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

Abstract

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces 𝒮₀(ℝ^𝑛) and 𝒮(ℍ^𝑛+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.

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∗(ℝ^𝑛) satisfy the size estimate |𝒲_𝜓𝑓(𝑥,𝑦)|≤𝐶𝑦^𝛼, which, plus a side condition, essentially characterizes the space itself. We will replace the regularity measurement 𝑦^𝛼 by weights from the interesting class of regularly varying functions [7, 8]. Familiar functions such as 𝑦^𝛼,𝑦^𝛼|log𝑦|^𝛽,𝑦^𝛼|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(ℝ^𝑛) and 𝒮(ℍ^𝑛+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 Section 4.1, 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∗(ℝ^𝑛), 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 spaces 𝐶^𝛼,𝐿(ℝ^𝑛) and 𝐶^∗𝛼,𝐿(ℝ^𝑛), 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: 𝐶^𝛽(𝐶ℝ^𝑛)⊂𝐶^𝛼,𝐿(ℝ^𝑛)⊂𝐶^𝛾(ℝ^𝑛),when0<𝛾<𝛼<𝛽,_𝛽∗(ℝ^𝑛)⊂𝐶^∗𝛼,𝐿(ℝ^𝑛)⊂𝐶^𝛾∗(ℝ^𝑛),if𝛾<𝛼<𝛽.(1.1) 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 Section 2 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 by ℍ^𝑛+1=ℝ^𝑛×ℝ₊ the upper half space. If 𝑥∈ℝ^𝑛 and 𝑚∈ℕ^𝑛, then |𝑥| denotes the euclidean norm, 𝑥^𝑚=𝑥^𝑚1₁⋯𝑥^𝑚𝑛_𝑛,  𝜕^𝑚=𝜕^𝑚𝑥=𝜕^𝑚1_𝑥1⋯𝜕^𝑚𝑛_𝑥𝑛,  𝑚!=𝑚₁!𝑚₂!⋯𝑚_𝑛! and |𝑚|=𝑚₁+⋯+𝑚_𝑛. 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 ℝ^𝑛 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(ℝ^𝑛). We will fix constants in the Fourier transform as 𝜑(𝜉)=∫_ℝ𝑛𝜑(𝑡)𝑒^{−𝑖𝜉⋅𝑡}d𝑡. The moments of 𝜑∈𝒮(ℝ^𝑛) are denoted by 𝜇_𝑚(𝜑)=∫_ℝ𝑛𝑡^𝑚𝜑(𝑡)d𝑡, 𝑚∈ℕ^𝑛.

Following [1], the space of highly time-frequency localized functions 𝒮₀(ℝ^𝑛) is defined as 𝒮₀(ℝ^𝑛)={𝜑∈𝒮(ℝ^𝑛)∶𝜇_𝑚(𝜑)=0,forall𝑚∈ℕ^𝑛}; it is provided with the relative topology inhered from 𝒮(ℝ^𝑛). In [1], the topology of 𝒮₀(ℝ^𝑛) is introduced in an apparently different way; however, both approaches are equivalent in view of the open mapping theorem. Observe that 𝒮₀(ℝ^𝑛) is a closed subspace of 𝒮(ℝ^𝑛) and that 𝜑∈𝒮₀(ℝ^𝑛) if and only if 𝜕^𝑚𝜑(0)=0, for all 𝑚∈ℕ^𝑛. It is important to point out that 𝒮₀(ℝ^𝑛) is also well known in the literature as the Lizorkin space of test functions (cf. [13]).

The space 𝒮(ℍ^𝑛+1) of highly localized functions on the half space [1] consists of those Φ∈𝐶^∞(ℍ^𝑛+1) for which 𝜌^{0𝑙,𝑘,𝜈,𝑚}(Φ)=sup_{(𝑥,𝑦)∈ℍ𝑛+1}𝑦^𝑙+1𝑦^𝑙1+|𝑥|²^𝑘/2||𝜕^𝜈𝑦𝜕^𝑚𝑥Φ(𝑥,𝑦)||<∞,(2.1) for all 𝑙,𝑘,𝜈∈ℕ, and 𝑚∈ℕ^𝑛. The canonical topology on 𝒮(ℍ^𝑛+1) is induced by this family of seminorms [1]. For later use, we will denote by 𝜌_𝑘,𝑚 the corresponding seminorms in 𝒮(ℝ^𝑛), namely, 𝜌_𝑘,𝑚(𝜑)=sup_{𝑡∈ℝ𝑛}1+|𝑡|²^𝑘/2||𝜕^𝑚𝜑(𝑡)||,𝑘∈ℕ,𝑚∈ℕ^𝑛.(2.2)

The corresponding duals of these three spaces are 𝒮^(ℝ^𝑛), 𝒮^0(ℝ^𝑛), and 𝒮^(ℍ^𝑛+1). They are, respectively, the spaces of tempered distributions, Lizorkin distributions, and distributions of slow growth on ℍ^𝑛+1. Since the elements of 𝒮₀(ℝ^𝑛) are orthogonal to every polynomial, 𝒮^0(ℝ^𝑛) can be canonically identified with the quotient space of 𝒮^(ℝ^𝑛) 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^(ℝ^𝑛,𝑋)=𝐿_𝑏(𝒮(ℝ^𝑛),𝑋), namely, the space of continuous linear mappings from 𝒮(ℝ^𝑛) into 𝑋.

2.2. Wavelet Transform

In this paper a wavelet simply means a function 𝜓∈𝒮(ℝ^𝑛) that satisfies 𝜇₀(𝜓)=∫_ℝ𝑛𝜓(𝑡)d𝑡=0.

The wavelet transform of 𝑓∈𝒮^(ℝ^𝑛) with respect to the wavelet 𝜓∈𝒮(ℝ^𝑛) is defined as 𝒲_𝜓𝑓(𝑥,𝑦)=𝑓(𝑡),1𝑦^𝑛𝜓𝑡−𝑥𝑦=𝑦1^𝑛_ℝ𝑛𝑓(𝑡)𝜓𝑡−𝑥𝑦d𝑡,(2.3) where (𝑥,𝑦)∈ℍ^𝑛+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∈𝒮^0(ℝ^𝑛) if the wavelet 𝜓∈𝒮₀(ℝ^𝑛). The wavelet transform can be defined exactly in the same way for vector-valued distributions.

