new classes of weighted holder-zygmund spaces¨ and the...

Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2012, Article ID 815475, 18 pagesdoi:10.1155/2012/815475

Research ArticleNew Classes of Weighted Hölder-Zygmund Spacesand the Wavelet Transform

Stevan Pilipović,1 Dušan Rakić,2 and Jasson Vindas3

1 Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4,21000 Novi Sad, Serbia

2 Faculty of Technology, University of Novi Sad, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia3 Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000 Gent, Belgium

Correspondence should be addressed to Dušan Rakić, [email protected]

Received 14 November 2011; Accepted 27 May 2012

Academic Editor: Hans G. Feichtinger

Copyright q 2012 Stevan Pilipović et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inversewavelet transforms on the spaces S0�Rn� and S�Hn�1�. We then introduce and study a new classof weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. Theanalysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paleypairs.

1. Introduction

The purpose of this article is twofold. The main one is to define and analyze a new classof weighted Hölder-Zygmund spaces via the wavelet transform �1–3�. It is well known�1, 4–6� that the wavelet transforms of elements of the classical Zygmund space Cα∗ �R

n�satisfy the size estimate |Wψf�x, y�| ≤ Cyα, which, plus a side condition, essentiallycharacterizes the space itself. We will replace the regularity measurement yα by weightsfrom the interesting class of regularly varying functions �7, 8�. Familiar functions such asyα, yα| logy|β, yα| log | logy||β, . . ., are regularly varying.

The continuity of the wavelet transform and its left inverse on test function spaces �9�play a very important role when analyzingmany function and distribution spaces �1�, such asthe ones introduced in this paper. Our second aim is to provide a new proof of the continuitytheorem, originally obtained in �9�, for these transforms on the function spaces S0�Rn� andS�Hn�1�. Our approach to the proof is completely elementary and substantially simplifies themuch longer original proof from �9� �see also �1, Chapter 1��.

2 Journal of Function Spaces and Applications

The definition of our weighted Zygmund spaces is based on the useful conceptof �generalized� Littlewood-Paley pairs, introduced in Section 4.1, which generalizes thefamiliar notion of �continuous� Littlewood-Paley decomposition of the unity �5�. In addition,an important tool in our analysis is the use of pointwise weak regularity properties vector-valued distributions and their �tauberian� characterizations in terms of the wavelet transform�10, 11�. Even in the classical case Cα∗ �R

n�, our analysis provides a new approach to the studyof Hölder-Zygmund spaces. It is then very likely that this kind of arguments might also beapplied to study other types of smooth spaces, such as Besov-type spaces.

Our new classes of spacesCα,L�Rn� andCα,L∗ �Rn�, the L-Hölder and L-Zygmund spacesof exponent α that will be introduced in Section 5, contribute to the analysis of local regularityof functions or distributions by refining the regularity scale provided by the classical Hölder-Zygmund spaces. In fact, as explained in Section 5, they satisfy the following useful inclusionrelations:

Cβ�Rn� ⊂ Cα,L�Rn� ⊂ Cγ�Rn�, when 0 < γ < α < β,

Cβ∗�Rn� ⊂ Cα,L∗ �Rn� ⊂ Cγ∗�Rn�, if γ < α < β.

�1.1�

Situations in which these kinds of refinements are essential often occur in the literature, andthey have already shown to be meaningful in applications. The particular instance L�y� �| logy|β has been extensively studied �see, e.g., �1, page 276��. Our analysis will treat moregeneral weights, specifically, the important case when L is a slowly varying function �7, 8�.

The paper is organized as follows.We review in Section 2 basic facts about test functionspaces, the wavelet transform, and its left-inverse, namely, the wavelet synthesis operator. InSection 3, we will provide the announced new proof of the continuity theorem for the waveletand wavelet synthesis transforms when acting on test function spaces. We then explain inSection 4 some useful concepts that will be applied to the analysis of our weighted versionsof theHölder-Zygmund spaces; in particular, wewill discuss there the notion of �generalized�Littlewood-Paley pairs and some results concerning pointwise weak regularity of vector-valued distributions. Finally, we give the definition and study relevant properties of the newclass of Hölder-Zygmund spaces in Section 5.

