Anisotropic Weak Hardy Spaces and Wavelets

Abstract

We characterize the anisotropic weak Hardy spaces 𝐻^𝐴𝑝,∞(ℝ^𝑛) associated with an expansive matrix 𝐴 by using square functions involving wavelets coefficients.

1. Introduction

Bownik, in a series of papers [1–5], studied anisotropic function spaces associated with dilations. In the monography [1] he investigated anisotropic Hardy spaces. Suppose that 𝐴 is an expansive matrix (also called dilation) in ℝ^𝑛, that is, 𝐴 is an 𝑛×𝑛-matrix all of whose eigenvalues 𝜆 satisfy |𝜆|>1. If 𝜙 is a function in the Schwartz class 𝑆(ℝ^𝑛) such that ∫_ℝ𝑛𝜙(𝑥)𝑑𝑥≠0 and 𝑓 is a distribution in 𝑆(ℝ^𝑛)^, the dual space of 𝑆(ℝ^𝑛), the radial maximal function 𝑀_𝜙(𝑓) is defined by𝑀_𝜙(𝑓)=sup_𝑘∈ℤ||𝑓∗𝜙_𝑘||,(1.1) where 𝜙_𝑘(𝑥)=|det𝐴|^−𝑘𝜙(𝐴^−𝑘𝑥),𝑥∈ℝ^𝑛, and 𝑘∈ℤ. For every 0<𝑝<∞, the Hardy space 𝐻^𝑝𝐴(ℝ^𝑛) associated with 𝐴 consists of all those 𝑓∈𝑆(ℝ^𝑛)′ such that 𝑀_𝜙(𝑓)∈𝐿^𝑝(ℝ^𝑛). The space 𝐻^𝑝𝐴(ℝ^𝑛) does not depend on the function 𝜙 and the ‖⋅‖_𝐻^𝑝𝐴_(ℝ^𝑛₎-norm is defined by‖𝑓‖_𝐻^𝑝𝐴_(ℝ)^𝑛=‖𝑀_𝜙(𝑓)‖_𝑝,𝑓∈𝐻^𝑝𝐴(ℝ^𝑛).(1.2)

Like in the classical case [6], the anisotropic Hardy space 𝐻^𝑝𝐴(ℝ^𝑛) can be characterized by nontangential or grand maximal functions [1, Theorem 7.1, page 42]. Also, atomic representations of the distributions in 𝐻^𝑝𝐴(ℝ^𝑛) are obtained in [1, Theorem 6.5, page 39]. Wavelets for a dilation 𝐴 are studied in [1, Chapter 2]. The author proved that 𝑟-wavelets associated with the expansive matrix 𝐴 form an unconditional basis for the anisotropic Hardy spaces defined by 𝐴. Recently, Bownik et al. [7, 8] have investigated weighted anisotropic Hardy spaces.

The weak anisotropic Hardy space 𝐻^𝐴𝑝,∞(ℝ^𝑛),0<𝑝≤1, was introduced by Ding and Lan [9]. A distribution 𝑓∈𝑆(ℝ^𝑛)′ is in 𝐻^𝐴𝑝,∞(ℝ^𝑛) if and only if 𝑀_𝜙(𝑓)∈𝐿^𝑝,∞(ℝ^𝑛), where 𝐿^𝑝,∞(ℝ^𝑛) denotes the weak 𝐿^𝑝-space. We define‖𝑓‖_𝐻^𝐴𝑝,∞_(ℝ^𝑛₎=‖‖𝑀_𝜙𝑓‖‖_{𝐿𝑝,∞(ℝ𝑛)},𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛).(1.3) As the case 𝐻^𝑝𝐴(ℝ^𝑛), the space 𝐻^𝐴𝑝,∞(ℝ^𝑛) does not depend on the election of 𝜙 and it can be described by nontangential and grand maximal functions. Atomic representations of distributions in 𝐻^𝐴𝑝,∞(ℝ^𝑛) were established in [9, Theorem 1.1]. In this paper we study wavelets for 𝐻^𝐴𝑝,∞(ℝ^𝑛). We characterize in Theorem 2.2 bellow the distributions in 𝐻^𝐴𝑝,∞(ℝ^𝑛) by square functions involving wavelets coefficients. Our result can be seen as an anisotropic version of the one showed in [10] (see also [11]).

This paper is organized as follows. In Section 2 we recall the main definitions and properties about the anisotropic setting that we need throughout the paper. We also state our result (Theorem 2.2). The proof of Theorem 2.2 is presented in Section 3.

2. Preliminaries and Results

We now recall the main definitions and properties concerning the analysis in the anisotropic setting. We refer the reader to [1] where the anisotropic theory associated with expansive matrixes was developed. Suppose that 𝐴 is an expansive matrix in ℝ^𝑛. We denote by Δ an ellipsoid for 𝐴 such that the Lebesgue measure |Δ| of Δ is equal to 1. For every 𝑘∈ℤ, we define 𝐵_𝑘=𝐴^𝑘Δ. We consider the mapping 𝜌∶ℝ^𝑛→[0,∞) given by𝜌(𝑥)=||det𝐴||^𝑗0,,𝑥𝑥∈=𝐵0._𝑗+1⧵𝐵_𝑗,𝑗∈ℤ,(2.1) Thus, 𝜌 is a homogeneous quasinorm associated with 𝐴 in the sense of [1, Definition 2.3, page 6]. This means that 𝜌 satisfies the following properties:(a)𝜌(𝑥)>0, 𝑥∈ℝ^𝑛⧵{0},(b)𝜌(𝐴𝑥)=|det𝐴|𝜌(𝑥), 𝑥∈ℝ^𝑛,(c)𝜌(𝑥+𝑦)≤𝐻_𝜌(𝜌(𝑥)+𝜌(𝑦)), 𝑥,𝑦∈ℝ^𝑛,

where 𝐻_𝜌≥1. In [1, Lemma 2.4, page 6] it was proved that if 𝜌₁ and 𝜌₂ are two homogeneous quasinorms associated with 𝐴 then,1𝐶𝜌¹(𝑥)≤𝜌₂(𝑥)≤𝐶𝜌₁(𝑥),𝑥∈ℝ^𝑛,(2.2) for a certain 𝐶>0. As it was mentioned above a tempered distribution 𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛),  0<𝑝≤1, when 𝑀_𝜙(𝑓)∈𝐿^𝑝,∞(ℝ^𝑛), where 𝜙∈𝑆(ℝ^𝑛) such that ∫_ℝ𝑛𝜙(𝑥)𝑑𝑥≠0. In [9, Theorem 1.1] Ding and Lan proved the following atomic representation for the distributions in 𝐻^𝐴𝑝,∞(ℝ^𝑛) that will be useful in the sequel.

