Maximal Function Characterizations of Hardy Spaces on ${\mathbb{R}}^{\mathit{n}}$ with Pointwise Variable Anisotropy

Abstract

In 2011, Dekel et al. developed highly geometric Hardy spaces $H^p(\Theta )$, for the full range $0<p\le 1$, which were constructed by continuous multi-level ellipsoid covers $\Theta$ of ${\mathbb{R}}^n$ with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in $\Theta$ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of $H^p(\Theta )$ in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.

1. Introduction

As a generalization of the classical isotropic Hardy spaces $H^p\left({\mathbb{R}}^n\right)$[1], anisotropic Hardy spaces $H_A^p\left({\mathbb{R}}^n\right)$ were introduced and investigated by Bownik [2] in 2003. These spaces were defined on ${\mathbb{R}}^n$, associated with a fixed expansive matrix, which acts on an ellipsoid instead of Euclidean balls. In [3,4,5,6,7,8], many authors also studied Bownik’s anisotropic Hardy spaces. In 2011, Dekel et al. [9] further generalized Bownik’s spaces by constructing Hardy spaces with pointwise variable anisotropy $H^p(\Theta ),0<p\le 1$, associated with an ellipsoid cover $\Theta$. The anisotropy in Bownik’s Hardy spaces is the same one at each point in ${\mathbb{R}}^n$, while the anisotropy in $H^p(\Theta )$ can change rapidly from point to point and from level to level. Moreover, the ellipsoid cover $\Theta$ is a very general setting that includes the classical isotropic setting, non-isotropic setting of Calderón and Torchinsky [10], and the anisotropic setting of Bownik [2] as special cases; see more details in ([2], pp. 2–3) and ([11], p. 157).

On the other hand, maximal function characterizations are very fundamental characterizations of Hardy spaces, and they are crucial to conveniently apply the real-variable theory of Hardy spaces $H^p\left({\mathbb{R}}^n\right)$ with $p\in (0,1]$. Maximal function characterizations were first shown for the classical isotropic Hardy spaces $H^p\left({\mathbb{R}}^n\right)$ by Fefferman and Stein in their fundamental work [1], ([12], Chapter III). Analogous results were shown by Calder$\acute{\mathrm{o}}$n and Torchinsky [10,13] for parabolic $H^p$ spaces and Uchiyama [14] for $H^p$ on a homogeneous-type space. In 2003, Bownik ([2], p. 42) obtained the maximal function characterizations of the anisotropic Hardy space $H_A^p\left({\mathbb{R}}^n\right)$. This was further extended to anisotropic Hardy spaces of the Musielak–Orlicz type in [15], to anisotropic Hardy–Lorentz spaces in [16], to variable anisotropic Hardy spaces in [17], and to anisotropic mixed-norm Hardy spaces in [18].

Motivated by the abovementioned facts, a natural question arises: Do the maximal function characterizations still hold for Hardy spaces $H^p(\Theta )$ with variable anisotropy? In this article, we answer this question affirmatively in the sense that the ellipsoids in $\Theta$ can change shape rapidly from level to level, which is a variable anisotropic extension of Bownik’s [2].

This article is organized as follows.

In Section 2, we recall some notation and definitions concerning anisotropic continuous ellipsoid cover $\Theta$, several maximal functions, and anisotropic Hardy spaces $H^p(\Theta )$ defined via the grand radial maximal function. We also give some propositions about $H^p(\Theta )$, several classes of variable anisotropic maximal functions, and Schwartz functions since they provide tools for further work. In Section 3, we first state the main result: if the ellipsoids in $\Theta$ can rapidly change shape from level to level (see Definition 1), denoted as ${\Theta }_t$, we may obtain some real-variable characterizations of $H^p\left({\Theta }_t\right)$ in terms of the radial, the non-tangential, and the tangential maximal functions (see Theorem 1). Then, we present several lemmas that are isotropic extensions in the setting of variable anisotropy, and finally, we show the proof for the main result.

In the process of proving the main result, we used the methods from Stein [1] and Bownik [2]. However, it is worth pointing out that these ellipsoids of Bownik were images of the unit ball by powers of a fixed expansive matrix, whereas in our case, the ellipsoids of Dekel are images of the unit ball by powers of a group of matrices satisfying some “shape condition”. This makes the proof complicated and needs many subtle estimates such as Propositions 5 and 6, and Lemma 1.

However, this article left an open question: if the maximal function characterizations of $H^p(\Theta )$ still hold true in the sense that the ellipsoids of $\Theta$ change rapidly from level to level and from point to point?

Finally, we note some conventions on notation. Let ${\mathbb{N}}_0:=\{0,1,2,\dots \}$ and $\lceil t\rceil$ be the smallest integer no less than t. For any $\alpha :=({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}_0^n$, $\left|\alpha \right|:={\alpha }_1+\cdots +{\alpha }_n$ and ${\partial }^{\alpha }:={\left(\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\cdots {\left(\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}$. Throughout the whole paper, we denote by C a positive constant that is independent on the main parameters but may vary from line to line. For any sets $E,F\subset {\mathbb{R}}^n$, we use $E^{\complement }$ to denote the set ${\mathbb{R}}^n\backslash E$. If there are no special instructions, any space $\mathcal{X}\left({\mathbb{R}}^n\right)$ is denoted simply by $\mathcal{X}$. Denote by $\mathcal{S}$ the space of all Schwartz functions and ${\mathcal{S}}^{\prime }$ the space of all tempered distributions.

2. Preliminary and Some Basic Propositions

In this section, we first recall the notion of continuous ellipsoid covers $\Theta$ and we introduce the pointwise continuity for $\Theta$. An ellipsoid $\xi$ in ${\mathbb{R}}^n$ is an image of the Euclidean unit ball ${\mathbb{B}}^n:=\{x\in {\mathbb{R}}^n:\left|x\right|<1\}$ under an affine transform, i.e.,

$$\xi :=M_{\xi }\left({\mathbb{B}}^n\right)+c_{\xi },$$

where $M_{\xi }$ is a non-singular matrix and $c_{\xi }\in {\mathbb{R}}^n$ is the center.

Let us begin with the definition of continuous ellipsoid covers, which was introduced in ([11], Definition 2.4).

Definition 1.

We say that

$$\Theta :=\{\theta (x,t):x\in {\mathbb{R}}^n,t\in \mathbb{R}\}$$

is a continuous ellipsoid cover of ${\mathbb{R}}^n$ or, in short, an ellipsoid cover if there exist positive constants $p(\Theta ):=\{a_1,\dots ,a_6\}$ such that

(i)

For every $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$, there exists an ellipsoid $\theta (x,t):=M_{x,t}\left({\mathbb{B}}^n\right)+x$ satisfying

$$a_1{2}^{-t}\le \left|\theta (x,t)\right|\le a_2{2}^{-t}.$$

(1)

(ii)

Intersecting ellipsoids from Θ satisfy a “shape condition”, i.e., for any $x,y\in {\mathbb{R}}^n$, $t\in \mathbb{R}$ and $s\ge 0$, if $\theta (x,t)\cap \theta \left(y,t+s\right)\ne Ø$, then

$$a_3{2}^{-a_{4}s}\le \frac{1}{\parallel {\left(M_{y,t+s}\right)}^{-1}{M}_{x,t}\parallel }\le \parallel {\left(M_{x,t}\right)}^{-1}{M}_{y,t+s}\parallel \le a_5{2}^{-a_{6}s}.$$

(2)

where $\parallel ·\parallel$ is the matrix norm given by $\parallel M\parallel :={max}_{\left|x\right|=1}\left|Mx\right|$ for an $n\times n$ real matrix M.

Particularly, for any $\theta (x,t)\in \Theta$, when the related matrix function $M_{x,t}$ of $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$ is reduced to the matrix function $M_t$ of $t\in \mathbb{R}$, we call a cover Θ a t-continuous ellipsoid cover, denoted as ${\Theta }_t$.

The word continuous refers to the fact that ellipsoids $\theta x,t$ are defined for all values of $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$, and we say that a continuous ellipsoid cover Θ is pointwise continuous if, for every $t\in \mathbb{R}$, the matrix valued function $x\mapsto M_{x,t}$ is continuous:

$$\begin{array}{c}\hfill \parallel M_{x^{\prime },t}-M_{x,t}\parallel \to 0as{x}^{\prime }\to x.\end{array}$$

(3)

Remark 1.

By ([19], Theorem 2.2), we know that the pointwise continuous assumption is not necessary since it is always possible to construct an equivalent ellipsoid cover

$$\Xi :=\{\zeta x,t:x\in {\mathbb{R}}^n,t\in \mathbb{R}\}$$

such that Ξ is pointwise continuous and Ξ is equivalent to Θ. Here, we say that two ellipsoid covers Θ and Ξ are equivalent if there exists a constant $C>0$ such that, for any $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$, we have

$$\frac{1}{C}\zeta x,t\subset \theta x,t\subset C\zeta x,t.$$

Taking $M_{y,t+s}=M_{x,t}$ in (2), we have

$$\begin{array}{c}\hfill a_3\le 1\mathrm{and}{a}_5\ge 1.\end{array}$$

(4)

For more properties about ellipsoid covers, see [9,11].

For any $N,\tilde{N}\in {\mathbb{N}}_0$ with $N\le \tilde{N}$, let

$$\begin{array}{c}\hfill {\mathcal{S}}_{N,\tilde{N}}:=\left\{\psi \in \mathcal{S}:{∥\psi ∥}_{{\mathcal{S}}_{N,\tilde{N}}}:=\underset{\alpha \in {\mathbb{N}}_0^n,\left|\alpha \right|\le N}{max}\underset{y\in {\mathbb{R}}^n}{sup}{(1+|y\left|\right)}^{\tilde{N}}\left|{\partial }^{\alpha }\psi \left(y\right)\right|\le 1\right\}.\end{array}$$

For any $\phi \in \mathcal{S}$, $x\in {\mathbb{R}}^n$, $t\in \mathbb{R}$ and $\theta (x,t)=M_{x,t}\left({\mathbb{B}}^n\right)+x$, denote

$${\phi }_{x,t}\left(y\right):=\left|\det \left(M_{x,t}^{-1}\right)\right|\phi \left(M_{x,t}^{-1}y\right),y\in {\mathbb{R}}^n.$$

Particularly, when the matrix $M_{x,t}$ is reduced to $M_t$, ${\phi }_{x,t}\left(y\right)$ is simply denoted as ${\phi }_t\left(y\right)$.

Now, we give the notions of anisotropic variants of the non-tangential, the grand non-tangential, the radial, the grand radial, and the tangential maximal functions.

Definition 2.

