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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 809121, 17 pages
http://dx.doi.org/10.1155/2012/809121
Research Article

Anisotropic Weak Hardy Spaces and Wavelets

1Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Matemáticas (ICMAT, CSIC-UAM, UC3M, UCM), C/Nicolás Cabrera 15, 28049 Madrid, Spain
2Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avenida Astrofísico Francisco Sánchez s/n, Santa Cruz de Tenerife, 38271 La Laguna, Spain

Received 17 January 2012; Accepted 5 April 2012

Academic Editor: Quanhua Xu

Copyright © 2012 B. Barrios and J. J. Betancor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction

Bownik, in a series of papers [15], 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 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(𝑛) such that 𝑛𝜙(𝑥)𝑑𝑥0 and 𝑓 is a distribution in 𝑆(𝑛), the dual space of 𝑆(𝑛), the radial maximal function 𝑀𝜙(𝑓) is defined by𝑀𝜙(𝑓)=sup𝑘||𝑓𝜙𝑘||,(1.1) where 𝜙𝑘(𝑥)=|det𝐴|𝑘𝜙(𝐴𝑘𝑥),𝑥𝑛, and 𝑘. For every 0<𝑝<, the Hardy space 𝐻𝑝𝐴(𝑛) associated with 𝐴 consists of all those 𝑓𝑆(𝑛) such that 𝑀𝜙(𝑓)𝐿𝑝(𝑛). The space 𝐻𝑝𝐴(𝑛) does not depend on the function 𝜙 and the 𝐻𝑝𝐴(𝑛)-norm is defined by𝑓𝐻𝑝𝐴()𝑛=𝑀𝜙(𝑓)𝑝,𝑓𝐻𝑝𝐴(𝑛).(1.2)

Like in the classical case [6], the anisotropic Hardy space 𝐻𝑝𝐴(𝑛) can be characterized by nontangential or grand maximal functions [1, Theorem 7.1, page 42]. Also, atomic representations of the distributions in 𝐻𝑝𝐴(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 𝐻𝐴𝑝,(𝑛),0<𝑝1, was introduced by Ding and Lan [9]. A distribution 𝑓𝑆(𝑛) is in 𝐻𝐴𝑝,(𝑛) if and only if 𝑀𝜙(𝑓)𝐿𝑝,(𝑛), where 𝐿𝑝,(𝑛) denotes the weak 𝐿𝑝-space. We define𝑓𝐻𝐴𝑝,(𝑛)=𝑀𝜙𝑓𝐿𝑝,(𝑛),𝑓𝐻𝐴𝑝,(𝑛).(1.3) As the case 𝐻𝑝𝐴(𝑛), the space 𝐻𝐴𝑝,(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 𝐻𝐴𝑝,(𝑛) were established in [9, Theorem 1.1]. In this paper we study wavelets for 𝐻𝐴𝑝,(𝑛). We characterize in Theorem 2.2 bellow the distributions in 𝐻𝐴𝑝,(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 𝑛. We denote by Δ an ellipsoid for 𝐴 such that the Lebesgue measure |Δ| of Δ is equal to 1. For every 𝑘, we define 𝐵𝑘=𝐴𝑘Δ. We consider the mapping 𝜌𝑛[0,) given by||||𝜌(𝑥)=det𝐴𝑗,𝑥𝐵𝑗+1𝐵𝑗,𝑗,0,𝑥=0.(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, 𝑥𝑛{0},(b)𝜌(𝐴𝑥)=|det𝐴|𝜌(𝑥), 𝑥𝑛,(c)𝜌(𝑥+𝑦)𝐻𝜌(𝜌(𝑥)+𝜌(𝑦)), 𝑥,𝑦𝑛,

where 𝐻𝜌1. 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),𝑥𝑛,(2.2) for a certain 𝐶>0. As it was mentioned above a tempered distribution 𝑓𝐻𝐴𝑝,(𝑛),  0<𝑝1, when 𝑀𝜙(𝑓)𝐿𝑝,(𝑛), where 𝜙𝑆(𝑛) such that 𝑛𝜙(𝑥)𝑑𝑥0. In [9, Theorem 1.1] Ding and Lan proved the following atomic representation for the distributions in 𝐻𝐴𝑝,(𝑛) that will be useful in the sequel.

