Open Access Published by De Gruyter Open Access July 8, 2020

# The special atom space and Haar wavelets in higher dimensions

• Eddy Kwessi , Geraldo de Souza , Ngalla Djitte and Mariama Ndiaye
From the journal Demonstratio Mathematica

## 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.

MSC 2010: 42B05; 42B30; 30B50; 30E20

## 1 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].

## Definition 1

Let 0<p1 be a real number and J be an interval of . An atom is a function b defined on the interval J and satisfying

1. |b(ξ)|1|J|1/p;

2. ξkb(ξ)dξ=0, for 0k1p1, where [x] is the integer part of x.

From this definition, special atoms for p1 were introduced as:

## Definition 2

A special atom of type 1 is a function b:J such that

b(t)=1onJorb(t)=1|I|1/p{χL(t)χR(t)},

where I is an interval contained in J, L and R are the halves of I such that I=LR, and |I| is the length of I (Figure 1a).

## Definition 3

A special atom of type 2 is a function a:J such that

a(t)=1|I|1/p{χI(t)},

where I is an interval contained in J (Figure 1b).

## Definition 4

For 1p<, the special atom space is defined as:

Type 1:

Bp=f:J;f(t)=n=0αnbn(t);n=0|αn|<,

where bn(t) are special atoms of type 1. Bp is endowed with the norm

fBp=infn=0|αn|,

where the infimum is taken over all representations of f.

Type 2:

Cp=f:J;f(t)=n=0βnan(t);n=0|βn|<,

where an(t) are special atoms of type 2. Cp is endowed with the norm

fCp=infn=0|βn|,

where the infimum is taken over all representations of f.

## Remark 1

For p=1, it is worth noting that the space C1 contains all simple functions. That is, if f is a simple function with f(x)=n=0kαnχIn(x), then fC1. Also, every element in C1 is the limit of a sequence of simple functions. Indeed C1L1.

## Theorem 1

(Bp,Bp)and(Cp,Cp)are Banach spaces.

## 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 H1, 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 Λ defined on J=[ah,a+h] as

Λ=f:J:fΛ=supξJh>0f(ξ+h)+f(ξh)2f(ξ)h<.

This result led to a simple proof that the Hardy space H1 indeed contains functions whose Fourier series diverge almost everywhere by observing that the Hardy space H1 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 B1 is a Banach equivalent to the space of analytic functions F on the complex unit disc for which F(z)=12π0102πeiξ+zeiξzf(ξ)dξ, where 0102π|F(z)|dz<. This analytic characterization also led to the lacunary characterization of functions in Bp, with 1p<. 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 Lp,q. Indeed, one property of the special atom space (type 2) is that it is the atomic decomposition of the Lorentz space Lp,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 L2(2), and generally in L2(k), k2. In the last section, we provide some applications of these facts.

## 3 Extension to high dimensions

Now consider the square J=[0,1]×[0,1]. We extend the definition of the function bn(t) to the plane as follows: consider an integer n, and real numbers an,bn,hn, and kn such that hn,kn>0 and liminfnhnkn>0. Consider a sub-rectangle Jn of J as

Jn=[anhn,an+hn]×[bnkn,bn+kn].

## Definition 5

Let 1p< and consider

Ln,1=[anhn,an]×[bnkn,bn],Ln,2=[anhn,an]×[bn,bn+kn],Rn,1=[an,an+hn]×[bnkn,bn),Rn,2=[an,an+hn]×(bn,bn+kn].

Let

Ln=Ln,1Rn,2andRn=Ln,2Rn,1.

We define the function Bn(x,y) in the plane as:

(3.1)Bn(x,y)=1|Jn|1/pχRn(x,y)χLn(x,y).

## Remark 2

1. Note that the subsets Ln and Rn of Jn are nonempty and disjoint so that Bn(x,y) can be written as:

Bn(x,y)=1|Jn|1/p{χLn,2(x,y)+χRn,1(x,y)χLn,1(x,y)χRn,2(x,y)},

where || represents the Lebesgue measure or the area to be more precise.