Let $f\in {\mathcal{S}}^{\prime }$, $\phi \in \mathcal{S}$ and $N,\tilde{N}\in {\mathbb{N}}_0$ with $N\le \tilde{N}$. We define the non-tangential, the grand non-tangential, the radial, the rand radial, and the tangential maximal functions, respectively as

$$M_{\phi }f\left(x\right):=\underset{t\in \mathbb{R}}{sup}\underset{y\in \theta (x,t)}{sup}|f\ast {\phi }_{x,t}\left(y\right)|,x\in {\mathbb{R}}^n,$$

$$M_{N,\tilde{N}}f\left(x\right):=\underset{\phi \in {\mathcal{S}}_{N,\tilde{N}}}{sup}{M}_{\phi }f\left(x\right),x\in {\mathbb{R}}^n,$$

$$M_{\phi }^{0}f\left(x\right):=\underset{t\in \mathbb{R}}{sup}|f\ast {\phi }_{x,t}\left(x\right)|,x\in {\mathbb{R}}^n,$$

$$M_{N,\tilde{N}}^{0}f\left(x\right):=\underset{\phi \in {\mathcal{S}}_{N,\tilde{N}}}{sup}{M}_{\phi }^{\circ }f\left(x\right),x\in {\mathbb{R}}^n,$$

$$T_{\phi }^{N}f\left(x\right):=\underset{t\in \mathbb{R}}{sup}\underset{y\in {\mathbb{R}}^n}{sup}|f\ast {\phi }_{x,t}\left(y\right)|{\left(1+\left|M_{x,t}^{-1}(x-y)\right|\right)}^{-N},x\in {\mathbb{R}}^n.$$

Here and hereafter, the symbol "∗" always represents a convolution.

Remark 2.

We immediately have the following pointwise estimate among the radial, the non-tangential, and the tangential maximal functions:

$$M_{\phi }^{0}f\left(x\right)\le M_{\phi }f\left(x\right)\le 2^N{T}_{\phi }^{N}f\left(x\right),x\in {\mathbb{R}}^n.$$

Next, we recall the definition of Hardy spaces with pointwise variable anisotropy ([9], Definition 3.6) via the grand radial maximal function.

Let $\Theta$ be an ellipsoid cover of ${\mathbb{R}}^n$ with parameters $p(\Theta )=\{a_1,\cdots ,a_6\}$ and $0<p\le 1$. We define $N_p(\Theta )$ as the minimal integer satisfying

$$N_p:=N_p(\Theta )>\frac{max(1,a_4)n+1}{a_{6}p},$$

(5)

and then ${\tilde{N}}_p(\Theta )$ as the minimal integer satisfying

$${\tilde{N}}_p:={\tilde{N}}_p(\Theta )>\frac{a_4{N}_p(\Theta )+1}{a_6}.$$

(6)

Definition 3.

Let Θ be a continuous ellipsoid cover and $0<p\le 1$. Define $M^0:=M_{N_p,{\tilde{N}}_p}^0$, and the anisotropic Hardy space is defined as

$$H_{N_p,{\tilde{N}}_p}^p(\Theta ):=\{f\in {\mathcal{S}}^{\prime }:M^{0}f\in L^p\}$$

with the (quasi-)norm ${\parallel f\parallel }_{H^p(\Theta )}:={\parallel M^{0}f\parallel }_{L^p}$.

Remark 3.

By Remark 1, we know that, for every continuous ellipsoid cover Θ, there exists an equivalent pointwise continuous ellipsoid cover Ξ. This implies that their corresponding (quasi-)norms ${\rho }_{\Theta }(·,·)$ and ${\rho }_{\Xi }(·,·)$ are also equivalent, and hence, the corresponding Hardy spaces $H^p(\Theta )=H^p(\Xi )(0<p\le 1)$ with equivalent (quasi-)norms (see ([9], Theorem 5.8)). Therefore, here and hereafter, we always consider Θ of $H^p(\Theta )$ to be a pointwise continuous ellipsoid cover.

Proposition 1.

Let Θ be an ellipsoid cover, $0<p\le 1\le q\le \infty$, $p<q$ and $l\ge N_p$ with $N_p$ as in (5). If $N\ge N_p$ and $\tilde{N}\ge \left(a_{4}N+1\right)/a_6$, then

$$\begin{array}{c}\hfill H_{N_p,{\tilde{N}}_p}^p(\Theta )=H_{q,l}^p(\Theta )=H_{N,\tilde{N}}^p(\Theta )\end{array}$$

with equivalent (quasi-)norms, where $H_{q,l}^p(\Theta )$ denotes the atomic Hardy space with pointwise variable anisotropy; see ([9], Definition 4.2).

Proof. 

This proposition is a corollary of ([9], Theorems 4.4 and 4.19). Indeed, by Definition 3, we obtain that, for any $N\ge N_p$ and $\tilde{N}\ge \left(a_{4}N+1\right)/a_6$,

$$H_{N_p,{\tilde{N}}_p}^p(\Theta )\subseteq H_{N,\tilde{N}}^p(\Theta ).$$

Combining this and $H_{q,l}^p(\Theta )\subseteq H_{N_p,{\tilde{N}}_p}^p(\Theta )$ (see ([9], Theorem 4.4)), we obtain

$$\begin{array}{c}\hfill H_{q,l}^p(\Theta )\subseteq H_{N,\tilde{N}}^p(\Theta ).\end{array}$$

(7)

By checking the definition of anisotropic $(p,q,l)$-atom (see ([9], Definition 4.1)), we know that every $(p,\infty ,l)$-atom is also a $(p,q,l)$-atom and hence

$$H_{\infty ,l}^p(\Theta )\subseteq H_{q,l}^p(\Theta ).$$

Let $l^{\prime }\ge max(l,N)$. By a similar argument to the proof of ([9], Theorem 4.19), we obtain

$$\begin{array}{c}\hfill H_{N,\tilde{N}}^p(\Theta )\subseteq H_{\infty ,l^{\prime }}^p(\Theta ),\end{array}$$

where $N\ge N_p$ and $\tilde{N}\ge \left(a_{4}N+1\right)/a_6$. Thus,

$$\begin{array}{c}\hfill H_{N,\tilde{N}}^p(\Theta )\subseteq H_{\infty ,l^{\prime }}^p(\Theta )\subseteq H_{\infty ,l}^p(\Theta )\subseteq H_{q,l}^p(\Theta ).\end{array}$$

(8)

Combining (7) and (8), we conclude that

$$\begin{array}{c}\hfill H_{N_p,{\tilde{N}}_p}^p(\Theta )=H_{q,l}^p(\Theta )=H_{N,\tilde{N}}^p(\Theta )\end{array}$$

with equivalent (quasi-)norms. □

Remark 4.

From Proposition 1, we deduce that, for any integers $N\ge N_p$ and $\tilde{N}\ge \left(a_{4}N+1\right)/a_6$, the definition of $H_{N,\tilde{N}}^p(\Theta )$ is independent of N and $\tilde{N}$. Therefore, from now on, we denote $H_{N,\tilde{N}}^p(\Theta )$ with $N\ge N_p$ and $\tilde{N}\ge \left(a_{4}N+1\right)/a_6$ simply by $H^p(\Theta )$.

Proposition 2

([9], Lemma 2.3). Let Θ be an ellipsoid cover. Then, there exists a constant $J:=J\left(p\right(\Theta \left)\right)\ge 1$ such that, for any $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$,

$$2M_{x,t}\left(\mathbb{B}\right)+x\subset \theta (x,t-J).$$

Here and hereafter, let J always be as in Proposition 2.

Definition 4

([9], Definition 3.1). Let Θ be an ellipsoid cover. For any locally integrable function f, the maximal function of the Hardy–Littlewood type of f is defined by

$$M_{\Theta }f\left(x\right):=\underset{t\in \mathbb{R}}{sup}\frac{1}{\left|\theta \right(x,t\left)\right|}{\int }_{\theta (x,t)}\left|f\left(y\right)\right|dy,x\in {\mathbb{R}}^n.$$

Proposition 3

([9], Theorem 3.3). Let Θ be an ellipsoid cover. Then,

(i) 

There exists a constant C depending only on $p(\Theta )$ and n such that for all $f\in L^1$ and $\alpha >0$,

$$\begin{array}{c}\hfill |\{x:M_{\Theta }f\left(x\right)>\alpha \}|\le C{\alpha }^{-1}{\parallel f\parallel }_{L^1};\end{array}$$

(9)

(ii) 

For $1<p<\infty$, there exists a constant $C_p$ depending only on C and p such that, for all $f\in L^p$,

$$\begin{array}{c}\hfill \parallel M_{\Theta }{f\parallel }_{L^p}\le C_p{\parallel f\parallel }_{L^p}.\end{array}$$

(10)

We give some useful results about variable anisotropic maximal functions with different apertures. They also play important roles in obtaining the maximal function characterizations of $H^p(\Theta )$. For any given $x\in {\mathbb{R}}^n$, suppose that $F:{\mathbb{R}}^n\times \mathbb{R}\to (0,\infty )$ is a Lebesgue measurable function. Let $\Theta$ be an ellipsoid cover. For fixed $l\in \mathbb{Z}$ and $t_0<0$, define the maximal function of F with aperture l as

$$\begin{array}{c}\hfill F_l^{\ast t_0}\left(x\right):=\underset{t\ge t_0}{sup}\underset{y\in \theta (x,t-lJ)}{sup}F(y,t).\end{array}$$

(11)

Proposition 4.

For any $l\in \mathbb{Z}$ and $t_0<0$, let $F_l^{\ast t_0}$ be as in (11). If the ellipsoid cover Θ is pointwise continuous, then $F_l^{\ast t_0}:{\mathbb{R}}^n\to (0,\infty ]$ is lower semi-continuous, i.e.,

$$\{x\in {\mathbb{R}}^n:F_l^{\ast t_0}\left(x\right)>\lambda \}\mathit{is}\mathit{open}\mathit{for}\mathit{any}\lambda >0.$$

Proof. 

If $F_l^{\ast t_0}\left(x\right)>\lambda$ for some $x\in {\mathbb{R}}^n$, then there exist $t\ge t_0$ and $y\in \theta (x,t-lJ)$ such that $F(y,t)>\lambda$. Since $\theta \left(x,t\right)$ is continuous for variable x (see Remark 1), there exists ${\delta }_1>0$ such that, for any $x^{\prime }\in U(x,\delta ):=\{z\in {\mathbb{R}}^n:|z-x|<\delta \}$, $y\in \theta (x^{\prime },t-lJ)$ and hence $F_l^{\ast t_0}\left(x^{\prime }\right)>\lambda$. □

By Proposition 4, we obtain that $\{x\in {\mathbb{R}}^n:F_l^{\ast t_0}\left(x\right)>\lambda \}$ is Lebesgue measurable. Based on this and inspired by ([2], Lemma 7.2), the following Proposition 5 shows some estimates for maximal function $F_l^{\ast t_0}$.

