$, $n\in\left\{0,1,\ldots,j\right\}$, and denote by $x_{n}$ and $\ell_{n}$ respectively the center and side-length of $Q^{n}$. We have

[TABLE]

For $x\notin\widetilde{Q^{n}}$, we have

[TABLE]

by the vanishing condition of the atom $\textbf{a}_{n}$. Since, $|x-x_{n}|>2|y-x_{n}|$, for all $y\in Q^{n}$, it comes from (4.2) that

[TABLE]

But,

[TABLE]

Hence

[TABLE]

Therefore,

[TABLE]

for all $x\in\mathbb{R}^{d}$, so that

[TABLE]

Set

[TABLE]

Let us fix $0<\eta<q$. Proceeding as in the proof of Theorem 4.6 in [1], we obtain

[TABLE]

and

[TABLE]

since $1<\frac{q}{\eta}\leq\frac{p}{\eta}<\frac{r}{\eta}$ , $\text{supp}\left(|T(\textbf{a}_{n})|\chi_{{}_{\widetilde{Q^{n}}}}\right)^{\eta}\subset\widetilde{Q^{n}}$ and

[TABLE]

Hence

[TABLE]

It follows that

[TABLE]

By Theorem 2.5. Therefore, $T$ is bounded from $\mathcal{H}_{fin}^{(q,p)}$ to $(L^{q},\ell^{p})$ and the density of $\mathcal{H}_{fin}^{(q,p)}$ in $\mathcal{H}^{(q,p)}$ with respect to the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$ yields the result. ∎

If $T$ is a Calderón-Zymund operator with a sufficiently smooth standard kernel $K$, then the previous result (Theorem 4.2) can be improved in the sense of values of $q$. More precisely, we have the following.

Theorem 4.3**.**

Let $T$ be a Calderón-Zygmund operator with kernel $K\in\mathcal{SK}(\mu,A)$. Suppose that there exist an integer $m>0$ and a constant $C_{m}>0$ such that

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq m$ and all $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle$. Then, for $\frac{d}{d+m}<q\leq 1$, $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$.

Proof.

It is similar to the one of Theorem 4.2. The only difference is that we use Taylor’s formula to obtain Estimate (4.7), with $\mu$ replaced by $m$. More precisely, let $r>\max\left\{2;p\right\}$ be a real and $\delta\geq m-1$ be an integer and a a $(q,r,\delta)$-atom supported in $Q(x_{0},\ell)$. For $x\notin\widetilde{Q}=2\sqrt{d}Q$, we have

[TABLE]

with

[TABLE]

according to Taylor’s formula and the vanishing condition of the atom a. But then,

[TABLE]

so that

[TABLE]

∎

In Theorem 4.3, we do not yet know whether (4.8) can be replaced by the following condition:

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq m$ and all $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle$.

A consequence of Theorem 4.3 is the following result.

Corollary 4.4**.**

If a Calderón-Zygmund operator $T$ satisfies the condition (4.8) for any integer $m>0$, then $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$, for all $0<q\leq 1$.

Proof.

For any $0<q\leq 1$, by taking an integer $m>d\left(\frac{1}{q}-1\right)$, the result immediately follows from Theorem 4.3. ∎

We can see that Proposition 5.4 in [1] is a consequence of Corollary 4.4. In fact, it’s easy to verify that for any integer $m>0$, the kernel $K$ of the above mentioned proposition, satisfies

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq m$ and all $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle$, where $C(m)>0$ is a constant independent of $\beta$ and $(x,y)$.

In the next lemma, we give some sufficient smoothness conditions for the kernel $K$ of a Calderón-Zygmund operator $T$, under which this operator can be extended to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$.

Lemma 4.5**.**

Let $T$ be a Calderón-Zygmund operator with the following properties:

1. (1)

There exist an integer $\delta\geq 0$ and a constant $C_{\delta}>0$ such that

[TABLE]

for all $\beta$ with $|\beta|\leq d+2\delta+3$ and all $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\backslash\triangle$. 2. (2)

If $f\in L_{\mathrm{comp}}^{2}(\mathbb{R}^{d})$ and $\int_{\mathbb{R}^{d}}x^{\beta}f(x)dx=0$, for all multi-indexes $\beta$ with $|\beta|\leq d+2\delta+2$, then

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq\delta$.

If $\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor\leq\delta$, then, for any $(a,Q)\in\mathcal{A}(q,\infty,d+2\delta+2)$ and any $1<r<+\infty$,

[TABLE]

where $c_{1}>0$ is a constant independent of the atom $a$.

Proof.