2. We also note that by restricting Bn(x,y) to the real line, we recover the definition (Definition 2) above, using the fact that B1 is equivalent to the space of functions f(x)=n=0αnbn(x), where bn's are special atoms of type 1, see [4].

3. We observe that the role of 1/p and by extension that of p is to extend the definition above to Lp spaces and as such, |J|1/p is a normalizing constant so that BnLp=1.

Figure 2 is an illustration of Ln and Rn in the plane.

In the next definition, we will drop the index n in Ln,i and Rn,i, i=1,2 respectively, for the sake of clarity. Consider UJ, measurable such that U=LR, with L=L1R2 and R=L2R1, for some sub-rectangles L1,L2,R1,R2 of U similar to those in Figure 2. Now we can define the special atom space on the plane. Let J=[0,1]×[0,1] and let a real number 1p<.

## Definition 6

We define a special atom of Type 1 on J as:

(3.2)Bn(x,y)=1|Jn|1/p{χRn(x,y)χLn(x,y)},

where |Jn|=4hnkn. Note that the subsets Ln and Rn are nonempty and disjoints so that

Bn(x,y)=1|Jn|1/p{χLn,2(x,y)+χRn,1(x,y)χLn,1(x,y)χRn,2(x,y)}.

## Definition 7

We define the special atom of Type 2 on J as

An(x,y)=1|Jn|1/pχJn(x,y).

## Definition 8

The special atom space on J (or 2) is defined as:

Type 1:

Bp=f:J;f(x,y)=n=0αnBn(x,y);n=0|αn|<,

where Jn’s are subsets in J, αn are real numbers, Bn(x,y) are special atoms of type 1. Bp is endowed with the norm:

fBp=infn=0|αn|,

where the infimum is taken over all representations of f.

Type 2:

Cp=f:J;f(x,y)=n=0βnAn(x,y);n=0|βn|<,

where An(x,y) are special atoms of type 2 and βn are real numbers. Cp is endowed with the norm

fCp=infn=0|βn|,

where the infimum is taken over all representations of f.

Figure 1

An illustration of the special atom of types 1 and 2 for p=1. (a) Type 1, (b) Type 2.

Figure 2

An illustration of Ln and Rn in the plane.

In the next definition, U and V represent measurable subsets of J, where || is the Lebesgue measure.

## Definition 9

Let1p<. We define the following auxiliary spaces.

Dp=f:J;fL1(J),fDp=supUJ0<|U|<11|U|1/pUf(x,y)dxdy<,
Ep=f:J;fL1(J),fEp=supVJV=LR0<|V|<11|V|1/pRf(x,y)dxdyLf(x,y)dxdy<.

These spaces will be shown later to have interesting connections to one another. We note that Bp and Cp are special atomic decomposition spaces in the plane, whereas the auxiliary spaces Dp and Ep are their dual counterparts. We have the following results.

## Theorem 2

For1p<, (Cp,Cp),(Bp,Bp),(Dp,Dp), and(Ep,Ep)are all Banach spaces.

## 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 0<α<1 is defined as:

Lipα=f:J:fLipα=suph>0ξJ|f(ξ+h)f(ξ)|hα<.

Recall that the generalized Lipchitz space of order 0<α2 is defined as:

Λα=f:J:fΛα=suph>0ξJ|f(ξ+h)f(ξ+h)2f(ξ)|(2h)α<.

Note that for 0<α<1, the spaces Lipα and Λα are the same and for α=1, Λα is the Zygmund space Λ. Suppose J=[0,1]×[0,1] and let gDp such that g(x,y)=f(x) for some differentiable function f defined on [0,1]. Let U=[ξ,ξ+h]×[0,1]J. Then, |U|=h and

gDp=supUJ0<|U|<11|U|1pUg(x,y)dxdy=suph>0ξJ1h1/pξξ+hf(x)dx=suph>0ξJ|f(ξ+h)f(ξ)|h1/p=fLip1/p.

Suppose J=[0,1]×[0,1] and let gEp such that g(x,y)=f(x) for some differentiable function f defined on [0,1].