Proposition 5.

Let Θ be an ellipsoid cover, $F_l^{\ast t_0}$ and $F_{l^{\prime }}^{\ast t_0}$ as in (11) with integers $l>l^{\prime }$ and $t_0<0$. Then, there exists a constant $C>0$ that depends on parameters $p(\Theta )$ such that, for any functions $F_l^{\ast t_0}$, $F_{l^{\prime }}^{\ast t_0}$ and $\lambda >0$, we have

$$\begin{array}{c}\hfill \left|\left\{x\in {\mathbb{R}}^n:F_l^{\ast t_0}\left(x\right)>\lambda \right\}\right|\le C{2}^{(l-l^{\prime })J}\left|\left\{x\in {\mathbb{R}}^n:F_{l^{\prime }}^{\ast t_0}\left(x\right)>\lambda \right\}\right|\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{F}_l^{\ast t_0}\left(x\right)dx\le C{2}^{(l-l^{\prime })J}{\int }_{{\mathbb{R}}^n}{F}_{l^{\prime }}^{\ast t_0}\left(x\right)dx.\end{array}$$

(13)

Proof. 

Let $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^n:F_{l^{\prime }}^{\ast t_0}\left(x\right)>\lambda \}$. We claim that

$$\begin{array}{c}\hfill \left\{x\in {\mathbb{R}}^n:F_l^{\ast t_0}\left(x\right)>\lambda \right\}\subset \left\{x\in {\mathbb{R}}^n:M_{\Theta }\left({\chi }_{\mathrm{\Omega }}\right)\left(x\right)\ge C_1{2}^{(l^{\prime }-l)J}\right\},\end{array}$$

(14)

where $C_1$ is a positive constant to be fixed later. Assuming that the claim holds for the moment, from this and a weak type (1,1) of $M_{\Theta }$ (see (9)), we deduce

$$\begin{array}{cc}\hfill \left|\left\{x\in {\mathbb{R}}^n:F_l^{\ast t_0}\left(x\right)>\lambda \right\}\right|& \le \left|\left\{x\in {\mathbb{R}}^n:M_{\Theta }\left({\chi }_{\mathrm{\Omega }}\right)\left(x\right)\ge C_1{2}^{(l^{\prime }-l)J}\right\}\right|\hfill \\ \hfill & \le C_1^{-1}{2}^{(l-l^{\prime })J}\parallel {\chi }_{\mathrm{\Omega }}{\parallel }_{L^1}\le C{2}^{(l-l^{\prime })J}\left|\mathrm{\Omega }\right|\hfill \end{array}$$

and hence (12) holds true, where $C:=1/C_1.$ Furthermore, integrating (12) on $(0,\infty )$ with respect to $\lambda$ yields (13). Therefore, (14) remains to be shown.

Suppose $F_l^{\ast t_0}\left(x\right)>\lambda$ for some $x\in {\mathbb{R}}^n$. Then, there exist t with $t\ge t_0$ and $y\in \theta (x,t-lJ)$ such that $F(y,t)>\lambda$. For any $l,l^{\prime }\in \mathbb{Z}$ and $l\ge l^{\prime }$, we first prove that the following holds true:

$$\begin{array}{c}\hfill {a_5}^{-1}\theta (y,t-l^{\prime }J)\subseteq \theta (x,t-(l+1)J)\cap \mathrm{\Omega }.\end{array}$$

(15)

For any $z\in {a_5}^{-1}\theta (y,t-l^{\prime }J)$, by (4), we have $z\in \theta (y,t-l^{\prime }J)$ and hence

$$\theta (z,t-l^{\prime }J)\cap \theta (y,t-l^{\prime }J)\ne Ø.$$

Thus, by (2), we have

$$∥M_{z,t-l^{\prime }J}^{-1}{M}_{y,t-l^{\prime }J}∥\le a_5.$$

From this, it follows that

$${a_5}^{-1}{M}_{z,t-l^{\prime }J}^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)\subseteq {\mathbb{B}}^n$$

and hence

$$\begin{array}{c}\hfill {a_5}^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)\subseteq M_{z,t-l^{\prime }J}\left({\mathbb{B}}^n\right).\end{array}$$

By this and $y\in a_5^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)+z$, we obtain $y\in \theta (z,t-l^{\prime }J)$. From this and $F(y,t)>\lambda$ with $t\ge t_0$, we deduce that $F_{l^{\prime }}^{\ast t_0}\left(z\right)>\lambda$, and hence, $z\in \mathrm{\Omega }$, which implies

$$\begin{array}{c}\hfill a_5^{-1}\theta (y,t-l^{\prime }J)\subseteq \mathrm{\Omega }.\end{array}$$

(16)

Moreover, by $y\in \theta (x,t-lJ)$, (2), and $l\ge l^{\prime }$, we have

$$∥M_{x,t-lJ}^{-1}{M}_{y,t-l^{\prime }J}∥\le a_5{2}^{-a_6(l-l^{\prime })J}\le a_5.$$

From this, it follows that

$${a_5}^{-1}{M}_{x,t-lJ}^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)\subseteq {\mathbb{B}}^n$$

and hence

$$\begin{array}{c}\hfill {a_5}^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)\subseteq M_{x,t-lJ}\left({\mathbb{B}}^n\right).\end{array}$$

By this, (4), $y\in \theta (x,t-lJ)$, and Proposition 2, we obtain

$$\begin{array}{c}\hfill {a_5}^{-1}{M}_{y,t-l^{\prime }J}\left({\mathbb{B}}^n\right)+y\subseteq 2M_{x,t-lJ}\left({\mathbb{B}}^n\right)+x\subseteq \theta (x,t-(l+1)J).\end{array}$$

From this and (16), we deduce that (15) holds true.

Next, let us prove (14). By (15) and (1), we obtain

$$\begin{array}{cc}\hfill \left|\theta \right(x,t-(l+1)J)\cap \mathrm{\Omega }|& \ge {\left(a_5\right)}^{-n}\left|\theta (y,t-l^{\prime }J)\right|\hfill \\ \hfill & \ge \frac{a_1}{{\left(a_5\right)}^n}{2}^{l^{\prime }J-t}.\hfill \end{array}$$

(17)

Taking $b_0:=t-(l+1)J$, by (1) and (17), we have

$$\begin{array}{cc}\hfill \frac{1}{\left|\theta \right(x,b_0\left)\right|}{\int }_{\theta (x,b_0)}\left|{\chi }_{\mathrm{\Omega }}\left(y\right)\right|dy& \ge {a_2}^{-1}{2}^{b_0}|\theta (x,b_0)\cap \mathrm{\Omega }|\ge \frac{a_1}{{\left(a_5\right)}^n{a}_2}{2}^{(l^{\prime }-l-1)J},\hfill \end{array}$$

which implies $M_{\Theta }\left({\chi }_{\mathrm{\Omega }}\right)\left(x\right)\ge C_1{2}^{(l^{\prime }-l)J}$ and hence (14) holds true, where $C_1:=2^{-J}{a}_1/\left[{\left(a_5\right)}^n{a}_2\right]$. □

The following result enables us to pass from one function in $\mathcal{S}$ to the sum of dilates of another function in $\mathcal{S}$ with nonzero mean, which is a variable anisotropic extension of ([12], p. 93, Lemma 2) of Stein and ([2], Lemma 7.3) of Bownik.

Proposition 6.

Let Θ be an ellipsoid cover of ${\mathbb{R}}^n$ and $\phi \in \mathcal{S}$, with ${\int }_{{\mathbb{R}}^n}\phi \left(x\right)dx\ne 0$. Then, for any $\psi \in \mathcal{S}$, $x\in {\mathbb{R}}^n$, and $t\in \mathbb{R}$, there exists a sequence ${\left\{{\eta }^k\right\}}_{k=0}^{\infty }$ and ${\eta }^k\in \mathcal{S}$, such that

$$\begin{array}{c}\hfill \psi =\sum _{k=0}^{\infty }{\eta }^k\ast {\phi }^k\end{array}$$

(18)

converges in $\mathcal{S}$, where

$${\phi }^k:=\left|\det \left(M_{x,t+kJ}^{-1}{M}_{x,t}\right)\right|\phi (M_{x,t+kJ}^{-1}{M}_{x,t}·),k>0,$$

where $J>0$ is as in Proposition 2.

Furthermore, for any positive integers $N,\tilde{N}$ and L, there exists a constant $C>0$ depending on φ, L, N, $\tilde{N}$, and $p(\Theta )$ but not ψ, such that

$$\begin{array}{c}\hfill {\parallel {\eta }^k\parallel }_{\mathcal{S}N,\tilde{N}}\le C{2}^{-kL}{\parallel \psi \parallel }_{{\mathcal{S}}_{N+n+1+\lceil L/\left(a_{6}J\right)\rceil ,\tilde{N}+n+1}}.\end{array}$$

(19)

Proof. 