Suppose that $\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor\leq\delta$. Let $(a,Q)\in\mathcal{A}(q,\infty,d+2\delta+2)$ and $1<r<+\infty$. Set $\widetilde{Q}:=2\sqrt{d}Q$ and denote by $x_{Q}$ the center of $Q$. Since $\mathcal{A}(q,\infty,d+2\delta+2)\subset\mathcal{A}(q,r,d+2\delta+2)$, we have

[TABLE]

Let us estimate $|T(a)(x)|$, for $x\notin\widetilde{Q}$. Fix $x\notin\widetilde{Q}$. Since $a\in L^{2}$ and $x\notin Q$, we have, by Taylor’s formula and the vanishing condition of the atom $a$,

[TABLE]

with

[TABLE]

and $\delta^{\prime}=d+2\delta+3$. Moreover, since

[TABLE]

we have

[TABLE]

since $|x-x_{Q}|\approx\ell(Q)+|x-x_{Q}|$. Thus,

[TABLE]

for all $x\notin\widetilde{Q}$. Therefore, there exists a positive constant $c_{1}$ independent of the atom $a$ such that

[TABLE]

and

[TABLE]

for all $x\notin\widetilde{Q}$, by (4.14) and (4.15). Also, it’s clear that

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq\delta$, by (4.13). Hence $\frac{1}{c_{1}}T(a)$ is a $(q,r,\delta)$-molecule centered at $Q$. ∎

Remark 4.6**.**

Lemma 4.5 also holds, for any $(a,Q)\in\mathcal{A}(q,s,d+2\delta+2)$ and any $1<r<\infty$, provided that $s\geq\max\left\{2;r\right\}$.

Theorem 4.7**.**

Let $T$ be a Calderón-Zygmund operator satisfying (4.12) and (4.13). If $\frac{d}{d+\delta+1}<q\leq 1$, then $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$.

Proof.

Noticing that $\frac{d}{d+\delta+1}<q\leq 1$ implies $\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor\leq\delta$, we consider the space $\mathcal{H}_{fin}^{(q,p)}$ of finite linear combination of $(q,\infty,d+2\delta+2)$-atoms, and $f\in\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$. Then, there exist a finite sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n=0}^{j}$ of elements of $\mathcal{A}(q,\infty,d+2\delta+2)$ and a finite sequence of scalars $\left\{{\lambda}_{n}\right\}_{n=0}^{j}$ such that $f=\sum_{n=0}^{j}\lambda_{n}\textbf{a}_{n}$. Let $\max\left\{1;p\right\}<r<+\infty$. By Lemma 4.5, $\frac{1}{c_{1}}T(\textbf{a}_{n})$ is a $(q,r,\delta)$-molecule centered at $Q^{n}$, for every $n\in\left\{0,1,\ldots,j\right\}$. Let us fix $0<\eta<q$. Clearly, we have

[TABLE]

Hence

[TABLE]

and

[TABLE]

by Theorem 2.7. It follows that

[TABLE]

by Theorem 2.5. Therefore, $T$ is bounded from $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$ to $\mathcal{H}^{(q,p)}$ and the density of $\mathcal{H}_{fin}^{(q,p)}\cap\mathcal{C}(\mathbb{R}^{d})$ in $\mathcal{H}^{(q,p)}$ with respect to the quasi-norm $\left\|\cdot\right\|_{\mathcal{H}^{(q,p)}}$ yields the result. ∎

By Remark 4.6, in the previous proof, we can also use the set $\mathcal{A}(q,r,d+2\delta+2)$, with $\max\left\{2;p\right\}<r<+\infty$, instead of $\mathcal{A}(q,\infty,d+2\delta+2)$.

Corollary 4.8**.**

Let $T$ be a Calderón-Zygmund operator which satisfies (4.12) and (4.13), for any integer $\delta\geq 0$. Then, for all $0<q\leq 1$, $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$.

4.2. Convolution Operator

Given a function $k$ defined and locally integrable on $\mathbb{R}^{d}\backslash\left\{0\right\}$, we say that a tempered distribution $K$ in $\mathbb{R}^{d}$ ($K\in\mathcal{S^{\prime}}:=\mathcal{S^{\prime}}(\mathbb{R}^{d})$) coincides with the function $k$ on $\mathbb{R}^{d}\backslash\left\{0\right\}$, if

[TABLE]

for all $\psi\in\mathcal{S}$, with $\text{supp}(\psi)\subset\mathbb{R}^{d}\backslash\left\{0\right\}$. It’s clear that (4.16) is not valid for any $\psi\in\mathcal{S}$. Here, we are interested in tempered distributions $K$ in $\mathbb{R}^{d}$ that coincide with a function $k$ on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and that have the form

[TABLE]

for some sequence $\sigma_{j}\downarrow 0$ as $j\rightarrow+\infty$ and independent of $\psi$. Also, we consider convolution operators $T$: $T(f)=K\ast f$, $f\in\mathcal{S}$. Thus, when $\widehat{K}\in L^{\infty}$, we have

[TABLE]

for all $f\in L^{2}$ with compact support and all $x\notin\text{supp}(f)$. For (4.18), see [20], Chap. 3, section 3, p. 113. From now on, unless otherwise specified, the letter $K$ stands both for the tempered distribution $K$ and the associated function $k$. We recall the following theorem.