2.3. Wavelet Synthesis Operator

Let 𝜂∈𝒮₀(ℝ^𝑛). The wavelet synthesis transform of Φ∈𝒮(ℍ^𝑛+1) with respect to the wavelet 𝜂 is defined as ℳ_𝜂Φ(𝑡)=^∞0_ℝ𝑛Φ(𝑥,𝑦)1𝑦^𝑛𝜂𝑡−𝑥𝑦d𝑥𝑦d𝑦,𝑡∈ℝ^𝑛.(2.4) Observe that the operator ℳ_𝜂 may be extended to act on the space 𝒮^(ℍ^𝑛+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(ℍ^𝑛+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 that 𝜓∈𝒮₀(ℝ^𝑛) admits a reconstruction wavelet 𝜂∈𝒮₀(ℝ^𝑛). More precisely, it means that the constant 𝑐_𝜓,𝜂=𝑐_𝜓,𝜂(𝜔)=^∞0𝜓(𝑟𝜔)𝜂̂(𝑟𝜔)d𝑟𝑟,𝜔∈𝕊^𝑛−1(2.5) is different from zero and independent of the direction 𝜔. Then, a straightforward calculation [1] shows that Id_𝒮0(ℝ^𝑛₎=1𝑐_𝜓,𝜂ℳ_𝜂𝒲_𝜓.(2.6) It is worth pointing out that (2.6) is also valid [1, 10] when 𝒲_𝜓 and ℳ_𝜂 act on the spaces 𝒮^0(ℝ^𝑛) and 𝒮^(ℍ^𝑛+1), respectively.

Furthermore, it is very important to emphasize that a wavelet 𝜓 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 function 𝜑∈𝒮(ℝ^𝑛) is said to be nondegenerate if for any 𝜔∈𝕊^𝑛−1 the function of one variable 𝑅_𝜔(𝑟)=𝜑(𝑟𝜔)∈𝐶^∞[0,∞) is not identically zero, that is, supp𝑅_𝜔≠∅, for each 𝜔∈𝕊^𝑛−1.

3. The Wavelet Transform of Test Functions

The wavelet and wavelet synthesis transforms induce the following bilinear mappings: 𝒲∶(𝜓,𝜑)⟼𝒲_𝜓𝜑,ℳ∶(𝜂,Φ)⟼ℳ_𝜂Φ.(3.1) 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(ℝ^𝑛)×𝒮₀(ℝ^𝑛)→𝒮(ℍ^𝑛+1), (ii)ℳ∶𝒮₀(ℝ^𝑛)×𝒮(ℍ^𝑛+1)→𝒮₀(ℝ^𝑛)are continuous.