2. Notation and Notions

We denote by Hn�1 � Rn × R� the upper half space. If x ∈ Rn and m ∈ Nn, then |x| denotesthe euclidean norm, xm � xm11 · · ·xmnn , ∂m � ∂mx � ∂m1x1 · · · ∂mnxn , m! � m1!m2! · · ·mn! and |m| �m1 � · · · �mn. If the jth coordinate of m is one and the others vanish, we then write ∂j � ∂mx .The set B�0, r� is the euclidean ball in Rn of radius r. In the sequel, we use C and C′ to denotepositive constants which may be different in various occurrences.

2.1. Function and Distribution Spaces

The well-known �12� Schwartz space of rapidly decreasing smooth test functions is denotedby S�Rn�. We will fix constants in the Fourier transform as ϕ̂�ξ� � ∫

Rnϕ�t�e−iξ·tdt. The mo-

ments of ϕ ∈ S�Rn� are denoted by μm�ϕ� �∫

Rntmϕ�t�dt,m ∈ Nn.

Following �1�, the space of highly time-frequency localized functionsS0�Rn� is definedas S0�Rn� � {ϕ ∈ S�Rn� : μm�ϕ� � 0, for all m ∈ Nn}; it is provided with the relative topology

Journal of Function Spaces and Applications 3

inhered from S�Rn�. In �1�, the topology of S0�Rn� is introduced in an apparently differentway; however, both approaches are equivalent in view of the open mapping theorem.Observe that S0�Rn� is a closed subspace of S�Rn� and that ϕ ∈ S0�Rn� if and only if∂mϕ̂�0� � 0, for all m ∈ Nn. It is important to point out that S0�Rn� is also well known inthe literature as the Lizorkin space of test functions �cf. �13��.

The space S�Hn�1� of highly localized functions on the half space �1� consists of thoseΦ ∈ C∞�Hn�1� for which

ρ0l,k,ν,m�Φ� � sup�x,y�∈Hn�1

(

yl �1yl

)

(

1 � |x|2)k/2∣

∣

∣∂νy∂mx Φ(

x, y)

∣

∣

∣


2.3. Wavelet Synthesis Operator

Let η ∈ S0�Rn�. The wavelet synthesis transform of Φ ∈ S�Hn�1� with respect to the waveletη is defined as

MηΦ�t� �∫∞

0

∫

Rn

Φ(

x, y) 1ynη

(

t − xy

)

dxdyy

, t ∈ Rn. �2.4�

Observe that the operator Mη may be extended to act on the space S′�Hn�1� via dualityarguments, see �1� for details �cf. �10� for the vector-valued case�. In this paper we restrictour attention to its action on the test function space S�Hn�1�.

The importance of the wavelet synthesis operator lies in fact that it can be used toconstruct a left inverse for the wavelet transform, whenever the wavelet possesses nicereconstruction properties. Indeed, assume that ψ ∈ S0�Rn� admits a reconstruction waveletη ∈ S0�Rn�. More precisely, it means that the constant

cψ,η � cψ,η�ω� �∫∞

0ψ̂�rω�η̂�rω�

drr, ω ∈ Sn−1 �2.5�

is different from zero and independent of the directionω. Then, a straightforward calculation�1� shows that

IdS0�Rn� �1cψ,η

MηWψ. �2.6�

It is worth pointing out that �2.6� is also valid �1, 10� when Wψ and Mη act on the spacesS′0�Rn� and S′�Hn�1�, respectively.

Furthermore, it is very important to emphasize that a wavelet ψ admits a reconstruc-tion wavelet η if and only if it is nondegenerate in the sense of the following definition �10�.

Definition 2.1. A test function ϕ ∈ S�Rn� is said to be nondegenerate if for any ω ∈ Sn−1 thefunction of one variable Rω�r� � ϕ̂�rω� ∈ C∞�0,∞� is not identically zero, that is, suppRω /� ∅,for each ω ∈ Sn−1.

3. The Wavelet Transform of Test Functions

The wavelet and wavelet synthesis transforms induce the following bilinear mappings:

W : (ψ, ϕ) −→ Wψϕ, M :(

η,Φ) −→ MηΦ. �3.1�

Our first main result is a new proof of the continuity theorem for these two bilinear mappingswhen acting on test function spaces. Such a result was originally obtained by Holschneider�1, 9�. Our proof is elementary and significantly simpler than the one given in �1�.


