We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces 𝒮0(𝑛) and 𝒮(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 [13]. It is well known [1, 46] that the wavelet transforms of elements of the classical Zygmund space 𝐶𝛼(𝑛) 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(𝑛) and 𝒮(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 𝐶𝛼(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 𝐶𝛼,𝐿(𝑛) and 𝐶𝛼,𝐿(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: 𝐶𝛽(𝑛)𝐶𝛼,𝐿(𝑛)𝐶𝛾(𝑛𝐶),when0<𝛾<𝛼<𝛽,𝛽(𝑛)𝐶𝛼,𝐿(𝑛)𝐶𝛾(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 𝑛+1=𝑛×+ the upper half space. If 𝑥𝑛 and 𝑚𝑛, then |𝑥| denotes the euclidean norm, 𝑥𝑚=𝑥𝑚11𝑥𝑚𝑛𝑛,  𝜕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 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(𝑛). We will fix constants in the Fourier transform as 𝜑(𝜉)=𝑛𝜑(𝑡)𝑒𝑖𝜉𝑡d𝑡. The moments of 𝜑𝒮(𝑛) are denoted by 𝜇𝑚(𝜑)=𝑛𝑡𝑚𝜑(𝑡)d𝑡, 𝑚𝑛.

Following [1], the space of highly time-frequency localized functions 𝒮0(𝑛) is defined as 𝒮0(𝑛)={𝜑𝒮(𝑛)𝜇𝑚(𝜑)=0,forall𝑚𝑛}; it is provided with the relative topology inhered from 𝒮(𝑛). In [1], the topology of 𝒮0(𝑛) is introduced in an apparently different way; however, both approaches are equivalent in view of the open mapping theorem. Observe that 𝒮0(𝑛) is a closed subspace of 𝒮(𝑛) and that 𝜑𝒮0(𝑛) if and only if 𝜕𝑚𝜑(0)=0, for all 𝑚𝑛. It is important to point out that 𝒮0(𝑛) is also well known in the literature as the Lizorkin space of test functions (cf. [13]).

The space 𝒮(𝑛+1) of highly localized functions on the half space [1] consists of those Φ𝐶(𝑛+1) for which 𝜌0𝑙,𝑘,𝜈,𝑚(Φ)=sup(𝑥,𝑦)𝑛+1𝑦𝑙+1𝑦𝑙1+|𝑥|2𝑘/2||𝜕𝜈𝑦𝜕𝑚𝑥||Φ(𝑥,𝑦)<,(2.1) for all 𝑙,𝑘,𝜈, and 𝑚𝑛. The canonical topology on 𝒮(𝑛+1) is induced by this family of seminorms [1]. For later use, we will denote by 𝜌𝑘,𝑚 the corresponding seminorms in 𝒮(𝑛), namely, 𝜌𝑘,𝑚(𝜑)=sup𝑡𝑛1+|𝑡|2𝑘/2||𝜕𝑚||𝜑(𝑡),𝑘,𝑚𝑛.(2.2)

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

2.2. Wavelet Transform

In this paper a wavelet simply means a function 𝜓𝒮(𝑛) that satisfies 𝜇0(𝜓)=𝑛𝜓(𝑡)d𝑡=0.

The wavelet transform of 𝑓𝒮(𝑛) with respect to the wavelet 𝜓𝒮(𝑛) is defined as 𝒲𝜓1𝑓(𝑥,𝑦)=𝑓(𝑡),𝑦𝑛𝜓𝑡𝑥𝑦=1𝑦𝑛𝑛𝑓(𝑡)𝜓𝑡𝑥𝑦d𝑡,(2.3) where (𝑥,𝑦)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 𝑓𝒮0(𝑛) if the wavelet 𝜓𝒮0(𝑛). The wavelet transform can be defined exactly in the same way for vector-valued distributions.

