# New Classes of Weighted Hölder-Zygmund Spaces and the Wavelet Transform

Authors: Stevan Pilipović; Dušan Rakić; Jasson Vindas
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces𝒮0(ℝn) and 𝒮(ℍn+1). We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.


---

## Body

## 1. Introduction

The purpose of this article is twofold. The main one is to define and analyze a new class of 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 C*α(ℝn) satisfy the size estimate |𝒲ψf(x,y)|≤Cyα, which, plus a side condition, essentially characterizes the space itself. We will replace the regularity measurement yα by weights from the interesting class of regularly varying functions [7, 8]. Familiar functions such as yα,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 analyzing many function and distribution spaces [1], such as the ones introduced in this paper. Our second aim is to provide a new proof of the continuity theorem, originally obtained in [9], for these transforms on the function spaces 𝒮0(ℝn) and 𝒮(ℍn+1). Our approach to the proof is completely elementary and substantially simplifies the much longer original proof from [9] (see also [1, Chapter 1]).The definition of our weighted Zygmund spaces is based on the useful concept of (generalized) Littlewood-Paley pairs, introduced in Section4.1, which generalizes the familiar 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 C*α(ℝn), our analysis provides a new approach to the study of Hölder-Zygmund spaces. It is then very likely that this kind of arguments might also be applied to study other types of smooth spaces, such as Besov-type spaces.Our new classes of spacesCα,L(ℝn) and C*α,L(ℝn), the L-Hölder and L-Zygmund spaces of exponent α that will be introduced in Section 5, contribute to the analysis of local regularity of 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 inclusion relations:
(1.1)Cβ(ℝn)⊂Cα,L(ℝn)⊂Cγ(ℝn),when0<γ<α<β,C*β(Rn)⊂C*α,L(Rn)⊂C*γ(Rn),ifγ<α<β.
Situations in which these kinds of refinements are essential often occur in the literature, and they 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 more general weights, specifically, the important case when L is a slowly varying function [7, 8].The paper is organized as follows. We review in Section2 basic facts about test function spaces, the wavelet transform, and its left-inverse, namely, the wavelet synthesis operator. In Section 3, we will provide the announced new proof of the continuity theorem for the wavelet and wavelet synthesis transforms when acting on test function spaces. We then explain in Section 4 some useful concepts that will be applied to the analysis of our weighted versions of the Hölder-Zygmund spaces; in particular, we will 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 new class of Hölder-Zygmund spaces in Section 5.


## 2. Notation and Notions

We denote byℍn+1=ℝn×ℝ+ the upper half space. If x∈ℝn and m∈ℕn, then |x| denotes the euclidean norm, xm=x1m1⋯xnmn,  ∂m=∂xm=∂x1m1⋯∂xnmn,  m!=m1!m2!⋯mn! and |m|=m1+⋯+mn. If the jth coordinate of m is one and the others vanish, we then write ∂j=∂xm. The set B(0,r) is the euclidean ball in ℝn of radius r. In the sequel, we use C and C′ to denote positive 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 denoted by 𝒮(ℝn). We will fix constants in the Fourier transform as φ^(ξ)=∫ℝn‍φ(t)e-iξ·tdt. The moments of φ∈𝒮(ℝn) are denoted by μm(φ)=∫ℝn‍tmφ(t)dt, m∈ℕn.Following [1], the space of highly time-frequency localized functions 𝒮0(ℝn) is defined as 𝒮0(ℝn)={φ∈𝒮(ℝn):μm(φ)=0,forallm∈ℕn}; it is provided with the relative topology inhered from 𝒮(ℝn). In [1], the topology of 𝒮0(ℝn) is introduced in an apparently different way; 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 ∂mφ^(0)=0, for all m∈ℕn. It is important to point out that 𝒮0(ℝn) is also well known in the literature as the Lizorkin space of test functions (cf. [13]).The space𝒮(ℍn+1) of highly localized functions on the half space [1] consists of those Φ∈C∞(ℍn+1) for which
(2.1)ρl,k,ν,m0(Φ)=sup⁡(x,y)∈Hn+1(yl+1yl)(1+|x|2)k/2|∂yν∂xmΦ(x,y)|<∞,
for all l,k,ν∈ℕ, and m∈ℕn. The canonical topology on 𝒮(ℍn+1) is induced by this family of seminorms [1]. For later use, we will denote by ρk,m the corresponding seminorms in 𝒮(ℝn), namely,
(2.2)ρk,m(φ)=sup⁡t∈Rn(1+|t|2)k/2|∂mφ(t)|,k∈N,m∈Nn.The corresponding duals of these three spaces are𝒮′(ℝn), 𝒮0′(ℝn), and 𝒮′(ℍn+1). They are, respectively, the spaces of tempered distributions, Lizorkin distributions, and distributions of slow growth on ℍn+1. Since the elements of 𝒮0(ℝn) are orthogonal to every polynomial, 𝒮0′(ℝn) can be canonically identified with the quotient space of 𝒮′(ℝn) modulo polynomials.Finally, we will also make use of spaces of vector-valued tempered distributions [14, 15]. If X is a locally convex topological vector space [15], then the space of X-valued tempered distributions is 𝒮′(ℝn,X)=Lb(𝒮(ℝn),X), namely, the space of continuous linear mappings from 𝒮(ℝn) into X.


### 2.2. Wavelet Transform

In this paper a wavelet simply means a functionψ∈𝒮(ℝn) that satisfies μ0(ψ)=∫ℝn‍ψ(t)dt=0.The wavelet transform off∈𝒮′(ℝn) with respect to the wavelet ψ∈𝒮(ℝn) is defined as
(2.3)Wψf(x,y)=〈f(t),1ynψ¯(t-xy)〉=1yn∫Rn‍f(t)ψ¯(t-xy)dt,
where (x,y)∈ℍn+1. The very last integral formula is a formal notation which makes sense when f is a function of tempered growth. Notice that the wavelet transform is also well defined via (2.3) for f∈𝒮0′(ℝn) if the wavelet ψ∈𝒮0(ℝn). The wavelet transform can be defined exactly in the same way for vector-valued distributions.


### 2.3. Wavelet Synthesis Operator

Letη∈𝒮0(ℝn). The wavelet synthesis transform of Φ∈𝒮(ℍn+1) with respect to the wavelet η is defined as
(2.4)MηΦ(t)=∫0∞‍∫Rn‍Φ(x,y)1ynη(t-xy)dxdyy,t∈Rn.
Observe that the operator ℳη may be extended to act on the space 𝒮′(ℍn+1) via duality arguments, see [1] for details (cf. [10] for the vector-valued case). In this paper we restrict our 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 to construct a left inverse for the wavelet transform, whenever the wavelet possesses nice reconstruction properties. Indeed, assume thatψ∈𝒮0(ℝn) admits a reconstruction wavelet η∈𝒮0(ℝn). More precisely, it means that the constant
(2.5)cψ,η=cψ,η(ω)=∫0∞‍ψ^¯(rω)η^(rω)drr,ω∈Sn-1
is different from zero and independent of the direction ω. Then, a straightforward calculation [1] shows that
(2.6)IdS0(Rn)=1cψ,ηMηWψ.
It is worth pointing out that (2.6) is also valid [1, 10] when 𝒲ψ and ℳη act on the spaces 𝒮0′(ℝn) and 𝒮′(ℍn+1), respectively.Furthermore, it is very important to emphasize that a waveletψ admits a reconstruction wavelet η if and only if it is nondegenerate in the sense of the following definition [10].Definition 2.1.
A test functionφ∈𝒮(ℝn) is said to be nondegenerate if for any ω∈𝕊n-1 the function of one variable Rω(r)=φ^(rω)∈C∞[0,∞) is not identically zero, that is, suppRω≠∅, for each ω∈𝕊n-1.


