Summability in anisotropic mixed-norm Hardy spaces

1.   Introduction

Let $A$ be a general expansive matrix on $\mathbb{R}^n$. In 2003, Bownik ^[1] investigated the anisotropic Hardy space $H^p_A({\mathbb{R}^n})$ with $p\in(0, {\infty})$, which includes both the classical Hardy space and the parabolic Hardy space of Calderón and Torchinsky ^[2] as special cases. Recently, Huang et al. ^[3] introduced the anisotropic mixed-norm Hardy space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ with respect to $\vec{p}\in(0, {\infty})^n$ and a general expansive matrix $A$, and established its various real-variable characterizations. This extends the real-variable theory of the Hardy space $H^p_A({\mathbb{R}^n})$ from ^[1]. For more information on mixed-norm function spaces, we refer the reader to ^{[4,5,6,7,8,9,10,11]}.

On the other hand, it is well known that Stein, Taibleson and Weiss ^[12] proved for the Bochner–Riesz summability that the maximal operator $\sigma_\ast^\theta$ of the $\theta$-means is bounded from the classical Hardy $H^p({{\mathbb{R}^n}})$ to the Lebesgue space $L^p({{\mathbb{R}^n}})$ with the index $p$ greater than some constant $p_0$. This result has been extended to many other Hardy-type and other summability methods. For more progress about this topic, we refer the reader to ^{[13,14,15,16,17,18]} and references therein. In particular, Weisz ^[18] proved that the maximal operator, induced by the so-called $\theta$-means, is bounded from the isotropic mixed-norm Hardy space $H^{\vec{p}}(\mathbb{R}^n)$ to the mixed-norm Lebesgue space $L^{\vec{p}}(\mathbb{R}^n)$. However, the corresponding conclusion of summability in anisotropic mixed-norm Hardy space $H_A^{\vec{p}}(\mathbb{R}^n)$ is still unknown.

In this paper, under some conditions on $\theta$ and $\vec{p}$, we prove that the maximal operator $\sigma_\ast^\theta$ is bounded from ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ to $L^{\vec{p}}(\mathbb{R}^n)$. As a consequence, we prove some norm and almost everywhere convergence results for the $\theta$-means. Moreover, sa special cases of the $\theta$-means, we consider the well-known Bochner–Riesz and Weierstrass summations. This paper is organized as follows: As a preliminary, in Section 2, we recall some definitions of expansive matrices, mixed-norm Lebesgue spaces ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$ and anisotropic mixed-norm Hardy spaces ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$. In Section 3, via borrowing some ideas from [18,Theorem 3] and [13,Theorem 7.4] as well as [14,Theorem 2.17], we prove our main result by using the known finite atomic characterization of $H_A^{\vec{p}}(\mathbb{R}^n)$ and a criterion on the boundedness of sublinear operators from ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ into ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$. Section 4 is aimed to consider two special summability methods, namely, the Bochner–Riesz and Weierstrass summations.

Finally, we make some conventions on notation. Let ${{\mathbb N}}: = \{1, 2, \ldots\}$, ${{\mathbb Z}}_+: = \{0\}\cup{{\mathbb N}}$ and ${\bf 0}$ be the origin of ${{\mathbb{R}^n}}$. For any ${\gamma}: = ({\gamma}_1, \ldots, {\gamma}_n)\in({{\mathbb Z}}_+)^n = :{{\mathbb Z}}_+^n$, let $|{\gamma}|: = {\gamma}_1+\cdots+{\gamma}_n$ and $\partial^{{\gamma}} : = (\frac{\partial}{\partial x_1})^{{\gamma}_1}\cdots(\frac{\partial}{\partial x_n})^{{\gamma}_n}.$ We use $C$ to denote a positive constant which is independent of the main parameters, but its value may change from line to line. In addition, we use $f{\lesssim} g$ to denote $f\leq Cg$ and, if $f{\lesssim} g{\lesssim} f$, we then write $f\sim g$. Moreover, for a given set $\Omega\subset{{\mathbb{R}^n}}$, we denote its characteristic function by ${\bf 1}_\Omega$, the set ${{\mathbb{R}^n}}\setminus \Omega$ by $\Omega^\complement$ and its n-dimensional Lebesgue measure by $|\Omega|$. For any $t\in\mathbb{R}$, The symbol $\lfloor t\rfloor$ denotes the largest integer not greater than $t$. For each $r\in[1, {\infty}]$, we denote by $r'$ its conjugate index, namely, $1/r+1/r' = 1$. Moreover, if $\vec{r}: = (r_1, \ldots, r_n)\in[1, {\infty}]^n$, we denote by $\vec{r}': = (r_1', \ldots, r_n')$ its conjugate index.