Theorem 4.9** ([7], Theorem 5.1).**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ which coincides with a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

[TABLE]

then, for $1<r<+\infty$,

[TABLE]

and

[TABLE]

Notice that $C_{r}:=C(d,r,A,B)$ (see [7], p. 110) and (4.19) is understood as $|\widehat{K}(\xi)|\leq A$ for almost all $\xi\in\mathbb{R}^{d}$. For our results, we will need conditions stronger than (4.20) which are recalled in the following two remarks.

Remark 4.10** ([7], Proposition 5.2).**

The Hörmander condition (4.20) holds, if for every $x\neq 0$

[TABLE]

where $C>0$ is a constant independent of $x$.

Remark 4.11**.**

If there exist an integer $\delta>0$ and a constant $C_{\delta}>0$ such that

[TABLE]

for all multi-indexes $\beta$ with $|\beta|\leq\delta$ and all $x\neq 0$, then (4.21) holds. Therefore, (4.20) is also satisfied.

This follows from the definition of $|\nabla K(x)|^{2}$.

Theorem 4.12**.**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ which coincides wich a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

[TABLE]

for all $x\neq 0$. If $\frac{d}{d+1}<q\leq 1$, then the operator $T(f)=K\ast f$, for all $f\in\mathcal{S}$, extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$.

Proof.

Let $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor$ be an integer, $r>\max\left\{2;p\right\}$ be a real and $\mathcal{H}_{fin}^{(q,p)}$ be the space of finite linear combination of $(q,r,\delta)$-atoms. Fix $f\in\mathcal{H}_{fin}^{(q,p)}$. Then, there exist a finite sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n=0}^{j}$ of elements of $\mathcal{A}(q,r,\delta)$ and a finite sequence of scalars $\left\{{\lambda}_{n}\right\}_{n=0}^{j}$ such that $f=\sum_{n=0}^{j}\lambda_{n}\textbf{a}_{n}$. Moreover, $T$ is bounded on $L^{r}$, by Remark 4.10 and Theorem 4.9. Set $\widetilde{Q^{n}}:=2\sqrt{d}Q^{n}$, $n\in\left\{0,1,\ldots,j\right\}$, and denote by $x_{n}$ and $\ell_{n}$ respectively the center and side-length of $Q^{n}$. We have

[TABLE]

Set

[TABLE]

Let us fix $0<\eta<q$. Arguing as in the proof of Theorem 4.2, we have

[TABLE]

To estimate $J$, it suffices to show that

[TABLE]

for all $x\notin\widetilde{Q^{n}}$. For this, we consider $x\notin\widetilde{Q^{n}}$. It’s clear that $x\notin Q^{n}$. Hence $x-y\neq 0$ and $K(x-y)$ is well defined, for all $y\in Q^{n}$. Thus,

[TABLE]

by the vanishing condition of the atom $\textbf{a}_{n}$. Furthermore, by the mean value Theorem,

[TABLE]

where $\xi=\theta(x-x_{n})+(1-\theta)(x-y)$, for some $0<\theta<1$. Also, $|x-x_{n}|>2|y-x_{n}|$ and $|\xi|>\frac{1}{2}|x-x_{n}|>0$, for all $y\in Q^{n}$. Therefore,

[TABLE]

Thus, arguing as in the proof of Theorem 4.2, we obtain

[TABLE]

This proves (4.24). Proceeding as in the proof of Theorem 4.6 in [1], we have

[TABLE]

Hence

[TABLE]

And we finish the proof as the one of Theorem 4.2. ∎

If $K$ is sufficiently smooth, the conclusion of Theorem 4.12 can be improved in the sense of values of $q$. More precisely, we have the following result which is proved as Theorem 4.12 with appropriate Taylor formula. The proof is not given.

Theorem 4.13**.**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ that coincides with a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

and, there exist an integer $\delta>0$ and a constant $B>0$ such that

[TABLE]

for all $x\neq 0$ and all multi-indexes $\beta$ with $|\beta|\leq\delta$. If $\frac{d}{d+\delta}<q\leq 1$, then the operator $T(f)=K\ast f$, for all $f\in\mathcal{S}$, extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$.