Theorem 2.1 (see [9, Theorem 1.1]). Suppose that 𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛), 0<𝑝≤1, and 𝑠≥[(1/𝑝−1)log|det𝐴|/log𝑚₁], where 𝑚₁=min{|𝜆|∶𝜆isaneigenvalueof𝐴}. Then, there exists a sequence of bounded functions {𝑓_𝑘}_𝑘∈ℤ satisfying, for a certain 𝐶>0,(a)𝑓=∑^{∞𝑘=−∞}𝑓_𝑘 in 𝑆(ℝ^𝑛)^, and ‖𝑓_𝑘‖_∞≤𝐶2^𝑘, for every 𝑘∈ℤ,(b)for every 𝑘∈ℤ, 𝑓_𝑘=∑^∞𝑖=0𝛽^𝑘𝑖, in 𝑆(ℝ^𝑛)′, where, for each 𝑖∈ℕ,(i)there exist 𝑥_𝑖,𝑘∈ℝ^𝑛 and 𝑗_𝑖,𝑘∈ℤ such that supp𝛽^𝑘𝑖⊂𝐵^𝑘𝑖=𝑥_𝑖,𝑘+𝐴^{𝑗𝑖,𝑘}Δ, and ^∞_𝑖=0|||𝐵^𝑘𝑖|||≤𝐶2_−𝑘𝑝‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)},(2.3)(ii)‖𝛽^𝑘𝑖‖_∞≤𝐶2^𝑘, (iii)∫_ℝ𝑛𝛽^𝑘𝑖(𝑥)𝑃(𝑥)𝑑𝑥=0, for every polynomial 𝑃 whose degree is less or equal to 𝑠.Conversely, if 𝑓∈𝑆(ℝ^𝑛)′ satisfying (a) and (b) where in (i)𝐶‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)} is replacing by 𝑀>0, then 𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛) and 𝑀∶‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}.

Theorem 2.1 is an anisotropic version of the isotropic result established by Fefferman and Soria [12, Proposition, page 8].

Let 𝜓∈𝐿²(ℝ^𝑛). For every 𝑗∈ℤ and 𝑘∈ℤ^𝑛, we define𝜓_𝑗,𝑘(𝑥)=||det𝐴||^𝑗/2𝜓𝐴^𝑗𝑥−𝑘,𝑥∈ℝ^𝑛.(2.4)

We say that 𝜓 is a Bessel 𝐴-wavelet if there exists 𝐶>0 such that, for every 𝑓∈𝐿²(ℝ^𝑛),_𝑗∈ℤ_{,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²≤𝐶‖𝑓‖²².(2.5)  𝜓 is a frame 𝐴-wavelet when, for a certain 𝐶>0,1𝐶‖𝑓‖²²≤_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²≤𝐶‖𝑓‖²²,𝑓∈𝐿²(ℝ^𝑛).(2.6) We say that 𝜓 is a tight frame 𝐴-wavelet when (2.6) holds with 𝐶=1.

As usual if 𝑔∈𝐿²(ℝ^𝑛) we denote by 𝑔̂ the Fourier transform of 𝑔.

We now establish our result where the distributions in 𝐻^𝑝𝐴(ℝ^𝑛), 0<𝑝≤1, are characterized by using square functions involving 𝐴-wavelets.

Theorem 2.2. Let 𝑓∈𝑆(ℝ^𝑛)^ and 0<𝑝≤1. Assume that 𝜓∈𝑆(ℝ^𝑛) is a tight frame 𝐴-wavelet where 𝐴 is an expansive matrix in ℝ^𝑛, such that 𝜓(0)≠0, supp𝜓 is compact, and 0∉supp𝜓. Then, the following properties are equivalent:(a)𝑊(𝑓)=(∑_{𝑗∈ℤ,𝑘∈ℤ𝑛}|⟨𝑓,𝜓_𝑗,𝑘⟩|²|𝜓_𝑗,𝑘|²)^1/2∈𝐿^𝑝,∞(ℝ^𝑛),(b)𝐺(𝑓)=(∑_{𝑗∈ℤ,𝑘∈ℤ𝑛}|⟨𝑓,𝜓_𝑗,𝑘⟩|²𝜒_{𝑄𝑗,𝑘}|𝑄_𝑗,𝑘|⁻¹)^1/2∈𝐿^𝑝,∞(ℝ^𝑛),where 𝑄_𝑗,𝑘=𝐴^−𝑗([0,1]^𝑛+𝑘), 𝑗∈ℤ, and 𝑘∈ℤ^𝑛,(c)𝑆(𝑓)=(∑_{𝑗∈ℤ,𝑘∈ℤ𝑛}|⟨𝑓,𝜓_𝑗,𝑘⟩|²𝜒_{𝑅𝑗,𝑘}|𝑄_𝑗,𝑘|⁻¹)^1/2∈𝐿^𝑝,∞(ℝ^𝑛),where for every 𝑗∈ℤ and 𝑘∈ℤ^𝑛, 𝑅_𝑗,𝑘 is a measurable set such that 𝑅_𝑗,𝑘⊂𝑄_𝑗,𝑘 and |𝑅_𝑗,𝑘|≥𝛾|𝑄_𝑗,𝑘|, for a certain 𝛾∈(0,1),(d)the distribution 𝑓=∑_{𝑗∈ℤ,𝑘∈ℤ𝑛}⟨𝑓,𝜓_𝑗,𝑘⟩𝜓_𝑗,𝑘 is in 𝐻^𝐴𝑝,∞(ℝ^𝑛).Moreover, if one of (and then all) the above conditions is satisfied, then ‖𝑓‖_𝐻^𝐴𝑝,∞_(ℝ^𝑛₎∼‖𝑊(𝑓)‖_𝐿^𝑝,∞_(ℝ^𝑛₎∼‖𝐺(𝑓)‖_𝐿^𝑝,∞_(ℝ^𝑛₎∼‖𝑆(𝑓)‖_𝐿^𝑝,∞_(ℝ^𝑛₎.(2.7)