Proof. Continuity of the Wavelet Mapping. We will prove that for arbitrary 𝑙,𝑘,𝜈∈ℕ,𝑚∈ℕ^𝑛, there exist 𝑁∈ℕ and 𝐶>0 such that 𝜌^{0𝑙,𝑘,𝜈,𝑚}𝒲_𝜓𝜑≤𝐶_{𝑖′,|𝑗′|,𝑖,|𝑗|≤𝑁}𝜌_{𝑖′,𝑗′}(𝜓)𝜌_𝑖,𝑗(𝜑).(3.2) We begin by making some reductions. Observe that, for constants 𝑐_𝑗 which do not depend on 𝜑 and 𝜓, 𝜕^𝜈𝑦𝜕^𝑚𝑥𝒲_𝜓𝜑(𝑥,𝑦)=𝜕^𝜈𝑦𝜕^𝑚𝑥_ℝ𝑛=𝜑(𝑦𝑡+𝑥)𝜓(𝑡)d𝑡_{|𝑗|≤𝜈}𝑐_𝑗_ℝ𝑛=𝜕^𝑚+𝑗𝜑(𝑦𝑡+𝑥)𝑡^𝑗𝜓(𝑡)d𝑡_{|𝑗|≤𝜈}𝑐_𝑗𝒲_𝜓𝑗𝜑_𝑚+𝑗(𝑥,𝑦),(3.3) where 𝜓_𝑗(𝑡)=𝑡^𝑗𝜓(𝑡)∈𝒮₀(ℝ^𝑛) and 𝜑_𝑚+𝑗=𝜕^𝑚+𝑗𝜑∈𝒮₀(ℝ^𝑛). 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 𝑐_𝑟,𝑠 independent of 𝜓 and 𝜑, 1+|𝑥|²^𝑘/2𝒲_𝜓𝜑(𝑥,𝑦)=1(2𝜋)^𝑛(1−Δ^𝜉=1)^𝑘/2𝑒^{𝑖𝜉⋅𝑥},𝜑(𝜉)𝜓(𝑦𝜉)(2𝜋)^𝑛_{|𝑟|+|𝑠|≤𝑘}𝑐_𝑟,𝑠𝑦^|𝑠|_ℝ𝑛=1𝑒^{𝑖𝜉⋅𝑥}𝜕^𝑟𝜑(𝜉)𝜕^𝑠𝜉𝜓(𝑦𝜉)d𝜉(2𝜋)^𝑛_{|𝑟|+|𝑠|≤𝑘}𝑐_𝑟,𝑠𝑦^|𝑠|𝒲_𝜓𝑠𝜑_𝑟(𝑥,𝑦),(3.4) where 𝜑_𝑟(𝑡)=(−𝑖𝑡)^𝑟𝜑(𝑡) and 𝜓_𝑠(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 term 𝑦^𝑙|𝒲_𝜓𝜑(𝑥,𝑦)|. Since 𝜕^𝑗𝜑(0)=0 for every 𝑗∈ℕ^𝑛, we can apply the Taylor formula to obtain 𝜑(𝜉)=_{||𝑗||=𝑙−𝑛}𝜕^𝑗𝜑𝑧_𝜉𝑗!𝜉^𝑗,forsome𝑧_𝜉inthelinesegment[0,𝜉].(3.5) Hence, 𝑦^𝑙||𝒲_𝜓𝜑(𝑥,𝑦)||=𝑦^𝑙(2𝜋)^𝑛|||𝑒_{𝑖𝜉⋅𝑥}𝜑(𝜉),𝜓(𝑦𝜉)|||≤1(2𝜋)^𝑛_ℝ𝑛≤1||𝜑(𝜉)||𝑦^𝑙||𝜓(𝑦𝜉)||d𝜉(2𝜋)^𝑛_{||𝑗||=𝑙−𝑛}1𝑗!_ℝ^𝑛≤1||𝜕^𝑗𝜑𝑧_𝜉||𝑦^𝑛||(𝑦𝜉)^𝑗𝜓(𝑦𝜉)||d𝜉(2𝜋)^𝑛_{||𝑗||=𝑙−𝑛}𝜌_0,𝑗𝜑𝑗!_ℝ^𝑛≤1||𝜉^𝑗𝜓(𝜉)||d𝜉(2𝜋)^𝑛_{||𝑗||=𝑙−𝑛}𝜌_0,𝑗𝜑𝜌_𝑙+𝑛,0𝜓𝑗!_ℝ^𝑛d𝜉1+||𝜉||²^𝑛≤𝐶𝜌_𝑙+𝑛,0(𝜑)_{||𝑗||≤2𝑙+2𝑛}𝜌_2𝑛,𝑗(𝜓),∀(𝑥,𝑦)∈ℍ^𝑛+1.(3.6)
It remains to estimate 𝑦^−𝑙|𝒲_𝜓𝜑(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 𝑧_𝑡=𝑧(𝑡,𝑥,𝑦) in the line segment [𝑥,𝑦𝑡], 1𝑦^𝑙||𝒲_𝜓𝜑(𝑥,𝑦)||=1𝑦^𝑙||||_ℝ𝑛=1𝜑(𝑦𝑡+𝑥)𝜓(𝑡)d𝑡||||𝑦^𝑙|||||_ℝ𝑛𝜓(𝑡)_|𝑗|<𝑙𝜕^𝑗𝜑(𝑥)𝑗!(𝑦𝑡)^𝑗+_|𝑗|=𝑙𝜕^𝑗𝜑𝑧_𝑡≤𝑗!(𝑦𝑡)^𝑗d𝑡_{||𝑗||=𝑙}|||||𝜌_0,𝑗(𝜑)𝑗!_ℝ^𝑛1+|𝑡|²^𝑙/2≤𝐶𝜌_𝑙+2𝑛,0||𝜓(𝑡)||d𝑡(𝜓)_{||𝑗||=𝑙}𝜌_0,𝑗(𝜑).(3.7) 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∈ℕ^𝑛 there exist 𝑁∈ℕ and 𝐶>0 such that 𝜌_𝑘,𝜅ℳ_𝜂Φ≤𝐶_{𝑘1,𝑘2||𝑗||,𝑙,|𝑚|≤𝑁𝑖,≤𝑁}𝜌_𝑖,𝑗(𝜂)𝜌^0𝑘_{1,𝑘2,𝑙,𝑚}(Φ).(3.8) Since 𝜕^𝜅𝑡ℳ_𝜂Φ=ℳ_𝜂𝜕^𝜅𝑥Φ, 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⋅𝑥}d𝑥. We may assume that 𝑘 is even. We then have (1+|𝑡|²)^𝑘/2||ℳ_𝜂Φ(𝑡)||=(1+|𝑡|²)^𝑘/2||||_ℝ+_ℝ𝑛Φ(𝑡−𝑥,𝑦)1𝑦^𝑛𝜂𝑥=(1+|𝑡|₂𝑦d𝑥𝑦d𝑦||||)^𝑘/2(2𝜋)^𝑛|||||_ℝ+_ℝ𝑛(1−Δ_𝜉)^𝑘/2𝑒^𝑖𝑡𝜉(1+|𝑡|²)^𝑘/2Φ(𝜉,𝑦)𝜂̂(𝑦𝜉)d𝜉d𝑦≤1𝑦|||||(2𝜋)^𝑛_ℝ+_ℝ𝑛|||(1−Δ_𝜉)^𝑘/2Φ(𝜉,𝑦)𝜂̂(𝑦𝜉)|||d𝜉d𝑦≤1𝑦(2𝜋)^𝑛_{|𝑟|+|𝑠|≤𝑘}𝑐_𝑟,𝑠_ℝ+_ℝ𝑛≤1||𝜕^𝑟𝜉Φ(𝜉,𝑦)||𝑦^|𝑠|−1||𝜕^𝑠𝜂̂(𝑦𝜉)||d𝜉d𝑦(2𝜋)^𝑛_{|𝑟|+|𝑠|≤𝑘}𝑐_𝑟,𝑠𝜌_0,𝑠(𝜂̂)_ℝ+_ℝ𝑛≤𝐶_𝑦^|𝑠|−1||𝜕^𝑟𝜉Φ(𝜉,𝑦)||d𝜉d𝑦_|𝑟|+_{|𝑠|≤𝑘}𝜌_0,𝑠(𝜂̂)𝜌^{0|𝑠|−1,2𝑛,0,𝑟}Φ+𝜌^{0|𝑠|+1,2𝑛,0,𝑟}Φ_ℝ+_ℝ𝑛d𝜉(1+|𝜉|²)^𝑛d𝑦1+𝑦²≤𝐶_{||𝑗|||𝑟|+|𝑠|≤𝑘≤2𝑛}𝜌_{|𝑠|+2𝑛,0}(𝜂)𝜌^{0|𝑠|−1,|𝑟|+2𝑛,0,𝑗}(Φ)+𝜌^{0|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 bilinear mapping ℳ∶(𝜂,Φ)↦ℳ_𝜂Φ to act on ℳ∶𝒮(ℝ^𝑛)×𝒮ℍ^𝑛+1⟶𝒮(ℝ^𝑛),(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 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. Let 𝜓∈𝒮(ℝ^𝑛) be a nondegenerate wavelet. Its index of nondegenerateness is the (finite) number 𝜏=inf𝑟∈ℝ₊∶supp𝑅_𝜔∩[0,𝑟]≠∅,∀𝜔∈𝕊^𝑛−1,(4.1) where 𝑅_𝜔 are the functions of one variable 𝑅_𝜔(𝑟)=𝜓(𝑟𝜔).