We give some consequences of Theorem 4.13.

Corollary 4.14**.**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ which coincides with a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

and, for any integer $\delta>0$, satisfies (4.25). Then, for all $0<q\leq 1$, the operator $T(f)=K\ast f$, for all $f\in\mathcal{S}$, extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$.

Corollary 4.15**.**

For $d=1$, the Hilbert transform $H$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$, for all $0<q\leq 1$.

Proof.

We recall that the Hilbert transform $H(f)$ of $f\in\mathcal{S}$ is defined by

[TABLE]

where $K(x)=\frac{1}{\pi}\frac{1}{x}$ and

[TABLE]

See [11], Definition 4.1.1, p. 250. Also,

[TABLE]

for $\xi\in\mathbb{R}$, where \text{sgn}(\xi)=\left\{\begin{array}[]{lll}\frac{\xi}{|\xi|}\ ,&\text{ if }&\xi\neq 0\\ 0\ ,&\text{ if }&\xi=0.\end{array}\right.

For (4.27), see [11], (4.1.11), p. 253. Hence $|\widehat{K}(\xi)|\leq 1,$ for all $\xi\in\mathbb{R}$, and, for any integer $\delta>0$,

[TABLE]

for all $x\neq 0$ and all $\beta\leq\delta$. Thus, the distribution $K$ satisfies the conditions of Corollary 4.14 and the result follows. ∎

Corollary 4.16**.**

The Riesz transforms $R_{j}$, $1\leq j\leq d$, extend to bounded operators from $\mathcal{H}^{(q,p)}$ to $(L^{q},\ell^{p})$, for every $0<q\leq 1$.

Proof.

We recall that the Riesz transform $R_{j}(f)$, $1\leq j\leq d$, of $f\in\mathcal{S}$ is given by

[TABLE]

where $K_{j}(x)=\frac{\Gamma\left(\frac{d+1}{2}\right)}{\pi^{\frac{d+1}{2}}}\frac{x_{j}}{|x|^{d+1}}$ and

[TABLE]

See [11], Definition 4.1.13, p. 259. Also

[TABLE]

for $\xi\neq 0$. For (4.29), see [11], Proposition 4.1.14, p. 260 and [7], Chap. 4, (4.8), p. 76. Hence

[TABLE]

and, for any integer $\delta>0$,

[TABLE]

for all $x\neq 0$ and all multi-indexes $\beta$ with $|\beta|\leq\delta$. For (4.31), see [15], p. 120. Thus, the distribution $K_{j}$ satisfies the conditions of Corollary 4.14 and the result follows. ∎

In the next result, we prove that the convolution operator defined in Theorem 4.13 is bounded in Hardy-amalgam spaces.

Theorem 4.17**.**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ that coincides with a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

and, there exist an integer $\delta>0$ and a constant $B>0$ such that

[TABLE]

for all $x\neq 0$ and all multi-indexes $\beta$ with $|\beta|\leq\delta$. If $\frac{d}{d+\delta}<q\leq 1$, then the operator $T(f)=K\ast f$, for all $f\in\mathcal{S}$, extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$.

Proof.

Under the assumption that $\frac{d}{d+\delta}<q\leq 1$, we have $\delta\geq\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor+1$. Let $r>\max\left\{2;p\right\}$ be a real. We consider the space $\mathcal{H}_{fin}^{(q,p)}$ of finite linear combination of $(q,r,\delta)$-atoms. For $f\in\mathcal{H}_{fin}^{(q,p)}$, there exist a finite sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n=0}^{j}$ of elements of $\mathcal{A}(q,r,\delta)$ and a finite sequence of scalars $\left\{{\lambda}_{n}\right\}_{n=0}^{j}$ such that $f=\sum_{n=0}^{j}\lambda_{n}\textbf{a}_{n}$. Set $\widetilde{Q^{n}}:=2\sqrt{d}Q^{n}$, $n\in\left\{0,1,\ldots,j\right\}$, and denote by $x_{n}$ and $\ell_{n}$ respectively the center and side-length of $Q^{n}$. We have, for all $x\in\mathbb{R}^{d}$,

[TABLE]

Therefore,

[TABLE]

with

[TABLE]

Let us fix $0<\eta<q$. Since $\text{supp}\left(\mathcal{M}_{0}(T(\textbf{a}_{n}))\chi_{{}_{\widetilde{Q^{n}}}}\right)^{\eta}\subset\widetilde{Q^{n}}$ , $1<\frac{q}{\eta}\leq\frac{p}{\eta}<\frac{r}{\eta}$ and

