The special atom space and Haar wavelets in higher dimensions

Abstract In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.


Introduction
In our case, we start by recalling the definition of the special atom space on the interval = [ ] J 0, 1 . The definitions over general interval [ ] a b , and over follow along similar lines. This is done for the sake of understanding the transition from [ ] 0, 1 to [ ] × [ ] 0, 1 0, 1 and from to × . First, let us recall the definition of general atom proposed in [1].
1 be a real number and J be an interval of . An atom is a function b defined on the interval J and satisfying From this definition, special atoms for ≥ p 1 were introduced as: where I is an interval contained in J, L and R are the halves of I such that = ∪ I L R, and | | I is the length of I (Figure 1a).  Proof. The proof can be found in [2]. □ 2 Motivation for the need of high dimension special atom spaces The special atom was introduced by Geraldo de Souza in PhD thesis (see [2]) partly to answer one main criticism of atoms, in that, they are too general and so far their main application was to prove that the dual of the Hardy space H 1 , unknown at the time, was indeed the Space of Bounded Means Oscillations. Unbeknownst to the community at the time was that special cases of the atomic decomposition of Hardy's space would prove very beneficial in addressing unsolved problems. For instance, the special atom space as introduced by Geraldo de Souza has for dual space the space of derivatives (in the sense of distributions) of functions belonging to the Zygmund space This result led to a simple proof that the Hardy space H 1 indeed contains functions whose Fourier series diverge almost everywhere by observing that the Hardy space H 1 is a superspace of the special atom space, and such functions actually exist in the special atom spaces, see for instance [3]. Moreover, the special atom space B 1 is a Banach equivalent to the space of analytic functions F on the complex unit disc for which ∫ ∫ . This analytic characterization also led to the lacunary characterization of functions in B p , with ≤ < ∞ p 1 . The question that was later raised by Brett Wick (Washington University, St. Louis, USA) was whether such a characterization could be achieved in the bidisk or even tridisk. To entertain such a question, a rigorous definition of special atom spaces in higher dimension is needed that would also preserve the key properties of the underlying space. Another important aspect of the special atom spaces that needs special care in high dimension is that of its connections with the Lorentz spaces L p q , . Indeed, one property of the special atom space (type 2) is that it is the atomic decomposition of the Lorentz space L p,1 , which facilitates the study of operators such as the composition and multiplication operators. It is still not clear how to deal with these operators in higher dimensions but with a rigorous definition of special atoms, this could be possible. The extension of the special atom space we propose in the sequel leads to a natural definition of Haar wavelets in higher dimensions. Indeed, Haar wavelets are still preferred by certain practitioners for their simplicity and relatively ease of use. They can prove very useful in physical problems like heat transfer where the solution can be found relatively fast even at low resolution. The definition we propose allows us to easily prove that the Haar system forms an orthonormal basis in ( ) L 2 2 , and generally in ( ) In the last section, we provide some applications of these facts. Consider a sub-rectangle J n of J as

Extension to high dimensions
, .
n n n n n n n n n Definition 5. Let ≤ < ∞ p 1 and consider We define the function ( ) B x y , n in the plane as: 1. Note that the subsets L n and R n of J n are nonempty and disjoint so that ( ) B x y , n can be written as: where |⋅| represents the Lebesgue measure or the area to be more precise. 2. We also note that by restricting ( ) B x y , n to the real line, we recover the definition (Definition 2) above, using the fact that B 1 is equivalent to the space of functions ( ) = ∑ ( ) = ∞ f x αb x n n n 0 , where b n 's are special atoms of type 1, see [4]. 3. We observe that the role of /p 1 and by extension that of p is to extend the definition above to L p spaces and as such, Figure 2 is an illustration of L n and R n in the plane.
In the next definition, we will drop the index n in L n i , and R n i , , = i 1, 2 respectively, for the sake of clarity. Consider ⊆ U J, measurable such that = ∪ U L R, with = ∪ L L R Definition 6. We define a special atom of Type 1 on J as: where | | = J h k 4 n n n . Note that the subsets L n and R n are nonempty and disjoints so that ,1 ,1 ,2 a n − h n a n a n + h n where J n 's are subsets in J, α n are real numbers, ( ) B x y , n are special atoms of type 1. B p is endowed with the norm: where the infimum is taken over all representations of f. Type 2: n are special atoms of type 2 and β n are real numbers. C p is endowed with the norm where the infimum is taken over all representations of f.
In the next definition, U and V represent measurable subsets of J, where |⋅| is the Lebesgue measure.
. We define the following auxiliary spaces.
These spaces will be shown later to have interesting connections to one another. We note that B p and C p are special atomic decomposition spaces in the plane, whereas the auxiliary spaces D p and E p are their dual counterparts. We have the following results.
Proof. The proof is similar to the one defined on the real line, see for instance [4], and will be omitted here for the sake of brevity. □ Remark 3. Recall that the Lipchitz space of order < < α 0 1 is defined as: Recall that the generalized Lipchitz space of order < ≤ α 0 2 is defined as: Note that for < < α 0 1, the spaces Lip α and Λ α are the same and for = The above equalities show that the spaces D p and E p are, respectively, natural extensions of Lipschitz spaces Lip α of order < < α 0 1 and generalized Lipschitz spaces Λ α of order < < α 0 2 of functions in higher dimensions.
In the sequel, we will show the consequence of this extension to higher dimensions. We will in particular connect this extension to the proper definition of Haar systems and Haar wavelets.