Let V=[ξh,ξ+h]×[0,1]=[ξh,ξ)×[0,1][ξ,ξ+h]×[0,1]J. Then, |V|=2h. Let R=[ξ,ξ+h]×[0,1] and L=[ξh,ξ)×[0,1]. Then,

gEp=supVJV=LR0<|V|<11|V|1pRg(x,y)dxdyLg(x,y)dxdy=suph>0ξJ1h1/pξξ+hf(x)dxξhξf(x)dx=suph>0ξJ|f(ξ+h)+f(ξh)2f(ξ)|(2h)1/p=fΛ1/p.

The above equalities show that the spaces Dp and Ep are, respectively, natural extensions of Lipschitz spaces Lipα of order 0<α<1 and generalized Lipschitz spaces Λα of order 0<α<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.

## 4 Main results

### Theorem 3

For a real number1p<, the spaceBpis continuously contained inCp. Moreover, forfBp

fCp4fBp.

### Proof

Let fBp. Then, f has the atomic decomposition f(x,y)=n=0βnBn(x,y) with n=0|βn|<. For a given sub-rectangle Jn of J, we have that

Bn(x,y)=1|Jn|1/p{χRn(x,y)χLn(x,y)}=1|Jn|1/p{χLn,2(x,y)+χRn,1(x,y)χLn,1(x,y)χRn,2(x,y)}=Ln,2Jn1/p1Ln,21/pχLn,2(x,y)+Rn,1Jn1/p1Rn,11/pχRn,1(x,y)Ln,1Jn1/p1Ln,11/pχLn,1(x,y)Rn,2Jn1/p1Rn,21/pχRn,2(x,y).

Put

A1,n(x,y)=1Ln,21/pχLn,2(x,y),A2,n(x,y)=1Rn,11/pχRn,1(x,y),A3,n(x,y)=1Ln,11/pχLn,1(x,y),A4,n(x,y)=1Rn,21/pχRn,2(x,y).

Likewise, put

K1,n=Ln,2Jn1/p,K2,n=Rn,1Jn1/p,K3,n=Ln,1Jn1/p,K4,n=Rn,2Jn1/p.

It follows that

f(x,y)=f1(x,y,β)+f2(x,y)f3(x,y)f4(x,y),wherefi(x,y)=n=0βnKi,nAi,n(x,y).

With this notation, and considering the fact that Cp is a linear space, we can conclude that fCp. Moreover, since the sub-rectangles Ln,i and Rn,i for i=1,2 are contained in Jn, we have that Ki,n1, which implies that |Ki,n||βn||βn|. Hence,

fiCp=infn=0|βn||Ki,n|infn=0|βn|=fBp.

It follows that

fCp=f1+f2f3f3Cpf1Cp+f2Cp+f3Cp+f4Cp4fBp.

### 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 Cp into Bp, at this point, we can only conjecture that it may be true as well.

### Theorem 4

Consider1p<.

• The Lebesgue spaceLis continuously contained inDpandEp. That is,

1. LDpandgDpgL.

2. LEpandgEp2gL.

• The spaceBpis continuously contained in the Lebesgue spaceLp(J). That is,BpLpandgLpgBp.

### Proof

Let gL. Then, we know that |g(x,y)|ggL. Therefore, given UJ

Ug(x,y)dxdyUgdxdy=|U|g.

Hence, multiplying both sides of the above inequality by 1/|U|1/p and taking the supremum over all UJ such that |U|1, we have:

supUJ0<|U|<11|U|1/pUg(x,y)dxdyg.

In other words, gDp and gDpgL. The proof that LEp follows along the same lines.

Now let fBp such that f(x,y)=n=0αnBn(x,y) with n=0|αn|<. Then, fLp=n=0αnBnLp. We observe that by definition, BnLp=1. Hence, given an integer N>0, we have

n=0NαnBnLpn=0N|αn|BnLp=n=0n|αn|n=0|αn|.

Using the continuity of the norm in Lp, it follows that:

fLp=n=0αnBn(x,y)Lp=limNn=0NαnBnLp=limNn=0NαnBnLpn=0|αn|.

Taking the infimum of ever all representations of f, we have:

fLpfBp.