2.   Preliminaries

In this section, we recall the notions of expansive matrices, mixed-norm Lebesgue spaces and anisotropic mixed-norm Hardy spaces.

We begin with the following notion of mixed-norm Lebesgue spaces from ^[19].

Definition 2.1. Let ${\vec{p}}: = (p_1, \ldots, p_n)\in (0, {\infty}]^n$. The mixed-norm Lebesgue space ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$ is defined to be the set of all measurable functions $f$ such that

with the usual modifications made when $p_i = {\infty}$ for some $i\in \{1, \ldots, n\}$.

Recall also that the notions of expansive matrices and homogeneous quasi-norms were originally introduced by Bownik in ^[1].

Definition 2.2. A real $n\times n$ matrix $A$ is called an expansive matrix (shortly, a dilation) if

here and thereafter, $\sigma(A)$ denotes the collection of all eigenvalues of $A$.

Definition 2.3. Let $A$ be a dilation. A measurable mapping $\rho:\ {{\mathbb{R}^n}} \to [0, \infty)$ is called a homogeneous quasi-norm, associated with $A$, if

${\rm(i)}$ $x\neq{\bf 0}$ implies that $\rho(x)\in(0, {\infty})$;

${\rm{(ii)}}$ for each $x\in{{\mathbb{R}^n}}$, $\rho(Ax) = b\rho(x)$, here and below, $b: = |\det A|$;

${\rm{(iii)}}$ for any $x$, $y\in{{\mathbb{R}^n}}$, $\rho(x+y)\le c[\rho(x)+\rho(y)]$, where $c$ is a positive constant independent of $x$ and $y$.

For any given dilation $A$, it was proved in [1,Lemma 2.2] that there exists an open set $\Delta\subset{{\mathbb{R}^n}}$ which has the following property: $|\Delta| = 1$, and we can find a constant $\tau\in(1, \infty)$ such that $\Delta\subset \tau\Delta\subset A\Delta$. For any $i\in{{\mathbb Z}}$, we define $B_i: = A^i\Delta$. It is easy to check that $\{B_i\}_{i\in{{\mathbb Z}}}$ is a family of open sets around the origin, $B_i\subset \tau B_i\subset B_{i+1}$ and $|B_i| = b^i$. For any given dilation $A$, we use the symbol $\mathfrak{B}$ to denote the set of all dilated balls, namely,

and

By [1,Lemma 2.4], we know that any two homogeneous quasi-norms associated with the same fixed dilation $A$ are equivalent. Thus, in what follows, we always use the step homogeneous quasi-norm defined by setting, for each $x\in{{\mathbb{R}^n}}$,

for convenience. Let $\lambda_-$, $\lambda_+\in(1, {\infty})$ be two numbers such that

Throughout this article, the symbol ${{\mathcal S}}({{\mathbb{R}^n}})$ denotes the space of all Schwartz functions, namely, the set of all $C^\infty({{\mathbb{R}^n}})$ functions $\phi$ satisfying that, for any $k\in{{\mathbb Z}}_+$ and multi-index ${\beta}\in{{\mathbb Z}}_+^n$,

The topology of ${{\mathcal S}}({{\mathbb{R}^n}})$ is determined by $\{\|\cdot\|_{{\beta}, \, k}\}_{{\beta}\in{{\mathbb Z}}_+^n, \, k\in{{\mathbb Z}}_+}$. Moreover, we use ${{\mathcal S}}'({{\mathbb{R}^n}})$ to denote the dual space of ${{\mathcal S}}({{\mathbb{R}^n}})$, namely, the space of all tempered distributions on ${{\mathbb{R}^n}}$ equipped with the weak-$\ast$ topology. For any $N\in\mathbb{Z}_+$, let ${{\mathcal S}}_N({{\mathbb{R}^n}})$ denote the following set:

For an $n$-dimensional vector $\vec{p}: = (p_1, \ldots, p_n)\in{(0, {\infty}]}^n$, let

The following definition of anisotropic mixed-norm Hardy spaces was first introduced by Huang et. al ^[3].