2.3. Wavelet Synthesis Operator

Let 𝜂𝒮0(𝑛). The wavelet synthesis transform of Φ𝒮(𝑛+1) with respect to the wavelet 𝜂 is defined as 𝜂Φ(𝑡)=0𝑛1Φ(𝑥,𝑦)𝑦𝑛𝜂𝑡𝑥𝑦d𝑥d𝑦𝑦,𝑡𝑛.(2.4) Observe that the operator 𝜂 may be extended to act on the space 𝒮(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(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 𝜓𝒮0(𝑛) admits a reconstruction wavelet 𝜂𝒮0(𝑛). More precisely, it means that the constant 𝑐𝜓,𝜂=𝑐𝜓,𝜂(𝜔)=0𝜓(𝑟𝜔)̂𝜂(𝑟𝜔)d𝑟𝑟,𝜔𝕊𝑛1(2.5) is different from zero and independent of the direction 𝜔. Then, a straightforward calculation [1] shows that Id𝒮0(𝑛)=1𝑐𝜓,𝜂𝜂𝒲𝜓.(2.6) It is worth pointing out that (2.6) is also valid [1, 10] when 𝒲𝜓 and 𝜂 act on the spaces 𝒮0(𝑛) and 𝒮(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 𝜑𝒮(𝑛) is said to be nondegenerate if for any 𝜔𝕊𝑛1 the function of one variable 𝑅𝜔(𝑟)=𝜑(𝑟𝜔)𝐶[0,) is not identically zero, that is, supp𝑅𝜔, for each 𝜔𝕊𝑛1.

3. The Wavelet Transform of Test Functions

The wavelet and wavelet synthesis transforms induce the following bilinear mappings: 𝒲(𝜓,𝜑)𝒲𝜓𝜑,(𝜂,Φ)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)𝒲𝒮0(𝑛)×𝒮0(𝑛)𝒮(𝑛+1), (ii)𝒮0(𝑛)×𝒮(𝑛+1)𝒮0(𝑛)are continuous.