The following simplified proof is accomplished by Dekel. By scaling $\phi$, we can assume that ${\int }_{{\mathbb{R}}^n}\phi \left(x\right)dx=1$ and $|\widehat{\phi }\left(\xi \right)|\ge 1/2$, for $\left|\xi \right|\le 2$. This assumption only impacts the constant in (19). Let $\zeta \in \mathcal{S}$ such that $0\le \zeta \le 1$ on ${\mathbb{B}}^n$ and supp $\left(\zeta \right)\subset 2{\mathbb{B}}^n$. We fix $x\in {\mathbb{R}}^n$ and $t\in \mathbb{R}$, denote $M_k:=M_{x,t+kJ}$, and define the sequence of functions ${\left\{{\zeta }_k\right\}}_{k=0}^{\infty }$, where ${\zeta }_0:=\zeta$, and

$${\zeta }_k:=\zeta \left({\left(M_{x,t}^{-1}{M}_k\right)}^T·\right)-\zeta \left({\left(M_{x,t}^{-1}{M}_{k-1}\right)}^T·\right),k\ge 1,$$

where $M^T$ denotes the transpose of a matrix M. We claim that

$$\begin{array}{c}\hfill \mathrm{supp}\left({\zeta }_k\right)\subset \left\{\xi \in {\mathbb{R}}^n:a_5^{-1}{2}^{-a_{6}J}{2}^{a_{6}kJ}\le \left|\xi \right|\le 2a_3^{-1}{2}^{a_{4}kJ}\right\}.\end{array}$$

(20)

Indeed, by the properties of $\zeta$, Proposition 2 and (2),

$$\begin{array}{cc}\hfill \xi \in \mathrm{supp}\left({\zeta }_k\right)& \Rightarrow {\left(M_{x,t}^{-1}{M}_k\right)}^T\left(\xi \right)\in 2{\mathbb{B}}^n\vee {\left(M_{x,t}^{-1}{M}_{k-1}\right)}^T\left(\xi \right)\in 2{\mathbb{B}}^n\hfill \\ \hfill & \Rightarrow \xi \in 2{\left(M_k^{-1}{M}_{x,t}\right)}^T\left({\mathbb{B}}^n\right)\vee \xi \in 2{\left(M_{k-1}^{-1}{M}_{x,t}\right)}^T\left({\mathbb{B}}^n\right)\hfill \\ \hfill & \Rightarrow \xi \in 2a_3^{-1}{2}^{a_{4}kJ}{\mathbb{B}}^n.\hfill \end{array}$$

In the other direction, Proposition 2 and the properties of $\zeta$ yield

$$\begin{array}{cc}\hfill \xi \in {\left(M_{k-1}^{-1}{M}_{x,t}\right)}^T\left({\mathbb{B}}^n\right)& \Rightarrow {\left(M_{x,t}^{-1}{M}_k\right)}^T\left(\xi \right)\in {\mathbb{B}}^n,{\left(M_{x,t}^{-1}{M}_{k-1}\right)}^T\left(\xi \right)\in {\mathbb{B}}^n\hfill \\ \hfill & \Rightarrow {\zeta }_k\left(\xi \right)=0.\hfill \end{array}$$

Applying (2), we have

$$\xi \notin {\left(M_{k-1}^{-1}{M}_{x,t}\right)}^T\left({\mathbb{B}}^n\right)\Rightarrow \left|\xi \right|\ge 2a_5^{-1}{2}^{a_6(k-1)J}.$$

This proves (20). Additionally, by (2), for any $\xi \in {\mathbb{R}}^n$,

$$\begin{array}{c}\hfill \left|{\left(M_{x,t}^{-1}{M}_k\right)}^T\xi \right|\le ∥M_{x,t}^{-1}{M}_k∥\left|\xi \right|\le a_5{2}^{-a_{6}kJ}\left|\xi \right|\to 0,k\to \infty .\end{array}$$

From this, we deduce that, for any $\xi \in {\mathbb{R}}^n$, for a large enough k, ${\left(M_{x,t}^{-1}{M}_k\right)}^T\xi \in {\mathbb{B}}^n.$ This implies that

$$\sum _{k=0}^{\infty }{\zeta }_k\left(\xi \right)=1,\forall \xi \in {\mathbb{R}}^n.$$

Thus, formally, a Fourier transform of (18) is given by

$$\begin{array}{c}\hfill \widehat{\psi }=\sum _{k=0}^{\infty }\widehat{{\eta }^k}\widehat{\phi }\left({\left(M_{x,t}^{-1}{M}_k\right)}^T·\right),\widehat{{\eta }^k}:=\frac{{\zeta }_k}{\widehat{\phi }\left({\left(M_{x,t}^{-1}{M}_k\right)}^T·\right)}\widehat{\psi }.\end{array}$$

Observe that ${\eta }^k$ is well defined and in $\mathcal{S}$. Indeed, $\widehat{{\eta }^k}$ is well defined with $0/0:=0,$ since by our assumption on $\phi$,

$$\begin{array}{cc}\hfill \xi \in supp\left({\zeta }_k\right)& \Rightarrow \xi \in 2{\left(M_k^{-1}{M}_{x,t}\right)}^T\left({\mathbb{B}}^n\right)\hfill \\ \hfill & \Rightarrow \left|{\left(M_{x,t}^{-1}{M}_k\right)}^T\xi \right|\le 2\hfill \\ \hfill & \Rightarrow \widehat{\phi }\left({\left(M_{x,t}^{-1}{M}_k\right)}^T\xi \right)\ge \frac{1}{2}.\hfill \end{array}$$

From this, it is obvious that $\widehat{{\eta }^k}\in \mathcal{S}$, and therefore, ${\eta }^k\in \mathcal{S}$. We now proceed to prove (19). First, observe that, for any $\eta \in \mathcal{S},N,\tilde{N}\in \mathbb{N},$

$$\begin{array}{c}\hfill {\parallel \eta \parallel }_{{\mathcal{S}}_{N,\tilde{N}}}\le C(N,\tilde{N},n){\parallel \widehat{\eta }\parallel }_{{\mathcal{S}}_{\tilde{N},N+n+1}}.\end{array}$$

(21)

Next, we claim that, for any $K\in \mathbb{N},$

$$\begin{array}{c}\hfill \underset{\left|\alpha \right|\le K}{max}{∥{\partial }^{\alpha }\left({\zeta }_k/\widehat{\phi }\left({\left(M_k^{-1}{M}_{x,t}\right)}^T·\right)\right)∥}_{\infty }\le C(K,n,\phi ).\end{array}$$

(22)

Indeed, on its support, any partial derivative of ${\zeta }_k/\widehat{\phi }({\left(M_{x,t}^{-1}{M}_k\right)}^T·)$ has a denominator with its absolute value bounded from below and a numerator that is a superposition of compositions of partial derivatives of $\eta$ and $\phi$ with contractive matrices of the type ${\left(M_{x,t}^{-1}{M}_k\right)}^T$. Using (20)–(22), we obtain

$$\begin{array}{cc}\hfill {∥{\eta }^k∥}_{{\mathcal{S}}_{N,\tilde{N}}}& \le C{∥\widehat{{\eta }^k}∥}_{{\mathcal{S}}_{\tilde{N},N+n+1}}\hfill \\ \hfill & \le C\underset{\left|\xi \right|\ge a_5^{-1}{2}^{-a_{6}J}{2}^{a_{6}kJ}}{sup}\underset{\left|\alpha \right|\le \tilde{N}}{max}\left|{\partial }^{\alpha }\widehat{{\eta }^k}\left(\xi \right)\right|{(1+|\xi \left|\right)}^{N+n+1}\hfill \\ \hfill & \le C\underset{\left|\xi \right|\ge a_5^{-1}{2}^{-a_{6}J}{2}^{a_{6}kJ}}{sup}\underset{\left|\alpha \right|\le \tilde{N}}{max}\left|{\partial }^{\alpha }\widehat{\psi }\left(\xi \right)\right|{(1+|\xi \left|\right)}^{N+n+1}\hfill \\ \hfill & \le C\underset{\left|\xi \right|\ge a_5^{-1}{2}^{-a_{6}J}{2}^{a_{6}kJ}}{sup}\underset{\left|\alpha \right|\le \tilde{N}}{max}\left|{\partial }^{\alpha }\widehat{\psi }\left(\xi \right)\right|{(1+|\xi \left|\right)}^{N+n+1+\lceil L/\left(a_{6}J\right)\rceil }\hfill \\ \hfill & \times {(1+|\xi \left|\right)}^{-\lceil L/\left(a_{6}J\right)\rceil }\hfill \\ \hfill & \le C{2}^{-kL}{∥\widehat{\psi }∥}_{{\mathcal{S}}_{\tilde{N},N+n+1+\lceil L/\left(a_{6}J\right)\rceil }}\hfill \\ \hfill & \le C{2}^{-kL}{∥\psi ∥}_{{\mathcal{S}}_{N+n+1+\lceil L/\left(a_{6}J\right)\rceil ,\tilde{N}+n+1}}.\hfill \end{array}$$

□

3. Maximal Function Characterizations of ${\mathit{H}}^{\mathit{p}}\left({\Theta }_{\mathit{t}}\right)$

In this section, we show the maximal function characterizations of $H^p\left({\Theta }_t\right)$ using the radial, the non-tangential, and the tangential maximal functions of a single test function $\phi \in \mathcal{S}$.

Theorem 1.

Let ${\Theta }_t$ be a t-continuous ellipsoid cover, $0<p\le 1$, and $\phi \in \mathcal{S}$ satisfy ${\int }_{{\mathbb{R}}^n}\phi \left(x\right)dx\ne 0$. Then, for any $f\in {\mathcal{S}}^{\prime }$, the following are mutually equivalent:

$$\begin{array}{c}\hfill f\in H^p\left({\Theta }_t\right);\end{array}$$

(23)

$$\begin{array}{c}\hfill M_{\phi }f\in L^p;\end{array}$$

(24)

$$\begin{array}{c}\hfill M_{\phi }^{0}f\in L^p;\end{array}$$

(25)

$$\begin{array}{cc}\hfill T_{\phi }^{N}f\in L^p& ,N>\frac{1}{a_{6}p}.\hfill \end{array}$$

(26)

In this case,

$$\begin{array}{c}\hfill {\parallel f\parallel }_{H^p\left({\Theta }_t\right)}={∥M^{0}f∥}_{L^p}\le C_1{∥T_{\phi }^{N}f∥}_{L^p}\le C_2{∥M_{\phi }f∥}_{L^p}\le C_3{∥M_{\phi }^{0}f∥}_{L^p}\le C_4{\parallel f\parallel }_{H^p\left({\Theta }_t\right)},\end{array}$$

where the positive constants $C_1$, $C_2$, $C_3$ and $C_4$ are independent of f.

The framework to prove Theorem 1 is motivated by Fefferman and Stein [1], ([12], Chapter III), and Bownik ([2], p. 42, Theorem 7.1).