Theorem 3.1. The two bilinear mappings

�i� W : S0�Rn� × S0�Rn� → S�Hn�1�,�ii� M : S0�Rn� × S�Hn�1� → S0�Rn�

are continuous.

Proof. Continuity of the Wavelet Mapping. We will prove that for arbitrary l, k, ν ∈ N, m ∈ Nn,there existN ∈ N and C > 0 such that

ρ0l,k,ν,m(Wψϕ

) ≤ C∑

i′,|j ′ |,i,|j|≤Nρi′,j ′(

ψ)

ρi,j(

ϕ)

. �3.2�

We begin by making some reductions. Observe that, for constants cj which do not depend onϕ and ψ,

∂νy∂mx Wψϕ

(

x, y)

� ∂νy∂mx

∫

Rn

ϕ(

yt � x)

ψ�t�dt

�∑

|j|≤νcj

∫

Rn

∂m�jϕ(

yt � x)

tjψ�t�dt

�∑

|j|≤νcjWψjϕm�j

(

x, y)

,

�3.3�

where ψj�t� � tjψ�t� ∈ S0�Rn� and ϕm�j � ∂m�jϕ ∈ S0�Rn�. It is therefore enough to show �3.2�for ν � 0 and m � 0. Next, we may assume that k is even. We then have, for constants cr,sindependent of ψ and ϕ,

(

1 � |x|2)k/2Wψϕ

(

x, y)

�1

�2π�n〈

�1 −Δξ�k/2eiξ·x, ϕ̂�ξ�ψ̂(

yξ)

〉

�1

�2π�n∑

|r|�|s|≤kcr,sy

|s|∫

Rn

eiξ·x∂rϕ̂�ξ�∂sξψ̂(

yξ)

dξ

�1

�2π�n∑

|r|�|s|≤kcr,sy

|s|Wψsϕr(

x, y)

,

�3.4�

where ϕr�t� � �−it�rϕ�t� and ψs�t� � �it�sψ�t�. Thus, it clearly suffices to establish �3.2� fork � ν � 0 andm � 0. We may also assume that l ≥ n.

We first estimate the term yl|Wψϕ�x, y�|. Since ∂jϕ̂�0� � 0 for every j ∈ Nn, we canapply the Taylor formula to obtain

ϕ̂�ξ� �∑

|j|�l−n∂jϕ̂(

zξ)

j!ξj , for some zξ in the line segment �0, ξ�. �3.5�


Hence,

yl∣

∣Wψϕ(

x, y)∣

∣ �yl

�2π�n∣

∣

∣

〈

eiξ·xϕ̂�ξ�, ψ̂(

yξ)

〉∣

∣

∣ ≤ 1�2π�n

∫

Rn

∣

∣ϕ̂�ξ�∣

∣yl∣

∣ψ̂(

yξ)∣

∣dξ

≤ 1�2π�n

∑

|j|�l−n1j!

∫

Rn

∣

∣

∣∂jϕ̂(

zξ)

∣

∣

∣yn∣

∣

∣

(

yξ)jψ̂(

yξ)

∣

∣

∣dξ

≤ 1�2π�n

∑

|j|�l−nρ0,j(

ϕ̂)

j!

∫

Rn

∣

∣

∣ξj ψ̂�ξ�∣

∣

∣dξ

≤ 1�2π�n

∑

|j|�l−nρ0,j(

ϕ̂)

ρl�n,0(

ψ̂)

j!

∫

Rn

dξ(

1 � |ξ|2)n

≤ Cρl�n,0(

ϕ)∑

|j|≤2l�2nρ2n,j

(

ψ)

, ∀ (x, y) ∈ Hn�1.