[TABLE]

we obtain, as in the proof of Theorem 4.12,

[TABLE]

To estimate $J$, it suffices to prove that

[TABLE]

for all $x\notin\widetilde{Q^{n}}$. To show (4.33), we consider $x\notin\widetilde{Q^{n}}$. For every $t>0$, set $K^{(t)}=K\ast\varphi_{t}$. We have

[TABLE]

for all $z\in\mathbb{R}^{d}$, and

[TABLE]

for all $z\neq 0$ and all multi-indices $\beta$ with $|\beta|\leq\delta$. For (4.34) and (4.35), see [12], (6.7.22) and (6.7.23), pp. 100-101 (also see [20], Lemma, p. 117). Thus, using Taylor’s formula at order $\delta-1$ and the vanishing condition of the atom $\textbf{a}_{n}$, we obtain

[TABLE]

with

[TABLE]

for all $y\in Q^{n}$, by (4.35). Hence

[TABLE]

and consequently

[TABLE]

Finally,

[TABLE]

and we end the proof as the one of Theorem 4.2. ∎

We give some consequences of Theorem 4.17.

Corollary 4.18**.**

Let $K$ be a tempered distribution in $\mathbb{R}^{d}$ which coincides with a locally integrable function on $\mathbb{R}^{d}\backslash\left\{0\right\}$ and is such that

[TABLE]

and, for any integer $\delta>0$, satisfies (4.32). Then, the operator $T(f)=K\ast f$, for all $f\in\mathcal{S}$, extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$, for all $0<q\leq 1$.

Corollary 4.19**.**

The Riesz transforms $R_{j}$, $j\in\left\{1,2,\ldots,d\right\}$, extend to bounded operators on $\mathcal{H}^{(q,p)}$, for all $0<q\leq 1$. In particular, when $d=1$, the Hilbert transform $H$ extends to a bounded operator on $\mathcal{H}^{(q,p)}$, for all $0<q\leq 1$.

Proof.

Corollary 4.19 immediately follows from corollary 4.18. ∎

Another consequence of Corollary 4.18 is the following.

Corollary 4.20** ([1], Proposition 5.5).**

Let $k\in\mathcal{S}$ and $T$ be the operator defined by

[TABLE]

for every $f\in L^{2}$. Then, $T$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q,p)}$, for all $0<q\leq 1$.

Proof.

Since $k\in\mathcal{S}$, we have clearly $k\in\mathcal{S^{\prime}}$ and $\widehat{k}\in\mathcal{S}$. Therefore, there exists a constant $A>0$ such that

[TABLE]

for all $\xi\in\mathbb{R}^{d}$. Also, for any integer $\delta>0$, it’s easy to verify that there exists a constant $B>0$ such that

[TABLE]

for all $x\neq 0$ and all multi-indexes $\beta$ with $|\beta|\leq\delta$. With (4.37) and (4.38), the result immediately follows from Corollary 4.18. ∎

4.3. Riesz Potential operators

In this subsection, we study the boundedness of Riesz potential operators $I_{\alpha}$, $0<\alpha<d$, in $\mathcal{H}^{(q,p)}$ spaces. It’s well known that the operator $I_{\alpha}$ is bounded from $L^{r}$ to $L^{s}$, for $1<r<s<+\infty$ such that $\frac{1}{s}=\frac{1}{r}-\frac{\alpha}{d}$ (see [7], Theorem 4.18, p. 89, [12], Theorem 6.1.3, p. 3 and [19], Chap. V, Theorem 1, pp. 119-120). Also, $I_{\alpha}$ is bounded from $\mathcal{H}^{q}$ to $\mathcal{H}^{p}$, when $\frac{1}{p}=\frac{1}{q}-\frac{\alpha}{d}$ (see [15], Theorem 3.1, p. 105 and [20], Chap. III, p. 136). We recall that

[TABLE]

for all $\phi\in\mathcal{S}$, with $\gamma_{\alpha}=\pi^{\frac{d}{2}-\alpha}\frac{\Gamma\left(\frac{\alpha}{2}\right)}{\Gamma\left(\frac{d-\alpha}{2}\right)}\cdot$ For (4.39), see [7], Chap. 4, p. 88. Denoting by $K_{\alpha}$ the Riesz kernel, (4.39) writes

[TABLE]

with

[TABLE]

(4.40) is also equal in the sense of distributions to

[TABLE]

where

[TABLE]

for all $\xi\neq 0$. For (4.41) and (4.42), see respectively [7], Chap. 4, p. 88 and (4.3), p. 71.

For our third main result, we will need the following lemma.