If we only know values of 𝒲_𝜓𝑓(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. Let 𝛼∈ℝ, 𝜙∈𝒮(ℝ^𝑛). Let 𝜓∈𝒮(ℝ^𝑛) 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 𝜇_𝑚(𝜓)=0 for all multi-index 𝑚∈ℕ^𝑛 with |𝑚|≤[𝛼].

Example 4.3. Let 𝜙∈𝒮(ℝ^𝑛) 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∗(ℝ^𝑛) (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(ℝ^𝑛) is a reconstruction wavelet for 𝜓 whose Fourier transform has support in 𝐵(0,𝜎) and 𝜑∈𝒟(ℝ^𝑛) is such that 𝜑(𝜉)=1 for 𝜉∈𝐵(0,𝜎) and supp𝜑⊂𝐵(0,𝑟), then, for all 𝑓∈𝒮^(ℝ^𝑛) and 𝜃∈𝒮(ℝ^𝑛)⟨𝑓,𝜃⟩=_ℝ𝑛(𝑓∗𝜙)(𝑡)𝜃₁(𝑡)d𝑡+1𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝒲_𝜓𝑓(𝑥,𝑦)𝒲_𝜂𝜃₂(𝑥,𝑦)d𝑥d𝑦𝑦,(4.2) where 𝜃̂₁(𝜉)=𝜃̂(𝜉)𝜑(𝜉)/𝜙(−𝜉) and 𝜃̂₂(𝜉)=𝜃̂(𝜉)(1−𝜑(𝜉)).

Proof. Observe that ⟨𝑓∗𝜙,𝜃₁⟩=(2𝜋)^−𝑛⟨𝑓(𝜉),𝜃̂(−𝜉)𝜑(−𝜉)⟩.(4.3) It is therefore enough to assume that 𝜃₁=0 so that 𝜃=𝜃₂. Our assumption over 𝜂 is that 𝜂∈𝒮₀(ℝ^𝑛), supp𝜂̂⊂𝐵(0,𝜎), and 𝑐_𝜓,𝜂=^∞0𝜓(𝑟𝜔)𝜂̂(𝑟𝜔)dr𝑟≠0(4.4) 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, 𝒲_𝜂𝜃(𝑥,𝑦)=0 for all (𝑥,𝑦)∈ℝ^𝑛×(1,∞). Exactly as in [1, page 66], the usual calculation shows that 𝜃(𝑡)=1𝑐_𝜓,𝜂ℳ_𝜓𝒲_𝜂𝜃(𝑡)=1𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝜓𝑡−𝑥𝑦𝒲^𝜂𝜃(𝑥,𝑦)d𝑥d𝑦𝑦.(4.5) Furthermore, since 𝒲_𝜂𝜃∈𝒮(ℍ^𝑛+1) (cf. Theorem 3.1), the last integral can be expressed as the limit in 𝒮(ℝ^𝑛) of Riemann sums. That justifies the exchange of dual pairing and integral in ⟨𝑓,𝜃⟩=𝑓(𝑡),1𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝜓𝑡−𝑥𝑦𝒲^𝜂𝜃(𝑥,𝑦)d𝑥d𝑦=1𝑦𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝒲_𝜓𝑓(𝑥,𝑦)𝒲_𝜂𝜃(𝑥,𝑦)d𝑥d𝑦𝑦.(4.6)

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, lim_𝜀→0+𝐿(𝑎𝜀)𝐿(𝜀)=1,foreach𝑎>0.(4.7) Familiar functions such as 1, |log𝜀|^𝛽, |log|log𝜀||^𝛽, …, are slowly varying. Regularly varying functions are then those that can be written as 𝜀^𝛼𝐿(𝜀), 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′(ℝ^𝑛,𝐸), we write 𝐟𝑥₀+𝜀𝑡=𝑂(𝜀^𝛼𝐿(𝜀))as𝜀⟶0⁺in𝒮^(ℝ^𝑛,𝐸),(4.8) if the order growth relation holds after evaluation at each test function, that is, ‖‖𝐟𝑥₀+𝜀𝑡,𝜑(𝑡)‖‖≤𝐶_𝜑𝜀^𝛼𝐿(𝜀),0<𝜀≤1,(4.9) for each test function 𝜑∈𝒮(ℝ^𝑛). 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^(ℝ^𝑛,𝐸) is said to have a value 𝐯∈𝐸 at the point 𝑥₀∈ℝ^𝑛 if lim_𝜀→0⁺𝐟(𝑥₀+𝜀𝑡)=𝐯, distributionally, that is, for each 𝜑∈𝒮(ℝ^𝑛)lim_𝜀→0+𝐟(𝑡),𝜀1^𝑛𝜑𝑡−𝑥⁰𝜀=𝐯_ℝ^𝑛𝜑(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 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 given 𝑥₀∈ℝ^𝑛 and 𝛼∈ℝ, the pointwise weak Hölder space [10] 𝐶^{𝑤𝛼,𝐿}(𝑥₀,𝐸) consists of those distributions 𝐟∈𝒮^(ℝ^𝑛,𝐸) for which there is an 𝐸-valued polynomial 𝐏 of degree less than 𝛼 such that (cf. Section 4.3) 𝐟𝑥₀+𝜀𝑡=𝐏(𝜀𝑡)+𝑂(𝜀^𝛼𝐿(𝜀))as𝜀⟶0⁺in𝒮^(ℝ^𝑛,𝐸).(4.11) 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” 𝐏(𝑡)=_|𝑚|<𝛼𝜕^𝑚𝐟𝑥₀𝑚!𝑡^𝑚.(4.12)

The pointwise weak Hölder space 𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸) of second type is defined as follows: 𝐟∈𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸) if (4.9) is just assumed to hold for each 𝜑∈𝒮(ℝ^𝑛) satisfying the requirement 𝜇_𝑚(𝜑)=0 for all multi-index |𝑚|≤𝛼. Naturally, the previous requirement is empty if 𝛼<0, thus, in such a case, 𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸)=𝐶^{𝑤𝛼,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 𝐶^{𝑤𝛼,𝐿}(𝑥₀,𝐸)⊊𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸)(cf. comments below Theorem 4.5). The usefulness of 𝐶^{𝛼,𝐿∗,𝑤}(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]). Let 𝛼∈ℝ and let 𝜓∈𝒮(ℝ^𝑛) be a nondegenerate wavelet with 𝜇_𝑚(𝜓)=0 for |𝑚|≤[𝛼]. Then, 𝐟∈𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸) if and only if there is 𝑘∈ℕ such that limsup_𝜀→0+sup_{|𝑥|2+𝑦2=1,𝑦>0}𝑦^𝑘𝜀^𝛼𝐿(𝜀)‖‖𝒲^𝜓𝐟𝑥₀+𝜀𝑥,𝜀𝑦‖‖<∞.(4.13)