Theorem 2.1 (see [9, Theorem 1.1]). Suppose that 𝑓𝐻𝐴𝑝,(𝑛), 0<𝑝1, and 𝑠[(1/𝑝1)log|det𝐴|/log𝑚1], where 𝑚1=min{|𝜆|𝜆isaneigenvalueof𝐴}. Then, there exists a sequence of bounded functions {𝑓𝑘}𝑘 satisfying, for a certain 𝐶>0,(a)𝑓=𝑘=𝑓𝑘 in 𝑆(𝑛), and 𝑓𝑘𝐶2𝑘, for every 𝑘,(b)for every 𝑘, 𝑓𝑘=𝑖=0𝛽𝑘𝑖, in 𝑆(𝑛), where, for each 𝑖,(i)there exist 𝑥𝑖,𝑘𝑛 and 𝑗𝑖,𝑘 such that supp𝛽𝑘𝑖𝐵𝑘𝑖=𝑥𝑖,𝑘+𝐴𝑗𝑖,𝑘Δ, and 𝑖=0|||𝐵𝑘𝑖|||𝐶2𝑘𝑝𝑓𝑝𝐻𝐴𝑝,(𝑛),(2.3)(ii)𝛽𝑘𝑖𝐶2𝑘, (iii)𝑛𝛽𝑘𝑖(𝑥)𝑃(𝑥)𝑑𝑥=0, for every polynomial 𝑃 whose degree is less or equal to 𝑠.Conversely, if 𝑓𝑆(𝑛) satisfying (a) and (b) where in (i)𝐶𝑓𝑝𝐻𝐴𝑝,(𝑛) is replacing by 𝑀>0, then 𝑓𝐻𝐴𝑝,(𝑛) and 𝑀𝑓𝑝𝐻𝐴𝑝,(𝑛).

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

Let 𝜓𝐿2(𝑛). For every 𝑗 and 𝑘𝑛, we define𝜓𝑗,𝑘||||(𝑥)=det𝐴𝑗/2𝜓𝐴𝑗𝑥𝑘,𝑥𝑛.(2.4)

We say that 𝜓 is a Bessel 𝐴-wavelet if there exists 𝐶>0 such that, for every 𝑓𝐿2(𝑛),𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2𝐶𝑓22.(2.5)𝜓 is a frame 𝐴-wavelet when, for a certain 𝐶>0,1𝐶𝑓22𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2𝐶𝑓22,𝑓𝐿2(𝑛).(2.6) We say that 𝜓 is a tight frame 𝐴-wavelet when (2.6) holds with 𝐶=1.

As usual if 𝑔𝐿2(𝑛) we denote by ̂𝑔 the Fourier transform of 𝑔.

We now establish our result where the distributions in 𝐻𝑝𝐴(𝑛), 0<𝑝1, are characterized by using square functions involving 𝐴-wavelets.

Theorem 2.2. Let 𝑓𝑆(𝑛) and 0<𝑝1. Assume that 𝜓𝑆(𝑛) is a tight frame 𝐴-wavelet where 𝐴 is an expansive matrix in 𝑛, such that 𝜓(0)0, supp𝜓 is compact, and 0supp𝜓. Then, the following properties are equivalent:(a)𝑊(𝑓)=(𝑗,𝑘𝑛|𝑓,𝜓𝑗,𝑘|2|𝜓𝑗,𝑘|2)1/2𝐿𝑝,(𝑛),(b)𝐺(𝑓)=(𝑗,𝑘𝑛|𝑓,𝜓𝑗,𝑘|2𝜒𝑄𝑗,𝑘|𝑄𝑗,𝑘|1)1/2𝐿𝑝,(𝑛),where 𝑄𝑗,𝑘=𝐴𝑗([0,1]𝑛+𝑘), 𝑗, and 𝑘𝑛,(c)𝑆(𝑓)=(𝑗,𝑘𝑛|𝑓,𝜓𝑗,𝑘|2𝜒𝑅𝑗,𝑘|𝑄𝑗,𝑘|1)1/2𝐿𝑝,(𝑛),where for every 𝑗 and 𝑘𝑛, 𝑅𝑗,𝑘 is a measurable set such that 𝑅𝑗,𝑘𝑄𝑗,𝑘 and |𝑅𝑗,𝑘|𝛾|𝑄𝑗,𝑘|, for a certain 𝛾(0,1),(d)the distribution 𝑓=𝑗,𝑘𝑛𝑓,𝜓𝑗,𝑘𝜓𝑗,𝑘 is in 𝐻𝐴𝑝,(𝑛).Moreover, if one of (and then all) the above conditions is satisfied, then 𝑓𝐻𝐴𝑝,(𝑛)𝑊(𝑓)𝐿𝑝,(𝑛)𝐺(𝑓)𝐿𝑝,(𝑛)𝑆(𝑓)𝐿𝑝,(𝑛).(2.7)