Lemma 4.21**.**

Let $0<\sigma<d$, $0<w\leq h<+\infty$ and $0<u\leq v<+\infty$ such that

[TABLE]

and

[TABLE]

Then, for all sequences of cubes $\left\{Q^{n}\right\}_{n\geq 0}$ and all sequences of nonnegative scalars $\left\{\delta_{n}\right\}_{n\geq 0}$, we have

[TABLE]

where $\ell_{n}:=\ell(Q^{n})$ and $C:=C(d,\sigma,w,h,u,v)>0$ is a constant independent of the sequences $\left\{Q^{n}\right\}_{n\geq 0}$ and $\left\{\delta_{n}\right\}_{n\geq 0}$.

Proof.

The idea of our proof is the one of [18], Lemma 5.2, p. 138. Let $\left\{Q^{n}\right\}_{n\geq 0}$ be a sequence of cubes and $\left\{\delta_{n}\right\}_{n\geq 0}$ a sequence of nonnegative scalars.

We first suppose that each $Q^{n}$ is a dyadic cube, namely,

[TABLE]

where $\upsilon_{n}\in\mathbb{Z}$ is the generation of $Q^{n}$ and $(m_{n,1},m_{n,2},\ldots,m_{n,d})\in\mathbb{Z}^{d}$. Since dyadic cubes form a grid, we can assume that $Q^{n}\neq Q^{m}$, for $0\leq n<m<+\infty$.

Let

[TABLE]

By homogeneity of quasi-norms, we can assume that $\left\|F\right\|_{w,h}=1$. Thus, we have to show that $\left\|G\right\|_{u,v}\leq C$, where $C>0$ is a constant independent of the sequences $\left\{Q^{n}\right\}_{n\geq 0}$ and $\left\{\delta_{n}\right\}_{n\geq 0}$. For this purpose, we will consider separately cubes with measure less or equal to 1 and those with measure greater or equal to 2.

$1^{rst}$ case : we consider cubes $Q^{m}$ such that $|Q^{m}|\leq 1$. For each of them, we have

[TABLE]

Moreover, for all $k=(k_{1},k_{2},\ldots,k_{d})\in\mathbb{Z}^{d}$, $Q_{k}$ is a dyadic cube of generation [math] (we recall that $Q_{k}=\prod_{i=1}^{d}[k_{i},k_{i}+1)$ ) and $\left\{Q_{k}\right\}_{k\in\mathbb{Z}^{d}}$ forms a partition of $\mathbb{R}^{d}$. Thus, since $Q^{m}$ is a dyadic cube of generation $\upsilon_{m}\geq 0$, we have

[TABLE]

for some $k_{0}\in\mathbb{Z}^{d}$. Fix $x\in\mathbb{R}^{d}$. It comes that

[TABLE]

Also,

[TABLE]

Hence

[TABLE]

by (4.46) and (4.47). Since the cubes $\left\{Q^{m}\right\}_{m\geq 0}$ are dyadic cubes with $Q^{m}\neq Q^{n}$ for $0\leq m<n<+\infty$, we obtain

[TABLE]

with $C=\frac{1}{1-2^{\sigma-\frac{d}{w}}}+\frac{1}{1-2^{-\sigma}}>0$, according to Relation (4.43). It follows that

[TABLE]

But, $h\leq\frac{w}{u}v$ by (4.43) and (4.44). Hence

[TABLE]

$2^{nd}$ case : we consider cubes $Q^{m}$ such that $|Q^{m}|\geq 2$. For each of these cubes, we have

[TABLE]

Thus we have

[TABLE]

since $Q^{m}$ is a dyadic cube of generation $\upsilon_{m}<0$. Therefore, for fixed $x\in\mathbb{R}^{d}$, we have

[TABLE]

Thus,

[TABLE]

by (4.47) and (4.48). Thanks to Relation (4.44), we obtain as in the first case,

[TABLE]

with $C=\frac{1}{1-2^{-\sigma}}+\frac{1}{1-2^{\sigma-\frac{d}{h}}}>0$. Then, it follows that

[TABLE]

since $\frac{h}{v}u\leq w$ according to Relations (4.43) and (4.44).

Combining both cases, we finally obtain

[TABLE]

The lemma is then proved in the case of dyadic cubes.

For the general case, we consider for each cube $Q^{n}$ (not necessarily dyadic), a dyadic cube $R^{n}:=2^{-\upsilon_{n}}\prod_{i=1}^{d}[m_{n,i}\ ,\ m_{n,i}+1)$ such that

[TABLE]

We have

[TABLE]

for all $\gamma>\max\left\{1,\frac{1}{u}\right\}$, by Proposition 2.2. Since (4.45) is proved for dyadic cubes, we obtain

[TABLE]

where $\gamma=1+\frac{1}{w}$ , by Proposition 2.2. This ends the proof of the lemma. ∎

Theorem 4.22**.**

Let $0<q\leq\min(1,p)<+\infty$, $0<\alpha<d$ and $0<q_{1}\leq p_{1}<+\infty$ such that

[TABLE]

and

[TABLE]

Then, $I_{\alpha}$ extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q_{1},p_{1})}$.