�3.6�

It remains to estimate y−l|Wψϕ�x, y�|. We will now use the fact that all the moments ofψ vanish. If we apply the Taylor formula, we have, for some zt � z�t, x, y� in the line segment�x, yt�,

1yl∣

∣Wψϕ(

x, y)∣

∣ �1yl

∣

∣

∣

∣

∫

Rn

ϕ(

yt � x)

ψ�t�dt∣

∣

∣

∣

�1yl

∣

∣

∣

∣

∣

∣

∫

Rn

ψ�t�

⎛

⎝

∑

|j| 0 such that

ρk,κ(MηΦ

) ≤ C∑

k1,k2,l,|m|≤Ni,|j|≤N

ρi,j(

η)

ρ0k1,k2,l,m�Φ�.�3.8�


Since ∂κtMηΦ � Mη∂κxΦ, it is enough to prove �3.8� for κ � 0. We denote below ̂Φ thepartial Fourier transform of Φ with respect to the space coordinate, that is, ̂Φ�ξ, y� �∫

RnΦ�x, y�e−iξ·xdx. We may assume that k is even. We then have

�1 � |t|2�k/2∣∣MηΦ�t�∣

∣ � �1 � |t|2�k/2∣

∣

∣

∣

∣

∫

R�

∫

Rn

Φ(

t − x, y) 1ynη

(

x

y

)

dxdyy

∣

∣

∣

∣

∣

��1 � |t|2�k/2

�2π�n

∣

∣

∣

∣

∣

∣

∫

R�

∫

Rn

�1 −Δξ�k/2eitξ

�1 � |t|2�k/2̂Φ(

ξ, y)

η̂(

yξ)dξ dy

y

∣

∣

∣

∣

∣

∣

≤ 1�2π�n

∫

R�

∫

Rn

∣

∣

∣�1 −Δξ�k/2(

̂Φ(

ξ, y)

η̂(

yξ)

)∣

∣

∣

dξ dyy

≤ 1�2π�n

∑

|r|�|s|≤kcr,s

∫

R�

∫

Rn

∣

∣

∣∂rξ̂Φ(

ξ, y)

∣

∣

∣y|s|−1∣

∣∂sη̂(

yξ)∣

∣dξ dy

≤ 1�2π�n

∑

|r|�|s|≤kcr,sρ0,s

(

η̂)

∫

R�

∫

Rn

y|s|−1∣

∣

∣∂rξ̂Φ(

ξ, y)

∣

∣

∣dξ dy

≤ C′∑

|r|�|s|≤kρ0,s(

η̂)

(

ρ0|s|−1,2n,0,r(

̂Φ)

� ρ0|s|�1,2n,0,r(

̂Φ))

∫

R�

∫

Rn

dξ

�1 � |ξ|2�ndy

1 � y2

≤ C∑

|r|�|s|≤k|j|≤2n

ρ|s|�2n,0(

η)

(

ρ0|s|−1,|r|�2n,0,j�Φ� � ρ0|s|�1,|r|�2n,0,j�Φ�

)

.

�3.9�

This completes the proof.

Remark 3.2. It follows from the proof of the continuity of M that we can extend the bilinearmapping M : �η,Φ� → MηΦ to act on

M : S�Rn� × S(

Hn�1)

−→ S�Rn�, �3.10�

and it is still continuous.

4. Further Notions

Our next task is to define and study the properties of a new class of weighted Hölder-Zygmund spaces. We postpone that for Section 5. In this section we collect some usefulconcepts that will play an important role in the next section.


4.1. Generalized Littlewood-Paley Pairs

In our definition of weighted Zygmund spaces, we will employ a generalized Littlewood-Paley pair �16�. They generalize those occurring in familiar �continuous� Littlewood-Paleydecompositions of the unity �cf. Example 4.3 below�.

Let us start by introducing the index of nondegenerateness of wavelets, as defined in�10�. Even if a wavelet is nondegenerate, in the sense of Definition 2.1, there may be a ball onwhich its Fourier transform “degenerates.” We measure in the next definition how big thatball is.

Definition 4.1. Let ψ ∈ S�Rn� be a nondegenerate wavelet. Its index of nondegenerateness isthe �finite� number

τ � inf{

r ∈ R� : suppRω ∩ �0, r�/� ∅, ∀ ω ∈ Sn−1}

, �4.1�

where Rω are the functions of one variable Rω�r� � ψ̂�rω�.

If we only know values of Wψf�x, y� at scale y < 1, then the wavelet transform canbe blind when analyzing certain distributions �cf. �10, Section 7.2��. The idea behind theintroduction of Littlewood-Paley pairs is to have an alternative way for recovering such apossible lost of information by employing additional data with respect to another function φ�cf. �16��.