## 2.1. Function and Distribution Spaces

The well-known [12] Schwartz space of rapidly decreasing smooth test functions is denoted by 𝒮(ℝn). We will fix constants in the Fourier transform as φ^(ξ)=∫ℝn‍φ(t)e-iξ·tdt. The moments of φ∈𝒮(ℝn) are denoted by μm(φ)=∫ℝn‍tmφ(t)dt, m∈ℕn.Following [1], the space of highly time-frequency localized functions 𝒮0(ℝn) is defined as 𝒮0(ℝn)={φ∈𝒮(ℝn):μm(φ)=0,forallm∈ℕn}; it is provided with the relative topology inhered from 𝒮(ℝn). In [1], the topology of 𝒮0(ℝn) is introduced in an apparently different way; 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 ∂mφ^(0)=0, for all m∈ℕn. It is important to point out that 𝒮0(ℝn) is also well known in the literature as the Lizorkin space of test functions (cf. [13]).The space𝒮(ℍn+1) of highly localized functions on the half space [1] consists of those Φ∈C∞(ℍn+1) for which
(2.1)ρl,k,ν,m0(Φ)=sup⁡(x,y)∈Hn+1(yl+1yl)(1+|x|2)k/2|∂yν∂xmΦ(x,y)|<∞,
for all l,k,ν∈ℕ, and m∈ℕn. The canonical topology on 𝒮(ℍn+1) is induced by this family of seminorms [1]. For later use, we will denote by ρk,m the corresponding seminorms in 𝒮(ℝn), namely,
(2.2)ρk,m(φ)=sup⁡t∈Rn(1+|t|2)k/2|∂mφ(t)|,k∈N,m∈Nn.The corresponding duals of these three spaces are𝒮′(ℝn), 𝒮0′(ℝn), and 𝒮′(ℍn+1). They are, respectively, the spaces of tempered distributions, Lizorkin distributions, and distributions of slow growth on ℍn+1. Since the elements of 𝒮0(ℝn) are orthogonal to every polynomial, 𝒮0′(ℝn) can be canonically identified with the quotient space of 𝒮′(ℝn) modulo polynomials.Finally, we will also make use of spaces of vector-valued tempered distributions [14, 15]. If X is a locally convex topological vector space [15], then the space of X-valued tempered distributions is 𝒮′(ℝn,X)=Lb(𝒮(ℝn),X), namely, the space of continuous linear mappings from 𝒮(ℝn) into X.


## 2.2. Wavelet Transform

In this paper a wavelet simply means a functionψ∈𝒮(ℝn) that satisfies μ0(ψ)=∫ℝn‍ψ(t)dt=0.The wavelet transform off∈𝒮′(ℝn) with respect to the wavelet ψ∈𝒮(ℝn) is defined as
(2.3)Wψf(x,y)=〈f(t),1ynψ¯(t-xy)〉=1yn∫Rn‍f(t)ψ¯(t-xy)dt,
where (x,y)∈ℍn+1. The very last integral formula is a formal notation which makes sense when f is a function of tempered growth. Notice that the wavelet transform is also well defined via (2.3) for f∈𝒮0′(ℝn) if the wavelet ψ∈𝒮0(ℝn). The wavelet transform can be defined exactly in the same way for vector-valued distributions.


## 2.3. Wavelet Synthesis Operator

Letη∈𝒮0(ℝn). The wavelet synthesis transform of Φ∈𝒮(ℍn+1) with respect to the wavelet η is defined as
(2.4)MηΦ(t)=∫0∞‍∫Rn‍Φ(x,y)1ynη(t-xy)dxdyy,t∈Rn.
Observe that the operator ℳη may be extended to act on the space 𝒮′(ℍn+1) via duality arguments, see [1] for details (cf. [10] for the vector-valued case). In this paper we restrict our 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 to construct a left inverse for the wavelet transform, whenever the wavelet possesses nice reconstruction properties. Indeed, assume thatψ∈𝒮0(ℝn) admits a reconstruction wavelet η∈𝒮0(ℝn). More precisely, it means that the constant
(2.5)cψ,η=cψ,η(ω)=∫0∞‍ψ^¯(rω)η^(rω)drr,ω∈Sn-1
is different from zero and independent of the direction ω. Then, a straightforward calculation [1] shows that
(2.6)IdS0(Rn)=1cψ,ηMηWψ.
It is worth pointing out that (2.6) is also valid [1, 10] when 𝒲ψ and ℳη act on the spaces 𝒮0′(ℝn) and 𝒮′(ℍn+1), respectively.Furthermore, it is very important to emphasize that a waveletψ admits a reconstruction wavelet η if and only if it is nondegenerate in the sense of the following definition [10].Definition 2.1.
A test functionφ∈𝒮(ℝn) is said to be nondegenerate if for any ω∈𝕊n-1 the function of one variable Rω(r)=φ^(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:(3.1)W:(ψ,φ)⟼Wψφ,M:(η,Φ)⟼MηΦ.
Our first main result is a new proof of the continuity theorem for these two bilinear mappings when 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 exist N∈ℕ and C>0 such that
(3.2)ρl,k,ν,m0(Wψφ)≤C∑i′,|j′|,i,|j|≤N‍ρi′,j′(ψ)ρi,j(φ).
We begin by making some reductions. Observe that, for constants cj which do not depend on φ and ψ,
(3.3)∂yν∂xmWψφ(x,y)=∂yν∂xm∫Rn‍φ(yt+x)ψ¯(t)dt=∑|j|≤ν‍cj∫Rn‍∂m+jφ(yt+x)tjψ¯(t)dt=∑|j|≤ν‍cjWψjφm+j(x,y),
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,s independent of ψ and φ,
(3.4)(1+|x|2)k/2Wψφ(x,y)=1(2π)n〈(1-Δξ)k/2eiξ⋅x,φ^(ξ)ψ^¯(yξ)〉=1(2π)n∑|r|+|s|≤k‍cr,sy|s|∫Rn‍eiξ⋅x∂rφ^(ξ)∂ξsψ^¯(yξ)dξ=1(2π)n∑|r|+|s|≤k‍cr,sy|s|Wψsφr(x,y),
where φr(t)=(-it)rφ(t) and ψs(t)=(it)sψ(t). Thus, it clearly suffices to establish (3.2) for k=ν=0 and m=0. We may also assume that l≥n.
We first estimate the termyl|𝒲ψφ(x,y)|. Since ∂jφ^(0)=0 for every j∈ℕn, we can apply the Taylor formula to obtain
(3.5)φ^(ξ)=∑|j|=l-n‍∂jφ^(zξ)j!ξj,forsomezξinthelinesegment[0,ξ].
Hence,
(3.6)yl|Wψφ(x,y)|=yl(2π)n|〈eiξ⋅xφ^(ξ),ψ^¯(yξ)〉|≤1(2π)n∫Rn‍|φ^(ξ)|yl|ψ^(yξ)|dξ≤1(2π)n∑|j|=l-n‍1j!∫Rn‍|∂jφ^(zξ)|yn|(yξ)jψ^(yξ)|dξ≤1(2π)n∑|j|=l-n‍ρ0,j(φ^)j!∫Rn‍|ξjψ^(ξ)|dξ≤1(2π)n∑|j|=l-n‍ρ0,j(φ^)ρl+n,0(ψ^)j!∫Rn‍dξ(1+|ξ|2)n≤Cρl+n,0(φ)∑|j|≤2l+2n‍ρ2n,j(ψ),∀(x,y)∈Hn+1.
It remains to estimatey-l|𝒲ψφ(x,y)|. We will now use the fact that all the moments of ψ vanish. If we apply the Taylor formula, we have, for some zt=z(t,x,y) in the line segment [x,yt],
(3.7)1yl|Wψφ(x,y)|=1yl|∫Rn‍φ(yt+x)ψ¯(t)dt|=1yl|∫Rn‍ψ¯(t)(∑|j|<l‍∂jφ(x)j!(yt)j+∑|j|=l‍∂jφ(zt)j!(yt)j)dt|≤∑|j|=l‍ρ0,j(φ)j!∫Rn‍(1+|t|2)l/2|ψ(t)|dt≤Cρl+2n,0(ψ)∑|j|=l‍ρ0,j(φ).
The result immediately follows on combining the previous two estimates.
Continuity of the Wavelet Synthesis Mapping. We should now prove that for arbitrary k∈ℕ and κ∈ℕn there exist N∈ℕ and C>0 such that
(3.8)ρk,κ(MηΦ)≤C∑k1,k2,l,|m|≤N‍i,|j|≤Nρi,j(η)ρk1,k2,l,m0(Φ).
Since ∂tκℳηΦ=ℳη∂xκΦ, it is enough to prove (3.8) for κ=0. We denote below Φ^ the partial 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
(3.9)(1+|t|2)k/2|MηΦ(t)|=(1+|t|2)k/2|∫R+‍∫Rn‍Φ(t-x,y)1ynη(xy)dxdyy|=(1+|t|2)k/2(2π)n|∫R+‍∫Rn‍(1-Δξ)k/2eitξ(1+|t|2)k/2Φ^(ξ,y)η^(yξ)dξdyy|≤1(2π)n∫R+‍∫Rn‍|(1-Δξ)k/2(Φ^(ξ,y)η^(yξ))|dξdyy≤1(2π)n∑|r|+|s|≤k‍cr,s∫R+‍∫Rn‍|∂ξrΦ^(ξ,y)|y|s|-1|∂sη^(yξ)|dξdy≤1(2π)n∑|r|+|s|≤k‍cr,sρ0,s(η^)∫R+‍∫Rn‍y|s|-1|∂ξrΦ^(ξ,y)|dξdy≤C′∑|r|+|s|≤kρ0,s(η^)(ρ|s|-1,2n,0,r0(Φ^)+ρ|s|+1,2n,0,r0(Φ^))∫R+‍∫Rn‍dξ(1+|ξ|2)ndy1+y2≤C∑|r|+|s|≤k‍|j|≤2nρ|s|+2n,0(η)(ρ|s|-1,|r|+2n,0,j0(Φ)+ρ|s|+1,|r|+2n,0,j0(Φ)).
This completes the proof.Remark 3.2.
It follows from the proof of the continuity ofℳ that we can extend the bilinear mapping ℳ:(η,Φ)↦ℳηΦ to act on
(3.10)M:S(Rn)×S(Hn+1)⟶S(Rn),
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 Section5. In this section we collect some useful concepts 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-Paley decompositions 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 on which its Fourier transform “degenerates.” We measure in the next definition how big that ball is.Definition 4.1.
Letψ∈𝒮(ℝn) be a nondegenerate wavelet. Its index of nondegenerateness is the (finite) number
(4.1)τ=inf⁡{r∈R+:supp⁡Rω∩[0,r]≠∅,∀ω∈Sn-1},
where Rω are the functions of one variable Rω(r)=ψ^(rω).If we only know values of𝒲ψf(x,y) at scale y<1, then the wavelet transform can be blind when analyzing certain distributions (cf. [10, Section 7.2]). The idea behind the introduction of Littlewood-Paley pairs is to have an alternative way for recovering such a possible lost of information by employing additional data with respect to another function ϕ (cf. [16]).Definition 4.2.
Letα∈ℝ, ϕ∈𝒮(ℝn). Let ψ∈𝒮(ℝn) be a nondegenerate wavelet with the index 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-index m∈ℕn with |m|≤[α].Example 4.3.
Letϕ∈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 ψ^(ξ)=-ξ·∇φ^(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 function spaces, such as the classical Zygmund space C*α(ℝn) (cf., e.g., [5]).We pointed out above that LP-pairs enjoy powerful reconstruction properties. Let us make this more precise.Proposition 4.4.
Let(ϕ,ψ) be an LP-pair, the wavelet ψ having index of nondegenerateness τ and r>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 a reconstruction wavelet for ψ whose Fourier transform has support in B(0,σ) and φ∈𝒟(ℝn) is such that φ(ξ)=1 for ξ∈B(0,σ) and suppφ⊂B(0,r), then, for all f∈𝒮′(ℝn) and θ∈𝒮(ℝn)(4.2)〈f,θ〉=∫Rn‍(f*ϕ)(t)θ1(t)dt+1cψ,η∫01‍∫Rn‍Wψf(x,y)Wη¯θ2(x,y)dxdyy,
where θ^1(ξ)=θ^(ξ)φ(ξ)/ϕ^(-ξ) and θ^2(ξ)=θ^(ξ)(1-φ(ξ)).Proof.
Observe that(4.3)〈f*ϕ,θ1〉=(2π)-n〈f^(ξ),θ^(-ξ)φ(-ξ)〉.
It is therefore enough to assume that θ1=0 so that θ=θ2. Our assumption over η is that η∈𝒮0(ℝn), suppη^⊂B(0,σ), and
(4.4)cψ,η=∫0∞‍ψ^¯(rω)η^(rω)drr≠0
does not depend on the direction ω. We remark that such a reconstruction wavelet can always be found (see the proof of [10, Theorem 7.7]). Therefore, 𝒲η¯θ(x,y)=0 for all (x,y)∈ℝn×(1,∞). Exactly as in [1, page 66], the usual calculation shows that
(4.5)θ(t)=1cψ,ηMψ¯(Wη¯θ)(t)=1cψ,η∫01‍∫Rn‍ψ¯(t-xy)Wη¯θ(x,y)dxdyy.
Furthermore, since 𝒲η¯θ∈𝒮(ℍn+1) (cf. Theorem 3.1), the last integral can be expressed as the limit in 𝒮(ℝn) of Riemann sums. That justifies the exchange of dual pairing and integral in
(4.6)〈f,θ〉=〈f(t),1cψ,η∫01‍∫Rn‍ψ¯(t-xy)Wη¯θ(x,y)dxdyy〉=1cψ,η∫01‍∫Rn‍Wψf(x,y)Wη¯θ(x,y)dxdyy.