Inspired by Fefferman and Stein ([12], p. 97), and Bownik ([2], p. 47), we now start with maximal functions obtained from truncation with an additional extra decay term. Namely, for $t_0<0$ representing the truncation level and real number $L\ge 0$ representing the decay level, we define the radial, the non-tangential, the tangential, the grand radial, and the grand non-tangential maximal functions, respectively, as

$$\begin{array}{cc}\hfill & M_{\phi }^{0(t_0,L)}f\left(x\right):=\underset{t\ge t_0}{sup}\left|(f\ast {\phi }_{x,t})\left(x\right)\right|{\left(1+\left|M_{x,t_0}^{-1}x\right|\right)}^{-L}{\left(1+2^{t+t_0}\right)}^{-L},\hfill \\ \hfill & M_{\phi }^{(t_0,L)}f\left(x\right):=\underset{t\ge t_0}{sup}\underset{y\in \theta (x,t)}{sup}\left|(f\ast {\phi }_{x,t})\left(y\right)\right|{\left(1+\left|M_{x,t_0}^{-1}y\right|\right)}^{-L}{\left(1+2^{t+t_0}\right)}^{-L},\hfill \\ \hfill & T_{\phi }^{N(t_0,L)}f\left(x\right):=\underset{t\ge t_0}{sup}\underset{y\in {\mathbb{R}}^n}{sup}\frac{|(f\ast {\phi }_{x,t})\left(y\right)|}{{\left[1+\left|M_{x,t}^{-1}(x-y)\right|\right]}^N}\frac{1}{{\left(1+2^{t+t_0}\right)}^L{\left(1+\left|M_{x,t_0}^{-1}y\right|\right)}^L},\hfill \\ \hfill & M_{N,\tilde{N}}^{0(t_0,L)}f\left(x\right):=\underset{\phi \in {\mathcal{S}}_{N,\tilde{N}}}{sup}{M}_{\phi }^{0(t_0,L)}f\left(x\right)\hfill \\ \hfill \mathrm{and}\\ \hfill & M_{N,\tilde{N}}^{(t_0,L)}f\left(x\right):=\underset{\phi \in {\mathcal{S}}_{N,\tilde{N}}}{sup}{M}_{\phi }^{(t_0,L)}f\left(x\right).\hfill \end{array}$$

The following Lemma 1 guarantees control of the tangential by the non-tangential maximal function in $L^p\left({\mathbb{R}}^n\right)$ independent of $t_0$ and L.

Lemma 1.

Let ${\Theta }_t$ be a t-continuous ellipsoid cover. Suppose $p>0,N>1/\left(a_{6}p\right)$, and $\phi \in \mathcal{S}$. Then, there exists a positive constant C such that, for any $t_0<0,L\ge 0$ and $f\in {\mathcal{S}}^{\prime }$,

$${∥T_{\phi }^{N(t_0,L)}f∥}_{L^p}\le C{∥M_{\phi }^{(t_0,L)}f∥}_{L^p}.$$

Proof. 

Consider the function $F:{\mathbb{R}}^n\times \mathbb{R}⟶[0,\infty )$ given by

$$F(y,t):=|(f\ast {\phi }_t)\left(y\right){|}^p{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-pL}{(1+2^{t+t_0})}^{-pL}.$$

Let $F_l^{\ast t_0}$ be as in (11) with $l=0$. When $y\in \theta (x,t)$, we have $M_t^{-1}(x-y)\in {\mathbb{B}}^n$ and hence $|M_t^{-1}(x-y)|<1$. If $t\ge t_0$, then

$$F(y,t){\left[1+\left|M_t^{-1}(x-y)\right|\right]}^{-pN}\le F_0^{\ast t_0}\left(x\right).$$

When $y\in \theta (x,t-kJ)\backslash \theta (x,t-(k-1\left)J\right)$ for some $k\ge 1$, we have

$$\begin{array}{c}\hfill M_t^{-1}(x-y)\notin M_t^{-1}{M}_{t-(k-1)J}\left({\mathbb{B}}^n\right).\end{array}$$

(27)

By (2), we obtain

$$∥M_{t-(k-1)J}^{-1}{M}_t∥\le a_5{2}^{-a_6(k-1)J}$$

and hence,

$$M_{t-(k-1)J}^{-1}{M}_t\left({\mathbb{B}}^n\right)\subseteq a_5{2}^{-a_6(k-1)J}{\mathbb{B}}^n,$$

which implies

$$(2^{a_6(k-1)J}/a_5){\mathbb{B}}^n\subseteq M_t^{-1}{M}_{t-(k-1)J}\left({\mathbb{B}}^n\right).$$

From this and (27), it follows that $|M_t^{-1}(x-y)|\ge 2^{a_6(k-1)J}/a_5$. Thus, for any $t\ge t_0$, we have

$$F(y,t){\left[1+\left|M_t^{-1}(x-y)\right|\right]}^{-pN}\le {a_5}^{pN}{2}^{-pN{a}_6(k-1)J}{F}_k^{\ast t_0}\left(x\right).$$

By taking the supremum over all $y\in {\mathbb{R}}^n$ and $t\ge t_0$, we know that

$$\begin{array}{c}\hfill {\left[T_{\phi }^{N(t_0,L)}f\left(x\right)\right]}^p\le {a_5}^{pN}\sum _{k=0}^{\infty }{2}^{-pN{a}_6(k-1)J}{F}_k^{\ast t_0}\left(x\right).\end{array}$$

Therefore, using this and Proposition 5, we obtain

$$\begin{array}{cc}\hfill {∥T_{\phi }^{N(t_0,L)}f∥}_{L^p\left({\mathbb{R}}^n\right)}^p& \le {a_5}^{pN}\sum _{k=0}^{\infty }{2}^{-pN{a}_6(k-1)J}{\int }_{{\mathbb{R}}^n}{F}_k^{\ast t_0}\left(x\right)dx\hfill \\ \hfill & \le C{a_5}^{pN}\sum _{k=0}^{\infty }{2}^{-pN{a}_6(k-1)J}{2}^{kJ}{\int }_{{\mathbb{R}}^n}{F}_0^{\ast t_0}\left(x\right)dx\hfill \\ \hfill & =C{}^{\prime }{∥M_{\phi }^{(t_0,L)}f∥}_{L^p\left({\mathbb{R}}^n\right)}^p,\hfill \end{array}$$

where $C{}^{\prime }:=C{a_5}^{pN}{2}^{pN{a}_{6}J}{\sum }_{k=0}^{\infty }{2}^{(1-pN{a}_6)kJ}=C{a_5}^{pN}{2}^J/(1-2^{(1-pN{a}_6)J})$. □

The following Lemma 2 gives the pointwise majorization of the grand radial maximal function by the tangential one, which is a variable anisotropic extension of ([2], Lemma 7.5).

Lemma 2.

Let Θ be an ellipsoid cover of ${\mathbb{R}}^n$, $\phi \in \mathcal{S}$, ${\int }_{{\mathbb{R}}^n}\phi \left(x\right)dx\ne 0$, and $f\in {\mathcal{S}}^{\prime }$. For any given positive integers N and L, there exist integers $0<U\le \tilde{U}$, $U\ge N_p$, and $\tilde{U}\ge \widetilde{N_p}$ that are large enough and constant $C>0$ such that, for any $t_0<0$,

$$M_{U,\tilde{U}}^{0(t_0,L)}f\left(x\right)\le C{T}_{\phi }^{N(t_0,L)}f\left(x\right),\forall x\in {\mathbb{R}}^n.$$

Proof. 

The simplified proof of this final version is from Dekel (Lemma 6.20). By Proposition 6, for any $\psi \in \mathcal{S}$, $x\in {\mathbb{R}}^n$, $t\in \mathbb{R}$, there exists a sequence ${\left\{{\eta }^k\right\}}_{k=0}^{\infty }$, ${\eta }^k\in \mathcal{S}$ that satisfies

$$\psi =\sum _{k=0}^{\infty }{\eta }^k\ast {\phi }^k$$

converging in $\mathcal{S}$, where

$${\phi }^k:=\left|\det \left(M_{x,t+kJ}^{-1}{M}_{x,t}\right)\right|\phi (M_{x,t+kJ}^{-1}{M}_{x,t}·),k\ge 0.$$

Furthermore, for any positive integers $U,\tilde{U}$ and V,

$$\begin{array}{c}\hfill {\parallel {\eta }^k\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}\le C{2}^{-kV}{\parallel \psi \parallel }_{{\mathcal{S}}_{U+n+1+\lceil V/\left(a_{6}J\right)\rceil ,\tilde{U}+n+1}}.\end{array}$$

(28)

where the constant depends on $\phi ,U,\tilde{U},V,p(\Theta )$ but not $\psi$. Denoting $M_k:=M_{x,t+kJ}$, for $t\ge t_0$, implies

$$\begin{array}{cc}\hfill |(f\ast {\psi }_{x,t})\left(x\right)|& =\left|\left[f\ast \sum _{k=0}^{\infty }{\left({\eta }^k\ast {\phi }^k\right)}_{x,t}\right]\left(x\right)\right|\hfill \\ \hfill & \le C\left|\left[f\ast \sum _{k=0}^{\infty }\left|\det \left(M_k^{-1}\right)\right|{\int }_{{\mathbb{R}}^n}{\eta }^k\left(y\right)\phi \left(M_k^{-1}(·-M_{x,t}y)\right)dy\right]\left(x\right)\right|\hfill \\ \hfill & =C\left|\left[f\ast \sum _{k=0}^{\infty }\left|\det \left(M_k^{-1}{M}_{x,t}^{-1}\right)\right|{\int }_{{\mathbb{R}}^n}{\eta }^k\left(M_{x,t}^{-1}y\right)\phi \left(M_k^{-1}(·-y)\right)dy\right]\left(x\right)\right|\hfill \\ \hfill & \le C\sum _{k=0}^{\infty }\left|\left[f\ast {\left({\eta }^k\right)}_{x,t}\ast {\phi }_{x,t+kJ}\right]\left(x\right)\right|\hfill \\ \hfill & \le C\sum _{k=0}^{\infty }{\int }_{{\mathbb{R}}^n}\left|f\ast {\phi }_{x,t+kJ}(x-y)\right|\left|{\left({\eta }^k\right)}_{x,t}\left(y\right)\right|dy\hfill \\ \hfill & \le C{T}_{\phi }^{N(t_0,L)}f\left(x\right)\sum _{k=0}^{\infty }{\int }_{{\mathbb{R}}^n}{\left(1+\left|M_k^{-1}y\right|\right)}^N\hfill \\ \hfill & \times {\left(1+\left|M_{x,t_0}^{-1}(x-y)\right|\right)}^L{\left(1+2^{t+t_0+kJ}\right)}^L\left|{\left({\eta }^k\right)}_{x,t}\left(y\right)\right|dy.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill M_{\psi }^{0(t_0,L)}f\left(x\right)& \le T_{\phi }^{N(t_0,L)}f\left(x\right)\underset{t\ge t_0}{sup}\sum _{k=0}^{\infty }{\int }_{{\mathbb{R}}^n}{\left(1+\left|M_k^{-1}y\right|\right)}^N\hfill \\ \hfill & \times \frac{{\left(1+\left|M_{x,t_0}^{-1}(x-y)\right|\right)}^L{\left(1+2^{t+t_0+kJ}\right)}^L}{{\left(1+\left|M_{x,t_0}^{-1}x\right|\right)}^L{\left(1+2^{t+t_0}\right)}^L}\left|{\left({\eta }^k\right)}_{x,t}\left(y\right)\right|dy\hfill \\ \hfill & =:T_{\phi }^{N(t_0,L)}f\left(x\right)\underset{t\ge t_0}{sup}\sum _{k=0}^{\infty }{\mathrm{I}}_{t,k}.\hfill \end{array}$$