Proof. Continuity of the Wavelet Mapping. We will prove that for arbitrary 𝑙,𝑘,𝜈,𝑚𝑛, there exist 𝑁 and 𝐶>0 such that 𝜌0𝑙,𝑘,𝜈,𝑚𝒲𝜓𝜑𝐶𝑖,|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 on 𝜑 and 𝜓, 𝜕𝜈𝑦𝜕𝑚𝑥𝒲𝜓𝜑(𝑥,𝑦)=𝜕𝜈𝑦𝜕𝑚𝑥𝑛𝜑(𝑦𝑡+𝑥)=𝜓(𝑡)d𝑡|𝑗|𝜈𝑐𝑗𝑛𝜕𝑚+𝑗𝜑(𝑦𝑡+𝑥)𝑡𝑗=𝜓(𝑡)d𝑡|𝑗|𝜈𝑐𝑗𝒲𝜓𝑗𝜑𝑚+𝑗(𝑥,𝑦),(3.3) where 𝜓𝑗(𝑡)=𝑡𝑗𝜓(𝑡)𝒮0(𝑛) and 𝜑𝑚+𝑗=𝜕𝑚+𝑗𝜑𝒮0(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 𝑐𝑟,𝑠 independent of 𝜓 and 𝜑, 1+|𝑥|2𝑘/2𝒲𝜓1𝜑(𝑥,𝑦)=(2𝜋)𝑛(1Δ𝜉)𝑘/2𝑒𝑖𝜉𝑥,𝜑(𝜉)=1𝜓(𝑦𝜉)(2𝜋)𝑛|𝑟|+|𝑠|𝑘𝑐𝑟,𝑠𝑦|𝑠|𝑛𝑒𝑖𝜉𝑥𝜕𝑟𝜑(𝜉)𝜕𝑠𝜉=1𝜓(𝑦𝜉)d𝜉(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) for 𝑘=𝜈=0 and 𝑚=0. We may also assume that 𝑙𝑛.
We first estimate the term 𝑦𝑙|𝒲𝜓𝜑(𝑥,𝑦)|. Since 𝜕𝑗𝜑(0)=0 for every 𝑗𝑛, we can apply the Taylor formula to obtain 𝜑(𝜉)=||𝑗||=𝑙𝑛𝜕𝑗𝑧𝜑𝜉𝜉𝑗!𝑗,forsome𝑧𝜉[].inthelinesegment0,𝜉(3.5) Hence, 𝑦𝑙||𝒲𝜓||=𝑦𝜑(𝑥,𝑦)𝑙(2𝜋)𝑛|||𝑒𝑖𝜉𝑥𝜑(𝜉),|||1𝜓(𝑦𝜉)(2𝜋)𝑛𝑛||||𝑦𝜑(𝜉)𝑙||||1𝜓(𝑦𝜉)d𝜉(2𝜋)𝑛||𝑗||=𝑙𝑛1𝑗!𝑛||𝜕𝑗𝑧𝜑𝜉||𝑦𝑛||(𝑦𝜉)𝑗||1𝜓(𝑦𝜉)d𝜉(2𝜋)𝑛||𝑗||=𝑙𝑛𝜌0,𝑗𝜑𝑗!𝑛||𝜉𝑗||1𝜓(𝜉)d𝜉(2𝜋)𝑛||𝑗||=𝑙𝑛𝜌0,𝑗𝜌𝜑𝑙+𝑛,0𝜓𝑗!𝑛d𝜉||𝜉||1+2𝑛𝐶𝜌𝑙+𝑛,0(𝜑)||𝑗||2𝑙+2𝑛𝜌2𝑛,𝑗(𝜓),(𝑥,𝑦)𝑛+1.(3.6)
It remains to estimate 𝑦𝑙|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], 1𝑦𝑙||𝒲𝜓||=1𝜑(𝑥,𝑦)𝑦𝑙||||𝑛𝜑(𝑦𝑡+𝑥)𝜓||||=1(𝑡)d𝑡𝑦𝑙|||||𝑛𝜓(𝑡)|𝑗|<𝑙𝜕𝑗𝜑(𝑥)𝑗!(𝑦𝑡)𝑗+|𝑗|=𝑙𝜕𝑗𝜑𝑧𝑡𝑗!(𝑦𝑡)𝑗|||||d𝑡||𝑗||=𝑙𝜌0,𝑗(𝜑)𝑗!𝑛1+|𝑡|2𝑙/2||||𝜓(𝑡)d𝑡𝐶𝜌𝑙+2𝑛,0(𝜓)||𝑗||=𝑙𝜌0,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 and 𝜅𝑛 there exist 𝑁 and 𝐶>0 such that 𝜌𝑘,𝜅𝜂Φ𝐶𝑘1,𝑘2||𝑗||,𝑙,|𝑚|𝑁𝑖,𝑁𝜌𝑖,𝑗(𝜂)𝜌0𝑘1,𝑘2,𝑙,𝑚(Φ).(3.8) Since 𝜕𝜅𝑡𝜂Φ=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)=𝑛Φ(𝑥,𝑦)𝑒𝑖𝜉𝑥d𝑥. We may assume that 𝑘 is even. We then have (1+|𝑡|2)𝑘/2||𝜂||Φ(𝑡)=(1+|𝑡|2)𝑘/2||||+𝑛1Φ(𝑡𝑥,𝑦)𝑦𝑛𝜂𝑥𝑦d𝑥d𝑦𝑦||||=(1+|𝑡|2)𝑘/2(2𝜋)𝑛|||||+𝑛(1Δ𝜉)𝑘/2𝑒𝑖𝑡𝜉(1+|𝑡|2)𝑘/2Φ(𝜉,𝑦)̂𝜂(𝑦𝜉)d𝜉d𝑦𝑦|||||1(2𝜋)𝑛+𝑛|||(1Δ𝜉)𝑘/2Φ|||(𝜉,𝑦)̂𝜂(𝑦𝜉)d𝜉d𝑦𝑦1(2𝜋)𝑛|𝑟|+|𝑠|𝑘𝑐𝑟,𝑠+𝑛||𝜕𝑟𝜉||𝑦Φ(𝜉,𝑦)|𝑠|1||𝜕𝑠||1̂𝜂(𝑦𝜉)d𝜉d𝑦(2𝜋)𝑛|𝑟|+|𝑠|𝑘𝑐𝑟,𝑠𝜌0,𝑠(̂𝜂)+𝑛𝑦|𝑠|1||𝜕𝑟𝜉Φ||(𝜉,𝑦)d𝜉d𝑦𝐶|𝑟|+|𝑠|𝑘𝜌0,𝑠𝜌(̂𝜂)0|𝑠|1,2𝑛,0,𝑟Φ+𝜌0|𝑠|+1,2𝑛,0,𝑟Φ+𝑛d𝜉(1+|𝜉|2)𝑛d𝑦1+𝑦2𝐶||𝑗|||𝑟|+|𝑠|𝑘2𝑛𝜌|𝑠|+2𝑛,0(𝜌𝜂)0|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 that we can extend the bilinear mapping (𝜂,Φ)𝜂Φ to act on 𝒮(𝑛)×𝒮𝑛+1𝒮(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 𝜓𝒮(𝑛) be a nondegenerate wavelet. Its index of nondegenerateness is the (finite) number 𝜏=inf𝑟+supp𝑅𝜔[]0,𝑟,𝜔𝕊𝑛1,(4.1) where 𝑅𝜔 are the functions of one variable 𝑅𝜔(𝑟)=𝜓(𝑟𝜔).