Definition 2.4. ${\rm{(i)}}$ Let $\phi\in{{\mathcal S}}({{\mathbb{R}^n}})$ and $f\in{{\mathcal S}}'({{\mathbb{R}^n}})$. The non-tangential maximal function $M_\phi(f)$ with respect to $\phi$ is defined by setting, for any $x\in{{\mathbb{R}^n}}$,

here and thereafter, for any $\phi\in{{\mathcal S}}({{\mathbb{R}^n}})$ and $k\in\mathbb{Z}$, let $\phi_k(\cdot): = b^{k}\phi(A^{k}\cdot)$. Moreover, for any given $N\in\mathbb{N}$, the non-tangential grand maximal function $M_N(f)$ of $f\in{{\mathcal S}}'({{\mathbb{R}^n}})$ is defined by setting, for any $x\in{{\mathbb{R}^n}}$,

${\rm{(ii)}}$ Let ${\vec{p}}\in (0, {\infty})^n$ and $N\in\mathbb{N}\cap[\lfloor(\frac1{\min\{1, p_-\}}-1)\frac{\ln b}{\ln \lambda_-}\rfloor+2, {\infty})$, where $p_-$ is as in (2.4). The anisotropic mixed-norm Hardy space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ is defined by setting

and, for any $f\in{{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$, let $\|f\|_{{{H_A^{\vec{p}}({{\mathbb{R}^n}})}}}: = \| M_N(f)\|_{{{L^{\vec{p}}({{\mathbb{R}^n}})}}}$.

Remark 1. ${\rm(i)}$ Observe that, in [3,Theorem 4.7], it was proved that the space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ is independent of the choice of $N$ as in Definition 2.4(ⅱ).

${\rm(ii)}$ Note that Cleanthous et al. ^[20] investigated another kind of anisotropic mixed-norm Hardy space ${H_{{\vec{a}}}^{{\vec{p}}}({{\mathbb{R}^n}})}$, where ${\vec{a}}\in [1, {\infty})^n$ and ${\vec{p}}\in (0, {\infty})^n$; see [20,Definition 3.3]. We should point out that [3,Proposition 4] shows that, if

with ${\vec{a}}: = (a_1, \ldots, a_n)\in [1, {\infty})^n$, then the Hardy space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ in Definition 2.4(ⅱ) and the anisotropic mixed-norm Hardy space ${H_{{\vec{a}}}^{{\vec{p}}}({{\mathbb{R}^n}})}$ from ^[20] coincide with equivalent quasi-norms.

3.   Summability in ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$

In this section, we study the so-called $\theta$-summability for multi-dimensional Fourier transforms in the anisotropic mixed-norm Hardy space $H_A^{\vec{p}}(\mathbb{R}^n)$.

We always use $L_{\rm loc}^1({{\mathbb{R}^n}})$ to denote the set of all locally integrable functions on ${{\mathbb{R}^n}}$. To present our main result, we need several technical lemmas as follows. First, the following Lemmas 3.1, 3.2 and 3.3 are just, respectively, [3,Lemmas 3.4,3.2 and 4.4], which show some properties of the ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$ quasi-norm, the boundedness of the anisotropic Hardy–Littlewood maximal operator $M_{\rm HL}$ on the space ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$ and the anisotropic Fefferman–Stein vector-valued inequality on ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$.