Bownik proved in [1, Theorem 4.2, page 94] that there exists a tight frame 𝐴-wavelet 𝜓𝑆(𝑛) satisfying the conditions in Theorem 2.2 such that 𝑛𝑓(𝑥)𝑃(𝑥)𝑑𝑥=0, for every polynomial 𝑃 in 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], 𝐴𝑗𝑥0, as 𝑗, uniformly in 𝑥[0,1]𝑛. Moreover, there exist 0<𝜂<1 and 𝛿>0 for which||||[]𝜓(𝑥)𝛿,𝑥0,𝜂𝑛.(3.1) Then, for every 𝑗 and 𝑘𝑛, we have||𝜓𝑗,𝑘||𝛿(𝑥)||𝑄𝑗,𝑘||,𝑥𝑅𝑗,𝑘,(3.2) where 𝑅𝑗,𝑘=𝐴𝑗([0,𝜂]𝑛+𝑘)𝑄𝑗,𝑘. Hence, we can write 𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2||𝜓𝑗,𝑘||(𝑥)21/2𝛿𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||21||𝑄𝑗,𝑘||𝜒𝑅𝑗,𝑘(𝑥)1/2.(3.3) Note that||𝑅𝑗,𝑘||=||𝐴𝑗[]0,𝜂𝑛||=||||det𝐴𝑗𝜂𝑛=||𝑄𝑗,𝑘||𝜂𝑛.(3.4) Thus we show that (a) implies (b).

(b)(c) It is clear.

(c)(b) Assume that (c) holds. For every 𝑘, we define𝐸𝑘=𝑥𝑛𝑆𝑓(𝑥)>2𝑘.(3.5) According to our assumption we have that sup𝑘2𝑘𝑝|𝐸𝑘|<. Fix 𝛽(0,𝛾) and 𝑘. We denote by 𝐷𝑘 the set defined by𝐷𝑘=(𝑗,𝑙)×𝑛||𝑄𝑗,𝑙𝐸𝑘||||𝑄𝛽𝑗,𝑙||.(3.6) By 𝐷𝑘,max we represent the set that consists of all (𝑗,𝑙)𝐷𝑘 such that 𝑄𝑗,𝑙 is maximal with respect to the order in 𝐷𝑘 introduced in [1, Definition 6.4, page 105].

It is clear that||𝑄𝑗,𝑙||1𝛽||𝑄𝑗,𝑙𝐸𝑘||𝐶2𝑘𝑝,(𝑗,𝑙)𝐷𝑘.(3.7)

Moreover, |𝑄𝑗,𝑙|=|det𝐴|𝑗,𝑗, and 𝑙𝑛. Hence, there exists 𝑗0 such that 𝑗𝑗0 provided that (𝑗,𝑙)𝐷𝑘. Then, for every (𝑗,𝑙)𝐷𝑘, there exists (𝑗1,𝑙1)𝐷𝑘,max for which 𝑄𝑗,𝑙𝐷𝑘𝑄𝑗1,𝑙1 (see [1, page 105]).

For every (𝑗,𝑙)𝐷𝑘,max we define𝐷𝑘(𝑗,𝑙)=(𝑚,𝑠)𝐷𝑘𝑄𝑚,𝑠𝐷𝑘𝑄𝑗,𝑙.(3.8) By [1, Lemma 6.5, page 105], we can find 𝜂 such that for every (𝑗,𝑙)𝐷𝑘,max we have(𝑚,𝑠)𝐷𝑘(𝑗,𝑙)𝑄𝑚,𝑠||𝑙𝑙1||<𝜂𝑄𝑗,𝑙1.(3.9) If 𝐸𝑘=(𝑗,𝑙)𝐷𝑘𝑄𝑗,𝑙, it follows that||𝐸𝑘||=|||||(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝐷𝑘(𝑗,𝑙)𝑄𝑚,𝑠|||||(𝑗,𝑙)𝐷𝑘,max|||||(𝑚,𝑠)𝐷𝑘(𝑗,𝑙)𝑄𝑚,𝑠|||||(𝑗,𝑙)𝐷𝑘,max|||||||𝑙𝑙1||<𝜂𝑄𝑗,𝑙1|||||(2𝜂+1)𝑛(𝑗,𝑙)𝐷𝑘,max||||det𝐴𝑗=(2𝜂+1)𝑛(𝑗,𝑙)𝐷𝑘,max||𝑄𝑗,𝑙||(2𝜂+1)𝑛𝛽(𝑗,𝑙)𝐷𝑘,max||𝑄𝑗,𝑙𝐸𝑘||(2𝜂+1)𝑛𝛽||𝐸𝑘||.(3.10) In the last inequality we have used |𝑄𝑗,𝑙𝑄𝑗1,𝑙1|=0 provided that (𝑗,𝑙), (𝑗1,𝑙1)𝐷𝑘,max, (𝑗,𝑙)(𝑗1,𝑙1).