(29)

Let us now estimate ${\mathrm{I}}_{t,k}$ for $t\ge t_0$, $k\ge 0$. We begin with the simple observations that

$$\begin{array}{c}\hfill \frac{1+2^{t+t_0+kJ}}{1+2^{t+t_0}}=\frac{2^{kJ}(2^{-kJ}+2^{t+t_0})}{1+2^{t+t_0}}\le C{2}^{kJ}\end{array}$$

and

$$\begin{array}{c}\hfill 1+|x+y|\le 1+\left|x\right|+\left|y\right|\le (1+|x\left|\right)(1+|y\left|\right),x,y\in {\mathbb{R}}^n.\end{array}$$

(30)

Therefore, we may obtain

$$\begin{array}{cc}\hfill {\mathrm{I}}_{t,k}& \le C{2}^{t+kJL}{\int }_{{\mathbb{R}}^n}{\left(1+\left|M_k^{-1}y\right|\right)}^N{\left(1+\left|M_{x,t_0}^{-1}y\right|\right)}^L\left|{\eta }^k\left(M_{x,t}^{-1}y\right)\right|dy\hfill \\ \hfill & \le C{2}^{kJL}{\int }_{{\mathbb{R}}^n}{\left(1+∥M_k^{-1}{M}_{x,t}∥\left|y\right|\right)}^N{\left(1+∥M_{x,t_0}^{-1}{M}_{x,t}∥\left|y\right|\right)}^L\left|{\eta }^k\left(y\right)\right|dy,\hfill \end{array}$$

which, together with

$$\parallel M_k^{-1}{M}_{x,t}\parallel \le a_3{2}^{a_{4}kJ}\mathrm{and}\parallel M_{x,t_0}^{-1}{M}_{x,t}\parallel \le a_5{2}^{-a_6(t-t_0)}\le a_5(\mathrm{by}t\ge t_0\mathrm{and}\left(2\right)),$$

further implies that

$$\begin{array}{cc}\hfill {\mathrm{I}}_{t,k}& \le C{2}^{kJ(L+a_{4}N)}{\int }_{{\mathbb{R}}^n}{\left(1+\left|y\right|\right)}^{N+L}\left|{\eta }^k\left(y\right)\right|dy\hfill \\ \hfill & \le C{2}^{kJ(L+a_{4}N)}{∥{\eta }^k∥}_{{\mathcal{S}}_{0,\tilde{N}+n+L}}.\hfill \end{array}$$

(31)

We now apply (28) with $V:=\lceil J(L+a_{4}N)\rceil +1$, which gives

$$\begin{array}{c}\hfill {\mathrm{I}}_{t,k}\le C{2}^{-kV}{\parallel \psi \parallel }_{{\mathcal{S}}_{n+1+\lceil V/\left(a_{6}J\right)\rceil ,N+L+2n+2}}.\end{array}$$

(32)

This yields for any $\psi \in {\mathcal{S}}_{U,\tilde{U}}$, $U:=max(N_p,n+1+\lceil V/\left(a_{6}J\right)\rceil )$, $\tilde{U}:=max({\tilde{N}}_p,N+L+2n+2)$

$$\begin{array}{cc}\hfill M_{U,\tilde{U}}^{0(t_0,L)}f\left(x\right)& =\underset{\psi \in {\mathcal{S}}_{U,\tilde{U}}}{sup}{M}_{\psi }^{0(t_0,L)}f\left(x\right)\le C{T}_{\phi }^{N(t_0,L)}f\left(x\right).\hfill \end{array}$$

This finishes the proof of Lemma 2. □

The following Lemma 3 shows that the radial and the grand non-tangential maximal functions are pointwise equivalent, which is a variable anisotropic extension of ([2], Proposition 3.10).

Lemma 3

([19], Theorem 3.4). For any $N,\tilde{N}\in \mathbb{N}$ with $N\le \tilde{N}$, there exists a positive constant $C:=C\left(\tilde{N}\right)$ such that, for any $f\in {\mathcal{S}}^{\prime }$,

$$M_{N,\tilde{N}}^{0}f\left(x\right)\le M_{N,\tilde{N}}f\left(x\right)\le C{M}_{N,\tilde{N}}^{0}f\left(x\right),x\in {\mathbb{R}}^n.$$

The following Lemma 4 is a variable anisotropic extension of ([2], p. 46, Lemma 7.6).

Lemma 4.

Let ${\Theta }_t$ be a t-continuous ellipsoid cover, $\phi \in \mathcal{S}$, and $f\in {\mathcal{S}}^{\prime }$. Then, for every $M>0$ and $t_0<0$, there exist $L>0$ and $N^{\prime }>0$ large enough such that

$$\begin{array}{c}\hfill M_{\phi }^{(t_0,L)}f\left(x\right)\le C{2}^{-t_0(2a_4{N}^{\prime }+2L+a_{4}L)}{(1+|x\left|\right)}^{-M},x\in {\mathbb{R}}^n,\end{array}$$

(33)

where C is a positive constant dependent on $p(\Theta )$, $N^{\prime }$, f, and φ.

Proof. 

For any $\phi \in \mathcal{S}$, there exist an integer $N>0$ and positive constant $C:=C\left(\phi \right)$ such that, for any $N^{\prime }\ge N$ and $y\in {\mathbb{R}}^n$,

$$\begin{array}{c}\hfill |f\ast \phi \left(y\right)|\le {C\parallel \phi \parallel }_{{\mathcal{S}}_{N,N^{\prime }}}{(1+|y\left|\right)}^{N^{\prime }}.\end{array}$$

(34)

Therefore, for any $t_0<0$, $t\ge t_0$ and $x\in {\mathbb{R}}^n$, by (34), we have

$$\begin{array}{cc}\hfill & |(f\ast {\phi }_t)\left(y\right)|{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-L}{\left(1+2^{t+t_0}\right)}^{-L}\hfill \\ \hfill & \le C{2}^{-L\left(t+t_0\right)}\parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}{(1+|y\left|\right)}^{N^{\prime }}{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-L}.\hfill \end{array}$$

(35)

Let us first estimate $\parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}$. By the chain rule and (1), we have

$$\begin{array}{cc}\hfill \parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}& =|\det M_t^{-1}|\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le N}{sup}{(1+|z\left|\right)}^{N^{\prime }}\left|{\partial }^{\alpha }\left(\phi \left(M_t^{-1}·\right)\right)\left(z\right)\right|\hfill \\ \hfill & \le C{2}^t\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le N}{sup}{(1+|z\left|\right)}^{N^{\prime }}{∥M_t^{-1}∥}^{\left|\alpha \right|}\left|\left({\partial }^{\alpha }\phi \right)\left(M_t^{-1}z\right)\right|\hfill \\ \hfill & \le C{2}^t\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le N}{sup}(1+|M_{t}z{\left|\right)}^{N^{\prime }}{∥M_t^{-1}∥}^{\left|\alpha \right|}\left|{\partial }^{\alpha }\phi \left(z\right)\right|.\hfill \end{array}$$

(36)

Now, let us further estimate (36) in the following two cases.

Case 1:$t\ge 0$. By (2), we have

$$\begin{array}{cc}\hfill ∥M_t^{-1}∥& =∥M_t^{-1}{M}_0{M}_0^{-1}∥\le ∥M_t^{-1}{M}_0∥∥M_0^{-1}∥\le ∥M_0^{-1}∥a_3^{-1}{2}^{a_{4}t}=C{2}^{a_{4}t}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |M_{t}z|& =\left|M_0{M}_0^{-1}{M}_{t}z\right|\le ∥M_0∥\left|M_0^{-1}{M}_{t}z\right|\le ∥M_0∥∥M_0^{-1}{M}_t∥\left|z\right|\hfill \\ \hfill & \le \parallel M_0\parallel a_5{2}^{-a_{6}t}\left|z\right|\le C\left|z\right|.\hfill \end{array}$$

Inserting the above two estimates into (36) with $t\ge 0$, we know that

$$\begin{array}{cc}\hfill \parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}& \le C{2}^t\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le N}{sup}(1+|M_{t}z{\left|\right)}^{N^{\prime }}{∥M_t^{-1}∥}^{\left|\alpha \right|}\left|{\partial }^{\alpha }\phi \left(z\right)\right|\hfill \\ \hfill & \le C{2}^t{2}^{a_{4}tN}{\parallel \phi \parallel }_{{\mathcal{S}}_{N,N^{\prime }}}.\hfill \end{array}$$

(37)

Case 2:$t_0\le t<0$. By (2), we have

$$\begin{array}{cc}\hfill ∥M_t^{-1}∥& =∥M_t^{-1}{M}_0{M}_0^{-1}∥\le ∥M_t^{-1}{M}_0∥∥M_0^{-1}∥\le ∥M_0^{-1}∥a_5{2}^{a_{6}t}\le C\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |M_{t}z|& =\left|M_0{M}_0^{-1}{M}_{t}z\right|\le \parallel M_0\parallel \left|M_0^{-1}{M}_{t}z\right|\le \parallel M_0\parallel ∥M_0^{-1}{M}_t∥\left|z\right|\hfill \\ \hfill & \le \parallel M_0\parallel a_3^{-1}{2}^{-a_{4}t}\left|z\right|=C{2}^{-a_4{t}_0}\left|z\right|.\hfill \end{array}$$

Inserting the above two estimates into (36) with $t_0\le t<0$, we know that

$$\begin{array}{cc}\hfill \parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}& \le C{2}^t\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le N}{sup}(1+|M_{t}z{\left|\right)}^{N^{\prime }}{∥M_t^{-1}∥}^{\left|\alpha \right|}\left|{\partial }^{\alpha }\phi \left(z\right)\right|\hfill \\ \hfill & \le C{2}^{-a_4{t}_0{N}^{\prime }}{\parallel \phi \parallel }_{{\mathcal{S}}_{N,N^{\prime }}}.\hfill \end{array}$$

(38)

For any $M>0$, let $L:=M+N^{\prime }$. For any $t_0<0$, $t\ge t_0$ and taking some integer $N^{\prime }>0$ large enough, by (37) and (38), we obtain

$$\begin{array}{c}\hfill 2^{-L\left(t+t_0\right)}\parallel {\phi }_t{\parallel }_{{\mathcal{S}}_{N,N^{\prime }}}\le C{2}^{-t_0(a_4{N}^{\prime }+2L)}{\parallel \phi \parallel }_{{\mathcal{S}}_{N,N^{\prime }}}.\end{array}$$

(39)