Definition 4.2. Let α ∈ R, φ ∈ S�Rn�. Let ψ ∈ S�Rn� be a nondegenerate wavelet with theindex of nondegenerateness τ . The pair �φ, ψ� is said to be a Littlewood-Paley pair �LP-pair�of order α if ̂φ�ξ�/� 0 for |ξ| ≤ τ and μm�ψ� � 0 for all multi-indexm ∈ Nn with |m| ≤ �α�.

Example 4.3. Let φ ∈ S�Rn� be a radial function such that ̂φ is nonnegative, ̂φ�ξ� � 1 for|ξ| < 1/2 and ̂φ�ξ� � 0 for |ξ| > 1. Set ψ̂�ξ� � −ξ · ∇ϕ̂�ξ�. The pair �φ, ψ� is then clearly an LP-pair of order∞. Observe that this well-known pair is the one used in the so-called Littlewood-Paley decompositions of the unity and plays a crucial role in the study of various functionspaces, such as the classical Zygmund space Cα∗ �R

n� �cf., e.g., �5��.

We pointed out above that LP-pairs enjoy powerful reconstruction properties. Let usmake this more precise.

Proposition 4.4. Let �φ, ψ� be an LP-pair, the wavelet ψ having index of nondegenerateness τ andr > τ being a number such that ̂φ�ξ�/� 0 for |ξ| < r. Pick any σ in between τ and r. If η ∈ S0�Rn� is areconstruction wavelet for ψ whose Fourier transform has support in B�0, σ� and ϕ ∈ D�Rn� is suchthat ϕ�ξ� � 1 for ξ ∈ B�0, σ� and suppϕ ⊂ B�0, r�, then, for all f ∈ S′�Rn� and θ ∈ S�Rn�

〈

f, θ〉

�∫

Rn

(

f ∗ φ)�t�θ1�t�dt � 1cψ,η

∫1

0

∫

Rn

Wψf(

x, y)Wηθ2

(

x, y)dxdy

y, �4.2�

where ̂θ1�ξ� � ̂θ�ξ�ϕ�ξ�/ ̂φ�−ξ� and ̂θ2�ξ� � ̂θ�ξ��1 − ϕ�ξ��.


Proof. Observe that

〈

f ∗ φ, θ1〉

� �2π�−n〈 ̂f�ξ�, ̂θ�−ξ�ϕ�−ξ�〉. �4.3�

It is therefore enough to assume that θ1 � 0 so that θ � θ2. Our assumption over η is thatη ∈ S0�Rn�, supp η̂ ⊂ B�0, σ�, and

cψ,η �∫∞

0ψ̂�rω�η̂�rω�

drr/� 0 �4.4�

does not depend on the directionω. We remark that such a reconstruction wavelet can alwaysbe found �see the proof of �10, Theorem 7.7��. Therefore, Wηθ�x, y� � 0 for all �x, y� ∈ Rn ×�1,∞�. Exactly as in �1, page 66�, the usual calculation shows that

θ�t� �1cψ,η

Mψ(Wηθ

)

�t� �1cψ,η

∫1

0

∫

Rn

ψ

(

t − xy

)

Wηθ(

x, y)dxdy

y. �4.5�

Furthermore, sinceWηθ ∈ S�Hn�1� �cf. Theorem 3.1�, the last integral can be expressed as thelimit in S�Rn� of Riemann sums. That justifies the exchange of dual pairing and integral in

〈

f, θ〉

�

〈

f�t�,1cψ,η

∫1

0

∫

Rn

ψ

(

t − xy

)

Wηθ(

x, y)dxdy

y

〉

�1cψ,η

∫1

0

∫

Rn

Wψf(

x, y)Wηθ

(

x, y)dxdy

y.