By proceeding as in [1, page 107] by induction on (𝑗,𝑙)𝐷𝑘,max we find, for every (𝑗,𝑙)𝐷𝑘,max, a set𝐼(𝑗,𝑙)𝐷𝑘(𝑗,𝑙),(3.11) satisfying the following properties:𝐷𝑘=(𝑗,𝑙)𝐷𝑘,max𝑗𝐼(𝑗,𝑙),𝐼(𝑗,𝑙)𝐼1,𝑙1𝑗=,(𝑗,𝑙),1,𝑙1𝐷𝑘,max𝑗,(𝑗,𝑙)1,𝑙1.(3.12) Note that 𝐷𝑘+1𝐷𝑘. We can consider the sets 𝑍𝑘=𝐷𝑘𝐷𝑘+1,𝑍𝑘(𝑗,𝑙)=𝑍𝑘𝐼(𝑗,𝑙) and 𝑀𝑘(𝑗,𝑙)=(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑄𝑚,𝑠, for every (𝑗,𝑙)𝐷𝑘,max.

Let (𝑗,𝑙)𝐷𝑘,max. We can write𝑀𝑘(𝑗,𝑙)𝐸𝑘+1||||𝑆(𝑓)(𝑥)2𝑑𝑥=𝑀𝑘(𝑗,𝑙)𝐸𝑘+1𝑚,𝑠𝑛||𝑓,𝜓𝑚,𝑠||21||𝑄𝑚,𝑠||𝜒𝑅𝑚,𝑠(𝑥)𝑑𝑥𝑀𝑘(𝑗,𝑙)𝐸𝑘+1(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21||𝑄𝑚,𝑠||𝜒𝑅𝑚,𝑠(𝑥)𝑑𝑥(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21||𝑄𝑚,𝑠||||𝑅𝑚,𝑠𝐸𝑐𝑘+1||.(3.13) Also, for every (𝑚,𝑠)𝑍𝑘(𝑗,𝑙), we get||𝑅𝑚,𝑠𝐸𝑐𝑘+1||=||𝑅𝑚,𝑠||||𝑅𝑚,𝑠𝐸𝑘+1||||𝑄(𝛾𝛽)𝑚,𝑠||,(3.14) because |𝑅𝑚,𝑠|𝛾|𝑄𝑚,𝑠|, and, since (𝑚,𝑠)𝐷𝑘+1,||𝑅𝑚,𝑠𝐸𝑘+1||||𝑄𝑚,𝑠𝐸𝑘+1||||𝑄<𝛽𝑚,𝑠||.(3.15) Hence𝑀𝑘(𝑗,𝑙)𝐸𝑘+1||||𝑆(𝑓)(𝑥)2𝑑𝑥(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||2(𝛾𝛽).(3.16) On the other hand, we have that𝑀𝑘(𝑗,𝑙)𝐸𝑘+1||||𝑆(𝑓)(𝑥)2𝑑𝑥4𝑘+1||𝑀𝑘(𝑗,𝑙)𝐸𝑘+1||.(3.17) We conclude that(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||24𝑘+1||𝑀𝛾𝛽𝑘||.(𝑗,𝑙)(3.18) Again according to [1, Lemma 6.5] (see (3.9)) it follows that𝑀𝑘(𝑗,𝑙)||||𝑠𝑙<𝜂𝑄𝑗,𝑠.(3.19) Then||𝑀𝑘||(𝑗,𝑙)||||𝑠𝑙<𝜂||𝑄𝑗,𝑠||(2𝜂+1)𝑛||||det𝐴𝑗.(3.20) Therefore(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||2(2𝜂+1)𝑛4𝛾𝛽𝑘+1||𝑄𝑗,𝑙||.(3.21) Let 𝜈>0. We choose 𝑘0 such that 2𝑘0𝜈<2𝑘0+1 and we define the functions 𝐺1 and 𝐺2 by𝐺1(𝑓)=𝑘0𝑘=(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21||𝑄𝑚,𝑠||𝜒𝑄𝑚,𝑠1/2,𝐺2(𝑓)=𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21||𝑄𝑚,𝑠||𝜒𝑄𝑚,𝑠1/2.