It is worth mentioning that the elements of 𝐶^{𝛼,𝐿∗,𝑤}(𝑥₀,𝐸) for 𝛼=𝑝∈ℕ can be characterized by pointwise weak-asymptotic expansions. We have [10] that 𝐟∈𝐶^{𝑝,𝐿∗,𝑤}(𝑥₀,𝐸) if and only if it admits the following weak expansion: 𝐟𝑥₀+𝜀𝑡=_|𝑚|<𝑝𝜕^𝑚𝐟𝑥₀𝑚!(𝜀𝑡)^𝑚+𝜀^𝑝_|𝑚_|=𝑝𝑡^𝑚𝐜_𝑚(𝜀)+𝑂(𝜀^𝑝𝐿(𝜀))in𝒮^(ℝ^𝑛,𝐸),(4.14) where 𝜕^𝑚𝐟(𝑥₀) are interpreted in the Łojasiewicz sense and the 𝐜_𝑚∶(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 that 𝐿 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 to 𝐿. They were already defined and studied in [10]. Let 𝛼∈ℝ₊⧵ℕ. We say that a function 𝑓 belongs to the space 𝐶^𝛼,𝐿(ℝ^𝑛) if 𝑓 has continuous derivatives up to order less than 𝛼 and ‖𝑓‖_𝐶^𝛼,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|)<∞.(5.1) When 𝛼=𝑝+1∈ℕ, we replace the previous requirement by ‖𝑓‖_𝐶^𝛼,𝐿∶=_{||𝑗||≤𝑝}sup_{𝑡∈ℝ𝑛}||𝜕^𝑗𝑓(𝑡)||+_|𝑚|=𝑝sup_{0<|𝑡−𝑥|≤1}||𝜕^𝑚𝑓(𝑡)−𝜕^𝑚𝑓(𝑥)|||𝑡−𝑥|𝐿(|𝑡−𝑥|)<∞.(5.2)

The space 𝐶^𝛼,𝐿(ℝ^𝑛) is clearly a Banach space with the above norm. The conditions imposed over 𝐿 ensure that 𝐶^𝛼,𝐿(ℝ^𝑛) depends only on the behavior of 𝐿 near 0, thus, it is invariant under dilations. When 𝐿≡1, this space reduces to 𝐶^𝛼,𝐿(ℝ^𝑛)=𝐶^𝛼(ℝ^𝑛), the usual global (inhomogeneous) Hölder space [2, 3, 5]. Consequently, as in [10], we call 𝐶^𝛼,𝐿(ℝ^𝑛) 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: 𝐶^𝛽(ℝ^𝑛)⊊𝐶^𝛼,𝐿(ℝ^𝑛)⊊𝐶^𝛾(ℝ^𝑛),whenever0<𝛾<𝛼<𝛽.(5.3)

5.2. 𝐿-Zygmund Spaces

We now proceed to define the weighted Zygmund space 𝐶^∗𝛼,𝐿(ℝ^𝑛). Let 𝛼∈ℝ and fix an LP-pair (𝜙,𝜓) of order 𝛼. A distribution 𝑓∈𝒮^(ℝ^𝑛) is said to belong to the 𝐿-Zygmund space 𝐶^∗𝛼,𝐿(ℝ^𝑛) of exponent 𝛼 if ‖𝑓‖_𝐶^∗𝛼,𝐿∶=‖𝑓∗𝜙‖_𝐿^∞+sup_{𝑥∈ℝ𝑛}sup_0<𝑦≤11𝑦^𝛼𝐿(𝑦)||𝒲^𝜓𝑓(𝑥,𝑦)||<∞.(5.4)

Observe that we clearly have 𝐶^𝛼,𝐿(ℝ^𝑛)⊆𝐶^∗𝛼,𝐿(ℝ^𝑛), for 𝛼>0. We will show that if 𝛼∈ℝ₊⧵ℕ, we actually have the equality 𝐶^𝛼,𝐿(ℝ^𝑛)=𝐶^∗𝛼,𝐿(ℝ^𝑛). When 𝛼 is a positive integer, we have in turn 𝐶^𝛼,𝐿(ℝ^𝑛)⊊C^∗𝛼,𝐿(ℝ^𝑛). 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: 𝐶^𝛽∗(ℝ^𝑛)⊊𝐶^∗𝛼,𝐿(ℝ^𝑛)⊊𝐶^𝛾∗(ℝ^𝑛),whenever𝛾<𝛼<𝛽.(5.5) The definition of 𝐶^∗𝛼,𝐿(ℝ^𝑛) 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 of 𝐶^∗𝛼,𝐿(ℝ^𝑛) 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 continuous Littlewood-Paley decomposition of the unity (cf. Example 4.3) in the definition of 𝐶^∗𝛼,𝐿(ℝ^𝑛). Therefore, when 𝐿≡1, we recover the classical Zygmund space 𝐶^∗𝛼,𝐿(ℝ^𝑛)=𝐶^𝛼∗(ℝ^𝑛) [5]. Proposition 5.1 follows at once from the following lemma.