### 4.2. Slowly Varying Functions

The weights in our weighted versions of Hölder-Zygmund spaces will be taken from the class of Karamata regularly varying functions. Such functions have been very much studied and have numerous applications in diverse areas of mathematics. We refer to [7, 8] for their properties. Let us recall that a positive measurable function L is called slowly varying (at the origin) if it is asymptotically invariant under rescaling, that is,
(4.7)lim⁡ɛ→0+L(aɛ)L(ɛ)=1,foreacha>0.
Familiar functions such as 1, |logɛ|β, |log|logɛ||β, …, are slowly varying. Regularly varying functions are then those that can be written as ɛαL(ɛ), where L is slowly varying and α∈ℝ.


### 4.3. Weak Asymptotics

We will use some notions from the theory of asymptotics of generalized functions [10, 17–19]. The weak asymptotics of distributions, also known as quasiasymptotics, measure pointwise scaling 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), we write
(4.8)f(x0+ɛt)=O(ɛαL(ɛ))asɛ⟶0+inS′(Rn,E),
if the order growth relation holds after evaluation at each test function, that is,
(4.9)‖〈f(x0+ɛt),φ(t)〉‖≤CφɛαL(ɛ),0<ɛ≤1,
for each test function φ∈𝒮(ℝn). Observe that weak asymptotics are directly involved in Meyer's notion of the scaling weak pointwise exponent, so useful in the study of pointwise regularity and oscillating properties of functions [3].One can also use these ideas to study exact pointwise scaling asymptotic properties of distributions (cf. [10, 17, 18, 20]). We restrict our attention here to the important notion of the value of a distribution at a point, introduced and studied by Łojasiewicz in [21, 22] (see also [23–25]). The vector-valued distribution f∈𝒮′(ℝn,E) is said to have a value v∈E at the point x0∈ℝn if limɛ→0+f(x0+ɛt)=v, distributionally, that is, for each φ∈𝒮(ℝn)(4.10)lim⁡ɛ→0+〈f(t),1ɛnφ(t-x0ɛ)〉=v∫Rn‍φ(t)dt∈E.
In such a case, we simply write f(x0)=v, distributionally.