Bownik proved in [1, Theorem 4.2, page 94] that there exists a tight frame 𝐴-wavelet 𝜓∈𝑆(ℝ^𝑛) satisfying the conditions in Theorem 2.2 such that ∫_ℝ𝑛𝑓(𝑥)𝑃(𝑥)𝑑𝑥=0, for every polynomial 𝑃 in ℝ^𝑛.

3. Proof of Theorem 2.2

In this section we present a proof of Theorem 2.2. Throughout this section with 𝐶 we denote a positive constant that can change in each occurrence.

(a)⇒(c) According to [1, equation (2.2), page 5], 𝐴^𝑗𝑥→0, as 𝑗→−∞, uniformly in 𝑥∈[0,1]^𝑛. Moreover, there exist 0<𝜂<1 and 𝛿>0 for which||𝜓(𝑥)||≥𝛿,𝑥∈[0,𝜂]^𝑛.(3.1) Then, for every 𝑗∈ℤ and 𝑘∈ℤ^𝑛, we have||𝜓_𝑗,𝑘(𝑥)||≥𝛿||𝑄_𝑗,𝑘||,𝑥∈𝑅^𝑗,𝑘,(3.2) where 𝑅_𝑗,𝑘=𝐴^−𝑗([0,𝜂]^𝑛+𝑘)⊆𝑄_𝑗,𝑘. Hence, we can write _{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²||𝜓_𝑗,𝑘(𝑥)||²^1/2≥𝛿_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²1||𝑄_𝑗,𝑘||𝜒^{𝑅𝑗,𝑘}(𝑥)^1/2.(3.3) Note that||𝑅_𝑗,𝑘||=||𝐴^−𝑗[0,𝜂]^𝑛||=||det𝐴||^−𝑗𝜂^𝑛=||𝑄_𝑗,𝑘||𝜂^𝑛.(3.4) Thus we show that (a) implies (b).

(b)⇒(c) It is clear.

(c)⇒(b) Assume that (c) holds. For every 𝑘∈ℤ, we define𝐸_𝑘=𝑥∈ℝ^𝑛∶𝑆𝑓(𝑥)>2^𝑘.(3.5) According to our assumption we have that sup_𝑘∈ℤ2^𝑘𝑝|𝐸_𝑘|<∞. Fix 𝛽∈(0,𝛾) and 𝑘∈ℕ. We denote by 𝐷_𝑘 the set defined by𝐷_𝑘=(𝑗,𝑙)∈ℤ×ℤ^𝑛∶||𝑄_−𝑗,𝑙∩𝐸_𝑘||≥𝛽||𝑄_−𝑗,𝑙||.(3.6) By 𝐷_𝑘,max we represent the set that consists of all (𝑗,𝑙)∈𝐷_𝑘 such that 𝑄_−𝑗,𝑙 is maximal with respect to the order in ≤_𝐷𝑘 introduced in [1, Definition 6.4, page 105].

It is clear that||𝑄_−𝑗,𝑙||≤1𝛽||𝑄^−𝑗,𝑙∩𝐸_𝑘||≤𝐶2^−𝑘𝑝,(𝑗,𝑙)∈𝐷_𝑘.(3.7)

Moreover, |𝑄_−𝑗,𝑙|=|det𝐴|^𝑗,𝑗∈ℤ, and 𝑙∈ℤ^𝑛. Hence, there exists 𝑗₀∈ℤ such that 𝑗≤𝑗₀ provided that (𝑗,𝑙)∈𝐷_𝑘. Then, for every (𝑗,𝑙)∈𝐷_𝑘, there exists (𝑗₁,𝑙₁)∈𝐷_𝑘,max for which 𝑄_−𝑗,𝑙≤_𝐷𝑘𝑄_{−𝑗1,𝑙1} (see [1, page 105]).

For every (𝑗,𝑙)∈𝐷_𝑘,max we define𝐷_𝑘(𝑗,𝑙)=(𝑚,𝑠)∈𝐷_𝑘∶𝑄_−𝑚,𝑠≤_𝐷𝑘𝑄_−𝑗,𝑙.(3.8) By [1, Lemma 6.5, page 105], we can find 𝜂∈ℕ such that for every (𝑗,𝑙)∈𝐷_𝑘,max we have_(𝑚,𝑠)_{∈𝐷𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠⊆_{||𝑙−𝑙1||<𝜂}𝑄_{−𝑗,𝑙1}.(3.9) If 𝐸^∗𝑘=⋃_{(𝑗,𝑙)∈𝐷𝑘}𝑄_−𝑗,𝑙, it follows that||𝐸^∗𝑘||=|||||_{(𝑗,𝑙)∈𝐷𝑘,max(𝑚,𝑠)}_{∈𝐷𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠≤_{(𝑗,𝑙)∈𝐷𝑘,max}||||||||||_{(𝑚,𝑠)∈𝐷𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠≤_{(𝑗,𝑙)∈𝐷𝑘,max}||||||||||_{||𝑙−𝑙1||<𝜂}𝑄_{−𝑗,𝑙1}≤(2𝜂+1)_𝑛|||||_(𝑗,𝑙)_{∈𝐷𝑘,max}||det𝐴||^𝑗=(2𝜂+1)^𝑛_(𝑗,𝑙)_{∈𝐷𝑘,max}≤(2𝜂+1)_𝑛||𝑄_−𝑗,𝑙||𝛽_{(𝑗,𝑙)∈𝐷𝑘,max}≤(2𝜂+1)_𝑛||𝑄_−𝑗,𝑙∩𝐸_𝑘||𝛽||𝐸^𝑘||.(3.10) In the last inequality we have used |𝑄_−𝑗,𝑙∩𝑄_{−𝑗1,𝑙1}|=0 provided that (𝑗,𝑙), (𝑗₁,𝑙₁)∈𝐷_𝑘,max, (𝑗,𝑙)≠(𝑗₁,𝑙₁).