(3.22) Note that if (𝑚,𝑠)×𝑛 and 𝑓,𝜓𝑚,𝑠0 then there exists 𝑘1 such that 𝑅𝑚,𝑠𝐸𝑘1. Hence (𝑚,𝑠)𝐷𝑘, for every 𝑘,𝑘𝑘1. Moreover, there exists 𝑘2 such that (𝑚,𝑠)𝐷𝑘, when 𝑘𝑘2. We deduce that 𝐺2=𝐺21+𝐺22. By (3.10) and (3.21), since (𝑗,𝑙)𝐷𝑘,max|𝑄𝑗,𝑙||𝐸𝑘|, we infer𝐺1𝑓22(2𝜂+1)𝑛𝛾𝛽𝑘0𝑘=(𝑗,𝑙)𝐷𝑘,max4𝑘+1||𝑄𝑗,𝑙||(2𝜂+1)𝑛𝛾𝛽𝑘0𝑘=4𝑘+1||𝐸𝑘||(2𝜂+1)2𝑛𝛽(𝛾𝛽)𝑘0𝑘=4𝑘+1||𝐸𝑘||4(2𝜂+1)2𝑛𝛽(𝛾𝛽)𝑘0𝑘=2(2𝑝)𝑘4(2𝜂+1)2𝑛2𝛽(𝛾𝛽)(2𝑝)𝑘012(2𝑝)𝐶𝜈2𝑝.(3.23) Moreover, since 𝐺2(𝑓)(𝑥)=0 when 𝑥𝑘𝑘0+1𝐸𝑘, by (3.10), it follows that||𝑥𝑛𝐺2(||𝑓)(𝑥)0𝑘=𝑘0+1||𝐸𝑘||(2𝜂+1)𝑛𝛽𝑘=𝑘0+1||𝐸𝑘||𝐶(2𝜂+1)𝑛𝛽𝑘=𝑘0+12𝑘𝑝𝐶𝜈𝑝.(3.24) Then, we conclude that||{𝑥𝑛|||||𝐺(𝑓)(𝑥)>𝜈}𝑥𝑛𝐺1𝜈(𝑓)(𝑥)>2|||+||𝑥𝑛𝐺2||(𝑓)(𝑥)0{𝑥𝑛𝐺1(𝑓)(𝑥)>𝜈/2}2𝐺1(𝑓)(𝑥)𝜈2𝑑𝑥+𝐶𝜈𝑝𝐶𝜈𝑝.(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 such that 2𝑘0𝜈<2𝑘0+1 and we define𝑓1=𝑘0𝑘=(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑓,𝜓𝑚,𝑠𝜓𝑚,𝑠.(3.26) By proceeding as in the proof of (c)(b) we get𝑘0𝑘=(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||2𝐶𝜈2𝑝.(3.27) Then, since 𝜓 is a tight frame 𝐴-wavelet, by using duality, we obtain that 𝑓122𝐶𝑘0𝑘=(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||2𝐶2𝑝.(3.28) We now consider𝑓2=𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑓,𝜓𝑚,𝑠𝜓𝑚,𝑠.(3.29) We choose 𝜑𝑆(𝑛) such that supp𝜑 is compact and bounded away from the origin and that 𝑙𝜑((𝐴)𝑙𝑦)=1,𝑦𝑛{0}. We are going to prove that ||||𝑥2sup𝛼||𝜑𝛼𝑓2||||||𝐶(𝑥)>𝜈𝜈𝑝.(3.30) For every (𝑗,𝑙)𝐷𝑘,max, we define𝐶(𝑗,𝑙)=||𝑙𝑙1||𝑏𝜂𝑄𝑗,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,max𝐶(𝑗,𝑙).(3.32) If (𝑗,𝑙)𝐷𝑘,max, then||||𝐶(𝑗,𝑙)||𝑙𝑙1||𝑏𝜂||𝑄𝑗,𝑙1||(2𝑏𝜂+1)𝑛||𝑄𝑗,𝑙||.(3.33) Hence, since (𝑗,𝑙)𝐷𝑘,max|𝑄𝑗,𝑙||𝐸𝑘|, by (3.10) it follows that||Ω||=|||||𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max|||||𝐶(𝑗,𝑙)(2𝑏𝜂+1)𝑛𝑘=𝑘0+1||𝐸𝑘||𝐶𝑘=𝑘0+1||𝐸𝑘||𝐶𝑘=𝑘0+12𝑘𝑝𝐶2𝑘0𝑝𝐶𝜈𝑝.(3.34) By (3.34), (3.30) will be proved when we obtain that||||𝑥Ωsup𝛼||𝜑𝛼𝑓2||||||𝐶(𝑥)>𝜈𝜈𝑝.(3.35) Since supp𝜓 and supp𝜑 are compact and 0supp𝜓supp𝜑, there exists 𝑀>0 such that supp𝜓𝑚,𝑠supp𝜑𝛼=, provided that |𝑚𝛼|>𝑀,𝑚,𝛼, and 𝑠𝑛.