### 4.4. Pointwise Weak Hölder Space

An important tool in Section5 will be the concept of pointwise weak Hölder spaces of vector-valued distributions and their intimate connection with boundary asymptotics of the wavelet transform. 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 close relatives of Bony's two-microlocal spaces [2, 3]. Again, we denote by E a Banach space, L is a slowly varying function at the origin.For a givenx0∈ℝn and α∈ℝ, the pointwise weak Hölder space [10] Cwα,L(x0,E) consists of those distributions f∈𝒮′(ℝn,E) for which there is an E-valued polynomial P of degree less than α such that (cf. Section 4.3)
(4.11)f(x0+ɛt)=P(ɛt)+O(ɛαL(ɛ))asɛ⟶0+inS′(Rn,E).
Observe that if α<0, then the polynomial is irrelevant. In addition, when α≥0, this polynomial 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”
(4.12)P(t)=∑|m|<α‍∂mf(x0)m!tm.The pointwise weak Hölder spaceC*,wα,L(x0,E) of second type is defined as follows: f∈C*,wα,L(x0,E) if (4.9) is just assumed to hold for each φ∈𝒮(ℝn) satisfying the requirement μm(φ)=0 for all multi-index |m|≤α. Naturally, the previous requirement is empty if α<0, thus, in such a case, C*,wα,L(x0,E)=Cwα,L(x0,E). One can also show that if α∉ℕ, the equality between these two spaces remains true [10]. On the other hand, when α∈ℕ, we have the strict inclusion Cwα,L(x0,E)⊊C*,wα,L(x0,E)(cf. comments below Theorem 4.5). The usefulness of C*,wα,L(x0,E) lies in the fact that it admits a precise wavelet characterization. The following theorem is shown in [10], it forms part of more general tauberian-type results that will not be discussed here.Theorem 4.5 (see [10]).
Letα∈ℝ and let ψ∈𝒮(ℝn) be a nondegenerate wavelet with μm(ψ)=0 for |m|≤[α]. Then, f∈C*,wα,L(x0,E) if and only if there is k∈ℕ such that
(4.13)limsup⁡ɛ→0+sup⁡|x|2+y2=1,y>0ykɛαL(ɛ)‖Wψf(x0+ɛx,ɛy)‖<∞.It is worth mentioning that the elements ofC*,wα,L(x0,E) for α=p∈ℕ can be characterized by pointwise weak-asymptotic expansions. We have [10] that f∈C*,wp,L(x0,E) if and only if it admits the following weak expansion:
(4.14)f(x0+ɛt)=∑|m|<p‍∂mf(x0)m!(ɛt)m+ɛp∑|m|=p‍tmcm(ɛ)+O(ɛpL(ɛ))inS′(Rn,E),
where ∂mf(x0) are interpreted in the Łojasiewicz sense and the cm:(0,∞)→E are continuous functions. Comparison of this weak expansion with (4.11) explains the difference between the two pointwise spaces when α=p∈ℕ.


## 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-Paley decompositions 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 on which its Fourier transform “degenerates.” We measure in the next definition how big that ball is.Definition 4.1.
Letψ∈𝒮(ℝn) be a nondegenerate wavelet. Its index of nondegenerateness is the (finite) number
(4.1)τ=inf⁡{r∈R+:supp⁡Rω∩[0,r]≠∅,∀ω∈Sn-1},
where Rω are the functions of one variable Rω(r)=ψ^(rω).If we only know values of𝒲ψf(x,y) at scale y<1, then the wavelet transform can be blind when analyzing certain distributions (cf. [10, Section 7.2]). The idea behind the introduction of Littlewood-Paley pairs is to have an alternative way for recovering such a possible lost of information by employing additional data with respect to another function ϕ (cf. [16]).Definition 4.2.
Letα∈ℝ, ϕ∈𝒮(ℝn). Let ψ∈𝒮(ℝn) be a nondegenerate wavelet with the index 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-index m∈ℕn with |m|≤[α].Example 4.3.
Letϕ∈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 ψ^(ξ)=-ξ·∇φ^(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 function spaces, such as the classical Zygmund space C*α(ℝn) (cf., e.g., [5]).We pointed out above that LP-pairs enjoy powerful reconstruction properties. Let us make this more precise.Proposition 4.4.
Let(ϕ,ψ) be an LP-pair, the wavelet ψ having index of nondegenerateness τ and r>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 a reconstruction wavelet for ψ whose Fourier transform has support in B(0,σ) and φ∈𝒟(ℝn) is such that φ(ξ)=1 for ξ∈B(0,σ) and suppφ⊂B(0,r), then, for all f∈𝒮′(ℝn) and θ∈𝒮(ℝn)(4.2)〈f,θ〉=∫Rn‍(f*ϕ)(t)θ1(t)dt+1cψ,η∫01‍∫Rn‍Wψf(x,y)Wη¯θ2(x,y)dxdyy,
where θ^1(ξ)=θ^(ξ)φ(ξ)/ϕ^(-ξ) and θ^2(ξ)=θ^(ξ)(1-φ(ξ)).Proof.
Observe that(4.3)〈f*ϕ,θ1〉=(2π)-n〈f^(ξ),θ^(-ξ)φ(-ξ)〉.
It is therefore enough to assume that θ1=0 so that θ=θ2. Our assumption over η is that η∈𝒮0(ℝn), suppη^⊂B(0,σ), and
(4.4)cψ,η=∫0∞‍ψ^¯(rω)η^(rω)drr≠0
does not depend on the direction ω. We remark that such a reconstruction wavelet can always be found (see the proof of [10, Theorem 7.7]). Therefore, 𝒲η¯θ(x,y)=0 for all (x,y)∈ℝn×(1,∞). Exactly as in [1, page 66], the usual calculation shows that
(4.5)θ(t)=1cψ,ηMψ¯(Wη¯θ)(t)=1cψ,η∫01‍∫Rn‍ψ¯(t-xy)Wη¯θ(x,y)dxdyy.
Furthermore, since 𝒲η¯θ∈𝒮(ℍn+1) (cf. Theorem 3.1), the last integral can be expressed as the limit in 𝒮(ℝn) of Riemann sums. That justifies the exchange of dual pairing and integral in
(4.6)〈f,θ〉=〈f(t),1cψ,η∫01‍∫Rn‍ψ¯(t-xy)Wη¯θ(x,y)dxdyy〉=1cψ,η∫01‍∫Rn‍Wψf(x,y)Wη¯θ(x,y)dxdyy.


## 4.2. Slowly Varying Functions

The weights in our weighted versions of Hölder-Zygmund spaces will be taken from the class of Karamata regularly varying functions. Such functions have been very much studied and have numerous applications in diverse areas of mathematics. We refer to [7, 8] for their properties. Let us recall that a positive measurable function L is called slowly varying (at the origin) if it is asymptotically invariant under rescaling, that is,
(4.7)lim⁡ɛ→0+L(aɛ)L(ɛ)=1,foreacha>0.
Familiar functions such as 1, |logɛ|β, |log|logɛ||β, …, are slowly varying. Regularly varying functions are then those that can be written as ɛαL(ɛ), where L is slowly varying and α∈ℝ.


## 4.3. Weak Asymptotics

We will use some notions from the theory of asymptotics of generalized functions [10, 17–19]. The weak asymptotics of distributions, also known as quasiasymptotics, measure pointwise scaling 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), we write
(4.8)f(x0+ɛt)=O(ɛαL(ɛ))asɛ⟶0+inS′(Rn,E),
if the order growth relation holds after evaluation at each test function, that is,
(4.9)‖〈f(x0+ɛt),φ(t)〉‖≤CφɛαL(ɛ),0<ɛ≤1,
for each test function φ∈𝒮(ℝn). Observe that weak asymptotics are directly involved in Meyer's notion of the scaling weak pointwise exponent, so useful in the study of pointwise regularity and oscillating properties of functions [3].One can also use these ideas to study exact pointwise scaling asymptotic properties of distributions (cf. [10, 17, 18, 20]). We restrict our attention here to the important notion of the value of a distribution at a point, introduced and studied by Łojasiewicz in [21, 22] (see also [23–25]). The vector-valued distribution f∈𝒮′(ℝn,E) is said to have a value v∈E at the point x0∈ℝn if limɛ→0+f(x0+ɛt)=v, distributionally, that is, for each φ∈𝒮(ℝn)(4.10)lim⁡ɛ→0+〈f(t),1ɛnφ(t-x0ɛ)〉=v∫Rn‍φ(t)dt∈E.
In such a case, we simply write f(x0)=v, distributionally.