By proceeding as in [1, page 107] by induction on (𝑗,𝑙)∈𝐷_𝑘,max we find, for every (𝑗,𝑙)∈𝐷_𝑘,max, a set𝐼(𝑗,𝑙)⊆𝐷_𝑘(𝑗,𝑙),(3.11) satisfying the following properties:𝐼(𝑗,𝑙)∩𝐼𝑗₁𝐷_𝑘=_{(𝑗,𝑙)∈𝐷𝑘,max}𝐼(𝑗,𝑙),,𝑙₁=∅,(𝑗,𝑙),𝑗₁,𝑙₁∈𝐷_𝑘,max,(𝑗,𝑙)≠𝑗₁,𝑙₁.(3.12) Note that 𝐷_𝑘+1⊆𝐷_𝑘. We can consider the sets 𝑍_𝑘=𝐷_𝑘⧵𝐷_𝑘+1,𝑍_𝑘(𝑗,𝑙)=𝑍_𝑘∩𝐼(𝑗,𝑙) and 𝑀_𝑘(𝑗,𝑙)=⋃_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠, for every (𝑗,𝑙)∈𝐷_𝑘,max.

Let (𝑗,𝑙)∈𝐷_𝑘,max. We can write_{𝑀𝑘(𝑗,𝑙)⧵𝐸𝑘+1}||𝑆(𝑓)(𝑥)||²𝑑𝑥=_{𝑀𝑘(𝑗,𝑙)⧵𝐸𝑘+1𝑚∈ℤ}_{,𝑠∈ℤ𝑛}||𝑓,𝜓_−𝑚,𝑠||²1||𝑄_−𝑚,𝑠||𝜒^{𝑅−𝑚,𝑠}≥(𝑥)𝑑𝑥_{𝑀𝑘(𝑗,𝑙)⧵𝐸𝑘+1(𝑚,𝑠)}_{∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²1||𝑄_−𝑚,𝑠||𝜒^{𝑅−𝑚,𝑠}≥(𝑥)𝑑𝑥_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²1||𝑄_−𝑚,𝑠||||𝑅^−𝑚,𝑠∩𝐸^𝑐𝑘+1||.(3.13) Also, for every (𝑚,𝑠)∈𝑍_𝑘(𝑗,𝑙), we get||𝑅_−𝑚,𝑠∩𝐸^𝑐𝑘+1||=||𝑅_−𝑚,𝑠||−||𝑅_−𝑚,𝑠∩𝐸_𝑘+1||≥(𝛾−𝛽)||𝑄_−𝑚,𝑠||,(3.14) because |𝑅_−𝑚,𝑠|≥𝛾|𝑄_−𝑚,𝑠|, and, since (𝑚,𝑠)∉𝐷_𝑘+1,||𝑅_−𝑚,𝑠∩𝐸_𝑘+1||≤||𝑄_−𝑚,𝑠∩𝐸_𝑘+1||<𝛽||𝑄_−𝑚,𝑠||.(3.15) Hence_{𝑀𝑘(𝑗,𝑙)⧵𝐸𝑘+1}||𝑆(𝑓)(𝑥)||²𝑑𝑥≥_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²(𝛾−𝛽).(3.16) On the other hand, we have that_{𝑀𝑘(𝑗,𝑙)⧵𝐸𝑘+1}||𝑆(𝑓)(𝑥)||²𝑑𝑥≤4^𝑘+1||𝑀_𝑘(𝑗,𝑙)⧵𝐸_𝑘+1||.(3.17) We conclude that_(𝑚,𝑠)_{∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²≤4^𝑘+1𝛾−𝛽||𝑀^𝑘(𝑗,𝑙)||.(3.18) Again according to [1, Lemma 6.5] (see (3.9)) it follows that𝑀_𝑘(𝑗,𝑙)⊆_{||||𝑠−𝑙<𝜂}𝑄_−𝑗,𝑠.(3.19) Then||𝑀_𝑘(𝑗,𝑙)||≤_{||||𝑠−𝑙<𝜂}||𝑄_−𝑗,𝑠||≤(2𝜂+1)^𝑛||det𝐴||^𝑗.(3.20) Therefore_(𝑚,𝑠)_{∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²≤(2𝜂+1)^𝑛𝛾−𝛽4^𝑘+1||𝑄_−𝑗,𝑙||.(3.21) Let 𝜈>0. We choose 𝑘₀∈ℕ such that 2^𝑘0≤𝜈<2^𝑘0+1 and we define the functions 𝐺₁ and 𝐺₂ by𝐺₁(𝑓)=⎛⎜⎜⎝^𝑘0_𝑘_=−∞_{(𝑗,𝑙)∈𝐷𝑘,max}_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²1||𝑄_−𝑚,𝑠||𝜒^{𝑄−𝑚,𝑠}⎞⎟⎟⎠^1/2𝐺₂,(𝑓)=⎛⎜⎜⎝^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²1||𝑄_−𝑚,𝑠||𝜒^{𝑄−𝑚,𝑠}⎞⎟⎟⎠^1/2.(3.22) Note that if (𝑚,𝑠)∈ℤ×ℤ^𝑛 and ⟨𝑓,𝜓_−𝑚,𝑠⟩≠0 then there exists 𝑘₁∈ℤ such that 𝑅_−𝑚,𝑠⊆𝐸_𝑘1. Hence (𝑚,𝑠)∈𝐷_𝑘, for every 𝑘∈ℤ,𝑘≤𝑘₁. Moreover, there exists 𝑘₂∈ℤ such that (𝑚,𝑠)∉𝐷_𝑘, when 𝑘≥𝑘₂. We deduce that 𝐺²=𝐺²¹+𝐺²². By (3.10) and (3.21), since ∑_{(𝑗,𝑙)∈𝐷𝑘,max}|𝑄_−𝑗,𝑙|≤|𝐸^∗𝑘|, we infer‖‖𝐺₁𝑓‖‖²²≤(2𝜂+1)^𝑛𝛾−𝛽^𝑘0_𝑘=−∞_{(𝑗,𝑙)∈𝐷𝑘,max}≤(2𝜂+1)_𝑛4^𝑘+1||𝑄_−𝑗,𝑙||𝛾−𝛽^𝑘0_𝑘=−∞≤(2𝜂+1)_2𝑛4^𝑘+1||𝐸^∗𝑘||𝛽(𝛾−𝛽)^𝑘0_𝑘=−∞≤4(2𝜂+1)_2𝑛4^𝑘+1||𝐸_𝑘||𝛽(𝛾−𝛽)^𝑘0_𝑘=−∞2^{(2−𝑝)𝑘}≤4(2𝜂+1)^2𝑛𝛽(𝛾−𝛽)12−2^{(2−𝑝)𝑘0−(2−𝑝)}≤𝐶𝜈^2−𝑝.(3.23) Moreover, since 𝐺₂(𝑓)(𝑥)=0 when 𝑥∉⋃_{𝑘≥𝑘0+1}𝐸^∗𝑘, by (3.10), it follows that||𝑥∈ℝ^𝑛∶𝐺₂(𝑓)(𝑥)≠0||≤^∞_𝑘=_𝑘0+1||𝐸^∗𝑘||≤(2𝜂+1)^𝑛𝛽^∞_𝑘=𝑘0+1≤𝐶(2𝜂+1)_𝑛||𝐸_𝑘||𝛽^∞_𝑘=𝑘0+12^−𝑘𝑝≤𝐶𝜈^−𝑝.(3.24) Then, we conclude that||{𝑥∈≤ℝ|||^𝑛𝑥∶∈𝐺ℝ(𝑓_𝑛)(𝑥)>𝜈}||∶𝐺₁(𝑓)(𝑥)>𝜈2|||+||𝑥∈ℝ^𝑛≤∶𝐺₂(𝑓)(𝑥)≠0_{{𝑥∈ℝ𝑛∶𝐺1(𝑓)(𝑥)>𝜈/2}}||2𝐺¹(𝑓)(𝑥)𝜈²𝑑𝑥+𝐶𝜈^−𝑝≤𝐶𝜈^−𝑝.(3.25) Thus (b) is proved.