Lemma 3.1. Let ${\vec{p}}\in (0, {\infty}]^n$. Then, for any $s\in (0, {\infty})$ and $f\in{{L^{\vec{p}}({{\mathbb{R}^n}})}}$,

here and below, for any ${\gamma}\in{{\mathbb R}}$, ${\gamma}{\vec{p}}: = ({\gamma} p_1, \ldots, {\gamma} p_n)$.In addition, for any ${\lambda}\in{\mathbb C}$, $r\in [0, \min\{1, p_-\}]$with $p_-$ as in (2.4) and $f, \ h\in{{L^{\vec{p}}({{\mathbb{R}^n}})}}$, $\|{\lambda} f\|_{{{L^{\vec{p}}({{\mathbb{R}^n}})}}} = |{\lambda}|\|f\|_{{{L^{\vec{p}}({{\mathbb{R}^n}})}}}$ and

Lemma 3.2. Let ${\vec{p}}\in (1, {\infty})^n$. Then there exists a positiveconstant $C$ such that, for any $f\in L_{\rm loc}^1({{\mathbb{R}^n}})$,

where $M_{{\rm HL}}$ denotes the anisotropic Hardy–Littlewood maximal operatordefined by setting, for any $f\in L^1_{{\rm loc}}({{\mathbb{R}^n}})$ and $x\in{{\mathbb{R}^n}}$,

with $\mathfrak{B}$ as in (2.1).

Lemma 3.3. Let ${\vec{p}}\in(1, {\infty})^n$ and $v\in(1, {\infty}]$. Then, for any sequence $\{f_k\}_{k\in{{\mathbb N}}}$ of measurable functions,

with the usual modification made when $v = {\infty}$, where $C$ is a positive constant independent of $\{f_k\}_{k\in{{\mathbb N}}}$.

Let ${\vec{p}}\in(0, {\infty})^n$ and $k\in{{\mathbb Z}}_+$. Then, by the fact that, for any dilated ball $B\in\mathfrak{B}$ and $q\in(0, \underline{p})$, ${\bf 1}_{A^kB}\le b^{k/q}[{M_{{{\rm{HL}}}}}({\bf 1}_B)]^{1/q}$ as well as Lemmas 3.1 and 3.3, we conclude that there exists a positive constant $C$ such that, for any sequence $\{B^{(i)}\}_{i\in{{\mathbb N}}}\subset\mathfrak{B}$,

The last inequality used Lemma 3.3 with the fact that ${\vec{p}}/q\in(1, {\infty})^n$.

For any given $p\in(0, {\infty}]$ and measurable set $\Omega\subset{{\mathbb{R}^n}}$. The Lebesgue space $L^p(\Omega)$ is defined to be the set of all measurable functions $f$ on $\Omega$ such that, when $p\in(0, {\infty})$,

and

We also need the following notions of anisotropic mixed-norm $({\vec{p}}, r, s)$-atoms and anisotropic mixed-norm finite atomic Hardy spaces ${{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$ from ^[3].

Definition 3.4. Let ${\vec{p}}\in(0, {\infty})^n$, $r\in(1, {\infty}]$ and

with $p_-$ as in (2.4).

${\rm (I)}$ A measurable function $a$ on ${{\mathbb{R}^n}}$ is called an anisotropic mixed-norm $({\vec{p}}, r, s)$-atom if

${\rm (I)}_1$ ${\,\mathrm{ supp }}\, a \subset B$ with some $B\in\mathfrak{B}$, where $\mathfrak{B}$ is as in (2.1);

${\rm (I)}_2$ $\|a\|_{L^r({{\mathbb{R}^n}})}\le \frac{|B|^{1/r}}{\|{\bf 1}_B\|_{{{L^{\vec{p}}({{\mathbb{R}^n}})}}}}$;

${\rm (I)}_3$ for any ${\alpha}\in{{\mathbb Z}}_+^n$ with $|{\alpha}|\le s$, $\int_{\mathbb R^n}a(x)x^{\alpha}\, dx = 0$.

$({\rm{II}})$ The anisotropic mixed-norm finite atomic Hardy space ${{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$ is defined to be the set of all $f\in{{\mathcal S}}'({{\mathbb{R}^n}})$ satisfying that there exist $I\in{{\mathbb N}}$, $\{{\lambda}_i\}_{i\in[1, I]\cap{{\mathbb N}}}\subset\mathbb{C}$ and a finite sequence of $({\vec{p}}, r, s)$-atoms, $\{a_i\}_{i\in[1, I]\cap{{\mathbb N}}}$, supported, respectively, in $\{B^{(i)}\}_{i\in[1, I]\cap{{\mathbb N}}}\subset\mathfrak{B}$ such that

Moreover, for any $f\in{{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$, let

where $\underline{p}$ is as in (2.4) and the infimum is taken over all decompositions of $f$ as above.