Relationship between B p and C p
. For a given sub-rectangle J n of J, we have that   Likewise, put It follows that i n n i n i n 1 2 3 4 0 , , With this notation, and considering the fact that C p is a linear space, we can conclude that ∈ f C p . Moreover, since the sub-rectangles L n i , and R n i , for = i 1, 2 are contained in J n , we have that Remark 4. It is important to note that the constant 4 in the theorem above is sharp which comes from the proof. As for the inclusion of C p into B p , at this point, we can only conjecture that it may be true as well.

Relationship with the Lebesgue spaces L ∞ and L p
Proof. Let ∈ ∞ g L . Then, we know that | ( )| ≤ ∥ ∥ ≔ ∥ ∥ ∞ ∞ g x y g g , L . Therefore, given ⊆ U J  Taking the infimum of ever all representations of f, we have: We observe that from the same token that C p is also a subspace of L p . The proof is similar to the one above and will not be given for the sake of brevity.

Hölder's-type inequalities
be a real number. 1. If ∈ g D p and ∈ f C p , then . The sequence F n is unbounded in general so that the Dominated Convergence Theorem will fail and thus the inequality above cannot be obtained.

Duality
be a real number. 1. The dual space ( ) * C p of C p is equivalent to D p with equivalent norms, that is, ∈ ( ) * φ B p if and only if there is a unique ∈ g D p such that The special atom space and Haar wavelets in higher dimensions  139 Likewise, the dual space ( ) * B p of B p is equivalent to E p with equivalent norms.
Proof. Fix ∈ g D p . Define a functional φ g on C p as: The linearity of the integral makes φ g a linear functional and using the Hölder's-type inequalities above, we have Now consider ∈ ( ) * φ C p . Then, there exists an absolute constant M such that Since J is a rectangle, we can define a σ-finite measure μ as follows: let ⊆ E J be a rectangle.
The latter implies that the measure μ is absolute continuous with respect to the Lebesgue measure |⋅|, which is also σ-finite. Therefore, by the Radon-Nykodym theorem, there exists a measurable function ∈ g L 1 such that The continuity of φ allows us to write it in the form In particular, for all ∈ n , we would have∥ ∥ < In other words, So dividing both sides by | | / E p 1 and taking the supremum over all ⊆ E J such that < | | < E 0 1, we have The latter inequality fails to be true once we chose . This and the inequality in A similar approach will yield the second part of the theorem. □ Remark 6. A more direct proof that ∥ ∥ ≥ ∥ ∥ φ K g D p is as follows: The result follows with = K 1.

Extension to the space
Consider the cube = J I 3 and let a sub-cube J n of J be defined, for a given integer n, real numbers a b c , , ,   We define the function ( ) B x y z , , n in the space as (see Figure 3 for an illustration) The eight cubes represent, respectively, L n i j , , for = i j , 1,2 defined above in the following way: x y x z y z L x y x z y z L x y x z y z R x y x z y z R x y x z y z R x y x z y z R x y x z y z The goal of the colors in the picture is to illustrate the fact that by restricting the definition to either plane, we will recover the definition of (⋅ ⋅) B , n in the bisphere as above, or by restricting it to either coordinate line, we will recover the original definition of the special atom. With this definition in hand, we see that the results of the previous section naturally extend to the trisphere space (more generally to the space, see Section 6). It can even be extended to the polysphere k for ≥ k 3 by observing that there will be − 2 k 1 intervals ⋅ L n, and − 2 k 1 intervals ⋅ R n, and by combining them adequately.

Discussion
The special atom space may have been understudied in the literature because of its relative simplicity. That simplicity seemingly hides deep connections to well-known spaces.
In their inception, functions in B p are given in their atomic decomposition forms. The space B p , however, has an analytic form using the following result.
This result means that B p is the real characterization (or the boundary value space) of S w in that With the definition of ( ) B I p k at hand and for an integer ≥ k 2, we have the following.
Conjecture 1. There exist weights ( ) z Φ and an holomorphic function F on k such that

Lacunary functions
In Section 6.1, we saw that the weighted special atom space has an analytic characterization as the weighted Bergman-Besov-Lipschitz space. Above, we also mentioned that B p can be extended to the bisphere or polysphere. However, to be able to define analytic functions on the bidisk and polydisk, we need to make sure that lacunary functions are properly characterized and removed. First, recall that Definition 11. A lacunary function F is an analytic function possessing the so-called Hadamard gaps, that is, This result essentially says that if we hope to extend the result of [5] to the space or to a higher dimension space, we need to discard lacunary functions. Better, we need to characterize the sub-space of lacunary functions. On the unit sphere, this was done in [8], where the space ( ) b I p of lacunary functions on I was characterized as: , is a positive weight function satisfying certain conditions. Basically, it was proved that if F is lacunary on , then The result in (6.4) relies heavily on the following theorem by A. Zygmund. Definition 13. Let In Figure 4, we show a representation of ( ) ψ x y , and ( ) ϕ x y , in the space.

9)
This system is illustrated in three dimension in Figure 5. 90 to construct f N P . We observe that even for low resolution level and relatively small number of points, we obtain a good approximation.
The special atom space and Haar wavelets in higher dimensions  149