If we only know values of 𝒲𝜓𝑓(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, 𝜙𝒮(𝑛). Let 𝜓𝒮(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 𝜇𝑚(𝜓)=0 for all multi-index 𝑚𝑛 with |𝑚|[𝛼].

Example 4.3. Let 𝜙𝒮(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)=𝜉𝜑(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 𝐶𝛼(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 𝜂𝒮0(𝑛) is a reconstruction wavelet for 𝜓 whose Fourier transform has support in 𝐵(0,𝜎) and 𝜑𝒟(𝑛) is such that 𝜑(𝜉)=1 for 𝜉𝐵(0,𝜎) and supp𝜑𝐵(0,𝑟), then, for all 𝑓𝒮(𝑛) and 𝜃𝒮(𝑛)𝑓,𝜃=𝑛(𝑓𝜙)(𝑡)𝜃11(𝑡)d𝑡+𝑐𝜓,𝜂10𝑛𝒲𝜓𝑓(𝑥,𝑦)𝒲𝜂𝜃2(𝑥,𝑦)d𝑥d𝑦𝑦,(4.2) where ̂𝜃1̂(𝜉)=𝜃(𝜉)𝜑(𝜉)/𝜙(𝜉) and ̂𝜃2̂(𝜉)=𝜃(𝜉)(1𝜑(𝜉)).

Proof. Observe that 𝑓𝜙,𝜃1=(2𝜋)𝑛𝑓̂𝜃(𝜉),(𝜉)𝜑(𝜉).(4.3) It is therefore enough to assume that 𝜃1=0 so that 𝜃=𝜃2. Our assumption over 𝜂 is that 𝜂𝒮0(𝑛), supp̂𝜂𝐵(0,𝜎), and 𝑐𝜓,𝜂=0𝜓(𝑟𝜔)̂𝜂(𝑟𝜔)dr𝑟0(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, 𝒲𝜂𝜃(𝑥,𝑦)=0 for all (𝑥,𝑦)𝑛×(1,). Exactly as in [1, page 66], the usual calculation shows that 1𝜃(𝑡)=𝑐𝜓,𝜂𝜓𝒲𝜂𝜃1(𝑡)=𝑐𝜓,𝜂10𝑛𝜓𝑡𝑥𝑦𝒲𝜂𝜃(𝑥,𝑦)d𝑥d𝑦𝑦.(4.5) Furthermore, since 𝒲𝜂𝜃𝒮(𝑛+1) (cf. Theorem 3.1), the last integral can be expressed as the limit in 𝒮(𝑛) of Riemann sums. That justifies the exchange of dual pairing and integral in 1𝑓,𝜃=𝑓(𝑡),𝑐𝜓,𝜂10𝑛𝜓𝑡𝑥𝑦𝒲𝜂𝜃(𝑥,𝑦)d𝑥d𝑦𝑦=1𝑐𝜓,𝜂10𝑛𝒲𝜓𝑓(𝑥,𝑦)𝒲𝜂𝜃(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, lim𝜀0+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 𝜀𝛼𝐿(𝜀), where 𝐿 is slowly varying and 𝛼.