Then, from [3,Theorem 8.1(ⅰ) and Remark 13], we immediately deduce the succeeding criterion on the boundedness of sublinear operators from ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ into ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$, which plays a key role in the proof of our main result.

Lemma 3.5. Let ${\vec{p}}\in (0, {\infty})^n$, $r\in(\max\{p_+, 1\}, {\infty})$with $p_+$ as in (2.4) and $s$ be as in (3.2).Assume that $T:\ {{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}\to{{L^{\vec{p}}({{\mathbb{R}^n}})}}$ is a sublinear operator satisfying thatthere exists a positive constant $C$ such that, for any $f\in {{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$,

Then $T$ uniquely extends to a bounded sublinearoperator from ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ into ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$.

Recall that, for any given $p\in [1, 2]$ and any $f\in L^p({{\mathbb{R}^n}})$, the Fourier inversion formula, namely,

holds true if $\widehat f \in L^1({{\mathbb{R}^n}})$, here and below, $\imath: = \sqrt{-1}$, $x\cdot t : = \sum_{k = 1}^n x_k t_k$ for any $x: = (x_1, \ldots, x_n)$, $t: = (t_1, \ldots, t_n)\in{{\mathbb{R}^n}}$, and $\widehat{f}$ denotes the Fourier transform of $f$, which is defined by setting, for any $t\in{{\mathbb{R}^n}}$,

This motivates the following definition of $\theta$-summability of Fourier transforms; see, for instance, ^{[15,16,17,18,21]} for the classical case and ^[13,14] for the anisotropic case. We always suppose that

where the symbol $C_0({{\mathbb R}})$ denotes the set of all continuous functions $f$ satisfying that $\lim_{|x|\to{\infty}}|f(x)| = 0$. Let $A$ be a given dilation, $m\in {{\mathbb Z}}$ and $p\in [1, 2]$. The $m$-th anisotropic $\theta$-mean, denoted by $\sigma_m^{\theta}$, is defined by setting, for any $f\in L^{p}({{\mathbb{R}^n}})$ and $x\in {{\mathbb{R}^n}}$,

where $A^*$ be the transposed matrix of $A$. This integral is well defined because $\theta\in L^p(\mathbb{R})$ with $p\in [1, 2]$ and $\widehat {f}\in L^{p'}({{\mathbb{R}^n}})$. Let $\theta_0(x): = \theta(|x|)$ for any $x\in{{\mathbb{R}^n}}$ and assume that

Moreover, by [14,(2.17)], we can rewrite $\sigma _m^{\theta} f(x)$ in (3.4) as

This definition of anisotropic $\theta$-means can be extended to any $f\in{{L^{\vec{p}}({{\mathbb{R}^n}})}}$ with $p_-\in[1, {\infty})$ by setting, for any $x\in{{\mathbb{R}^n}}$,

where $m\in{{\mathbb Z}}$. Furthermore, (3.6) induces the definition of maximal $\theta$-operators $\sigma_*^\theta$ as follows: for any $f\in {{L^{\vec{p}}({{\mathbb{R}^n}})}}$ with $p_-\in[1, {\infty})$,

Now we state the main result of this paper as follows.

Theorem 3.6. Let $\theta$ and $\theta_0$ be, respectively, as in (3.3) and (3.5) satisfying that there exists a positive constant $\beta\in(1, {\infty})$ such that, for any${\gamma}\in {{\mathbb Z}}_+^n$ and $x\in{{\mathbb{R}^n}}\setminus\{{\bf 0}\}$,

where $C$ is a positive constant independent of $x$.If ${\vec{p}}\in (0, {\infty})^n$,

with $\lambda_-$ as in (2.3), then there exists a positive constant $C_{(p_-, p_+)}$, with $p_-$ and $p_+$ as in (2.4), such that, for any $f\in{{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$,

Proof. By Lemma 3.5, to prove Theorem 3.6, we only need to show that, for any $f\in{{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$,

with $s$ being as in (3.2) large enough and $r\in(\max\{p_+, 1\}, {\infty})$ to be chosen later, where $p_+$ is as in (2.4).