�4.6�

4.2. Slowly Varying Functions

The weights in our weighted versions of Hölder-Zygmund spaces will be taken from theclass of Karamata regularly varying functions. Such functions have been very much studiedand have numerous applications in diverse areas of mathematics. We refer to �7, 8� for theirproperties. Let us recall that a positive measurable function L is called slowly varying �at theorigin� if it is asymptotically invariant under rescaling, that is,

limε→ 0�

L�aε�L�ε�

� 1, for each a > 0. �4.7�

Familiar functions such as 1, | log ε|β, | log | log ε||β, . . ., are slowly varying. Regularly varyingfunctions are then those that can be written as εαL�ε�, where L is slowly varying and α ∈ R.


4.3. Weak Asymptotics

Wewill use some notions from the theory of asymptotics of generalized functions �10, 17–19�.The weak asymptotics of distributions, also known as quasiasymptotics, measure pointwisescaling growth of distributions with respect to regularly varying functions in the weak sense.Let E be a Banach space with norm ‖ ‖ and let L be slowly varying. For f ∈ S′�Rn, E�, wewrite

f�x0 � εt� � O�εαL�ε�� as ε −→ 0� in S′�Rn, E�, �4.8�

if the order growth relation holds after evaluation at each test function, that is,

∥

∥

〈

f�x0 � εt�, ϕ�t�〉∥

∥ ≤ CϕεαL�ε�, 0 < ε ≤ 1, �4.9�

for each test function ϕ ∈ S�Rn�. Observe that weak asymptotics are directly involved inMeyer’s notion of the scaling weak pointwise exponent, so useful in the study of pointwiseregularity and oscillating properties of functions �3�.

One can also use these ideas to study exact pointwise scaling asymptotic propertiesof distributions �cf. �10, 17, 18, 20��. We restrict our attention here to the important notion ofthe value of a distribution at a point, introduced and studied by Łojasiewicz in �21, 22� �seealso �23–25��. The vector-valued distribution f ∈ S′�Rn, E� is said to have a value v ∈ E at thepoint x0 ∈ Rn if limε→ 0� f�x0 � εt� � v, distributionally, that is, for each ϕ ∈ S�Rn�

limε→ 0�

〈

f�t�,1εnϕ

(

t − x0ε

)〉

� v∫

Rn

ϕ�t�dt ∈ E. �4.10�

In such a case, we simply write f�x0� � v, distributionally.

4.4. Pointwise Weak Hölder Space

An important tool in Section 5 will be the concept of pointwise weak Hölder spaces of vector-valued distributions and their intimate connection with boundary asymptotics of the wavelettransform. These pointwise spaces have been recently introduced and investigated in �10�.They are extended versions of Meyer’s pointwise weak spaces from �3�. They are also closerelatives of Bony’s two-microlocal spaces �2, 3�. Again, we denote by E a Banach space, L is aslowly varying function at the origin.

For a given x0 ∈ Rn and α ∈ R, the pointwise weak Hölder space �10� Cα,Lw �x0, E�consists of those distributions f ∈ S′�Rn, E� for which there is an E-valued polynomial P ofdegree less than α such that �cf. Section 4.3�

f�x0 � εt� � P�εt� �O�εαL�ε�� as ε −→ 0� in S′�Rn, E�. �4.11�


Observe that if α < 0, then the polynomial is irrelevant. In addition, when α ≥ 0, this poly-nomial is unique; in fact �4.11� readily implies that the Łojasiewicz point values ∂mf�x0� exist,distributionally, for |m| < α and that P is the “Taylor polynomial”

P�t� �∑

|m|0

yk

εαL�ε�

∥

∥Wψf(

x0 � εx, εy)∥

∥


5.1. L-Hölder Spaces

We now introduce weighted Hölder spaces with respect to L. They were already defined andstudied in �10�. Let α ∈ R� \N. We say that a function f belongs to the space Cα,L�Rn� if f hascontinuous derivatives up to order less than α and

∥

∥f∥

∥

Cα,L :�∑

|j|≤�α�supt∈Rn

∣

∣

∣∂jf�t�∣

∣

∣ �∑

|m|��α�sup

0


Proposition 5.1. The definition of Cα,L∗ �Rn� does not depend on the choice of the LP-pair. Moreover,different LP-pairs lead to equivalent norms.

In view of Proposition 5.1, we may employ an LP-pair coming from a continuousLittlewood-Paley decomposition of the unity �cf. Example 4.3� in the definition of Cα,L∗ �Rn�.Therefore, when L ≡ 1, we recover the classical Zygmund space Cα,L∗ �Rn� � Cα∗ �Rn� �5�.Proposition 5.1 follows at once from the following lemma.