(c)⇒(d) Suppose that (c) holds. We keep the notation that it was used in the proof of (c)⇒(b).

Let 𝜈>0. We choose 𝑘₀∈ℤ such that 2^𝑘0≤𝜈<2^𝑘0+1 and we define𝑓₁=^𝑘0_𝑘_=−∞_{(𝑗,𝑙)∈𝐷𝑘,max}_{(−𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}⟨𝑓,𝜓_−𝑚,𝑠⟩𝜓_−𝑚,𝑠.(3.26) By proceeding as in the proof of (c)⇒(b) we get^𝑘⁰_𝑘=−∞_{(𝑗,𝑙)∈𝐷𝑘,max}_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²≤𝐶𝜈^2−𝑝.(3.27) Then, since 𝜓 is a tight frame 𝐴-wavelet, by using duality, we obtain that ‖‖𝑓₁‖‖²²≤𝐶^𝑘0_𝑘_=−∞_{(𝑗,𝑙)∈𝐷𝑘,max}_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²≤𝐶^2−𝑝.(3.28) We now consider𝑓₂=^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑓,𝜓_−𝑚,𝑠𝜓_−𝑚,𝑠.(3.29) We choose 𝜑∈𝑆(ℝ^𝑛) such that supp𝜑 is compact and bounded away from the origin and that ∑_𝑙∈ℤ𝜑((𝐴^∗)^𝑙𝑦)=1,𝑦∈ℝ^𝑛⧵{0}. We are going to prove that ||||𝑥∉ℝ₂∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₂(𝑥)||>𝜈||||≤𝐶𝜈^𝑝.(3.30) For every (𝑗,𝑙)∈𝐷_𝑘,max, we define𝐶(𝑗,𝑙)=_{||𝑙−𝑙1||≤𝑏𝜂}𝑄_{−𝑗,𝑙1},(3.31) where 𝜂>0 is the one appeared in (3.9) and 𝑏>2 will be chosen later. We also considerΩ=^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}𝐶(𝑗,𝑙).(3.32) If (𝑗,𝑙)∈𝐷_𝑘,max, then||𝐶(𝑗,𝑙)||≤_{||𝑙−𝑙1||≤𝑏𝜂}||𝑄_{−𝑗,𝑙1}||≤(2𝑏𝜂+1)^𝑛||𝑄_−𝑗,𝑙||.(3.33) Hence, since ∑_{(𝑗,𝑙)∈𝐷𝑘,max}|𝑄_−𝑗,𝑙|≤|𝐸^∗𝑘|, by (3.10) it follows that||Ω||=|||||^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}𝐶(𝑗,𝑙)|||||≤(2𝑏𝜂+1)^𝑛∞_𝑘=_𝑘0+1||𝐸^∗𝑘||≤𝐶^∞_𝑘=_𝑘0+1≤𝐶_∞||𝐸_𝑘||_𝑘=_𝑘0+12^−𝑘𝑝≤𝐶2^−𝑘0𝑝≤𝐶𝜈^𝑝.(3.34) By (3.34), (3.30) will be proved when we obtain that||||𝑥∉Ω∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₂(𝑥)||>𝜈||||≤𝐶𝜈^𝑝.(3.35) Since supp𝜓 and supp𝜑 are compact and 0∉supp𝜓∪supp𝜑, there exists 𝑀>0 such that supp𝜓_−𝑚,𝑠∩supp𝜑_𝛼=∅, provided that |𝑚−𝛼|>𝑀,𝑚,𝛼∈ℤ, and 𝑠∈ℤ^𝑛.