4.3. Weak Asymptotics

We will use some notions from the theory of asymptotics of generalized functions [10, 1719]. 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 𝐟𝒮(𝑛,𝐸), we write 𝐟𝑥0+𝜀𝑡=𝑂(𝜀𝛼𝐿(𝜀))as𝜀0+in𝒮(𝑛,𝐸),(4.8) if the order growth relation holds after evaluation at each test function, that is, 𝐟𝑥0+𝜀𝑡,𝜑(𝑡)𝐶𝜑𝜀𝛼𝐿(𝜀),0<𝜀1,(4.9) for each test function 𝜑𝒮(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 [2325]). The vector-valued distribution 𝐟𝒮(𝑛,𝐸) is said to have a value 𝐯𝐸 at the point 𝑥0𝑛 if lim𝜀0+𝐟(𝑥0+𝜀𝑡)=𝐯, distributionally, that is, for each 𝜑𝒮(𝑛)lim𝜀0+1𝐟(𝑡),𝜀𝑛𝜑𝑡𝑥0𝜀=𝐯𝑛𝜑(𝑡)d𝑡𝐸.(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 𝑥0𝑛 and 𝛼, the pointwise weak Hölder space [10] 𝐶𝑤𝛼,𝐿(𝑥0,𝐸) consists of those distributions 𝐟𝒮(𝑛,𝐸) for which there is an 𝐸-valued polynomial 𝐏 of degree less than 𝛼 such that (cf. Section 4.3) 𝐟𝑥0+𝜀𝑡=𝐏(𝜀𝑡)+𝑂(𝜀𝛼𝐿(𝜀))as𝜀0+in𝒮(𝑛,𝐸).(4.11) Observe that if 𝛼<0, then the polynomial is irrelevant. In addition, when 𝛼0, 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𝜕𝑚𝐟𝑥0𝑡𝑚!𝑚.(4.12)

The pointwise weak Hölder space 𝐶𝛼,𝐿,𝑤(𝑥0,𝐸) of second type is defined as follows: 𝐟𝐶𝛼,𝐿,𝑤(𝑥0,𝐸) if (4.9) is just assumed to hold for each 𝜑𝒮(𝑛) 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, 𝐶𝛼,𝐿,𝑤(𝑥0,𝐸)=𝐶𝑤𝛼,𝐿(𝑥0,𝐸). One can also show that if 𝛼, the equality between these two spaces remains true [10]. On the other hand, when 𝛼, we have the strict inclusion 𝐶𝑤𝛼,𝐿(𝑥0,𝐸)𝐶𝛼,𝐿,𝑤(𝑥0,𝐸)(cf. comments below Theorem 4.5). The usefulness of 𝐶𝛼,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 and let 𝜓𝒮(𝑛) be a nondegenerate wavelet with 𝜇𝑚(𝜓)=0 for |𝑚|[𝛼]. Then, 𝐟𝐶𝛼,𝐿,𝑤(𝑥0,𝐸) if and only if there is 𝑘 such that limsup𝜀0+sup|𝑥|2+𝑦2=1,𝑦>0𝑦𝑘𝜀𝛼𝒲𝐿(𝜀)𝜓𝐟𝑥0+𝜀𝑥,𝜀𝑦<.(4.13)