Lemma 5.2. Let 𝑓∈𝐶^∗𝛼,𝐿(ℝ^𝑛), then for every 𝜃∈𝒮(ℝ^𝑛) there holds ‖𝑓∗𝜃‖_𝐿^∞≤𝐶‖𝑓‖_𝐶^∗𝛼,𝐿,(5.6) where ‖𝑓‖_𝐶^∗𝛼,𝐿 is given by (5.4). Furthermore, if 𝔅⊂𝒮(ℝ^𝑛) is a bounded set such that 𝜇_𝑚(𝜃)=0 for all 𝜃∈𝔅 and all multi-index 𝑚≤[𝛼], then sup_{𝑥∈ℝ𝑛}sup_0<𝑦≤11𝑦^𝛼𝐿(𝑦)||𝒲^𝜃𝑓(𝑥,𝑦)||≤𝐶‖𝑓‖_𝐶^∗𝛼,𝐿,∀𝜃∈𝔅.(5.7)

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 𝒲_𝜃𝑓(𝑥,𝑦)=𝐹₁(𝑥,𝑦)+𝐹₂(𝑥,𝑦)+𝐹₃(𝑥,𝑦),(5.8) where 𝐹₁(𝑥,𝑦)=1(2𝜋)^𝑛_ℝ𝑛𝐹𝑓∗𝜑(𝑥+𝑦𝑡)𝜃(𝑡)d𝑡,₂(𝑥,𝑦)=1𝑐_𝜓,𝜂^01/𝑦_ℝ𝑛𝒲_𝜓𝑓(𝑦𝑏+𝑥,𝑦𝑎)𝒲_𝜂𝜃_2,𝑦(𝑏,𝑎)d𝑏d𝑎𝐹₃𝑎,(𝑥,𝑦)=1𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝒲_𝜓𝑓(𝑦𝑏+𝑥,𝑦𝑎)𝒲_𝜂𝜃_3,𝑦(𝑏,𝑎)d𝑏d𝑎𝑎,(5.9) with 𝜃̂_2,𝑦(𝜉)=𝜃̂(𝜉)(1−𝜑(𝜉))(1−𝜑(𝜉/𝑦)) and 𝜃̂_3,𝑦(𝜉)=𝜃̂(𝜉)𝜑(𝜉)(1−𝜑(𝜉/𝑦)). To estimate 𝐹₁, we first observe that if 𝛼<0, then sup_0<𝑦≤1sup_{𝑥∈ℝ𝑛}||𝐹₁(𝑥,𝑦)||𝑦^𝛼𝐿(𝑦)≤𝐶||𝑓||^{𝐶∗𝛼,𝐿}‖𝜃‖_𝐿¹sup_0<𝑦≤11𝑦^𝛼𝐿(𝑦)≤𝐶^||𝑓||_{𝐶∗𝛼,𝐿},(5.10) because slowly varying functions satisfy the estimates 𝑦^𝜀<𝐶𝐿(𝑦) for any exponent 𝜀>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 𝜇_𝑚(𝜃)=0 for |𝑚|≤[𝛼], we obtain sup_0<𝑦≤1sup_{𝑥∈ℝ𝑛}||𝐹₁(𝑥,𝑦)||𝑦^𝛼𝐿(𝑦)≤sup_0<𝑦≤1𝐶‖𝑓‖_𝐶^∗𝛼,𝐿𝑦^{[𝛼]+1−𝛼}𝐿(𝑦)_ℝ^𝑛≤𝐶′‖𝑓‖_{𝐶∗𝛼,𝐿}|𝑡|^[𝛼]+1||𝜃(𝑡)||d𝑡.(5.11) We now bound 𝐹₂ and 𝐹₃. If 𝜀>0, Potter's estimate [7, page 25] gives the existence of a constant 𝐶=𝐶_𝜀 such that 𝐿(𝑎𝑦)𝐿(𝑦)<𝐶𝑎^𝜀+1𝑎^𝜀,∀0<𝑦<1,𝑎<1/𝑦.(5.12) Thus, ||𝐹₂(𝑥,𝑦)||𝑦^𝛼𝐿(𝑦)≤‖𝑓𝑐‖^{𝐶∗𝛼,𝐿}_𝜓,𝜂^01/𝑦_ℝ𝑛𝑎^𝛼−1𝐿(𝑎𝑦)𝐿(𝑦)||𝒲^𝜂≤𝐶_𝜃_2,𝑦(𝑏,𝑎)||d𝑏d𝑎‖𝑓‖_𝐶^∗𝛼,𝐿^∞0_ℝ𝑛𝑎^𝛼+𝑎^𝛼−2||𝒲_𝜂𝜃_2,𝑦(𝑏,𝑎)||d𝑏d𝑎.(5.13) Notice that {𝜃_2,𝑦∈𝒮₀(ℝ^𝑛)∶𝜃∈𝔅,𝑦∈(0,1]} is a bounded set in 𝒮₀(ℝ^𝑛) because the derivatives of 𝜑 are supported in {𝜉∶𝜎≤|𝜉|≤𝑟}. Thus, due to the continuity of 𝒲_𝜂 (cf. Theorem 3.1), {𝒲_𝜂𝜃_𝑦∶𝜃∈𝔅,𝑦∈(0,1]} is a bounded set in 𝒮(ℍ^𝑛+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 sup_0<𝑦≤1sup_{𝑥∈ℝ𝑛}||𝐹₂(𝑥,𝑦)||𝑦^𝛼𝐿(𝑦)≤𝐶‖𝑓‖^{𝐶∗𝛼,𝐿}.(5.14) Next, for 𝐹₃, we have ||𝐹₃(𝑥,𝑦)||𝑦^𝛼𝐿(𝑦)≤𝐶′‖𝑓‖^{𝐶∗𝛼,𝐿}¹⁰_ℝ𝑛𝑎^𝛼−3/2+𝑎^𝛼−1/2||𝒲_𝜂𝜃_3,𝑦(𝑏,𝑎)||d𝑏d𝑎.(5.15) 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(ℝ^𝑛)→𝐶^{∗𝛼−|𝑚|,𝐿}(ℝ^𝑛) is continuous, for any 𝑚∈ℕ^𝑛,(ii)if 𝑓∗𝜙∈𝐿^∞(ℝ^𝑛) and 𝜕_𝑗𝑓∈𝐶^{∗𝛼−1,𝐿}(ℝ^𝑛) for 𝑗=1,…,𝑛, then 𝑓∈𝐶^∗𝛼,𝐿(ℝ^𝑛),(iii)the mapping (1−Δ)^𝛽/2 is an isomorphism of the space 𝐶^∗𝛼,𝐿(ℝ^𝑛) onto 𝐶^{∗𝛼−𝛽,𝐿}(ℝ^𝑛), for arbitrary 𝛼,𝛽∈ℝ.