Let 𝛼∈ℤ. We can write𝜑_𝛼∗𝑓₂=^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}^𝛼+𝑀_{𝑚=𝛼−𝑀}_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑓,𝜓_−𝑚,𝑠𝜑_𝛼∗𝜓_−𝑚,𝑠.(3.36) Using Hölder’s inequality we get|||||_{𝛼+𝑀𝑚=}_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑓,𝜓_−𝑚,𝑠𝜑_𝛼≤⎛⎜⎜⎝∗𝜓_−𝑚,𝑠(𝑥)_𝛼+𝑀|||||_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²⎞⎟⎟⎠^1/2⎛⎜⎜⎝^𝛼+𝑀_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝜑_𝛼∗𝜓_−𝑚,𝑠(𝑥)||²⎞⎟⎟⎠^1/2.(3.37) From (3.21) we deduce that⎛⎜⎜⎝^𝛼+𝑚_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²⎞⎟⎟⎠^1/2≤⎛⎜⎜⎝_{(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝑓,𝜓_−𝑚,𝑠||²⎞⎟⎟⎠^1/2≤𝐶2^𝑘||𝑄_−𝑗,𝑙||^1/2.(3.38) According to [13, page 1482] for any 𝜆>1 there exists 𝐶=𝐶(𝜆)>0 for which||𝜑_𝛽∗𝜓_−𝑚,𝑠(𝑥)||≤𝐶||det𝐴||^−𝑚/21+𝜌_𝐴(𝐴^−𝑚𝑥−𝑠)^−𝜆,𝑥∈ℝ^𝑛,(3.39) when |𝛽−𝑚|≤𝑀.