Let 𝛼. We can write𝜑𝛼𝑓2=𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑓,𝜓𝑚,𝑠𝜑𝛼𝜓𝑚,𝑠.(3.36) Using Hölder’s inequality we get|||||𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑓,𝜓𝑚,𝑠𝜑𝛼𝜓𝑚,𝑠|||||(𝑥)𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21/2𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝜑𝛼𝜓𝑚,𝑠(||𝑥)21/2.(3.37) From (3.21) we deduce that𝛼+𝑚𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21/2(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝑓,𝜓𝑚,𝑠||21/2𝐶2𝑘||𝑄𝑗||,𝑙1/2.(3.38) According to [13, page 1482] for any 𝜆>1 there exists 𝐶=𝐶(𝜆)>0 for which||𝜑𝛽𝜓𝑚,𝑠||||||(𝑥)𝐶det𝐴𝑚/21+𝜌𝐴(𝐴𝑚𝑥𝑠)𝜆,𝑥𝑛,(3.39) when |𝛽𝑚|𝑀.

Let 𝜆>0 that will be fixed later. We can write𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝜑𝛼𝜓𝑚,𝑠(||𝑥)21/2𝐶𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||||det𝐴𝑚1+𝜌𝐴(𝐴𝑚𝑥𝑠)2𝜆1/2𝐶𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||||det𝐴𝑚/21+𝜌𝐴(𝐴𝑚𝑥𝑠)𝜆,𝑥𝑛.(3.40) Let 𝑘,𝑘>𝑘0, and (𝑗,𝑙)𝐷𝑘,max. Assume that 𝑥Ω. Then 𝑥𝐶(𝑗,𝑙). By (3.9) we know that(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑄𝑚,𝑠||𝑙𝑙1||<𝜂𝑄𝑗,𝑙1.(3.41) Let (𝑚,𝑠)𝑍𝑘(𝑗,𝑙). We define𝑥=𝐴𝑗𝑥,𝑥𝑚,𝑠=𝐴𝑚𝑠,𝑥𝑚,𝑠=𝐴𝑗𝑥𝑚,𝑠,𝑥𝑗,𝑙=𝐴𝑗𝑙,𝑥𝑗,𝑙=𝑙.(3.42) Note that𝑥||𝑙𝑙1||<𝑏𝜂[]0,1𝑛+𝑙1,𝑥𝑚,𝑠||𝑙𝑙1||<𝜂[]0,1𝑛+𝑙1,𝑥𝑗,𝑙||𝑙𝑙1||<𝜂[]0,1𝑛+𝑙1.(3.43) We have that𝜌𝐴𝑥𝑥𝑚,𝑠1𝐻𝜌𝐴𝑥𝑥𝑗,𝑙𝜌𝐴𝑥𝑚,𝑠𝑥𝑗,𝑙.(3.44) By [1, Lemma 3.2] it follows that, for certain 𝛾>0,𝜌𝐴𝑥𝑚,𝑠𝑥𝑗,𝑙𝐶𝜂𝛾,(3.45) and, for some 𝛾>0,𝜌𝐴𝑥𝑥𝑗,𝑙𝐶(𝑏𝜂)𝛾.(3.46) Choosing 𝑏 large enough we get that𝜌𝐴𝑥𝑥𝑚,𝑠1𝜌2𝐻𝐴𝑥𝑥𝑗,𝑙.(3.47) Then,𝜌𝐴𝑥𝑥𝑚,𝑠=𝜌𝐴𝐴𝑗𝑥𝑥𝑚,𝑠=||||det𝐴𝑗𝜌𝐴𝑥𝑥𝑚,𝑠||||det𝐴𝑗𝜌2𝐻𝐴𝑥𝑥𝑗,𝑙=1𝜌2𝐻𝐴𝑥𝑥𝑗,𝑙.