For this purpose, suppose now $f\in {{H_{A, \, {{\rm{fin}}}}^{\vec{p}, r, s}({{\mathbb{R}^n}})}}$. Then, by Definition 3.4(Ⅱ), we find that there exist some $I\in{{\mathbb N}}$, $\{{\lambda}_i\}_{i\in[1, I]\cap{{\mathbb N}}}\subset\mathbb{C}$ and a finite sequence of $({\vec{p}}, r, s)$-atoms, $\{a_i\}_{i\in[1, I]\cap{{\mathbb N}}}$, supported, respectively, in $\{B^{(i)}\}_{i\in[1, I]\cap{{\mathbb N}}}\subset\mathfrak{B}$ such that $f = \sum_{i = 1}^I\lambda_ia_i$ in ${{\mathcal S}}'({{\mathbb{R}^n}})$ and

where $\underline{p}$ is as in (2.4). Take two sequences $\{x_i\}_{i\in[1, I]\cap{{\mathbb N}}}\subset{{\mathbb{R}^n}}$ and $\{k_i\}_{i\in[1, I]\cap{{\mathbb N}}}\subset{{\mathbb Z}}$ such that, for any $i\in[1, I]\cap{{\mathbb N}}$, $x_i+B_{k_i} = B^{(i)}$. Then

where $\omega$ is as in (2.2).

For ${\rm J}_1$, take $u\in L^{({\vec{p}}/\underline{p})'}({{\mathbb{R}^n}})$ satisfying that $\|u\|_{L^{({\vec{p}}/\underline{p})'}({{\mathbb{R}^n}})}\leq 1$ and

From this and the Hölder inequality, it follows that, for any $t\in(1, {\infty})$ with $p_+ < t\underline{p} < r$,

By this, the fact that $\sigma_*^\theta$ is bounded on $L^q({{\mathbb{R}^n}})$ for any $1 < q < {\infty}$, and Definition 3.4(Ⅰ), we further conclude that

Note that the assumption $0 < p_+/\underline{p} < t$ implies that $t' < {\vec{p}}/\underline{p})'\le{\infty}$. From this, Lemmas 3.2 and 3.1, the fact that $\|u\|_{L^{({\vec{p}}/\underline{p})'}({{\mathbb{R}^n}})}\leq 1$, (3.1) and (3.10), we deduce that

We next deal with ${\rm J}_2$. To do this, we first claim that, for any $i\in[1, I]\cap{{\mathbb N}}$ and $x\in(x_i+A^\omega B_{k_i})^{\complement}$,

where ${\beta}$ is as in (3.7) and (3.8). Assume that (3.13) holds true for the time being. Then, by (3.8), Lemmas 3.1 and 3.3 as well as (3.10), we find that

This, together with (3.11) and (3.12), further implies that (3.9) holds true.

Thus, to complete the proof of Theorem 3.6, it suffices to verify (3.13). To this end, let $a$ be any $({\vec{p}}, r, s)$-atom supported in $x_0+B_k\in\mathfrak{B}$, where $x_0\in{{\mathbb{R}^n}}$, $k\in{{\mathbb Z}}$ and $\mathfrak{B}$ is as in (2.1). Without loss of generality, we may assume that $x_0 = {\bf 0}$. Suppose that $P$ is a polynomial of degree not more than $s$. Then, by (3.6), Definition 3.4(Ⅰ) and the Hölder inequality, we obtain that, for any $m\in{{\mathbb Z}}$ and $x\in(B_{k+\omega})^{\complement}$,

Assume that $x\in B_{k+\omega+\nu+1}\setminus B_{k+\omega+\nu}$ for some $\nu\in{{\mathbb Z}}_+$. Then, using [1,(2.11)], we have

In addition, the Taylor remainder theorem implies that

where $N\in[0, s+1]\cap{{\mathbb Z}}_+$.