We observe that from the same token that Cp is also a subspace of Lp. The proof is similar to the one above and will not be given for the sake of brevity.

### Theorem 5

Let1<p<be a real number.

1. IfgDpandfCp, then

Jf(x,y)g(x,y)dxdyfCpgDp.
2. IfgEpandfBp, then

Jf(x,y)g(x,y)dxdyfBpgEp.

### Proof

Let 1<p< be a real number, let fCp such that f(x,y)=n=0αnAn(x,y) with n=0|αn|<. Consider gDp. We know from the above result that both f,gL1. Since n=0|αn|<, α=supn|αn| exists. Let l=liminfn|Jn|. Let Fn(x,y)=αnAn(x,y)g(x,y). Then, for all n and (x,y)J,

|Fn(x,y)|=αn1|Jn|1/pχJn(x,y)g(x,y)αl1/p|g(x,y)|L1.

By the Dominated Convergence Theorem, we have

Jf(x,y)g(x,y)dxdy=Jn=0αnAn(x,y)g(x,y)dxdy=Jn=0αn1|Jn|1/pχJn(x,y)g(x,y)dxdy=n=0αn1|Jn|1/pJχJn(x,y)g(x,y)dxdy=n=0αn1|Jn|1/pJng(x,y)dxdy.

Taking the absolute value on both sides of the above equality, we have

Jf(x,y)g(x,y)dxdyn=0|αn|1|Jn|1/pJng(x,y)dxdy.

Taking the supremum over all subsets Jn of J such that 0<|Jn|<1, we have

Jf(x,y)g(x,y)dxdyn=0|αn|supJnJ0<|Jn|<11|Jn|1/pJng(x,y)dxdy=n=0αngDp.

To conclude, we take the infimum over all representations of f and we obtain

Jf(x,y)g(x,y)dxdyfCpgDp.

The proof of the second part of the theorem is similar by noting that in this case for all n and (x,y)J, we have

|Fn(x,y)|=βnBn(x,y)g(x,y)2αl1/p|g(x,y)|L1,

so that the Dominated Convergence Theorem can still be used.□

### Remark 5

This proof illustrates the importance of choosing sequences hn and kn for which liminfnhnkn>0. Indeed, suppose hn=kn=12n. Then, |Jn|=hnkn=1n2 and liminfnhnkn=0. In this case,

Jf(x,y)g(x,y)dxdy=Jn=0Fn(x,y)dxdy,

with Fn(x,y)=αnn2/pχJn(x,y)g(x,y). The sequence Fn is unbounded in general so that the Dominated Convergence Theorem will fail and thus the inequality above cannot be obtained.

### Theorem 6

Let1<p<be a real number.

1. The dual space(Cp)ofCpis equivalent toDpwith equivalent norms, that is,φ(Bp)if and only if there is a uniquegDpsuch that

φg(f)=Jf(x,y)g(x,y)dxdy.

Moreover,

φg(Cp)gDp.
2. Likewise, the dual space(Bp)ofBpis equivalent toEpwith equivalent norms.

### Proof

Fix gDp. Define a functional φg on Cp as: φg:Cp with

φg(f)=Jf(x,y)g(x,y)dxdy.

The linearity of the integral makes φg a linear functional and using the Hölder’s-type inequalities above, we have

|φg(f)|=Jf(x,y)g(x,y)dxdyfCpgDp.

It follows that

(4.1)φg(Cp)=supfCp=1|φg(f)|gDp.

Now consider φ(Cp). Then, there exists an absolute constant M such that

|φ(f)|MfCp,fCp.

Since J is a rectangle, we can define a σ-finite measure μ as follows: let EJ be a rectangle. Put μ(E)=φ(χE). Since χE=|E|1/p1|E|1/pχE, it follows that χECp with χECp=|E|1/p. Moreover, |μ(E)|=|φ(χE)|M|E|1/p. 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 gL1 such that

μ(E)=Eg(x,y)dxdy.

It remains to show that gDp and that there exists a constant K>0 such that φ(Cp)KgDp.

We note that

|μ(E)|=|φ(χE)|=Eg(x,y)dxdyM|E|1/p.