It is worth mentioning that the elements of 𝐶𝛼,𝐿,𝑤(𝑥0,𝐸) for 𝛼=𝑝 can be characterized by pointwise weak-asymptotic expansions. We have [10] that 𝐟𝐶𝑝,𝐿,𝑤(𝑥0,𝐸) if and only if it admits the following weak expansion: 𝐟𝑥0=+𝜀𝑡|𝑚|<𝑝𝜕𝑚𝐟𝑥0𝑚!(𝜀𝑡)𝑚+𝜀𝑝|𝑚|=𝑝𝑡𝑚𝐜𝑚(𝜀)+𝑂(𝜀𝑝𝐿(𝜀))in𝒮(𝑛,𝐸),(4.14) 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.

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+. We say that a function 𝑓 belongs to the space 𝐶𝛼,𝐿(𝑛) if 𝑓 has continuous derivatives up to order less than 𝛼 and 𝑓𝐶𝛼,𝐿=||𝑗||[𝛼]sup𝑡𝑛||𝜕𝑗||+𝑓(𝑡)|𝑚|=[𝛼]sup0<|𝑡𝑥|1||𝜕𝑚𝑓(𝑡)𝜕𝑚||𝑓(𝑥)|𝑡𝑥|𝛼[𝛼]𝐿(|𝑡𝑥|)<.(5.1) When 𝛼=𝑝+1, we replace the previous requirement by 𝑓𝐶𝛼,𝐿=||𝑗||𝑝sup𝑡𝑛||𝜕𝑗||+𝑓(𝑡)|𝑚|=𝑝sup0<|𝑡𝑥|1||𝜕𝑚𝑓(𝑡)𝜕𝑚||𝑓(𝑥)|𝑡𝑥|𝐿(|𝑡𝑥|)<.(5.2)

The space 𝐶𝛼,𝐿(𝑛) is clearly a Banach space with the above norm. The conditions imposed over 𝐿 ensure that 𝐶𝛼,𝐿(𝑛) depends only on the behavior of 𝐿 near 0, thus, it is invariant under dilations. When 𝐿1, this space reduces to 𝐶𝛼,𝐿(𝑛)=𝐶𝛼(𝑛), the usual global (inhomogeneous) Hölder space [2, 3, 5]. Consequently, as in [10], we call 𝐶𝛼,𝐿(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: 𝐶𝛽(𝑛)𝐶𝛼,𝐿(𝑛)𝐶𝛾(𝑛),whenever0<𝛾<𝛼<𝛽.(5.3)

5.2. 𝐿-Zygmund Spaces

We now proceed to define the weighted Zygmund space 𝐶𝛼,𝐿(𝑛). Let 𝛼 and fix an LP-pair (𝜙,𝜓) of order 𝛼. A distribution 𝑓𝒮(𝑛) is said to belong to the 𝐿-Zygmund space 𝐶𝛼,𝐿(𝑛) of exponent 𝛼 if 𝑓𝐶𝛼,𝐿=𝑓𝜙𝐿+sup𝑥𝑛sup0<𝑦11𝑦𝛼𝐿||𝒲(𝑦)𝜓||𝑓(𝑥,𝑦)<.(5.4)

Observe that we clearly have 𝐶𝛼,𝐿(𝑛)𝐶𝛼,𝐿(𝑛), for 𝛼>0. We will show that if 𝛼+, we actually have the equality 𝐶𝛼,𝐿(𝑛)=𝐶𝛼,𝐿(𝑛). When 𝛼 is a positive integer, we have in turn 𝐶𝛼,𝐿(𝑛)C𝛼,𝐿(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: 𝐶𝛽(𝑛)𝐶𝛼,𝐿(𝑛)𝐶𝛾(𝑛),whenever𝛾<𝛼<𝛽.(5.5) The definition of 𝐶𝛼,𝐿(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 𝐶𝛼,𝐿(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 𝐶𝛼,𝐿(𝑛). Therefore, when 𝐿1, we recover the classical Zygmund space 𝐶𝛼,𝐿(𝑛)=𝐶𝛼(𝑛) [5]. Proposition 5.1 follows at once from the following lemma.