Proof.

Let $\delta\geq\max\left\{\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor;\alpha\right\}$ be an integer and $r>\max\left\{p_{1};\frac{d}{d-\alpha}\right\}$ be a real. Let $\mathcal{H}_{fin}^{(q,p)}$ be the space of finite linear combinations of $(q,r,\delta)$-atoms and $f\in\mathcal{H}_{fin}^{(q,p)}$. Then, there exist a finite sequence $\left\{(\textbf{a}_{n},Q^{n})\right\}_{n=0}^{j}$ in $\mathcal{A}(q,r,\delta)$ and a finite sequence of scalars $\left\{{\lambda}_{n}\right\}_{n=0}^{j}$ such that $f=\sum_{n=0}^{j}\lambda_{n}\textbf{a}_{n}$. Set $\widetilde{Q^{n}}:=2\sqrt{d}Q^{n}$, $n\in\left\{0,1,\ldots,j\right\}$, and denote by $x_{n}$ and $\ell_{n}$ respectively the center and the side-length of $Q^{n}$.

Now, we distinguish the cases: $1<q_{1}<+\infty$ and $0<q_{1}\leq 1$.

Case 1: $1<q_{1}<+\infty$. Then, $\mathcal{H}^{(q_{1},p_{1})}=(L^{q_{1}},\ell^{p_{1}})$ with norms equivalence, by [1], Theorem 3.2, p. 1905. With this in mind, we have

[TABLE]

for almost all $x\in\mathbb{R}^{d}$. Hence

[TABLE]

Fix $0<\eta<q$. Set

[TABLE]

Since $r>\frac{d}{d-\alpha}$, there exists $1<s<r$ such that

[TABLE]

and

[TABLE]

by (4.49). Thus,

[TABLE]

by (4.51), the fact that $I_{\alpha}$ is bounded from $L^{s}$ to $L^{r}$, $\mathcal{A}(q,r,\delta)\subset\mathcal{A}(q,s,\delta)$ and (4.52). Furthermore, $1<q_{1}\leq p_{1}<r<+\infty$ and $\text{supp}(|I_{\alpha}(\textbf{a}_{n})|\chi_{{}_{\widetilde{Q^{n}}}})\subset\widetilde{Q^{n}}$. Hence

[TABLE]

by Proposition 4.5 in [1] and the proof of Theorem 4.6 in [1]. But,

[TABLE]

Hence

[TABLE]

by Lemma 4.21. To estimate $J$, notice that

[TABLE]

for all $x\in\mathbb{R}^{d}\backslash\left\{0\right\}$ and all multi-indices $\beta$ with $|\beta|\leq\delta+1$, where $C_{d,\alpha,\delta}>0$ is a constant independent of $\beta$ and $x$. Consider $x\notin\widetilde{Q^{n}}$. We have $x-y\neq 0$, for all $y\in Q^{n}$. Thus, using Taylor’s formula and the vanishing condition of the atom $\textbf{a}_{n}$ , we obtain

[TABLE]

with

[TABLE]

for all $y\in Q^{n}$, where $\delta^{\prime}=\delta+1$, by (4.54). Hence

[TABLE]

where $\vartheta=\frac{d-\alpha+\delta+1}{d}\cdot$ Moreover, since $\delta\geq\max\left\{\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor;\alpha\right\}$, we have $1<\vartheta$ and $1<q_{1}\vartheta\leq p_{1}\vartheta<+\infty$. Therefore,

[TABLE]

by the proof of Theorem 4.6 in [1] and Lemma 4.21. Finally, we obtain

[TABLE]

Thus,

[TABLE]

by Theorem 2.5, since (4.50) implies that $r>p_{1}>p$. Hence $I_{\alpha}$ is bounded from $\mathcal{H}_{fin}^{(q,p)}$ to $(L^{q_{1}},\ell^{p_{1}})$. The density of $\mathcal{H}_{fin}^{(q,p)}$ in $\mathcal{H}^{(q,p)}$ and the fact that $(L^{q_{1}},\ell^{p_{1}})=\mathcal{H}^{(q_{1},p_{1})}$ with norms equivalence yield the result.