## 4.4. Pointwise Weak Hölder Space

An important tool in Section5 will be the concept of pointwise weak Hölder spaces of vector-valued distributions and their intimate connection with boundary asymptotics of the wavelet transform. 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 close relatives of Bony's two-microlocal spaces [2, 3]. Again, we denote by E a Banach space, L is a slowly varying function at the origin.For a givenx0∈ℝn and α∈ℝ, the pointwise weak Hölder space [10] Cwα,L(x0,E) consists of those distributions f∈𝒮′(ℝn,E) for which there is an E-valued polynomial P of degree less than α such that (cf. Section 4.3)
(4.11)f(x0+ɛt)=P(ɛt)+O(ɛαL(ɛ))asɛ⟶0+inS′(Rn,E).
Observe that if α<0, then the polynomial is irrelevant. In addition, when α≥0, this polynomial 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”
(4.12)P(t)=∑|m|<α‍∂mf(x0)m!tm.The pointwise weak Hölder spaceC*,wα,L(x0,E) of second type is defined as follows: f∈C*,wα,L(x0,E) if (4.9) is just assumed to hold for each φ∈𝒮(ℝn) satisfying the requirement μm(φ)=0 for all multi-index |m|≤α. Naturally, the previous requirement is empty if α<0, thus, in such a case, C*,wα,L(x0,E)=Cwα,L(x0,E). One can also show that if α∉ℕ, the equality between these two spaces remains true [10]. On the other hand, when α∈ℕ, we have the strict inclusion Cwα,L(x0,E)⊊C*,wα,L(x0,E)(cf. comments below Theorem 4.5). The usefulness of C*,wα,L(x0,E) lies in the fact that it admits a precise wavelet characterization. The following theorem is shown in [10], it forms part of more general tauberian-type results that will not be discussed here.Theorem 4.5 (see [10]).
Letα∈ℝ and let ψ∈𝒮(ℝn) be a nondegenerate wavelet with μm(ψ)=0 for |m|≤[α]. Then, f∈C*,wα,L(x0,E) if and only if there is k∈ℕ such that
(4.13)limsup⁡ɛ→0+sup⁡|x|2+y2=1,y>0ykɛαL(ɛ)‖Wψf(x0+ɛx,ɛy)‖<∞.It is worth mentioning that the elements ofC*,wα,L(x0,E) for α=p∈ℕ can be characterized by pointwise weak-asymptotic expansions. We have [10] that f∈C*,wp,L(x0,E) if and only if it admits the following weak expansion:
(4.14)f(x0+ɛt)=∑|m|<p‍∂mf(x0)m!(ɛt)m+ɛp∑|m|=p‍tmcm(ɛ)+O(ɛpL(ɛ))inS′(Rn,E),
where ∂mf(x0) are interpreted in the Łojasiewicz sense and the cm:(0,∞)→E are continuous functions. Comparison of this weak expansion with (4.11) explains the difference between the two pointwise spaces when α=p∈ℕ.


## 5. New Class of Hölder-Zygmund Spaces

Throughout this section, we assume thatL is a slowly varying function such that L and 1/L are locally bounded on (0,1].


### 5.1.L-Hölder Spaces

We now introduce weighted Hölder spaces with respect toL. They were already defined and studied in [10]. Let α∈ℝ+∖ℕ. We say that a function f belongs to the space Cα,L(ℝn) if f has continuous derivatives up to order less than α and
(5.1)‖f‖Cα,L:=∑|j|≤[α]‍sup⁡t∈Rn|∂jf(t)|+∑|m|=[α]‍sup⁡0<|t-x|≤1|∂mf(t)-∂mf(x)||t-x|α-[α]L(|t-x|)<∞.
When α=p+1∈ℕ, we replace the previous requirement by
(5.2)‖f‖Cα,L:=∑|j|≤p‍sup⁡t∈Rn|∂jf(t)|+∑|m|=p‍sup⁡0<|t-x|≤1|∂mf(t)-∂mf(x)||t-x|L(|t-x|)<∞.The spaceCα,L(ℝn) is clearly a Banach space with the above norm. The conditions imposed over L ensure that Cα,L(ℝn) depends only on the behavior of L near 0, thus, it is invariant under dilations. When L≡1, this space reduces to Cα,L(ℝn)=Cα(ℝn), the usual global (inhomogeneous) Hölder space [2, 3, 5]. Consequently, as in [10], we call Cα,L(ℝn) the global Hölder space with respect to L. Note that, because of the properties of L [7, 8], we have the following interesting inclusion relations:
(5.3)Cβ(Rn)⊊Cα,L(Rn)⊊Cγ(Rn),whenever0<γ<α<β.