Proof. (i) It is enough to consider 𝜕_𝑗. We have that 𝜕_𝑗𝑓∗𝜙=𝑓∗𝜕_𝑗𝜙 and 𝒲_𝜓𝜕_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}(ℝ^𝑛). In view of Proposition 5.1, it remains to observe that 1𝑦^𝛼𝐿(𝑦)𝒲^Δ𝜓𝑓(𝑥,𝑦)=1𝑦^𝛼−2𝐿(𝑦)𝒲^𝜓(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/2 maps continuously 𝐶^∗𝛼,𝐿(ℝ^𝑛) into 𝐶^{∗𝛼−𝛽,𝐿}(ℝ^𝑛). 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 𝒲_𝜓(1−Δ)^𝛽/2𝑓(𝑥,𝑦)=1(2𝜋)^𝑛𝑦^𝛽𝑓(𝜉),𝑒^{𝑖𝑥⋅𝜉}𝑦²=1+||𝑦𝜉||²^𝛽/2𝜓(𝑦𝜉)𝑦^𝛽𝒲_𝜃𝑦𝑓(𝑥,𝑦),where𝜃_𝑦=𝑦²−Δ^𝛽/2𝜓.(5.17) Finally, we can apply (5.7) because 𝔅={(𝑦²−Δ)^𝛽/2𝜓}_𝑦∈(0,1] is a bounded set in 𝒮(ℝ^𝑛) and 𝜇_𝑚(𝜃_𝑦)=0 for each multi-index |𝑚|≤[𝛼].

We can also use Proposition 4.4 to show that 𝐶^∗𝛼,𝐿(ℝ^𝑛) is a Banach space, as stated in the following proposition.

Proposition 5.4. The space 𝐶^∗𝛼,𝐿(ℝ^𝑛) 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 {𝑓_𝑗}^∞𝑗=0 is a Cauchy sequence in 𝐶^∗𝛼,𝐿(ℝ^𝑛). Then, there exist continuous functions 𝑔∈𝐿^∞(ℝ^𝑛) and 𝐺 defined on ℝ^𝑛×(0,1] such that 𝑓_𝑗∗𝜙→𝑔 in 𝐿^∞(ℝ^𝑛) and lim_𝑗→∞sup_𝑦∈(0,1]sup_{𝑥∈ℝ𝑛}1𝑦^𝛼𝐿(𝑦)||𝒲^𝜓𝑓_𝑗(𝑥,𝑦)−𝐺(𝑥,𝑦)||=0.(5.18) We define the distribution 𝑓∈𝒮^(ℝ^𝑛) whose action on test functions 𝜃∈𝒮(ℝ^𝑛) is given by ⟨𝑓,𝜃⟩∶=_ℝ𝑛𝑔(𝑡)𝜃₁(𝑡)d𝑡+1𝑐_𝜓,𝜂¹⁰_ℝ𝑛𝐺(𝑥,𝑦)𝒲_𝜂𝜃₂(𝑥,𝑦)d𝑥d𝑦𝑦.(5.19) Since the 𝑓_𝑗 have the representation (4.2), we immediately see that 𝑓_𝑗→𝑓 in 𝒮^(ℝ^𝑛). Thus, pointwisely, 𝒲(𝑓∗𝜙)(𝑡)=lim_𝑗→∞𝑓_𝑗∗𝜙(𝑡)=𝑔(𝑡),_𝜓𝑓(𝑥,𝑦)=lim_𝑗→∞𝒲_𝜓𝑓_𝑗(𝑥,𝑦)=𝐺(𝑥,𝑦).(5.20) This implies that lim_𝑗→∞‖𝑓−𝑓_𝑗‖_{𝐶∗𝛼,𝐿}=0, and so 𝐶^∗𝛼,𝐿(ℝ^𝑛) is complete.

We have arrived to the main and last result of this section. It provides the 𝐿-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 𝐶_𝑏(ℝ^𝑛) the Banach space of continuous and bounded functions.

Theorem 5.5. Let 𝛼>0.(a)If 𝛼∉ℕ, then 𝐶^∗𝛼,𝐿(ℝ^𝑛)=𝐶^𝛼,𝐿(ℝ^𝑛). Moreover, the norms (5.4) and (5.1) are equivalent. (b)If 𝛼=𝑝+1∈ℕ, then 𝐶^{∗𝑝+1,𝐿}(ℝ^𝑛) consists of functions with continuous derivatives up to order 𝑝 such that _|𝑚_|≤𝑝‖𝜕^𝑚𝑓‖_𝐿^∞+_|𝑚|=𝑝sup_{𝑡∈ℝ𝑛0<|ℎ|≤1}||𝜕^𝑚𝑓(𝑡+ℎ)+𝜕^𝑚𝑓(𝑡−ℎ)−2𝜕^𝑚𝑓(𝑡)||||ℎ||𝐿||ℎ||<∞.(5.21)
In addition, (5.21) produces a norm that is equivalent to (5.4).

