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
We characterize the anisotropic weak Hardy spaces associated with an expansive matrix by using square functions involving wavelets coefficients.
Bownik, in a series of papers [1–5], studied anisotropic function spaces associated with dilations. In the monography  he investigated anisotropic Hardy spaces. Suppose that is an expansive matrix (also called dilation) in , that is, is an -matrix all of whose eigenvalues satisfy . If is a function in the Schwartz class such that and is a distribution in , the dual space of , the radial maximal function is defined by
where , and . For every , the Hardy space associated with consists of all those such that . The space does not depend on the function and the -norm is defined by
Like in the classical case , the anisotropic Hardy space can be characterized by nontangential or grand maximal functions [1, Theorem 7.1, page 42]. Also, atomic representations of the distributions in are obtained in [1, Theorem 6.5, page 39]. Wavelets for a dilation are studied in [1, Chapter 2]. The author proved that -wavelets associated with the expansive matrix form an unconditional basis for the anisotropic Hardy spaces defined by . Recently, Bownik et al. [7, 8] have investigated weighted anisotropic Hardy spaces.
The weak anisotropic Hardy space , was introduced by Ding and Lan . A distribution is in if and only if , where denotes the weak -space. We define
As the case , the space does not depend on the election of and it can be described by nontangential and grand maximal functions. Atomic representations of distributions in were established in [9, Theorem 1.1]. In this paper we study wavelets for . We characterize in Theorem 2.2 bellow the distributions in by square functions involving wavelets coefficients. Our result can be seen as an anisotropic version of the one showed in  (see also ).
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  where the anisotropic theory associated with expansive matrixes was developed. Suppose that is an expansive matrix in . We denote by an ellipsoid for such that the Lebesgue measure of is equal to 1. For every , we define . We consider the mapping given by
Thus, is a homogeneous quasinorm associated with in the sense of [1, Definition 2.3, page 6]. This means that satisfies the following properties:(a), ,(b), ,(c), ,
where . In [1, Lemma 2.4, page 6] it was proved that if and are two homogeneous quasinorms associated with then,
for a certain . As it was mentioned above a tempered distribution , , when , where such that . In [9, Theorem 1.1] Ding and Lan proved the following atomic representation for the distributions in that will be useful in the sequel.
Theorem 2.1 (see [9, Theorem 1.1]). Suppose that , , and , where . Then, there exists a sequence of bounded functions satisfying, for a certain ,(a) in , and , for every ,(b)for every , , in , where, for each ,(i)there exist and such that , and
(ii), (iii), for every polynomial whose degree is less or equal to .Conversely, if satisfying and where in is replacing by , then and .
Theorem 2.1 is an anisotropic version of the isotropic result established by Fefferman and Soria [12, Proposition, page 8].
Let . For every and , we define
We say that is a Bessel -wavelet if there exists such that, for every ,
is a frame -wavelet when, for a certain ,
We say that is a tight frame -wavelet when (2.6) holds with .
As usual if we denote by the Fourier transform of .
We now establish our result where the distributions in , , are characterized by using square functions involving -wavelets.
Theorem 2.2. Let and . Assume that is a tight frame -wavelet where is an expansive matrix in , such that , is compact, and . Then, the following properties are equivalent:(a),(b),where , , and ,(c),where for every and , is a measurable set such that and , for a certain ,(d)the distribution is in .Moreover, if one of (and then all) the above conditions is satisfied, then
Bownik proved in [1, Theorem 4.2, page 94] that there exists a tight frame -wavelet satisfying the conditions in Theorem 2.2 such that , for every polynomial in .
In this section we present a proof of Theorem 2.2. Throughout this section with we denote a positive constant that can change in each occurrence.
According to [1, equation (2.2), page 5], , as , uniformly in . Moreover, there exist and for which
Then, for every and , we have
where . Hence, we can write
Thus we show that implies .
It is clear.
Assume that holds. For every , we define
According to our assumption we have that . Fix and . We denote by the set defined by
By we represent the set that consists of all such that is maximal with respect to the order in introduced in [1, Definition 6.4, page 105].
It is clear that
Moreover, , and . Hence, there exists such that provided that . Then, for every , there exists for which (see [1, page 105]).
For every we define
By [1, Lemma 6.5, page 105], we can find such that for every we have
If , it follows that
In the last inequality we have used provided that , , .
By proceeding as in [1, page 107] by induction on we find, for every , a set
satisfying the following properties:
Note that . We can consider the sets and , for every .
Let . We can write
Also, for every , we get
because , and, since ,
On the other hand, we have that
We conclude that
Again according to [1, Lemma 6.5] (see (3.9)) it follows that
Let . We choose such that and we define the functions and by
Note that if and then there exists such that . Hence , for every . Moreover, there exists such that , when . We deduce that . By (3.10) and (3.21), since , we infer
Moreover, since when , by (3.10), it follows that
Then, we conclude that
Thus is proved.
Suppose that holds. We keep the notation that it was used in the proof of .
Let . We choose such that and we define
By proceeding as in the proof of we get
Then, since is a tight frame -wavelet, by using duality, we obtain that
We now consider
We choose such that is compact and bounded away from the origin and that . We are going to prove that
For every , we define
where is the one appeared in (3.9) and will be chosen later. We also consider
If , then
Hence, since , by (3.10) it follows that
By (3.34), (3.30) will be proved when we obtain that
Since and are compact and , there exists such that , provided that , and .
Let . We can write
Using Hölder’s inequality we get
From (3.21) we deduce that
According to [13, page 1482] for any there exists for which
Let that will be fixed later. We can write
Let , and . Assume that . Then . By (3.9) we know that
Let . We define
We have that
By [1, Lemma 3.2] it follows that, for certain ,
and, for some ,
Choosing large enough we get that
By (3.40) and (3.48) with we obtain that
In the last inequality we have used (3.20). From (3.37), (3.38), and (3.49) it follows that
By [2, equation (2.4)] we know that , and . Therefore,
Hence, according to [9, Lemma 2.1], by taking , we obtain
Combining (3.28), (3.34), and (3.55) we get
Thus we conclude that and, therefore, is established.
We define the sublinear operator as follows:
Since is a tight frame -wavelet, we have that
Hence, is a bounded operator from into itself.
According to [1, Theorem 6.7, page 109], for every and ,
Then, the operator defined as above is also bounded from the anisotropic Hardy space into , for every . Our proof will be finished when we see that the operator is bounded from into . In order to do this we proceed interpolating by using the ideas developed in the proof of [9, Theorem 3.4].
Let and . We choose such that . We take the atomic decomposition of given by . Here , , as in Theorem 2.1 with , where . We decompose as follows:
We have that
According to Theorem 2.1, there exists such that, for every and , is a -atom. Hence, for every , and
Since is bounded from into , it follows that
Therefore, using [9, Lemma 2.1] we get
Putting together the above estimates we obtain
B. Barrios is partially supported by MTM2010-16518 and J. J. Betancor is partially supported by MTM2010-17974.
M. Bownik, “Anisotropic Hardy spaces and wavelets,” Memoirs of the American Mathematical Society, vol. 164, no. 781, 2003.View at Zentralblatt MATH