### 5.2.L-Zygmund Spaces

We now proceed to define the weighted Zygmund spaceC*α,L(ℝn). Let α∈ℝ and fix an LP-pair (ϕ,ψ) of order α. A distribution f∈𝒮′(ℝn) is said to belong to the L-Zygmund space C*α,L(ℝn) of exponent α if
(5.4)‖f‖C*α,L:=‖f*ϕ‖L∞+sup⁡x∈Rnsup⁡0<y≤11yαL(y)|Wψf(x,y)|<∞.Observe that we clearly haveCα,L(ℝn)⊆C*α,L(ℝn), for α>0. We will show that if α∈ℝ+∖ℕ, we actually have the equality Cα,L(ℝn)=C*α,L(ℝn). When α is a positive integer, we have in turn Cα,L(ℝn)⊊C*α,L(ℝn). As in the case of L-Hölder spaces, our L-Zygmund spaces refine the scale of classical Zygmund spaces; more precisely, we have again the following inclusions:
(5.5)C*β(Rn)⊊C*α,L(Rn)⊊C*γ(Rn),wheneverγ<α<β.
The definition of C*α,L(ℝn) can give the impression that it might depend on the choice of the LP-pair; however, this is not the case, as shown by the ensuing result.Proposition 5.1.
The definition ofC*α,L(ℝn) does not depend on the choice of the LP-pair. Moreover, different LP-pairs lead to equivalent norms.In view of Proposition5.1, we may employ an LP-pair coming from a continuous Littlewood-Paley decomposition of the unity (cf. Example 4.3) in the definition of C*α,L(ℝn). Therefore, when L≡1, we recover the classical Zygmund space C*α,L(ℝn)=C*α(ℝn) [5]. Proposition 5.1 follows at once from the following lemma.Lemma 5.2.
Letf∈C*α,L(ℝn), then for every θ∈𝒮(ℝn) there holds
(5.6)‖f*θ‖L∞≤C‖f‖C*α,L,
where ∥f∥C*α,L is given by (5.4). Furthermore, if 𝔅⊂𝒮(ℝn) is a bounded set such that μm(θ)=0 for all θ∈𝔅 and all multi-index m≤[α], then
(5.7)sup⁡x∈Rnsup⁡0<y≤11yαL(y)|Wθf(x,y)|≤C‖f‖C*α,L,∀θ∈B.Proof.
The estimate (5.6) easily follows from the representation (4.2) of f from Proposition 4.4. Let us show (5.7). We retain the notation from the statement of Proposition 4.4. In view of (4.2), a quick calculation yields
(5.8)Wθf(x,y)=F1(x,y)+F2(x,y)+F3(x,y),
where
(5.9)F1(x,y)=1(2π)n∫ℝn‍(f*φ^)(x+yt)θ¯(t)dt,F2(x,y)=1cψ,η∫01/y‍∫Rn‍Wψf(yb+x,ya)Wη¯θ2,y(b,a)dbdaa,F3(x,y)=1cψ,η∫01‍∫ℝn‍𝒲ψf(yb+x,ya)𝒲η¯θ3,y(b,a)dbdaa,
with θ^2,y(ξ)=θ¯^(ξ)(1-φ(ξ))(1-φ(ξ/y)) and θ^3,y(ξ)=θ¯^(ξ)φ(ξ)(1-φ(ξ/y)). To estimate F1, we first observe that if α<0, then
(5.10)sup⁡0<y≤1sup⁡x∈Rn|F1(x,y)|yαL(y)≤C|f|C*α,L‖θ‖L1sup⁡0<y≤11yαL(y)≤C′|f|C*α,L,
because slowly varying functions satisfy the estimates yɛ<CL(y) for any exponent ɛ>0 [7, 8]. When α≥0, we have that (f*φ^) is a C∞-function with bounded derivatives of any order. Then, by the Taylor formula, (5.6) (with θ=∂mφ^), and the assumption μm(θ)=0 for |m|≤[α], we obtain
(5.11)sup⁡0<y≤1sup⁡x∈Rn|F1(x,y)|yαL(y)≤sup⁡0<y≤1C‖f‖C*α,Ly[α]+1-αL(y)∫Rn‍|t|[α]+1|θ(t)|dt≤C′‖f‖C*α,L.
We now bound F2 and F3. If ɛ>0, Potter's estimate [7, page 25] gives the existence of a constant C=Cɛ such that
(5.12)L(ay)L(y)<C(aɛ+1aɛ),∀0<y<1,a<1/y.
Thus,
(5.13)|F2(x,y)|yαL(y)≤‖f‖C*α,Lcψ,η∫01/y‍∫Rn‍aα-1L(ay)L(y)|Wη¯θ2,y(b,a)|dbda≤C′‖f‖C*α,L∫0∞‍∫Rn‍(aα+aα-2)|Wη¯θ2,y(b,a)|dbda.
Notice that {θ2,y∈𝒮0(ℝn):θ∈𝔅,y∈(0,1]} is a bounded set in 𝒮0(ℝn) because the derivatives of φ are supported in {ξ:σ≤|ξ|≤r}. Thus, due to the continuity of 𝒲η¯ (cf. Theorem 3.1), {𝒲η¯θy:θ∈𝔅,y∈(0,1]} is a bounded set in 𝒮(ℍn+1). This implies that the integrals involved in the very last estimate are uniformly bounded for θ∈𝔅 and y∈(0,1]. Consequently, we obtain that
(5.14)sup⁡0<y≤1sup⁡x∈Rn|F2(x,y)|yαL(y)≤C‖f‖C*α,L.
Next, for F3, we have
(5.15)|F3(x,y)|yαL(y)≤C′‖f‖C*α,L∫01‍∫Rn‍(aα-3/2+aα-1/2)|Wη¯θ3,y(b,a)|dbda.
As in the proof of Theorem 3.1, the above integrand can be uniformly estimated by C(1+|b|2)-n. This completes the proof of (5.7).We obtain the following useful properties.Corollary 5.3.
The following properties hold:(i)
∂m:C*α,L(ℝn)→C*α-|m|,L(ℝn) is continuous, for any m∈ℕn,(ii)
iff*ϕ∈L∞(ℝn) and ∂jf∈C*α-1,L(ℝn) for j=1,…,n, then f∈C*α,L(ℝn),(iii)
the mapping(1-Δ)β/2 is an isomorphism of the space C*α,L(ℝn) onto C*α-β,L(ℝn), for arbitrary α,β∈ℝ.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 that Δf∈C*α-2,L(ℝn). In view of Proposition 5.1, it remains to observe that
(5.16)1yαL(y)WΔψf(x,y)=1yα-2L(y)Wψ(Δf)(x,y).
(iii) Since(1-Δ)-β/2 is the inverse of (1-Δ)β/2, it suffices to show that (1-Δ)β/2 maps continuously C*α,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∥C*α,L. We also have
(5.17)Wψ(1-Δ)β/2f(x,y)=1(2π)nyβ〈f^(ξ),eix⋅ξ(y2+|yξ|2)β/2ψ^¯(yξ)〉=1yβWθyf(x,y),whereθy=(y2-Δ)β/2ψ.
Finally, we can apply (5.7) because 𝔅={(y2-Δ)β/2ψ}y∈(0,1] is a bounded set in 𝒮(ℝn) and μm(θy)=0 for each multi-index |m|≤[α].We can also use Proposition4.4 to show that C*α,L(ℝn) is a Banach space, as stated in the following proposition.Proposition 5.4.
The spaceC*α,L(ℝn) is a Banach space when provided with the norm (5.4).Proof.
Letη,  φ,  θ1,  θ2 be as in the statement of Proposition 4.4. Suppose that {fj}j=0∞ is a Cauchy sequence in C*α,L(ℝn). Then, there exist continuous functions g∈L∞(ℝn) and G defined on ℝn×(0,1] such that fj*ϕ→g in L∞(ℝn) and
(5.18)lim⁡j→∞sup⁡y∈(0,1]sup⁡x∈Rn1yαL(y)|Wψfj(x,y)-G(x,y)|=0.
We define the distribution f∈𝒮′(ℝn) whose action on test functions θ∈𝒮(ℝn) is given by
(5.19)〈f,θ〉:=∫Rn‍g(t)θ1(t)dt+1cψ,η∫01‍∫Rn‍G(x,y)Wη¯θ2(x,y)dxdyy.
Since the fj have the representation (4.2), we immediately see that fj→f in 𝒮′(ℝn). Thus, pointwisely,
(5.20)(f*ϕ)(t)=limj→∞(fj*ϕ)(t)=g(t),Wψf(x,y)=lim⁡j→∞Wψfj(x,y)=G(x,y).
This implies that limj→∞∥f-fj∥C*α,L=0, and so C*α,L(ℝn) is complete.We have arrived to the main and last result of this section. It provides theL-Hölderian characterization of the L-Zygmund spaces of positive exponent. We will use in its proof a technique based on the Tauberian theorem for pointwise weak regularity of vector-valued distributions, explained in Section 4.4. We denote below by Cb(ℝn) the Banach space of continuous and bounded functions.Theorem 5.5.
Letα>0.(a)
Ifα∉ℕ, then C*α,L(ℝn)=Cα,L(ℝn). Moreover, the norms (5.4) and (5.1) are equivalent.(b)
Ifα=p+1∈ℕ, then C*p+1,L(ℝn) consists of functions with continuous derivatives up to order p such that
(5.21)∑|m|≤p‍‖∂mf‖L∞+∑|m|=p‍sup⁡t∈Rn0<|h|≤1|∂mf(t+h)+∂mf(t-h)-2∂mf(t)||h|L(|h|)<∞.
In addition, (5.21) produces a norm that is equivalent to (5.4).Proof.
Observe that theL-Hölderian type norm (resp. (5.21)) is clearly stronger than (5.4). Thus, if we show the equality of the spaces in (a) and (b), the equivalence of norms would be a direct consequence of the open mapping theorem.
Suppose thatf∈C*α,L(ℝn). Consider the Cb(ℝn)-valued distribution f∈𝒮′(ℝtn,Cb(ℝξn)) given by f(t)(ξ):=f(t+ξ), that is, the one whose action on test functions is given by
(5.22)〈f(t),θ(t)〉(ξ)=〈f(t+ξ),θ(t)〉=(f*θˇ)(ξ),θ∈S(Rn),ξ∈Rn.
It does take values in Cb(ℝn) because of (5.6). Clearly, 𝒲ψf(x,y)(ξ)=𝒲ψf(x+ξ,y). By (5.4) and Potter's estimate [7, page 25], we have that
(5.23)‖Wψf(ɛx,ɛy)‖Cb(Rn)≤CɛαL(ɛ)yα-1∀ɛ∈(0,1),(x,y)∈Rn×(0,1].
Therefore, the Tauberian Theorem 4.5 yields f∈C*,wα,L(0,Cb(ℝn)). Now, the Łojasiewicz point values ∂mf(0)=vm∈Cb(ℝn) exist, distributionally, for |m|<α. It explicitly means that for all θ∈𝒮(ℝn)(5.24)lim⁡ɛ→0+∂mf*θˇɛ=lim⁡ɛ→0+〈∂mf(ɛt),θ(t)〉=μ0(θ)vminCb(Rξn),
where θˇɛ(t)=ɛ-nθ(-t/ɛ). If we now take θ with μ0(θ)=1, we then conclude that ∂mf=vm∈Cb(ℝn) for each |m|<α. It remains in both cases to deal with the estimates for ∂mf; notice that ∂mf∈C*,wα-[α],L(0,Cb(ℝn)) when |m|=[α] and ∂mf∈C*,w1,L(0,Cb(ℝn)) when |m|=p. We now divide the proof into two cases.
Case
α
∉
ℕ. Fix a multi-index |m|=[α]. It suffices to show
(5.25)sup⁡0<|x-t|<1|∂mf(x)-∂mf(t)||x-t|α-[α]L(|x-t|)<∞.
We had already seen that ∂mf(t)(ξ)=∂mf(t+ξ)∈Cwα-[α],L(0,Cb(ℝξn))=C*,wα-[α],L(0,Cb(ℝξn)), that is,
(5.26)μ0(θ)∂mf(ξ)-∫Rn‍∂mf(ξ+ɛt)θ(t)dt=O(ɛα-[α]L(ɛ)),ɛ⟶0+,
in the space Cb(ℝξn), 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
(5.27)sup⁡ξ∈Rn|∂mf(ξ+h)-∂mf(ξ)|≤2sup⁡ξ∈Rn|∂mf(ξ)-∫Rn‍∂mf(|h|t+ξ)θ(t)dt|+sup⁡ξ∈Rn|∫Rn‍∂mf(ξ+|h|t)(θ(t)-θ(t-|h|-1h))dt|=O(|h|α-[α]L(|h|)),
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∈C*,w1,L(0,Cb(ℝn)), which, as commented in Section 4.4, yields the distributional expansion
(5.28)∂mf(ɛt)(ξ)=∂mf(ξ)+ɛ∑j=1n‍tjcj(ɛ,ξ)+O(ɛL(ɛ)),0<ɛ≤1,
in 𝒮′(ℝtn,Cb(ℝξn), where the cj(ɛ,·) are continuous Cb(ℝξn)-valued functions in ɛ. 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
(5.29)∂mf(ξ)=∫Rn‍∂mf(ξ+|h|t)θ(t)dt+O(|h|L(|h|)),0<|h|≤1,
uniformly in ξ∈ℝn. Since {θω:=θ(·+ω)+θ(·-ω)-2θ:|ω|=1} is compact in 𝒮(ℝn) and μm(θω)=0 for |m|≤1, the relations (5.28) and (5.29) give
(5.30)sup⁡ξ∈Rn|∂mf(ξ+h)+∂mf(ξ-h)-2∂mf(ξ)|≤3sup⁡ξ∈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,
as claimed.