That is,

1|E|1/pEg(x,y)dxdyM.

Taking the supremum over all rectangles EJ such that 0<|E|<1, we have

supEJ0<|E|<11|E|1/pEg(x,y)dxdyM<.

This means simply by definition that gDp. Now let us note that

φ(χE)=Eg(x,y)dxdy.

By linearity of φ, we have that

φ1|E|1/pχE=J1|E|1/pχE(x,y)g(x,y)dxdy.

The continuity of φ allows us to write it in the form

φ(f)=φg(f)=Jf(x,y)g(x,y)dxdy.

Now suppose that for all constants K>0, φg(Cp)<KgDp. In particular, for all n, we would have φg(Cp)<1ngDp. Now consider f0=1|E|1/pχE. We observe that f0Cp with f0Cp=1. So, in particular, we have that

|E|1/p|φg(f0)|=|φg(χE)|supfCp=1|φg(f)|=φg(Cp)<1ngDp.

In other words,

n|φg(χE)|=nEg(x,y)dxdy<gDp.

So dividing both sides by |E|1/p and taking the supremum over all EJ such that 0<|E|<1, we have

ngDp<|E|1/pgDp.

The latter inequality fails to be true once we chose n=[|E|1/p]+1. Therefore, there must exist K>0 such that φg(Cp)KgDp. This and the inequality in (4.1) prove that φg(Cp)gDp. A similar approach will yield the second part of the theorem.□

### Remark 6

A more direct proof that φKgDp is as follows:

φg(Cp)=supfCp=1|φg(f)||φg(f0)|=1|E|1/pEg(x,y)dxdygDp.

The result follows with K=1.

## 5 Extension to the space

Consider the cube J=I3 and let a sub-cube Jn of J be defined, for a given integer n, real numbers an,bn,cn,hn,kn, and mn with hn,kn,mn>0 as

Jn=[anhn,an+hn]×[bnkn,bn+kn]×[cnmn,cn+mn].

## Definition 10

Let

Ln,1,1=[anhn,an]×[bnkn,bn]×[cnmn,cn),Ln,1,2=[anhn,an]×[bnkn,bn]×[cn,cn+mn],Ln,2,1=[anhn,an]×[bn,bn+kn]×[cnmn,cn),Ln,2,2=[anhn,an]×[bn,bn+kn]×[cn,cn+mn],Rn,1,1=[an,an+hn]×[bnkn,bn)×[cnmn,cn),Rn,1,2=[an,an+hn]×[bnkn,bn)×[cn,cn+mn],Rn,2,1=[an,an+hn]×(bn,bn+kn]×[cnmn,cn),Rn,2,2=[an,an+hn]×(bn,bn+kn]×[cn,cn+mn].

Let

Ln=Ln,1,1Ln,1,2Rn,2,1Rn,2,2andRn=Ln,2,1Ln,2,2Rn,1,1Rn,1,2.

We define the function Bn(x,y,z) in the space as (see Figure 3 for an illustration)

(5.1)Bn(x,y,z)=1|Jn|1/p{χRn(x,y,z)χLn(x,y,z)}.

The eight cubes represent, respectively, Ln,i,j for i,j=1,2 defined above in the following way:

Ln,1,1xy-yellow,xz-green,yz-blue,Ln,1,2xy-yellow,xz-magenta,yz-brown,Ln,2,1xy-red,xz-green,yz-brown,Ln,2,2xy-red,xz-magenta,yz-blue,Rn,1,1xy-red,xz-magenta,yz-blue,Rn,1,2xy-red,xz-green,yz-brown,Rn,2,1xy-yellow,xz-magenta,yz-brown,Rn,2,2xy-yellow,xz-green,yz-blue.

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 Bn(,) 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 Tk for k3 by observing that there will be 2k1 intervals Ln, and 2k1 intervals Rn, and by combining them adequately.

## 6 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.

### 6.1 Relationship with the weighted Bergman spaces

Let D={z:|z|<1} be the unit disk and T={zC:|z|=1} be the unit circle. Given k>1, the polydisk is defined as Dk={(z1,z2,,zk)k:|zi|<1,1ik} and the polysphere is given as Tk={(z1,z2,,zk)k:|zi|=1,1ik}. Also, in this section, I=[0,1].