Inserting (39) into (35), we further obtain

$$\begin{array}{cc}\hfill & |(f\ast {\phi }_t)\left(y\right)|{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-L}{\left(1+2^{t+t_0}\right)}^{-L}\hfill \\ \hfill & \le C{2}^{-t_0(a_4{N}^{\prime }+2L)}{\parallel \phi \parallel }_{{\mathcal{S}}_{N,N^{\prime }}}{(1+|y\left|\right)}^{N^{\prime }}{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-L}.\hfill \end{array}$$

(40)

For any $y\in \theta (x,t)$, there exists $z\in {\mathbb{B}}^n$ such that $y=x+M_{t}z$. By (30), we have

$$\begin{array}{c}\hfill 1+\left|y\right|=1+|x+M_{t}z|\le (1+\left|x\right|\left)\right(1+|M_{t}z\left|\right).\end{array}$$

(41)

If $t\ge 0$, by (2), then

$$\begin{array}{cc}\hfill |M_{t}z|& =\left|M_0{M}_0^{-1}{M}_{t}z\right|\le ∥M_0∥\left|M_0^{-1}{M}_{t}z\right|\le ∥M_0∥∥M_0^{-1}{M}_t∥\left|z\right|\hfill \\ \hfill & \le \parallel M_0\parallel a_5{2}^{-a_{6}t}\left|z\right|\le C.\hfill \end{array}$$

If $t_0\le t<0$, by (2), then

$$\begin{array}{cc}\hfill |M_{t}z|& =\left|M_0{M}_0^{-1}{M}_{t}z\right|\le \parallel M_0\parallel \left|M_0^{-1}{M}_{t}z\right|\le \parallel M_0\parallel ∥M_0^{-1}{M}_t∥\left|z\right|\hfill \\ \hfill & \le \parallel M_0\parallel a_3^{-1}{2}^{-a_{4}t}\left|z\right|=C{2}^{-a_4{t}_0}.\hfill \end{array}$$

Therefore, for any $t\ge t_0$, by using the above two estimates, we have

$$\begin{array}{c}\hfill |M_{t}z|\le C{2}^{-a_4{t}_0}.\end{array}$$

From this and (41), it follows that

$$\begin{array}{c}\hfill (1+|y\left|\right)\le C{2}^{-a_4{t}_0}(1+|x\left|\right).\end{array}$$

(42)

Moreover, for any $t_0<0$, by (2), we have

$$\begin{array}{c}\hfill 1+\left|x\right|\le 1+\parallel M_0\parallel ∥M_0^{-1}{M}_{t_0}∥\left|M_{t_0}^{-1}x\right|\le C{2}^{-a_4{t}_0}\left(1+\left|M_{t_0}^{-1}x\right|\right).\end{array}$$

Furthermore, for any $y\in \theta (x,t)$, we have $x\in M_t\left({\mathbb{B}}^n\right)+y$. Thus, there exists $z\in {\mathbb{B}}^n$ such that $x=M_{t}z+y$. Hence, for any $t\ge t_0$, by (30) and (2), we obtain

$$\begin{array}{cc}\hfill \left(1+\left|M_{t_0}^{-1}x\right|\right)& =\left(1+\left|M_{t_0}^{-1}(y+M_{t}z)\right|\right)\le \left(1+\left|M_{t_0}^{-1}y\right|\right)\left(1+∥M_{t_0}^{-1}{M}_t∥\left|z\right|\right)\hfill \\ \hfill & \le \left(1+\left|M_{t_0}^{-1}y\right|\right)\left(1+a_5{2}^{-a_6(t-t_0)}\left|z\right|\right)\le C\left(1+\left|M_{t_0}^{-1}y\right|\right).\hfill \end{array}$$

Combining with the above two inequalities, we have

$$\begin{array}{c}\hfill (1+|M_{t_0}^{-1}y\left|\right)\ge C{2}^{a_4{t}_0}(1+|x\left|\right).\end{array}$$

(43)

Thus, for any $t\ge t_0$ and $y\in \theta (x,t)$, inserting (42) and (43) into (40) with $L=M+N^{\prime }$, we obtain

$$\begin{array}{cc}\hfill & |(f\ast {\phi }_t)\left(y\right)|{\left(1+\left|M_{t_0}^{-1}y\right|\right)}^{-L}{\left(1+2^{t+t_0}\right)}^{-L}\le C{2}^{-t_0(2a_4{N}^{\prime }+2L+a_{4}L)}{(1+|x\left|\right)}^{-M},\hfill \end{array}$$

which implies that (33) holds true and hence completes the proof of Lemma 4. □

Note that the above argument gives the same estimate for the truncated grand maximal function $M_{N,\tilde{N}}^{0(t_0,L)}f\left(x\right)$. As a consequence of Lemma 4, we obtain that, for any choice of $t_0<0$ and any $f\in {\mathcal{S}}^{\prime }$, we can find an appropriate $L>0$ so that the maximal function, say $M_{\phi }^{(t_0,L)}f$, is bounded and belongs to $L^p\left({\mathbb{R}}^n\right)$. This becomes crucial in the proof of Theorem 1, where we work with truncated maximal functions, The complexity of the preceding argument stems from the fact that, a priori, we do not know whether $M_{\phi }^{0}f\in L^p$ implies $M_{\phi }f\in L^p$. Instead, we must work with variants of maximal functions for which this is satisfied.

Proof of Theorem 1.

Suppose that ${\Theta }_t$ is a t-continuous ellipsoid cover and $\phi \in \mathcal{S}$ satisfying ${\int }_{{\mathbb{R}}^n}\phi \left(x\right)dx$$\ne 0$. From Remark 2 and the definition of the grand radial maximal function, it follows that

$$\left(26\right)\Rightarrow \left(24\right)\Rightarrow \left(25\right)$$

and

$$\left(23\right)\Rightarrow \left(25\right).$$

By Lemma 1 applied for $L=0$, we have

$${∥T_{\phi }^{N(t_0,0)}f∥}_{L^p}\le C{∥M_{\phi }^{(t_0,0)}f∥}_{L^p}\mathrm{for}\mathrm{any}f\in {\mathcal{S}}^{\prime }\mathrm{and}{t}_0<0.$$

As $t_0\to -\infty$, by the monotone convergence theorem, we obtain

$${∥T_{\phi }^{N}f∥}_{L^p}\le C{∥M_{\phi }f∥}_{L^p},$$

which shows $\left(24\right)\Rightarrow \left(26\right)$.

Combining Lemma 2 applied for $N>1/\left(a_{6}p\right)$ and $L=0$ and Lemma 1 applied for $L=0$, we conclude that there exist integers $0<U\le \tilde{U}$, $U>N_p$, $\tilde{U}\ge \widetilde{N_p}$ that are large enough and a positive constant C such that

$${∥M_{U,\tilde{U}}^{0(t_0,0)}f∥}_{L^p}\le C{∥M_{\phi }^{(t_0,0)}f∥}_{L^p}\mathrm{for}\mathrm{any}f\in {\mathcal{S}}^{\prime }\mathrm{and}{t}_0<0.$$

As $t_0\to -\infty$, by the monotone convergence theorem, we obtain

$${∥M_{U,\tilde{U}}^{0}f∥}_{L^p}\le C{∥M_{\phi }f∥}_{L^p}.$$

From this and Proposition 1, we deduce that

$${\parallel f\parallel }_{H^p\left({\Theta }_t\right)}={∥M_{N_p,{\tilde{N}}_p}^{0}f∥}_{L^p}\le C{∥M_{U,\tilde{U}}^{0}f∥}_{L^p}\le C{∥M_{\phi }f∥}_{L^p}$$

and hence $\left(24\right)\Rightarrow \left(23\right).$ $\left(25\right)\Rightarrow \left(24\right)$ remain to be shown.

Suppose now $M_{\phi }^{\circ }f\in L^p$. By Lemma 4, we can find a $L>0$ large enough such that (33) holds true, which implies $M_{\phi }^{(t_0,L)}f\in L^p$ for all $t_0<0$. Combining Lemmas 1 and 2, we obtain that there exist $0<U\le \tilde{U}$, $U>N_p$, and $\tilde{U}\ge \widetilde{N_p}$ large enough such that

$$\begin{array}{c}\hfill {∥M_{U,\tilde{U}}^{0(t_0,L)}f∥}_p\le C_1{∥M_{\phi }^{(t_0,L)}f∥}_p,\end{array}$$

(44)

where constant $C_1$ is independent of $t_0<0$. For a given $t_0<0$, let

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_{t_0}:=\left\{x\in {\mathbb{R}}^n:M_{U,\tilde{U}}^{0(t_0,L)}f\left(x\right)\le C_2{M}_{\phi }^{(t_0,L)}f\left(x\right)\right\},\end{array}$$

(45)

where $C_2:=2^{1/p}{C}_1$. We claim that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^n}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx\le 2{\int }_{{\mathrm{\Omega }}_{t_0}}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx.\end{array}$$

(46)

Indeed, this follows from (44), $M_{\phi }^{(t_0,L)}f\in L^p$ and

$$\begin{array}{c}\hfill {\int }_{{\mathrm{\Omega }}_{t_0}^c}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx\le C_2^{-p}{\int }_{{\mathrm{\Omega }}_{t_0}^c}{\left[M_{U,\tilde{U}}^{0(t_0,L)}f\left(x\right)\right]}^{p}dx\le {(C_1/C_2)}^p{\int }_{{\mathbb{R}}^n}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx,\end{array}$$

where ${(C_1/C_2)}^p=1/2$.