## 5.1.L-Hölder Spaces

We now introduce weighted Hölder spaces with respect toL. They were already defined and studied in [10]. Let α∈ℝ+∖ℕ. We say that a function f belongs to the space Cα,L(ℝn) if f has continuous derivatives up to order less than α and
(5.1)‖f‖Cα,L:=∑|j|≤[α]‍sup⁡t∈Rn|∂jf(t)|+∑|m|=[α]‍sup⁡0<|t-x|≤1|∂mf(t)-∂mf(x)||t-x|α-[α]L(|t-x|)<∞.
When α=p+1∈ℕ, we replace the previous requirement by
(5.2)‖f‖Cα,L:=∑|j|≤p‍sup⁡t∈Rn|∂jf(t)|+∑|m|=p‍sup⁡0<|t-x|≤1|∂mf(t)-∂mf(x)||t-x|L(|t-x|)<∞.The spaceCα,L(ℝn) is clearly a Banach space with the above norm. The conditions imposed over L ensure that Cα,L(ℝn) depends only on the behavior of L near 0, thus, it is invariant under dilations. When L≡1, this space reduces to Cα,L(ℝn)=Cα(ℝn), the usual global (inhomogeneous) Hölder space [2, 3, 5]. Consequently, as in [10], we call Cα,L(ℝn) the global Hölder space with respect to L. Note that, because of the properties of L [7, 8], we have the following interesting inclusion relations:
(5.3)Cβ(Rn)⊊Cα,L(Rn)⊊Cγ(Rn),whenever0<γ<α<β.