(3.48) By (3.40) and (3.48) with 𝜆>3/2 we obtain that𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||𝜑𝛼𝜓𝑚,𝑠(||𝑥)21/2𝐶𝜌𝐴𝑥𝑥𝑗,𝑙𝜆𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||||det𝐴𝑚(1/2𝜆)𝐶𝜌𝐴𝑥𝑥𝑗,𝑙𝜆𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)||||det𝐴𝑚1/2+𝜆𝐶𝜌𝐴𝑥𝑥𝑗,𝑙𝜆𝛼+𝑀𝑚=𝛼𝑀|||||𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑄𝑚,𝑠|||||1/2+𝜆||𝑄𝐶𝑗,𝑙||1/2+𝜆𝜌𝐴𝑥𝑥𝑗,𝑙𝜆.(3.49) In the last inequality we have used (3.20). From (3.37), (3.38), and (3.49) it follows that|||||𝛼+𝑀𝑚=𝛼𝑀𝑠(𝑚,𝑠)𝑍𝑘(𝑗,𝑙)𝑓,𝜓𝑚,𝑠𝜑𝛼𝜓𝑚,𝑠|||||2(𝑥)𝐶𝑘||𝑄𝑗,𝑙||𝜆𝜌𝐴𝑥𝑥𝑗,𝑙𝜆3,𝜆2.(3.50) Whence||||𝑥Ωsup𝛼||𝜑𝛼𝑓2|||||||||||(𝑥)>𝜈𝑥Ω𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max𝐶2𝑘||𝑄𝑗,𝑙||𝜆𝜌𝐴𝑥𝑥𝑗,𝑙𝜆||||||||||>𝜈𝑥Ω𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max𝑐𝑘𝑗,𝑙𝑔𝑘𝑗,𝑙|||||,>𝜈(3.51) where 𝜆>3/2,𝑐𝑘𝑗,𝑙=𝐶2𝑘|𝑄𝑗,𝑙|𝜆 and𝑔𝑘𝑗,𝑙𝜒(𝑥)=Ω𝑐(𝑥)𝜌𝐴𝑥𝑥𝑗,𝑙𝜆,𝑥𝑛.(3.52) By [2, equation (2.4)] we know that |𝐵𝜌𝐴(𝑎,𝑟)|𝑟,𝑎𝑛, and 𝑟>0. Therefore,|||𝑥𝑛||𝑔𝑘𝑗,𝑙|||||=||(𝑥)>𝜔𝑥Ω𝑐𝜌𝐴𝑥𝑥𝑗,𝑙<𝜔1/𝜆||𝐶𝜔1/𝜆,𝜔>0.(3.53) Hence, according to [9, Lemma 2.1], by taking 𝜆>1/𝑝, we obtain|||||𝑥Ω𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max𝐶2𝑘||𝑄𝑗,𝑙||𝜆𝜌𝐴𝑥𝑥𝑗,𝑙𝜆|||||1>𝜈𝐶𝜈1/𝜆𝑘=𝑘0+1(𝑗,𝑙)𝐷𝑘,max2𝑘/𝜆||𝑄𝑗,𝑙||1𝐶𝜈1/𝜆𝑘=𝑘0+12𝑘/𝜆||𝐸𝑘||1𝐶𝜈1/𝜆𝑘=𝑘0+12𝑘/𝜆||𝐸𝑘||1𝐶𝜈1/𝜆𝑘=𝑘0+12𝑘(1/𝜆𝑝)1𝐶𝜈1/𝜆2𝑘0(1/𝜆𝑝)121/𝜆𝑝𝐶𝜈𝑝.(3.54) Hence,||||𝑥Ωsup𝛼||𝜑𝛼𝑓2||||||𝐶(𝑥)>𝜈𝜈𝑝.(3.55) Combining (3.28), (3.34), and (3.55) we get||||𝑥𝑛sup𝛼||𝜑𝛼||||||||||𝑓(𝑥)>𝜈𝑥𝑛sup𝛼||𝜑𝛼𝑓1(||>𝜈𝑥)2||||+||Ω||+||||𝑥Ωsup𝛼||𝜑𝛼𝑓2(||>𝜈𝑥)2||||4𝜈2sup𝛼||𝜑𝛼𝑓1||(𝑥)22+𝐶𝜈𝑝1𝐶𝜈2𝑓122+1𝜈𝑝𝐶𝜈𝑝.(3.56) Thus we conclude that 𝑓𝐻𝐴𝑝,(𝑛) and, therefore, (d) is established.