Lemma 5.2. Let f ∈ Cα,L∗ �Rn�, then for every θ ∈ S�Rn� there holds

∥

∥f ∗ θ∥∥L∞ ≤ C∥

∥f∥

∥

Cα,L∗, �5.6�

where ‖f‖Cα,L∗ is given by �5.4�. Furthermore, if B ⊂ S�Rn� is a bounded set such that μm�θ� � 0 forall θ ∈ B and all multi-indexm ≤ �α�, then

supx∈Rn

sup0


order. Then, by the Taylor formula, �5.6� �with θ � ∂mϕ̂�, and the assumption μm�θ� � 0 for|m| ≤ �α�, we obtain

sup0


Proof. �i� It is enough to consider ∂j . We have that ∂jf ∗ φ � f ∗ ∂jφ and Wψ∂jf�x, y� �−y−1W∂jψf�x, y�. Thus, the result follows at once by applying �5.6� with θ � ∂jφ and �5.7�with θ � ∂jψ.

�ii� If �φ, ψ� is an LP-pair, so is �φ,Δψ�. Note that our assumption and �i� imply thatΔf ∈ Cα−2,L∗ �Rn�. In view of Proposition 5.1, it remains to observe that

1yαL(

y)WΔψf

(

x, y)

�1

yα−2L(

y)Wψ

(

Δf)(

x, y)

. �5.16�

�iii� Since �1 − Δ�−β/2 is the inverse of �1 − Δ�β/2, it suffices to show that �1 − Δ�β/2maps continuously Cα,L∗ �Rn� into C

α−β,L∗ �Rn�. Using �5.6�with θ � �1 −Δ�β/2φ, we obtain that

‖�1 −Δ�β/2f ∗ φ‖L∞ ≤ C‖f‖Cα,L∗ . We also have

Wψ�1 −Δ�β/2f(

x, y)

�1

�2π�nyβ

〈

̂f�ξ�, eix·ξ(

y2 �∣

∣yξ∣

∣

2)β/2

ψ̂(

yξ)

〉

�1yβ

Wθyf(

x, y)

, where θy �(

y2 −Δ)β/2

ψ.

�5.17�

Finally, we can apply �5.7� because B � {�y2 −Δ�β/2ψ}y∈�0,1� is a bounded set in S�Rn� andμm�θy� � 0 for each multi-index |m| ≤ �α�.

We can also use Proposition 4.4 to show that Cα,L∗ �Rn� is a Banach space, as stated inthe following proposition.

Proposition 5.4. The space Cα,L∗ �Rn� is a Banach space when provided with the norm �5.4�.

Proof. Let η, ϕ, θ1, θ2 be as in the statement of Proposition 4.4. Suppose that {fj}∞j�0 is aCauchy sequence in Cα,L∗ �Rn�. Then, there exist continuous functions g ∈ L∞�Rn� and Gdefined on Rn × �0, 1� such that fj ∗ φ → g in L∞�Rn� and

limj→∞

supy∈�0,1�

supx∈Rn

1yαL(

y)

∣

∣Wψfj(

x, y) −G(x, y)∣∣ � 0. �5.18�

We define the distribution f ∈ S′�Rn� whose action on test functions θ ∈ S�Rn� is given by

〈

f, θ〉

:�∫

Rn

g�t�θ1�t�dt �1cψ,η

∫1

0

∫

Rn

G(

x, y)Wηθ2

(

x, y)dxdy

y. �5.19�

Since the fj have the representation �4.2�, we immediately see that fj → f in S′�Rn�. Thus,pointwisely,

(

f ∗ φ)�t� � limj→∞

(

fj ∗ φ)

�t� � g�t�,

Wψf(

x, y)

� limj→∞

Wψfj(

x, y)

� G(

x, y)

.�5.20�

This implies that limj→∞‖f − fj‖Cα,L∗ � 0, and so Cα,L∗ �Rn� is complete.