In their inception, functions in Bp are given in their atomic decomposition forms. The space Bp, however, has an analytic form using the following result.

### Theorem 7

[5] LetfBpandw(t)=t1/p. Define an analytic functionF(z)=12πππeitzeit+zf(t)dt. LetA(D)be the space of analytic functions onD. Let

Sw=FA(D):FSw=1010F(reiθ)w(1r)1rdθdr<.
Then,Bpis continuously contained inSwwith
(6.1)fBpFSw.

This result means that Bp is the real characterization (or the boundary value space) of Sw in that

• If FSw, then fBp where f(θ)=limr1ReF(reiθ).

• If fBp, then FSw where F(z)=12πππeitzeit+zf(t)dt.

Now considering a weight w satisfying certain conditions, we can replace |Jn| in the definition of Bp by w(Jn) to obtain a weighted special atoms space Bw. It was shown in [5] that the aforementioned theorem can be extended to the weighted case.

Now, recall that weighted Bergman-Besov-Lipschitz spaces BBLw are defined for a weight function w (defined above) as:

BBLw=f:I=[0,1]:02π02π|f(x)f(y)||xy|w(xy)dxdy<.

We note that this space is just the weighted version of the generalized Hölder spaces Λ11p,1,1 with weight w(t)=t1/p defined as:

Λ11p,1,1=f:I:02π02π|f(x)f(y)||xy|21/pdxdy<.

It was shown in [6] that BwBLLw with equivalent norms, meaning that Bw is the atomic decomposition of BLLw.

From now on, we will write Bp(I) as Bp. Let q such that 1p+1q=1. Let z=x+iy=reiθD. Then, dθdr=2rdxdy=2|z|dxdy=2π|z|dA(z), where dA(z)=dxdyπ. Now define the function Φ(z)=|z|(1|z|)1p1=|z|w(1|z|)1|z| on D.

Then, (6.1) can be written as

(6.2)fBp(I)D|F(z)|Φ(z)dA(z).

Noting that ΦL1(D), the right-hand side of (6.2) means that FAΦ1(D)=LΦ1(D)(D), the weighted Bergman space with weight Φ, where (D) is the space of holomorphic functions on D. We note that the standard Bergman weights are given as Φ0(z)=(α+1)(1|z|)α for α>1. With the choice of p1, Φ(z) can be transformed into a standard weight since in that case α=1q>1. When k2, we note that z=(z1,z2,,zk) is a vector in k, with zj=rjejiθj for 1jk, which can also be written as zj=xj+iyj. Then, for a differentiable function F on k, we put F(z)=(f1(z),f2(z),,fk(z)), where fj(z)=F(z)zj,1jk. Note that when k=1, this becomes f1(z)=F(z). For k2, we have |F(z)|F2=j=1k|fj(z)|2.

With the definition of Bp(Ik) at hand and for an integer k2, we have the following.

### Conjecture 1

There exist weightsΦ(z)and an holomorphic function F onDksuch that

(6.3)fBp(Ik)Dk|F(z)|Φ(z)dA(z),
forz=(z1,z2,,zk)Dkwithzj=xj+iyjanddA(z)=1(π)kj=1kdxjdyj.

### 6.2 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 Bp 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,

F(z)=k=1nkakznksuchthatλ=infknk+1nk>1.

Let us mention the Ostrowski-Hadamard gap theorem, see for example [7].

### Theorem 8

(Ostrowski-Hadamard) Suppose F is a lacunary function with radius of convergence 1. Then, f cannot be analytically continued from the open discDto any larger open set, including even a single point of the boundaryTofD.

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 bp(I) of lacunary functions on I was characterized as:

bp(D)=f:DZ:f(z)=n=0anzn,n=02nK(n,p)kIn|ak|212<,K(n,p)>0,

where K(n,p) is a positive weight function satisfying certain conditions. Basically, it was proved that if F is lacunary on D, then

(6.4)FBp(I)Fbp(D).