Case 2: $0<q_{1}\leq 1$. We have

[TABLE]

for all $x\in\mathbb{R}^{d}$. Hence

[TABLE]

with

[TABLE]

Fix $0<\eta<q$. Let us consider $1<s<r$ as in (4.51). We have

[TABLE]

by (4.52). Moreover, $\text{supp}\left(\mathcal{M}_{0}(I_{\alpha}(\textbf{a}_{n}))\chi_{{}_{\widetilde{Q^{n}}}}\right)^{\eta}\subset\widetilde{Q^{n}}$ and $1<\frac{q_{1}}{\eta}\leq\frac{p_{1}}{\eta}<\frac{r}{\eta}\cdot$ Hence

[TABLE]

by Proposition 4.5 in [1], the proof of Theorem 4.6 in [1] and (4.53). Set $\rho=\alpha\eta$ , $u=\frac{q_{1}}{\eta}$ ; $v=\frac{p_{1}}{\eta}$ ; $w=\frac{q}{\eta}$ and $h=\frac{p}{\eta}\cdot$ We have $0<\rho<d$ , $0<w\leq h<+\infty$ and $0<u\leq v<+\infty$. Furthermore,

[TABLE]

by (4.49) and (4.50). Thus, setting $\delta_{n}=\left(\frac{|\lambda_{n}|}{\left\|\chi_{{}_{Q^{n}}}\right\|_{q}}\right)^{\eta}$, for $n=0,1,\ldots,j$ , we have

[TABLE]

by Lemma 4.21. Hence

[TABLE]

To estimate $J$, it suffices to show that

[TABLE]

for all $x\notin\widetilde{Q^{n}}$, where $\vartheta=\frac{d-\alpha+\delta+1}{d}\cdot$ To prove (4.56), we consider $x\notin\widetilde{Q^{n}}$. For every $t>0$, set $K_{\alpha}^{(t)}=K_{\alpha}\ast\varphi_{t}$. We have

[TABLE]

for all $z\in\mathbb{R}^{d}\backslash\left\{0\right\}$ and all multi-indexes $\beta$ with $|\beta|=\delta+1$. The proof of (4.57) is similar to the one of (4.35) in the proof of Theorem 4.17; therefore, details are left to the reader. With (4.57), by using Taylor’s formula and the vanishing condition of the atom $\textbf{a}_{n}$, we have obtain, as in Case 1,

[TABLE]

where $\vartheta=\frac{d-\alpha+\delta+1}{d}\cdot$ Hence

[TABLE]

with $\vartheta=\frac{d-\alpha+\delta+1}{d}\cdot$ This establishes (4.56).

Moreover, since $\delta\geq\max\left\{\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor;\alpha\right\}$, we have $1<\vartheta$ and $1<q_{1}\vartheta\leq p_{1}\vartheta<+\infty$, by (4.49). Thus, as in Case 1, we obtain

[TABLE]

Hence

[TABLE]

Thus,

[TABLE]

by Theorem 2.5, since (4.50) implies that $r>p_{1}>p$. Therefore, $I_{\alpha}$ is bounded from $\mathcal{H}_{fin}^{(q,p)}$ to $\mathcal{H}^{(q_{1},p_{1})}$ and the density of $\mathcal{H}_{fin}^{(q,p)}$ in $\mathcal{H}^{(q,p)}$ yields the result. This ends the proof of Theorem 4.22. ∎

Remark 4.23**.**

The proof of Theorem 4.22 in the second case ($0<q_{1}\leq 1$) is still valid whenever $1<q_{1}<+\infty$.

We mention that one can use molecular characterization of $\mathcal{H}^{(q_{1},p_{1})}$ in the second case ($0<q_{1}\leq 1$) as defined in [1], for the proof.

Corollary 4.24**.**

$I_{\alpha}$* extends to a bounded operator from $\mathcal{H}^{(q,p)}$ to $\mathcal{H}^{(q_{2},p_{2})}$, if*

[TABLE]

Proof.

(4.58) implies that $q_{2}\leq q_{1}$ and $p_{1}\leq p_{2}$. Hence, by Theorem 4.22,

[TABLE]

for all $f\in\mathcal{H}_{fin}^{(q,p)}$, where $\mathcal{H}_{fin}^{(q,p)}$ is the subspace of $\mathcal{H}^{(q,p)}$ consisting of finite linear combinations of $(q,r,\delta)$-atoms with $\delta\geq\max\left\{\left\lfloor d\left(\frac{1}{q}-1\right)\right\rfloor;\alpha\right\}$ and $\max\left\{p_{1};\frac{d}{d-\alpha}\right\}<r<+\infty$ fixed. Then, the density of $\mathcal{H}_{fin}^{(q,p)}$ in $\mathcal{H}^{(q,p)}$ yields the result. ∎