We have arrived to the main and last result of this section. It provides the L-Hölderiancharacterization of the L-Zygmund spaces of positive exponent. We will use in its proof atechnique based on the Tauberian theorem for pointwise weak regularity of vector-valueddistributions, explained in Section 4.4. We denote below by Cb�Rn� the Banach space ofcontinuous and bounded functions.

Theorem 5.5. Let α > 0.

�a� If α /∈ N, then Cα,L∗ �Rn� � Cα,L�Rn�. Moreover, the norms �5.4� and �5.1� are equivalent.�b� If α � p � 1 ∈ N, then Cp�1,L∗ �Rn� consists of functions with continuous derivatives up to

order p such that

∑

|m|≤p

∥

∥∂mf∥

∥

L∞ �∑

|m|�psupt∈Rn

0


We had already seen that ∂mf�t��ξ� � ∂mf�t � ξ� ∈ Cα−�α�,Lw �0, Cb�Rnξ �� Cα−�α�,L∗,w �0, Cb�Rnξ ��,

that is,

μ0�θ�∂mf�ξ� −∫

Rn

∂mf�ξ � εt�θ�t�dt � O(

εα−�α�L�ε�)

, ε −→ 0�, �5.26�

in the space Cb�Rnξ �, for each θ ∈ S�Rn�. Hence, if 0 < |h| ≤ 1, we choose θ as before �μ0�θ� �1�, and we use the fact that {θ − θ�· −ω� : |ω| � 1} is compact in S�Rn�; we then have

supξ∈Rn

∣

∣∂mf�ξ � h� − ∂mf�ξ�∣∣ ≤ 2supξ∈Rn

∣

∣

∣

∣

∂mf�ξ� −∫

Rn

∂mf�|h|t � ξ�θ�t�dt∣

∣

∣

∣

� supξ∈Rn

∣

∣

∣

∣

∫

Rn

∂mf�ξ � |h|t�(

θ�t� − θ(

t − |h|−1h))

dt∣

∣

∣

∣

� O(

|h|α−�α�L�|h|�)

,

�5.27�

and this completes the proof of �a�.Case α � p � 1 ∈ N. The proof is similar to that of �a�. Fix now |m| � p. We now

have ∂mf ∈ C1,L∗,w�0, Cb�Rn��, which, as commented in Section 4.4, yields the distributionalexpansion

∂mf�εt��ξ� � ∂mf�ξ� � εn∑

j�1

tjcj�ε, ξ� �O�εL�ε��, 0 < ε ≤ 1, �5.28�

in S′�Rnt , Cb�Rnξ �, where the cj�ε, · � are continuous Cb�Rnξ �-valued functions in ε. We apply�5.28� on a test function θ ∈ S�Rn�, with μ0�θ� � 1, and

∫

Rntjθ�t�dt � 0 for j � 1, . . . , n, so we

get

∂mf�ξ� �∫

Rn

∂mf�ξ � |h|t�θ�t�dt �O�|h|L�|h|��, 0 < |h| ≤ 1, �5.29�

uniformly in ξ ∈ Rn. Since {θω :� θ�· � ω� � θ�· − ω� − 2θ : |ω| � 1} is compact in S�Rn� andμm�θω� � 0 for |m| ≤ 1, the relations �5.28� and �5.29� give

supξ∈Rn

∣

∣∂mf�ξ � h� � ∂mf�ξ − h� − 2∂mf�ξ�∣∣

≤ 3supξ∈Rn

∣

∣

∣

∣

∂mf�ξ� −∫

Rn

∂mf�|h|t � ξ�θ�t�dt∣

∣

∣

∣

�∣

∣

∣

∣

∫

Rn

∂mf�ξ � |h|t�(

θ(

t � |h|−1h)

� θ(

t − |h|−1h)

− 2θ�t�)

dt∣

∣

∣

∣

� O�|h|L�|h|��, 0 < |h| ≤ 1,

�5.30�

as claimed.


Acknowledgments

S. Pilipović acknowledges support by Project 174024 of the Serbian Ministry of Educationand Sciences. D. Rakić acknowledges support by Project III44006 of the Serbian Ministry ofEducation and Sciences and by Project 114-451-2167 of the Provincial Secretariat for Scienceand Technological Development. J. Vindas acknowledges support by a postdoctoral fellow-ship of the Research Foundation-Flanders �FWO, Belgium�.

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