The result in (6.4) relies heavily on the following theorem by A. Zygmund.

### Theorem 9

[9] Letf(z)=k=1akznkbe a lacunary function defined on the unit diskD. Then, there is an absolute constant c independent of f such that

k=1|ak|212cfL1(D).

Paley extended the result of A. Zygmund in [9] to the polydisk of H1(Dm).

### Definition 12

A function f(z)=kmak1akmz1nk1zmnkm defined on the polydisk Dm will be called lacunary if

λ=minλ1,λ2,,λm>1,whereλj=infkjnkj+1nkj>1, 1jm.

### Theorem 10

(Paley 1960) Letf(z)=kmak1akmz1nk1zmnkmbe a lacunary function defined on the polydiskDm. Then, there is an absolute constant c independent of f such that

km|ak1akm|212cfH1(Dm).

Now, with the definition of the special atom space in high dimension, we claim the following.

### Conjecture 2

There exists a weight functionK(n,p)characterizingbp(Dk)such that if F is lacunary on the polydiskDk,

FBp(Ik)Fbp(Dk).

The result in (6.4) also means that if Bp(I) is the space of functions defined on D having an analytic continuation on D, then Bp(I)=H1(D)\bp(D), where H1(D) is the Hardy’s space consisting of functions such that

fH1(D)=sup0<r<1T|f(reiθ)|dθ<.

Now that we have an extension of the special atom space to higher dimensions, then the same endeavor could be carried out in higher dimensions:

### Conjecture 3

IfBp(Ik)is the space of functions defined onDkhaving an analytic continuation onDk, thenBp(Ik)bp(Dk)=H1(Dk).

### 6.3 Relationship with Haar wavelets

We will define a Haar wavelet based on the special atom space in high dimensions given above and we show that it is an extension of the classical Haar wavelet in L2(I).

### Definition 13

Let

(6.5)ψ(x,y)=χR(x,y)χL(x,y),andϕ(x,y)=χJ(x,y),

where

R=0,12×0,1212,1×12,1,L=12,1×0,120,12×12,1,J=[0,1]×[0,1].

In Figure 4, we show a representation of ψ(x,y) and ϕ(x,y) in the space.

#### 6.3.1 Haar functions and Haar systems

For k=0,1,,2n1 and n and for j=0,1,,2m1 and m, let

(6.6)Jn,km,j=k2n,k+12n×j2m,j+12m,
(6.7)Ln,kh=k2n,2k+12n+1,Rn,kh=2k+12n+1,k+12n,Lm,jv=j2m,2j+12m+1,Rm,jv=2j+12m+1,j+12m.

Now define

(6.8)Ln,m,j,k=Ln,kh×Rm,jvLm,jv×Rn,kh,Rn,m,j,k=Lm,jv×Ln,khRm,jv×Rn,kh.

Now put

hn,km,j(x,y)=χLn,kh×Rm,jv(x,y)+χRn,kh×Lm,jv(x,y)χLn,kh×Lm,jv(x,y)χRn,kh×Rm,jv(x,y).

We normalize the later function on L2(J) to obtain the Haar System

(6.9)Hn,km,j(x,y)=2n+m2hn,km,j(x,y).

This system is illustrated in three dimension in Figure 5.

#### Theorem 11

The Haar system defined in (6.9) can be generated by a single functionψdefined in (6.5) as

Hn,km,j(x,y)=2n+m2ψ(2nxk,2myj).
Moreover, the family{Hn,km,j}is an orthonormal system inL2(J).

#### Proof

The proof can easily be obtained by noticing that by construction, if we project Hn,km,j to the x-axis or to the y-axis, we obtain an orthonormal basis of L2[0,1].

#### Definition 14

Let J=[0,1]×[0,1]. A wavelet on J is a function ψL2(J) such that for integers m,n,k,j, the family

Hn,km,j=2n+m2ψ(2nxk,2myj)

is an orthonormal basis in L2(J). A similar definition applies to L2(J).

#### Remark 7

• We observe that by restricting (or projecting) Hn,km,j(x,y) to the real line, we will obtain the Haar function or the special atom.