(d)(a) We define the sublinear operator 𝑊 as follows:𝑊𝑓=𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2||𝜓𝑗,𝑘||(𝑥)21/2,𝑓𝐿2(𝑛).(3.57) Since 𝜓 is a tight frame 𝐴-wavelet, we have that𝑊𝑓22=𝑛𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2||𝜓𝑗,𝑘(||𝑥)2=𝑑𝑥𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2||||det𝐴𝑗𝑛||𝜓𝐴𝑗||𝑥𝑘2=𝑑𝑥𝑛||||𝜓(𝑥)2𝑑𝑥𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2=𝑓22𝑛||||𝜓(𝑥)2𝑑𝑥,𝑓𝐿2(𝑛).(3.58) Hence, 𝑊 is a bounded operator from 𝐿2(𝑛) into itself.

According to [1, Theorem 6.7, page 109], for every 0<𝑞1 and 𝑓𝐻𝑞𝐴(𝑛),𝑓𝐻𝑞𝐴(𝑛)𝑗,𝑘𝑛||𝑓,𝜓𝑗,𝑘||2||𝜓𝑗,𝑘||(𝑥)21/2𝑞.(3.59) Then, the operator 𝑊 defined as above is also bounded from the anisotropic Hardy space 𝐻𝑞𝐴(𝑛) into 𝐿𝑞(𝑛), for every 0<𝑞1. Our proof will be finished when we see that the operator 𝑇 is bounded from 𝐻𝐴𝑝,(𝑛) into 𝐿𝑝,(𝑛). In order to do this we proceed interpolating by using the ideas developed in the proof of [9, Theorem 3.4].

Let 𝑓𝐻𝐴𝑝,(𝑛) and 𝜈>0. We choose 𝑘0 such that 2𝑘0𝜈<2𝑘0+1. We take the atomic decomposition of 𝑓 given by 𝑓=𝑘=𝑓𝑘. Here 𝑓𝑘=𝑖=0𝛽𝑘𝑖, 𝑘, as in Theorem 2.1 with 𝑠=[(2/𝑝1)((log|det𝐴|)/log𝑚)], where 𝑚=min{|𝜆|𝜆isaneigenvalueof𝐴}. We decompose 𝑓 as follows:𝑓=𝑘0𝑘=𝑓𝑘+𝑘=𝑘0+1𝑓𝑘=𝐹1+𝐹2.(3.60) We have that𝐹12𝑘0𝑘=𝑓𝑘2𝐶𝑘0𝑘=2𝑘𝑖=0|||𝐵𝑘𝑖|||1/2𝐶𝑘0𝑘=2𝑘(1𝑝/2)𝑓𝐻𝑝/2𝐴𝑝,(𝑛)𝐶2𝑘0(1𝑝/2)𝑓𝐻𝑝/2𝐴𝑝,(𝑛).(3.61) Then,||𝑥𝑛||𝑊𝐹1(||||𝐶𝑥)>𝜈𝜈2𝐹1222𝐶𝑘0(2𝑝)𝜈2𝑓𝑝𝐻𝐴𝑝,(𝑛)𝐶𝜈𝑝𝑓𝑝𝐻𝐴𝑝,(𝑛).(3.62)

According to Theorem 2.1, there exists 𝐶>0 such that, for every 𝑘 and 𝑖, 𝐶2𝑘|𝐵𝑘𝑖|2/𝑝𝛽𝑘𝑖 is a (𝑝/2,,𝑠)𝐻𝐴𝑝/2(𝑛)-atom. Hence, for every 𝑘, 𝑓𝑘𝐻𝐴𝑝/2(𝑛) and𝑓𝑘𝐻𝑝/2𝐴𝑝/2(𝑛)𝐶𝑖=02𝑘𝑝/2|||𝐵𝑘𝑖|||𝐶2𝑘𝑝/2𝑓𝑝𝐻𝐴𝑝,(𝑛).(3.63) Since 𝑊 is bounded from 𝐻𝐴𝑝/2(𝑛) into 𝐿𝑝/2(𝑛), it follows that||𝑥𝑛||𝑊𝑓𝑘||||(𝑥)>𝜈𝜈𝑝/2𝑊(𝑓𝑘)𝑝/2𝑝/2𝐶𝜈𝑝/2𝑓𝑘𝐻𝑝/2𝐴𝑝/2(𝑛),𝑘.(3.64)

Therefore, using [9, Lemma 2.1] we get||𝑥𝑛||𝑊𝐹2|||||||||(𝑥)>𝜈𝑥𝑛𝑘=𝑘0+1||𝑊𝑓𝑘|||||||=|||||(𝑥)>𝜈𝑥𝑛𝑘=𝑘0+1𝑓𝑘𝐻𝐴𝑝/2(𝑛)|||||𝑊𝑓𝑘𝑓𝑘𝐻𝐴𝑝/2(𝑛)||||||||||(𝑥)>𝜈𝐶𝜈𝑝/2𝑘=𝑘0+1𝑓𝑘𝐻𝑝/2𝐴𝑝/2(𝑛)𝐶𝜈𝑝/2𝑘=𝑘0+12𝑘𝑝/2𝑓𝑝𝐻𝐴𝑝,(𝑛)𝐶𝜈𝑝/22𝑘0𝑝/2𝑓𝑝𝐻𝐴𝑝,(𝑛)𝐶𝜈𝑝𝑓𝑝𝐻𝐴𝑝,(𝑛).(3.65) Putting together the above estimates we obtain||𝑥𝑛|||||||||𝑊(𝑓)(𝑥)>𝜈𝑥𝑛||𝑊𝐹1||>𝜈(𝑥)2|||+|||𝑥𝑛||𝑊𝐹2||>𝜈(𝑥)2|||𝐶𝜈𝑝𝑓𝑝𝐻𝐴𝑝,(𝑛).(3.66)

Acknowledgment

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

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