For the case when $k+m\in{{\mathbb Z}}\setminus{{\mathbb Z}}_+$ and $k+m+\nu\in{{\mathbb Z}}_+$, by (3.16), [1,(2.2)] and (3.7), it is easy to see that

with ${\beta}$ as in (3.7) and (3.8). From this, (3.15) and [1,(3.2)], it follows that

Therefore,

Choosing $N$ larger than $\beta$, we have

For the case when $k+m\in{{\mathbb Z}}\setminus{{\mathbb Z}}_+$ and $k+m+\nu\in{{\mathbb Z}}\setminus{{\mathbb Z}}_+$, similarly to (3.17), it is easy to check that, for any $x\in B_{k+\omega+\nu+1}\setminus B_{k+\omega+\nu}$,

This further implies that (3.18) holds true for this case by choosing $N$ large enough such that

Finally, for the case when $k+m\in{{\mathbb Z}}_+$, we choose $P\equiv0$. Then, by (3.14), (3.15) and the fact that $\beta\in(\ln b/\ln \lambda_-, {\infty})$, we conclude that, for any $x\in B_{k+\omega+\nu+1}\setminus B_{k+\omega+\nu}$,

and hence (3.18) also holds true for this case. This finishes the proof of (3.13) and hence of Theorem 3.6.

Remark 2. ${\rm (i)}$ If $A: = d\, {\rm I}_{n\times n}$ for some $d\in{{\mathbb R}}$ with $|d|\in(1, {\infty})$, where ${\rm I}_{n\times n}$ denotes the $n\times n$ unit matrix, then $\frac{\ln b}{\ln \lambda_-} = n$ and the Hardy space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ goes back to the isotropic mixed-norm Hardy space ${H^{{\vec{p}}}({{\mathbb{R}^n}})}$. In this case, Theorem 3.6 was obtained by Weisz in [18,Theorem 3]. Moreover, if ${\vec{p}}: = (\overbrace{p, \ldots, p}^{n\ \rm times})$ with some $p\in(0, {\infty})$, then Theorem 3.6 implies the well-known result, with $\beta\in(n, {\infty})$ and $p\in(n/\beta, \infty)$, for the classical Hardy space ${H^p({{\mathbb{R}^n}})}$ (see Weisz ^[16]). This classical result was also proved in a special case, namely, for the Bochner–Riesz means, in Stein et al. ^[12]. For the same case, a counterexample was also given in ^[12] to show that the same conclusion is not true for $p\in(0, n/\beta]$.

${\rm (ii)}$ When $A$ is as in (2.5), the space ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ is just the anisotropic mixed-norm Hardy space ${H_{{\vec{a}}}^{{\vec{p}}}({{\mathbb{R}^n}})}$; see Remark 1(ⅱ). We should point out that Theorem 3.6 is new even for this case.

As applications of Theorem 3.6, we next give some consequences about the convergence of $\sigma_m^\theta f$ as follows.

Corollary 3.7. With the same assumptions as in Theorem 3.6, if $f\in{{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$, then $\{\sigma_{m}^\theta f\}_{m\in{{\mathbb N}}}$ converges almost everywhereas well as in the ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$-norm as $m\to\infty$.

Proof. Let $h$ be a continuous function with compact support on ${{\mathbb{R}^n}}$. Then, by (3.6), we know that, for any $m\in{{\mathbb N}}$ and $x\in{{\mathbb{R}^n}}$,

Note that, for any $t\in{{\mathbb{R}^n}}$, $\lim_{m\to\infty}A^{-m}t = {\bf 0}$. By (3.5), the fact that $h$ is bounded and the Lebesgue dominated convergence theorem, we find that, for almost every $x\in{{\mathbb{R}^n}}$,

This convergence also holds true in the ${{L^{\vec{p}}({{\mathbb{R}^n}})}}$ quasi–norm since $h\in {{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ which implies $\sigma_*^\theta h\in {{L^{\vec{p}}({{\mathbb{R}^n}})}}$.

On the other hand, from [22,Lemma 9(ⅰ)], we deduce that, for any given $f\in {{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$ and $\epsilon\in(0, {\infty})$, there exists a continuous function with compact support $h$ such that

For any $K\in{{\mathbb N}}$ and $x\in{{\mathbb{R}^n}}$, let

To show Corollary 3.7, we only need to prove that $P = 0$ almost everywhere. To do this, observe that, for any $K\in{{\mathbb N}}$ and $x\in{{\mathbb{R}^n}}$,

Then we have

Combining this, Theorem 3.6 and (3.19), we conclude that

Note that $\epsilon\in(0, {\infty})$ is arbitrary. Then we immediately obtain $P = 0$ almost everywhere, which completes the proof of Corollary 3.7.