• For J=I×I×I or 3, we have the Haar system defined similarly and generated by a single function ψ as

Hn,km,j,l,q(x,y,z)=2n+m+l2ψ(2nxk,2myj,2lzq).
• In general, we can extend it to d for d2.

• The relationship between the Haar functions and the special atoms in higher dimension is very similar to the one in one-dimension.

Indeed, the special atom defined on the dyadic interval can be written in terms of the Haar function using Definition 5.1 with the notation

Bn,km,j(x,y)=1Jn,km,jχRn,m,k,j(x,y)χLn,m,k,j(x,y),

where Jn,km,j is the dyadic interval defined in (6.6) and Ln,m,k,j,Rn,m,k,j are defined in (6.8). Then,

Jn,km,j=12n12m=12n+msothat1Jn,km,j=2n+m.

Therefore,

Bn,km,j(x,y)=2n+m22n+m2χRn,m,k,j(x,y)χLn,m,k,j(x,y)=2n+m2hn,km,j(x,y).

Ultimately, the point of this discussion is to show what could be investigated using the extension of the special atom space proposed in this article.

Figure 3

To understand the picture, there are six colors: red and yellow on the xy-plane, green and magenta on the xz-plane, and brown and blue on the yz-plane. Each cube has two faces with the same color, by projection onto that plane.

Figure 4

An illustration of ψ(x,y) and ϕ(x,y).

Figure 5

Representation of Hn,km,j for n=m=1 and k=j=0 generated using a grid of 300 × 300 points over the rectangle 12,45×12,45. We observe that by projecting onto the xy-space, we obtain the sets Ln and Rn in Definition 5.

## 7 Applications

In this section, we show how to use the special atom above to estimate functions in the plane and space (Figures 6–9).

Figure 6

In the figures, we used N=17;P=90 to construct fNP. We observe that even for low resolution level and relatively small number of points, we obtain a good approximation.

Figure 7

(a) is a representation of f(x,y)=sin(πx2)sin(πy2), using a 12×12 grid of points. (b) is an estimate of f, using fNMPQ, for N,M=4;P=Q=12.

Figure 8

(a) is a representation of f(x,y)=sin(πx2)2sin(πy2)2, using a 12×12 grid of points. (b) is an estimate of f, using fNMPQ, for N,M=4;P=Q=12.

Figure 9

(a) is a representation of f(x,y)=12πsin(πx)2sin(πy)2, using a 12×12 grid of points. (b) is an estimate of f, using fNMPQ, for N,M=4;P=Q=12.

### 7.1 Applications in the plane

Let fL2(J). Then,

f(x)=n=0k=02n12n2αn,khn,k(x).

We know that αn,k=f,hn,k=Jf(x)hn,k(x)dx. Therefore, a consistent estimator of αn,k, for a fixed integer P is given as:

αn,k(P)=1Pi=1Pf(xi)hm,k(xi).

This also means that αn,k(P) is a Riemann sum of αn,k. In addition, fix an integer N (resolution level). An estimator of f is the sequence of functions

fNP(x)=n=0Nk=02n12n2αn,k(P)hn,k(x).

By construction, we have fNPfuniformlyasN,P.

### 7.2 Applications in the space

Let fL2(J). Then, since {hn,km,j} is an orthonormal basis in L2(J), we have

f(x,y)=n=0k=02n1m=0j=02m12n+m2αn,km,jhn,km,j(x,y).

We know that

αn,km,j=f,hn,km,j=Jf(x,y)hn,km,j(x,y)dxdy.

Therefore, a consistent estimator of αn,km,j, for fixed integers P and Q is given as:

αn,km,j(P,Q)=1PQi=1Pl=1Qf(xi,yl)hn,km,j(xi,yl).

In addition, fix two integers N,M (resolution levels). An estimator of f is the sequence of functions

fNMPQ(x,y)=n=0Nk=02n1m=0Mj=02m12n+m2αn,km,j(P,Q)hn,km,j(x,y).

By construction, we have

fNMPQfuniformlyasN,M,Q,P.

In the example below, we show that even for small values of N,M,P,Q the estimation of f using fNMPQ is quite good.

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