We also claim that, for $0<q<p$, there exists a constant $C_3>0$ such that, for any $t_0<0$,

$$\begin{array}{c}\hfill M_{\phi }^{(t_0,L)}f\left(x\right)\le C_3{\left[M_{\Theta }{\left(M_{\phi }^{0(t_0,L)}f\right)}^q\left(x\right)\right]}^{1/q},\end{array}$$

(47)

where $M_{\Theta }$ is as in Definition 4. Indeed, let $t\ge t_0$, $y\in \theta (x,t)$ and

$$\begin{array}{c}\hfill F(y,t):=|(f\ast {\phi }_t)\left(y\right)\left|\right(1+|M_{t_0}^{-1}y{\left|\right)}^{-L}{(1+2^{t+t_0})}^{-L}.\end{array}$$

Suppose that $x\in {\mathrm{\Omega }}_{t_0}$ and let $F_l^{\ast t_0}\left(x\right)$ be as in (11) with $l=0$. Then, there exist $t^{\prime }\in \mathbb{R}$ with $t^{\prime }\ge t_0$ and $y^{\prime }\in \theta (x,t^{\prime })$ such that

$$\begin{array}{c}\hfill F(y^{\prime },t^{\prime })\ge F_0^{\ast t_0}\left(x\right)/2=M_{\phi }^{(t_0,L)}f\left(x\right)/2.\end{array}$$

(48)

Consider $x^{\prime }\in y^{\prime }+M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)$ for some integer $l\ge 1$ to be specified later. Let $\mathrm{\Phi }\left(z\right):=\phi \left(z+M_{t^{\prime }}^{-1}(x^{\prime }-y^{\prime })\right)-\phi \left(z\right)$. Obviously, we have

$$\begin{array}{c}\hfill f\ast {\phi }_{t^{\prime }}\left(x^{\prime }\right)-f\ast {\phi }_{t^{\prime }}\left(y^{\prime }\right)=f\ast {\mathrm{\Phi }}_{t^{\prime }}\left(y^{\prime }\right).\end{array}$$

(49)

Let us first estimate ${\parallel \mathrm{\Phi }\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}$. From $x^{\prime }\in y^{\prime }+M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)$, we deduce that

$$M_{t^{\prime }}^{-1}(x^{\prime }-y^{\prime })\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right).$$

By this and the mean value theorem, we obtain

$$\begin{array}{cc}\hfill {\parallel \mathrm{\Phi }\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}& \le \underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}{\parallel \phi (·+h)-\phi (·)\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}\hfill \\ \hfill & =\underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le U}{sup}{(1+|z\left|\right)}^{\tilde{U}}|\left({\partial }^{\alpha }\phi \right)(z+h)-{\partial }^{\alpha }\phi \left(z\right)|\hfill \\ \hfill & \le C\underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le U+1}{sup}{(1+|z\left|\right)}^{\tilde{U}}\left|\left({\partial }^{\alpha }\phi \right)(z+h)\right|\hfill \\ \hfill & \times \underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}\left|h\right|.\hfill \end{array}$$

(50)

From (2), we deduce

$$\parallel M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\parallel \le a_5{2}^{-a_{6}lJ},$$

which implies

$$M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)\subset a_5{2}^{-a_{6}lJ}{\mathbb{B}}^n.$$

By this and $h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)$, we have $\left|h\right|\le a_5{2}^{-a_{6}lJ}$. From this and (30), we deduce that

$$\begin{array}{c}\hfill 1+\left|z\right|\le (1+|z+h\left|\right)(1+|h\left|\right)\le C(1+|z+h\left|\right),z\in {\mathbb{R}}^n.\end{array}$$

Applying this and $\left|h\right|\le a_5{2}^{-a_{6}lJ}$ in (50), we obtain

$$\begin{array}{cc}\hfill {\parallel \mathrm{\Phi }\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}& \le C\underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}\underset{z\in {\mathbb{R}}^n}{sup}\underset{\left|\alpha \right|\le U+1}{sup}{(1+|z+h\left|\right)}^{\tilde{U}}\left|\left({\partial }^{\alpha }\phi \right)(z+h)\right|\hfill \\ \hfill & \times \underset{h\in M_{t^{\prime }}^{-1}{M}_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{sup}\left|h\right|\le {C\parallel \phi \parallel }_{{\mathcal{S}}_{U+1,\tilde{U}}}{a}_5{2}^{-a_{6}lJ}\le C_4{2}^{-a_{6}lJ},\hfill \end{array}$$

(51)

where a positive constant $C_4$ does not depend on L.

Moreover, notice that, for any $x^{\prime }\in M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)+y^{\prime }$, there exists $z\in {\mathbb{B}}^n$ such that $x^{\prime }=M_{t^{\prime }+lJ}z+y^{\prime }$. By (30), (2), and $t^{\prime }\ge t_0$, we have

$$\begin{array}{cc}\hfill \left(1+\left|M_{t_0}^{-1}{x}^{\prime }\right|\right)& \le \left(1+\left|M_{t_0}^{-1}{y}^{\prime }\right|\right)\left(1+∥M_{t_0}^{-1}{M}_{t^{\prime }+lJ}∥\left|z\right|\right)\hfill \\ \hfill & \le \left(1+\left|M_{t_0}^{-1}{y}^{\prime }\right|\right)\left(1+a_5{2}^{-a_6(t^{\prime }-t_0+lJ)}\left|z\right|\right)\le 2a_5\left(1+\left|M_{t_0}^{-1}{y}^{\prime }\right|\right).\hfill \end{array}$$

(52)

Thus, for any $x\in {\mathrm{\Omega }}_{t_0}$, from (49), (52), (48), (51), Lemma 3, and (45), it follows that

$$\begin{array}{cc}\hfill 2^L{a}_5^{L}F(x^{\prime },t^{\prime })& =2^L{a}_5^L\left[|(f\ast {\phi }_{t^{\prime }})\left(x^{\prime }\right)\left|\right(1+|M_{t_0}^{-1}{x}^{\prime }{\left|\right)}^{-L}{(1+2^{t^{\prime }+t_0})}^{-L}\right]\hfill \\ \hfill & \ge \left[\right|f\ast {\phi }_{t^{\prime }}\left(y^{\prime }\right)|-|f\ast {\mathrm{\Phi }}_{t^{\prime }}\left(y^{\prime }\right)\left|\right]{\left(1+\left|M_{t_0}^{-1}{y}^{\prime }\right|\right)}^{-L}{\left(1+2^{t^{\prime }+t_0}\right)}^{-L}\hfill \\ \hfill & \ge F(y^{\prime },t^{\prime })-M_{U,\tilde{U}}^{(t_0,L)}f\left(x\right){\parallel \mathrm{\Phi }\parallel }_{{\mathcal{S}}_{U,\tilde{U}}}\hfill \\ \hfill & \ge M_{\phi }^{(t_0,L)}f\left(x\right)/2-C_4{2}^{-a_{6}lJ}C{M}_{U,\tilde{U}}^{0(t_0,L)}f\left(x\right)\hfill \\ \hfill & \ge M_{\phi }^{(t_0,L)}f\left(x\right)/2-C_4{C}_{2}C{2}^{-a_{6}lJ}{M}_{\phi }^{(t_0,L)}f\left(x\right).\hfill \end{array}$$

We choose an integer $l\ge 1$ large enough such that $C_4{C}_{2}C{2}^{-a_{6}lJ}\le 1/4$. Therefore, for any $x\in {\mathrm{\Omega }}_{t_0}$ and $x^{\prime }\in M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)+y^{\prime }$, we further have

$$\begin{array}{c}\hfill 2^L{a}_5^{L}F(x^{\prime },t^{\prime })\ge M_{\phi }^{(t_0,L)}f\left(x\right)/2-C_4{C}_{2}C{2}^{-a_{6}lJ}{M}_{\phi }^{(t_0,L)}f\left(x\right)\ge M_{\phi }^{(t_0,L)}f\left(x\right)/4.\end{array}$$

(53)

Moreover, by $y^{\prime }\in \theta (x,t^{\prime })$ and Proposition 2, we have

$$\begin{array}{cc}\hfill & M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)+y^{\prime }\subseteq M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)+M_{t^{\prime }}\left({\mathbb{B}}^n\right)+x\hfill \\ \hfill & \subseteq 2M_{t^{\prime }}\left({\mathbb{B}}^n\right)+x\subseteq \theta (x,t^{\prime }-J).\hfill \end{array}$$

(54)

Thus, for any $x\in {\mathrm{\Omega }}_{t_0}$ and $t\ge t_0$, by (53) and (54), we obtain

$$\begin{array}{cc}\hfill {\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^q& \le \frac{4^q{2}^{Lq}{a}_5^{Lq}}{|M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)|}{\int }_{y^{\prime }+M_{t^{\prime }+lJ}\left({\mathbb{B}}^n\right)}{\left[F(z,t^{\prime })\right]}^{q}dz\hfill \\ \hfill & \le C{4}^q{2}^{Lq}{a}_5^{Lq}\frac{2^{(l+1)J}}{\left|\theta (x,t^{\prime }-J)\right|}{\int }_{\theta (x,t^{\prime }-J)}{\left[M_{\phi }^{0(t_0,L)}f\left(z\right)\right]}^{q}dz\hfill \\ \hfill & \le C_3{M}_{\Theta }\left({\left(M_{\phi }^{0(t_0,L)}f\right)}^q\right)\left(x\right),\hfill \end{array}$$

which shows the above claim (47).

Consequently, by (46), (47), and Proposition 3 with $p/q>1$, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^n}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx& \le 2{\int }_{{\mathrm{\Omega }}_{t_0}}{\left[M_{\phi }^{(t_0,L)}f\left(x\right)\right]}^{p}dx\hfill \\ \hfill & \le 2{C_3}^p{\int }_{{\mathrm{\Omega }}_{t_0}}{\left[M_{\Theta }\left({\left(M_{\phi }^{0(t_0,L)}f\right)}^q\right)\left(x\right)\right]}^{p/q}dx\hfill \\ \hfill & \le C_5{\int }_{{\mathbb{R}}^n}{\left[M_{\phi }^{0(t_0,L)}f\left(x\right)\right]}^{p}dx,\hfill \end{array}$$

(55)

where the constant $C_5$ depends on $p/q>1$, $L\ge 0$ and $p(\Theta )$ but is independent of $t_0<0$. This inequality is crucial as it gives a bound of the non-tangential by the radial maximal function in $L^p$. The rest of the proof is immediate.

For any $x\in {\mathbb{R}}^n$, $y\in {\mathbb{R}}^n$ and $t<0$, by (2), we obtain

$$\begin{array}{cc}\hfill \left|M_t^{-1}y\right|& =\left|M_t^{-1}{M}_0{M}_0^{-1}y\right|\le ∥M_t^{-1}{M}_0∥∥M_0^{-1}∥\left|y\right|\hfill \\ \hfill & \le a_5{2}^{a_{6}t}∥M_0^{-1}∥\left|y\right|\to 0\mathrm{as}t\to -\infty .\hfill \end{array}$$

Hence, we obtain that $M_{\phi }^{(t_0,L)}f\left(x\right)$ converges pointwise and monotonically to $M_{\phi }f\left(x\right)$ for all $x\in {\mathbb{R}}^n$ as $t_0\to -\infty$, which together with (55) and the monotone convergence theorem, further implies that $M_{\phi }f\in L^p$. Therefore, we can now choose $L=0$, and again, by (55) and the monotone convergence theorem, we have $\parallel M_{\phi }{f\parallel }_p^p\le C_5{\parallel M_{\phi }^{0}f\parallel }_p^p$, where $C_5$ corresponds to $L=0$ and is independent of $f\in {\mathcal{S}}^{\prime }$. This finishes the proof of Theorem 1. □