The following Corollary 3.8 can be easily verified by Theorem 3.6 and an argument same as that used in the proof of [14,Corollary 2.20]; the details are omitted.

Corollary 3.8. With the same assumptions as in Theorem 3.6, if $f\in{{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$and there exists a subset $I\subset {{\mathbb{R}^n}}$ such that the restriction ${f|}_I \in L^{\vec{q}}(I)$ with $q_-\in[1, {\infty})$, then

Remark 3. Note that, if $p_-\in(1, {\infty})$, then ${{H_A^{\vec{p}}({{\mathbb{R}^n}})}} = {{L^{\vec{p}}({{\mathbb{R}^n}})}}$ with equivalent quasi-norm (see [3,Proposition 2]). Therefore, Corollary 3.8 further implies the following result: Under the same assumptions as in Theorem 3.6, if $f\in{{L^{\vec{p}}({{\mathbb{R}^n}})}}$ with $p_-\in(1, {\infty})$, then

4.   Two specific summability methods

As special cases, we consider two specific summability methods.

4.1.   Bochner–Riesz summation

For any ${\alpha}\in(0, {\infty})$ and $\gamma\in{{\mathbb N}}$, the Bochner-Riesz summation is defined by setting, for any $t\in{{\mathbb{R}^n}}$,

The next lemma comes from ^[16].

Lemma 4.1. Let $\theta_0$ be as in (4.1). If $\alpha\in(\frac{n-1}{2}, {\infty})$, then (3.3) and (3.5) hold trueand, for any ${\beta}\in{{\mathbb Z}}_+^n$, there exists a positive constant$C_{({\alpha}, \beta)}$, depending on ${\alpha}$ and ${\beta}$, such that, for any $x\in{{\mathbb{R}^n}}\setminus\{{\bf 0}\}$,

By Lemma 4.1 and Theorem 3.6, we have the following conclusion for the Bochner-Riesz summation; the details are omitted.

Theorem 4.2. Let $\theta_0$ be as in (4.1) and ${\vec{p}}\in(0, {\infty})^n$.Assume that

Then, for any $f\in {{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$,

where the positive constant $C_{(p_-, p_+)}$, with $p_-$ and $p_+$ as in (2.4), is independent of $f$.

Remark 4. Let $\theta_0$ be as in (4.1). Then, in this special case, the corresponding conclusions in Corollaries 3.7 and 3.8 hold true as well.

4.2.   Weierstrass summation

The Weierstrass summation is defined by setting, for any $t\in{{\mathbb{R}^n}}$,

The succeeding Lemma 4.3 is just [14,Lemma 2.27].

Lemma 4.3. Let $\theta_0$ be as in (4.2). Then (3.3) and (3.5) hold true and, for any $\beta\in(1, {\infty})$ and ${\alpha}\in {{\mathbb Z}}_+^n$, there exists a positive constant$\widetilde{C}_{({\alpha}, \beta)}$, depending on ${\alpha}$ and ${\beta}$, such that, for any $x\in{{\mathbb{R}^n}}\setminus\{{\bf 0}\}$,

By Lemma 4.3 and Theorem 3.6, we obtain the following result for the Weierstrass summation; the details are omitted.

Theorem 4.4. Let $\theta_0$ be as in (4.2) and ${\vec{p}}\in(0, {\infty})^n$.Then, for any $f\in {{H_A^{\vec{p}}({{\mathbb{R}^n}})}}$,

where the positive constant $\widetilde{C}_{(p_-, p_+)}$, with $p_-$ and $p_+$ as in (2.4), is independent of $f$.Moreover, the corresponding conclusions in Corollaries 3.7 and 3.8 hold true as well.

Acknowledgments

The author would like to thank the referees for their careful reading and helpful comments which indeed improved the presentation of this article.

Conflict of interest

The author declares that there is no conflict of interests in this manuscript.