Let 𝜆>0 that will be fixed later. We can write⎛⎜⎜⎝^𝛼+𝑀_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝜑_𝛼∗𝜓_−𝑚,𝑠(𝑥)||²⎞⎟⎟⎠^1/2≤𝐶⎛⎜⎜⎝^𝛼+𝑀_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||det𝐴||^−𝑚1+𝜌_𝐴(𝐴^−𝑚𝑥−𝑠)^2𝜆⎞⎟⎟⎠^1/2≤𝐶^𝛼+𝑀_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||det𝐴||^−𝑚/21+𝜌_𝐴(𝐴^−𝑚𝑥−𝑠)^𝜆,𝑥∈ℝ^𝑛.(3.40) Let 𝑘∈ℤ,𝑘>𝑘₀, and (𝑗,𝑙)∈𝐷_𝑘,max. Assume that 𝑥∉Ω. Then 𝑥∉𝐶(𝑗,𝑙). By (3.9) we know that_(𝑚,𝑠)_{∈𝑍𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠⊆_{||𝑙−𝑙1||<𝜂}𝑄_{−𝑗,𝑙1}.(3.41) Let (𝑚,𝑠)∈𝑍_𝑘(𝑗,𝑙). We define𝑥^′=𝐴^𝑗𝑥,𝑥_𝑚,𝑠=𝐴^𝑚𝑠,𝑥^𝑚,𝑠=𝐴^𝑗𝑥_𝑚,𝑠,𝑥_𝑗,𝑙=𝐴^−𝑗𝑙,𝑥^𝑗,𝑙=𝑙.(3.42) Note that𝑥𝑥^_𝑚,𝑠∉_{||𝑙−𝑙1||<𝑏𝜂}[0,1]^𝑛+𝑙₁,𝑥_𝑗,𝑙∈_{||𝑙−𝑙1||<𝜂}[0,1]^𝑛+𝑙₁,∈_{||𝑙−𝑙1||<𝜂}[0,1]^𝑛+𝑙₁.(3.43) We have that𝜌_𝐴𝑥^−𝑥^𝑚,𝑠≥1𝐻𝜌^𝐴𝑥^−𝑥^𝑗,𝑙−𝜌^𝐴𝑥^𝑚,𝑠−𝑥^𝑗,𝑙.(3.44) By [1, Lemma 3.2] it follows that, for certain 𝛾>0,𝜌_𝐴𝑥^𝑚,𝑠−𝑥^𝑗,𝑙≤𝐶𝜂^𝛾,(3.45) and, for some 𝛾′>0,𝜌_𝐴𝑥^−𝑥^𝑗,𝑙≥𝐶(𝑏𝜂)^𝛾′.(3.46) Choosing 𝑏 large enough we get that𝜌_𝐴𝑥^−𝑥^𝑚,𝑠≥12𝐻𝜌^𝐴𝑥^−𝑥^𝑗,𝑙.(3.47) Then,𝜌_𝐴𝑥−𝑥_𝑚,𝑠=≥𝜌||d_𝐴et𝐴𝐴||^−𝑗_−𝑗𝑥^−𝑥^𝑚,𝑠=||det𝐴||^−𝑗𝜌_𝐴𝑥^−𝑥^𝑚,𝑠2𝐻𝜌^𝐴𝑥^−𝑥^𝑗,𝑙=12𝐻𝜌^𝐴𝑥^−𝑥^𝑗,𝑙.(3.48) By (3.40) and (3.48) with 𝜆>3/2 we obtain that⎛⎜⎜⎝^𝛼+𝑀_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||𝜑_𝛼∗𝜓_−𝑚,𝑠(𝑥)||²⎞⎟⎟⎠^1/2≤𝐶𝜌_𝐴𝑥−𝑥_𝑗,𝑙^{𝜆𝛼+𝑀}_𝑚=_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||det𝐴||^{−𝑚(1/2−𝜆)}≤𝐶𝜌_𝐴𝑥−𝑥_𝑗,𝑙^{𝜆𝛼+𝑀}_𝑚=_𝛼−𝑀⎛⎜⎜⎝_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}||det𝐴||^𝑚⎞⎟⎟⎠^−1/2+𝜆≤𝐶𝜌_𝐴𝑥−𝑥_𝑗,𝑙^{𝜆𝛼+𝑀}_𝑚=_𝛼−𝑀|||||_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑄_−𝑚,𝑠|||||^−1/2+𝜆≤𝐶||𝑄^−𝑗,𝑙||^−1/2+𝜆𝜌_𝐴𝑥−𝑥_𝑗,𝑙^𝜆.(3.49) In the last inequality we have used (3.20). From (3.37), (3.38), and (3.49) it follows that|||||_{𝛼+𝑀𝑚=}_𝛼−𝑀_{𝑠∶(𝑚,𝑠)∈𝑍𝑘(𝑗,𝑙)}𝑓,𝜓_−𝑚,𝑠𝜑_𝛼∗𝜓_−𝑚,𝑠(𝑥)|||||≤𝐶2^𝑘||𝑄_−𝑗,𝑙||^𝜆𝜌_𝐴𝑥−𝑥_𝑗,𝑙^𝜆,𝜆≥32.(3.50) Whence||||𝑥∉Ω∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₂(𝑥)||>𝜈||||≤|||||⎧⎪⎨⎪⎩𝑥∉Ω∶^∞_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}𝐶2^𝑘||𝑄_−𝑗,𝑙||^𝜆𝜌_𝐴𝑥−𝑥_𝑗,𝑙^𝜆≤|||||⎧⎪⎨⎪⎩𝑥∉Ω∶_∞>𝜈⎫⎪⎬⎪⎭|||||_𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}𝑐^{𝑘𝑗,𝑙}𝑔^{𝑘𝑗,𝑙}>𝜈⎫⎪⎬⎪⎭|||||,(3.51) where 𝜆>3/2,𝑐^{𝑘𝑗,𝑙}=𝐶2^𝑘|𝑄_−𝑗,𝑙|^𝜆 and𝑔^{𝑘𝑗,𝑙}(𝑥)=𝜒^Ω𝑐(𝑥)𝜌_𝐴𝑥−𝑥_𝑗,𝑙^𝜆,𝑥∈ℝ^𝑛.(3.52) By [2, equation (2.4)] we know that |𝐵_𝜌𝐴(𝑎,𝑟)|≈𝑟,𝑎∈ℝ^𝑛, and 𝑟>0. Therefore,|||𝑥∈ℝ_𝑛∶||𝑔^{𝑘𝑗,𝑙}(𝑥)||>𝜔|||=||𝑥∈Ω^𝑐∶𝜌_𝐴𝑥−𝑥_𝑗,𝑙<𝜔^−1/𝜆||≤𝐶𝜔^−1/𝜆,𝜔>0.(3.53) Hence, according to [9, Lemma 2.1], by taking 𝜆>1/𝑝, we obtain|||||⎧⎪⎨⎪⎩𝑥∉Ω∶_∞𝑘=_𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}𝐶2^𝑘||𝑄_−𝑗,𝑙||^𝜆𝜌_𝐴𝑥−𝑥_𝑗,𝑙^𝜆≤𝐶1>𝜈⎫⎪⎬⎪⎭|||||𝜈^∞1/𝜆_𝑘=𝑘0+1_{(𝑗,𝑙)∈𝐷𝑘,max}≤𝐶12^𝑘/𝜆||𝑄_−𝑗,𝑙||𝜈^∞1/𝜆_𝑘=𝑘0+12^𝑘/𝜆||𝐸^∗𝑘||≤𝐶1𝜈^∞1/𝜆_𝑘=𝑘0+1≤𝐶12^𝑘/𝜆||𝐸_𝑘||𝜈^∞1/𝜆_𝑘=𝑘0+12^{𝑘(1/𝜆−𝑝)}≤𝐶1𝜈^1/𝜆2^{𝑘0(1/𝜆−𝑝)}1−2^{1/𝜆−𝑝}≤𝐶𝜈^𝑝.(3.54) Hence,||||𝑥∉Ω∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₂(𝑥)||>𝜈||||≤𝐶𝜈^𝑝.(3.55) Combining (3.28), (3.34), and (3.55) we get||||𝑥∈ℝ_𝑛≤||||𝑥∶∈suℝp_{𝛼∈ℤ𝑛}||𝜑_𝛼∗𝑓(𝑥)||>𝜈||||∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₁(𝑥)||>𝜈2||||+||Ω||+||||𝑥∉Ω∶sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₂(𝑥)||>𝜈≤42||||𝜈²‖‖‖sup_𝛼∈ℤ||𝜑_𝛼∗𝑓₁(𝑥)||‖‖‖²²+𝐶𝜈^𝑝≤𝐶1𝜈²‖‖𝑓₁‖‖²²+1≤𝐶𝜈^𝑝𝜈^𝑝.(3.56) Thus we conclude that 𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛) and, therefore, (d) is established.

(d)⇒(a) We define the sublinear operator 𝑊 as follows:𝑊𝑓=_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²||𝜓_𝑗,𝑘(𝑥)||²^1/2,𝑓∈𝐿²(ℝ^𝑛).(3.57) Since 𝜓 is a tight frame 𝐴-wavelet, we have that‖𝑊𝑓‖²²=_{ℝ𝑛𝑗∈ℤ}_{,𝑘∈ℤ𝑛}=||𝑓,𝜓_𝑗,𝑘||²||𝜓_𝑗,𝑘(𝑥)||²𝑑𝑥_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²||det𝐴||^𝑗_ℝ𝑛=||𝜓𝐴^𝑗𝑥−𝑘||²𝑑𝑥_ℝ𝑛||𝜓(𝑥)||²𝑑𝑥_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²=‖𝑓‖²²_ℝ𝑛||𝜓(𝑥)||²𝑑𝑥,𝑓∈𝐿²(ℝ^𝑛).(3.58) Hence, 𝑊 is a bounded operator from 𝐿²(ℝ^𝑛) into itself.