Proof. Observe that the 𝐿-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 that 𝑓∈𝐶^∗𝛼,𝐿(ℝ^𝑛). Consider the 𝐶_𝑏(ℝ^𝑛)-valued distribution 𝐟∈𝒮^(ℝ^𝑛𝑡,𝐶_𝑏(ℝ^𝑛𝜉)) given by 𝐟(𝑡)(𝜉)∶=𝑓(𝑡+𝜉), that is, the one whose action on test functions is given by ⟨𝐟(𝑡),𝜃(𝑡)⟩(𝜉)=⟨𝑓(𝑡+𝜉),𝜃(𝑡)⟩=𝑓∗𝜃̌(𝜉),𝜃∈𝒮(ℝ^𝑛),𝜉∈ℝ^𝑛.(5.22) It does take values in 𝐶_𝑏(ℝ^𝑛) because of (5.6). Clearly, 𝒲_𝜓𝐟(𝑥,𝑦)(𝜉)=𝒲_𝜓𝑓(𝑥+𝜉,𝑦). By (5.4) and Potter's estimate [7, page 25], we have that ‖‖𝒲_𝜓𝐟(𝜀𝑥,𝜀𝑦)‖‖_{𝐶𝑏(ℝ𝑛)}≤𝐶𝜀^𝛼𝐿(𝜀)𝑦^𝛼−1∀𝜀∈(0,1),(𝑥,𝑦)∈ℝ^𝑛×(0,1].(5.23) Therefore, the Tauberian Theorem 4.5 yields 𝐟∈𝐶^{𝛼,𝐿∗,𝑤}(0,𝐶_𝑏(ℝ^𝑛)). Now, the Łojasiewicz point values 𝜕^𝑚𝐟(0)=𝑣_𝑚∈𝐶_𝑏(ℝ^𝑛) exist, distributionally, for |𝑚|<𝛼. It explicitly means that for all 𝜃∈𝒮(ℝ^𝑛)lim_𝜀→0+𝜕^𝑚𝑓∗𝜃̌_𝜀=lim_𝜀→0+⟨𝜕^𝑚𝐟(𝜀𝑡),𝜃(𝑡)⟩=𝜇₀(𝜃)𝑣_𝑚in𝐶_𝑏ℝ^𝑛𝜉,(5.24) where 𝜃̌_𝜀(𝑡)=𝜀^−𝑛𝜃(−𝑡/𝜀). If we now take 𝜃 with 𝜇₀(𝜃)=1, we then conclude that 𝜕^𝑚𝑓=𝑣_𝑚∈𝐶_𝑏(ℝ^𝑛) for each |𝑚|<𝛼. It remains in both cases to deal with the estimates for 𝜕^𝑚𝑓; notice that 𝜕^𝑚𝐟∈𝐶^{𝛼−[𝛼],𝐿∗,𝑤}(0,𝐶_𝑏(ℝ^𝑛)) when |𝑚|=[𝛼] and 𝜕^𝑚𝐟∈𝐶^{1,𝐿∗,𝑤}(0,𝐶_𝑏(ℝ^𝑛)) when |𝑚|=𝑝. We now divide the proof into two cases.
Case 𝛼∉ℕ. Fix a multi-index |𝑚|=[𝛼]. It suffices to show sup_{0<|𝑥−𝑡|<1}||𝜕^𝑚𝑓(𝑥)−𝜕^𝑚𝑓(𝑡)|||𝑥−𝑡|^{𝛼−[𝛼]}𝐿(|𝑥−𝑡|)<∞.(5.25) We had already seen that 𝜕^𝑚𝐟(𝑡)(𝜉)=𝜕^𝑚𝑓(𝑡+𝜉)∈𝐶^{𝑤𝛼−[𝛼],𝐿}(0,𝐶_𝑏(ℝ^𝑛𝜉))=𝐶^{𝛼−[𝛼],𝐿∗,𝑤}(0,𝐶_𝑏(ℝ^𝑛𝜉)), that is, 𝜇₀(𝜃)𝜕^𝑚𝑓(𝜉)−_ℝ𝑛𝜕^𝑚𝑓(𝜉+𝜀𝑡)𝜃(𝑡)d𝑡=𝑂𝜀^{𝛼−[𝛼]}𝐿(𝜀),𝜀⟶0⁺,(5.26) in the space C_𝑏(ℝ^𝑛𝜉), for each 𝜃∈𝒮(ℝ^𝑛). 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(ℝ^𝑛); we then have sup_{𝜉∈ℝ𝑛}||𝜕^𝑚𝑓(𝜉+ℎ)−𝜕^𝑚𝑓(𝜉)||≤2sup_{𝜉∈ℝ𝑛}||||𝜕_𝑚+sup_{𝜉∈ℝ𝑛}𝑓(𝜉)−_ℝ𝑛𝜕^𝑚𝑓||ℎ||𝑡+𝜉𝜃(𝑡)d𝑡||||||||_ℝ𝑛=𝑂||ℎ||_{𝛼−[𝛼]}𝜕^𝑚𝑓𝜉+||ℎ||𝑡𝜃(𝑡)−𝜃𝑡−||ℎ||⁻¹ℎd𝑡||||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∈𝐶^{1,𝐿∗,𝑤}(0,𝐶_𝑏(ℝ^𝑛)), which, as commented in Section 4.4, yields the distributional expansion 𝜕^𝑚𝐟(𝜀𝑡)(𝜉)=𝜕^𝑚𝑓(𝜉)+𝜀^𝑛_𝑗=1𝑡_𝑗𝐜_𝑗(𝜀,𝜉)+𝑂(𝜀𝐿(𝜀)),0<𝜀≤1,(5.28) in 𝒮^(ℝ^𝑛𝑡,𝐶_𝑏(ℝ^𝑛𝜉), where the 𝐜_𝑗(𝜀,⋅) are continuous 𝐶_𝑏(ℝ^𝑛𝜉)-valued functions in 𝜀. We apply (5.28) on a test function 𝜃∈𝒮(ℝ^𝑛), with 𝜇₀(𝜃)=1, and ∫_ℝ𝑛𝑡_𝑗𝜃(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∈ℝ^𝑛. Since {𝜃_𝜔∶=𝜃(⋅+𝜔)+𝜃(⋅−𝜔)−2𝜃∶|𝜔|=1} is compact in 𝒮(ℝ^𝑛) and 𝜇_𝑚(𝜃_𝜔)=0 for |𝑚|≤1, the relations (5.28) and (5.29) give sup_{𝜉∈ℝ𝑛}||≤𝜕^𝑚3𝑓su(𝜉p_{𝜉∈ℝ𝑛}+ℎ)+𝜕^𝑚𝑓(𝜉−ℎ)−2𝜕^𝑚𝑓(𝜉)||||||𝜕_𝑚+||||𝑓(𝜉)−_ℝ𝑛𝜕^𝑚𝑓||ℎ||𝑡+𝜉𝜃(𝑡)d𝑡_ℝ𝑛||||=𝑂||ℎ||𝐿𝜕^𝑚𝑓||ℎ||𝜉+,||ℎ||0𝑡<||𝜃ℎ||𝑡≤+1||,ℎ||⁻¹ℎ+𝜃𝑡−||ℎ||⁻¹ℎ−2𝜃(𝑡)d𝑡||||(5.30) as claimed.

Acknowledgments

S. Pilipović acknowledges support by Project 174024 of the Serbian Ministry of Education and Sciences. D. Rakić acknowledges support by Project III44006 of the Serbian Ministry of Education and Sciences and by Project 114-451-2167 of the Provincial Secretariat for Science and Technological Development. J. Vindas acknowledges support by a postdoctoral fellowship of the Research Foundation-Flanders (FWO, Belgium).