## 5.2.L-Zygmund Spaces

We now proceed to define the weighted Zygmund spaceC*α,L(ℝn). Let α∈ℝ and fix an LP-pair (ϕ,ψ) of order α. A distribution f∈𝒮′(ℝn) is said to belong to the L-Zygmund space C*α,L(ℝn) of exponent α if
(5.4)‖f‖C*α,L:=‖f*ϕ‖L∞+sup⁡x∈Rnsup⁡0<y≤11yαL(y)|Wψf(x,y)|<∞.Observe that we clearly haveCα,L(ℝn)⊆C*α,L(ℝn), for α>0. We will show that if α∈ℝ+∖ℕ, we actually have the equality Cα,L(ℝn)=C*α,L(ℝn). When α is a positive integer, we have in turn Cα,L(ℝn)⊊C*α,L(ℝn). As in the case of L-Hölder spaces, our L-Zygmund spaces refine the scale of classical Zygmund spaces; more precisely, we have again the following inclusions:
(5.5)C*β(Rn)⊊C*α,L(Rn)⊊C*γ(Rn),wheneverγ<α<β.
The definition of C*α,L(ℝn) can give the impression that it might depend on the choice of the LP-pair; however, this is not the case, as shown by the ensuing result.Proposition 5.1.
The definition ofC*α,L(ℝn) does not depend on the choice of the LP-pair. Moreover, different LP-pairs lead to equivalent norms.In view of Proposition5.1, we may employ an LP-pair coming from a continuous Littlewood-Paley decomposition of the unity (cf. Example 4.3) in the definition of C*α,L(ℝn). Therefore, when L≡1, we recover the classical Zygmund space C*α,L(ℝn)=C*α(ℝn) [5]. Proposition 5.1 follows at once from the following lemma.Lemma 5.2.
Letf∈C*α,L(ℝn), then for every θ∈𝒮(ℝn) there holds
(5.6)‖f*θ‖L∞≤C‖f‖C*α,L,
where ∥f∥C*α,L is given by (5.4). Furthermore, if 𝔅⊂𝒮(ℝn) is a bounded set such that μm(θ)=0 for all θ∈𝔅 and all multi-index m≤[α], then
(5.7)sup⁡x∈Rnsup⁡0<y≤11yαL(y)|Wθf(x,y)|≤C‖f‖C*α,L,∀θ∈B.Proof.
The estimate (5.6) easily follows from the representation (4.2) of f from Proposition 4.4. Let us show (5.7). We retain the notation from the statement of Proposition 4.4. In view of (4.2), a quick calculation yields
(5.8)Wθf(x,y)=F1(x,y)+F2(x,y)+F3(x,y),
where
(5.9)F1(x,y)=1(2π)n∫ℝn‍(f*φ^)(x+yt)θ¯(t)dt,F2(x,y)=1cψ,η∫01/y‍∫Rn‍Wψf(yb+x,ya)Wη¯θ2,y(b,a)dbdaa,F3(x,y)=1cψ,η∫01‍∫ℝn‍𝒲ψf(yb+x,ya)𝒲η¯θ3,y(b,a)dbdaa,
with θ^2,y(ξ)=θ¯^(ξ)(1-φ(ξ))(1-φ(ξ/y)) and θ^3,y(ξ)=θ¯^(ξ)φ(ξ)(1-φ(ξ/y)). To estimate F1, we first observe that if α<0, then
(5.10)sup⁡0<y≤1sup⁡x∈Rn|F1(x,y)|yαL(y)≤C|f|C*α,L‖θ‖L1sup⁡0<y≤11yαL(y)≤C′|f|C*α,L,
because slowly varying functions satisfy the estimates yɛ<CL(y) for any exponent ɛ>0 [7, 8]. When α≥0, we have that (f*φ^) is a C∞-function with bounded derivatives of any order. Then, by the Taylor formula, (5.6) (with θ=∂mφ^), and the assumption μm(θ)=0 for |m|≤[α], we obtain
(5.11)sup⁡0<y≤1sup⁡x∈Rn|F1(x,y)|yαL(y)≤sup⁡0<y≤1C‖f‖C*α,Ly[α]+1-αL(y)∫Rn‍|t|[α]+1|θ(t)|dt≤C′‖f‖C*α,L.
We now bound F2 and F3. If ɛ>0, Potter's estimate [7, page 25] gives the existence of a constant C=Cɛ such that
(5.12)L(ay)L(y)<C(aɛ+1aɛ),∀0<y<1,a<1/y.
Thus,
(5.13)|F2(x,y)|yαL(y)≤‖f‖C*α,Lcψ,η∫01/y‍∫Rn‍aα-1L(ay)L(y)|Wη¯θ2,y(b,a)|dbda≤C′‖f‖C*α,L∫0∞‍∫Rn‍(aα+aα-2)|Wη¯θ2,y(b,a)|dbda.
Notice that {θ2,y∈𝒮0(ℝn):θ∈𝔅,y∈(0,1]} is a bounded set in 𝒮0(ℝn) because the derivatives of φ are supported in {ξ:σ≤|ξ|≤r}. Thus, due to the continuity of 𝒲η¯ (cf. Theorem 3.1), {𝒲η¯θy:θ∈𝔅,y∈(0,1]} is a bounded set in 𝒮(ℍn+1). This implies that the integrals involved in the very last estimate are uniformly bounded for θ∈𝔅 and y∈(0,1]. Consequently, we obtain that
(5.14)sup⁡0<y≤1sup⁡x∈Rn|F2(x,y)|yαL(y)≤C‖f‖C*α,L.
Next, for F3, we have
(5.15)|F3(x,y)|yαL(y)≤C′‖f‖C*α,L∫01‍∫Rn‍(aα-3/2+aα-1/2)|Wη¯θ3,y(b,a)|dbda.
As in the proof of Theorem 3.1, the above integrand can be uniformly estimated by C(1+|b|2)-n. This completes the proof of (5.7).We obtain the following useful properties.Corollary 5.3.
The following properties hold:(i)
∂m:C*α,L(ℝn)→C*α-|m|,L(ℝn) is continuous, for any m∈ℕn,(ii)
iff*ϕ∈L∞(ℝn) and ∂jf∈C*α-1,L(ℝn) for j=1,…,n, then f∈C*α,L(ℝn),(iii)
the mapping(1-Δ)β/2 is an isomorphism of the space C*α,L(ℝn) onto C*α-β,L(ℝn), for arbitrary α,β∈ℝ.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 that Δf∈C*α-2,L(ℝn). In view of Proposition 5.1, it remains to observe that
(5.16)1yαL(y)WΔψf(x,y)=1yα-2L(y)Wψ(Δf)(x,y).
(iii) Since(1-Δ)-β/2 is the inverse of (1-Δ)β/2, it suffices to show that (1-Δ)β/2 maps continuously C*α,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∥C*α,L. We also have
(5.17)Wψ(1-Δ)β/2f(x,y)=1(2π)nyβ〈f^(ξ),eix⋅ξ(y2+|yξ|2)β/2ψ^¯(yξ)〉=1yβWθyf(x,y),whereθy=(y2-Δ)β/2ψ.
Finally, we can apply (5.7) because 𝔅={(y2-Δ)β/2ψ}y∈(0,1] is a bounded set in 𝒮(ℝn) and μm(θy)=0 for each multi-index |m|≤[α].We can also use Proposition4.4 to show that C*α,L(ℝn) is a Banach space, as stated in the following proposition.Proposition 5.4.
The spaceC*α,L(ℝn) is a Banach space when provided with the norm (5.4).Proof.
Letη,  φ,  θ1,  θ2 be as in the statement of Proposition 4.4. Suppose that {fj}j=0∞ is a Cauchy sequence in C*α,L(ℝn). Then, there exist continuous functions g∈L∞(ℝn) and G defined on ℝn×(0,1] such that fj*ϕ→g in L∞(ℝn) and
(5.18)lim⁡j→∞sup⁡y∈(0,1]sup⁡x∈Rn1yαL(y)|Wψfj(x,y)-G(x,y)|=0.
We define the distribution f∈𝒮′(ℝn) whose action on test functions θ∈𝒮(ℝn) is given by
(5.19)〈f,θ〉:=∫Rn‍g(t)θ1(t)dt+1cψ,η∫01‍∫Rn‍G(x,y)Wη¯θ2(x,y)dxdyy.
Since the fj have the representation (4.2), we immediately see that fj→f in 𝒮′(ℝn). Thus, pointwisely,
(5.20)(f*ϕ)(t)=limj→∞(fj*ϕ)(t)=g(t),Wψf(x,y)=lim⁡j→∞Wψfj(x,y)=G(x,y).
This implies that limj→∞∥f-fj∥C*α,L=0, and so C*α,L(ℝn) is complete.We have arrived to the main and last result of this section. It provides theL-Hölderian characterization of the L-Zygmund spaces of positive exponent. We will use in its proof a technique based on the Tauberian theorem for pointwise weak regularity of vector-valued distributions, explained in Section 4.4. We denote below by Cb(ℝn) the Banach space of continuous and bounded functions.Theorem 5.5.
Letα>0.(a)
Ifα∉ℕ, then C*α,L(ℝn)=Cα,L(ℝn). Moreover, the norms (5.4) and (5.1) are equivalent.(b)
Ifα=p+1∈ℕ, then C*p+1,L(ℝn) consists of functions with continuous derivatives up to order p such that
(5.21)∑|m|≤p‍‖∂mf‖L∞+∑|m|=p‍sup⁡t∈Rn0<|h|≤1|∂mf(t+h)+∂mf(t-h)-2∂mf(t)||h|L(|h|)<∞.
In addition, (5.21) produces a norm that is equivalent to (5.4).Proof.
Observe that theL-Hölderian type norm (resp. (5.21)) is clearly stronger than (5.4). Thus, if we show the equality of the spaces in (a) and (b), the equivalence of norms would be a direct consequence of the open mapping theorem.
Suppose thatf∈C*α,L(ℝn). Consider the Cb(ℝn)-valued distribution f∈𝒮′(ℝtn,Cb(ℝξn)) given by f(t)(ξ):=f(t+ξ), that is, the one whose action on test functions is given by
(5.22)〈f(t),θ(t)〉(ξ)=〈f(t+ξ),θ(t)〉=(f*θˇ)(ξ),θ∈S(Rn),ξ∈Rn.
It does take values in Cb(ℝn) because of (5.6). Clearly, 𝒲ψf(x,y)(ξ)=𝒲ψf(x+ξ,y). By (5.4) and Potter's estimate [7, page 25], we have that
(5.23)‖Wψf(ɛx,ɛy)‖Cb(Rn)≤CɛαL(ɛ)yα-1∀ɛ∈(0,1),(x,y)∈Rn×(0,1].
Therefore, the Tauberian Theorem 4.5 yields f∈C*,wα,L(0,Cb(ℝn)). Now, the Łojasiewicz point values ∂mf(0)=vm∈Cb(ℝn) exist, distributionally, for |m|<α. It explicitly means that for all θ∈𝒮(ℝn)(5.24)lim⁡ɛ→0+∂mf*θˇɛ=lim⁡ɛ→0+〈∂mf(ɛt),θ(t)〉=μ0(θ)vminCb(Rξn),
where θˇɛ(t)=ɛ-nθ(-t/ɛ). If we now take θ with μ0(θ)=1, we then conclude that ∂mf=vm∈Cb(ℝn) for each |m|<α. It remains in both cases to deal with the estimates for ∂mf; notice that ∂mf∈C*,wα-[α],L(0,Cb(ℝn)) when |m|=[α] and ∂mf∈C*,w1,L(0,Cb(ℝn)) when |m|=p. We now divide the proof into two cases.
Case
α
∉
ℕ. Fix a multi-index |m|=[α]. It suffices to show
(5.25)sup⁡0<|x-t|<1|∂mf(x)-∂mf(t)||x-t|α-[α]L(|x-t|)<∞.
We had already seen that ∂mf(t)(ξ)=∂mf(t+ξ)∈Cwα-[α],L(0,Cb(ℝξn))=C*,wα-[α],L(0,Cb(ℝξn)), that is,
(5.26)μ0(θ)∂mf(ξ)-∫Rn‍∂mf(ξ+ɛt)θ(t)dt=O(ɛα-[α]L(ɛ)),ɛ⟶0+,
in the space Cb(ℝξn), 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
(5.27)sup⁡ξ∈Rn|∂mf(ξ+h)-∂mf(ξ)|≤2sup⁡ξ∈Rn|∂mf(ξ)-∫Rn‍∂mf(|h|t+ξ)θ(t)dt|+sup⁡ξ∈Rn|∫Rn‍∂mf(ξ+|h|t)(θ(t)-θ(t-|h|-1h))dt|=O(|h|α-[α]L(|h|)),
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∈C*,w1,L(0,Cb(ℝn)), which, as commented in Section 4.4, yields the distributional expansion
(5.28)∂mf(ɛt)(ξ)=∂mf(ξ)+ɛ∑j=1n‍tjcj(ɛ,ξ)+O(ɛL(ɛ)),0<ɛ≤1,
in 𝒮′(ℝtn,Cb(ℝξn), where the cj(ɛ,·) are continuous Cb(ℝξn)-valued functions in ɛ. 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
(5.29)∂mf(ξ)=∫Rn‍∂mf(ξ+|h|t)θ(t)dt+O(|h|L(|h|)),0<|h|≤1,
uniformly in ξ∈ℝn. Since {θω:=θ(·+ω)+θ(·-ω)-2θ:|ω|=1} is compact in 𝒮(ℝn) and μm(θω)=0 for |m|≤1, the relations (5.28) and (5.29) give
(5.30)sup⁡ξ∈Rn|∂mf(ξ+h)+∂mf(ξ-h)-2∂mf(ξ)|≤3sup⁡ξ∈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,
as claimed.



---

*Source: 815475-2012-07-14.xml*