According to [1, Theorem 6.7, page 109], for every 0<𝑞≤1 and 𝑓∈𝐻^𝑞𝐴(ℝ^𝑛),‖𝑓‖_𝐻^𝑞𝐴_(ℝ^𝑛₎∼‖‖‖‖_{𝑗∈ℤ,𝑘∈ℤ𝑛}||𝑓,𝜓_𝑗,𝑘||²||𝜓_𝑗,𝑘(𝑥)||²^1/2‖‖‖‖_𝑞.(3.59) Then, the operator 𝑊 defined as above is also bounded from the anisotropic Hardy space 𝐻^𝑞𝐴(ℝ^𝑛) into 𝐿^𝑞(ℝ^𝑛), for every 0<𝑞≤1. Our proof will be finished when we see that the operator 𝑇 is bounded from 𝐻^𝐴𝑝,∞(ℝ^𝑛) into 𝐿^𝑝,∞(ℝ^𝑛). In order to do this we proceed interpolating by using the ideas developed in the proof of [9, Theorem 3.4].

Let 𝑓∈𝐻^𝐴𝑝,∞(ℝ^𝑛) and 𝜈>0. We choose 𝑘₀∈ℤ such that 2^𝑘0≤𝜈<2^𝑘0+1. We take the atomic decomposition of 𝑓 given by 𝑓=∑^{∞𝑘=−∞}𝑓_𝑘. Here 𝑓_𝑘=∑^∞𝑖=0𝛽^𝑘𝑖, 𝑘∈ℤ, as in Theorem 2.1 with 𝑠=[(2/𝑝−1)((log|det𝐴|)/log𝑚)], where 𝑚=min{|𝜆|∶𝜆isaneigenvalueof𝐴}. We decompose 𝑓 as follows:𝑓=^𝑘0_𝑘_=−∞𝑓_𝑘+^∞_𝑘=_𝑘0+1𝑓_𝑘=𝐹₁+𝐹₂.(3.60) We have that‖‖𝐹₁‖‖₂≤^𝑘0_𝑘_=−∞‖‖𝑓_𝑘‖‖₂≤𝐶^𝑘0_𝑘_=−∞2^𝑘^∞_𝑖=0|||𝐵^𝑘𝑖|||^1/2≤𝐶^𝑘0_𝑘_=−∞2^{𝑘(1−𝑝/2)}‖𝑓‖^𝐻𝑝/2_{𝐴𝑝,∞(ℝ𝑛)}≤𝐶2^{𝑘0(1−𝑝/2)}‖𝑓‖^𝐻𝑝/2_{𝐴𝑝,∞(ℝ𝑛)}.(3.61) Then,||𝑥∈ℝ^𝑛∶||𝑊𝐹₁(𝑥)||>𝜈||≤𝐶𝜈²‖‖𝐹₁‖‖²²≤𝐶2^{𝑘0(2−𝑝)}𝜈²‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}≤𝐶𝜈^𝑝‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}.(3.62)

According to Theorem 2.1, there exists 𝐶>0 such that, for every 𝑘∈ℤ and 𝑖∈ℕ, 𝐶2^−𝑘|𝐵^𝑘𝑖|^−2/𝑝𝛽^𝑘𝑖 is a (𝑝/2,∞,𝑠)𝐻^𝐴𝑝/2(ℝ^𝑛)-atom. Hence, for every 𝑘∈ℤ, 𝑓_𝑘∈𝐻^𝐴𝑝/2(ℝ^𝑛) and‖‖𝑓_𝑘‖‖^𝐻𝑝/2_{𝐴𝑝/2(ℝ𝑛)}≤𝐶^∞_𝑖=02^𝑘𝑝/2|||𝐵^𝑘𝑖|||≤𝐶2_{−𝑘𝑝/2}‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}.(3.63) Since 𝑊 is bounded from 𝐻^𝐴𝑝/2(ℝ^𝑛) into 𝐿^𝑝/2(ℝ^𝑛), it follows that||𝑥∈ℝ^𝑛∶||𝑊𝑓_𝑘(𝑥)||>𝜈||≤𝜈^−𝑝/2‖‖𝑊(𝑓_𝑘)‖‖^𝑝/2𝑝/2≤𝐶𝜈^−𝑝/2‖‖𝑓_𝑘‖‖^𝐻𝑝/2_{𝐴𝑝/2(ℝ𝑛)},𝑘∈ℤ.(3.64)

Therefore, using [9, Lemma 2.1] we get||𝑥∈ℝ^𝑛∶||𝑊𝐹₂(𝑥)||>𝜈||≤|||||⎧⎪⎨⎪⎩𝑥∈ℝ^𝑛∶^∞_𝑘=_𝑘0+1=|||||⎧⎪⎨⎪⎩𝑥∈ℝ_𝑛||𝑊𝑓_𝑘(𝑥)||>𝜈⎫⎪⎬⎪⎭|||||∶^∞_𝑘=_𝑘0+1‖‖𝑓_𝑘‖‖_{𝐻𝐴𝑝/2(ℝ𝑛)}|||||𝑊𝑓_𝑘≤𝐶𝜈_∞−𝑝/2‖‖𝑓_𝑘‖‖_{𝐻𝐴𝑝/2(ℝ𝑛)}(𝑥)|||||>𝜈⎫⎪⎬⎪⎭|||||_𝑘=_𝑘0+1‖‖𝑓_𝑘‖‖^𝐻𝑝/2_{𝐴𝑝/2(ℝ𝑛)}≤𝐶𝜈^∞−𝑝/2_𝑘=_𝑘0+12^{−𝑘𝑝/2}‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}≤𝐶𝜈^−𝑝/22^{−𝑘0𝑝/2}‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}≤𝐶𝜈^𝑝‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}.(3.65) Putting together the above estimates we obtain||𝑥∈ℝ^𝑛∶||𝑊(𝑓)(𝑥)||>𝜈||≤|||𝑥∈ℝ^𝑛∶||𝑊𝐹₁(𝑥)||>𝜈2|||+|||𝑥∈ℝ^𝑛∶||𝑊𝐹₂(𝑥)||>𝜈≤𝐶2|||𝜈^𝑝‖𝑓‖^𝑝𝐻_{𝐴𝑝,∞(ℝ𝑛)}.(3.66)

Acknowledgment

B. Barrios is partially supported by MTM2010-16518 and J. J. Betancor is partially supported by MTM2010-17974.