Lemma 5.2. Let 𝑓𝐶𝛼,𝐿(𝑛), then for every 𝜃𝒮(𝑛) there holds 𝑓𝜃𝐿𝐶𝑓𝐶𝛼,𝐿,(5.6) where 𝑓𝐶𝛼,𝐿 is given by (5.4). Furthermore, if 𝔅𝒮(𝑛) is a bounded set such that 𝜇𝑚(𝜃)=0 for all 𝜃𝔅 and all multi-index 𝑚[𝛼], then sup𝑥𝑛sup0<𝑦11𝑦𝛼𝐿||𝒲(𝑦)𝜃||𝑓(𝑥,𝑦)𝐶𝑓𝐶𝛼,𝐿,𝜃𝔅.(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 𝒲𝜃𝑓(𝑥,𝑦)=𝐹1(𝑥,𝑦)+𝐹2(𝑥,𝑦)+𝐹3(𝑥,𝑦),(5.8) where 𝐹1(1𝑥,𝑦)=(2𝜋)𝑛𝑛(𝑓𝜑𝑥+𝑦𝑡)𝐹𝜃(𝑡)d𝑡,21(𝑥,𝑦)=𝑐𝜓,𝜂01/𝑦𝑛𝒲𝜓𝑓(𝑦𝑏+𝑥,𝑦𝑎)𝒲𝜂𝜃2,𝑦(𝑏,𝑎)d𝑏d𝑎𝑎,𝐹31(𝑥,𝑦)=𝑐𝜓,𝜂10𝑛𝒲𝜓𝑓(𝑦𝑏+𝑥,𝑦𝑎)𝒲𝜂𝜃3,𝑦(𝑏,𝑎)d𝑏d𝑎𝑎,(5.9) with ̂𝜃2,𝑦̂(𝜉)=𝜃(𝜉)(1𝜑(𝜉))(1𝜑(𝜉/𝑦)) and ̂𝜃3,𝑦̂(𝜉)=𝜃(𝜉)𝜑(𝜉)(1𝜑(𝜉/𝑦)). To estimate 𝐹1, we first observe that if 𝛼<0, then sup0<𝑦1sup𝑥𝑛||𝐹1||(𝑥,𝑦)𝑦𝛼||𝑓||𝐿(𝑦)𝐶𝐶𝛼,𝐿𝜃𝐿1sup0<𝑦11𝑦𝛼𝐿(𝑦)𝐶||𝑓||𝐶𝛼,𝐿,(5.10) because slowly varying functions satisfy the estimates 𝑦𝜀<𝐶𝐿(𝑦) for any exponent 𝜀>0 [7, 8]. When 𝛼0, we have that (𝑓𝜑) 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 sup0<𝑦1sup𝑥𝑛||𝐹1||(𝑥,𝑦)𝑦𝛼𝐿(𝑦)sup0<𝑦1𝐶𝑓𝐶𝛼,𝐿𝑦[𝛼]+1𝛼𝐿(𝑦)𝑛|𝑡|[𝛼]+1||||𝜃(𝑡)d𝑡𝐶𝑓𝐶𝛼,𝐿.(5.11) We now bound 𝐹2 and 𝐹3. If 𝜀>0, Potter's estimate [7, page 25] gives the existence of a constant 𝐶=𝐶𝜀 such that 𝐿(𝑎𝑦)𝑎𝐿(𝑦)<𝐶𝜀+1𝑎𝜀,0<𝑦<1,𝑎<1/𝑦.(5.12) Thus, ||𝐹2||(𝑥,𝑦)𝑦𝛼𝐿(𝑦)𝑓𝐶𝛼,𝐿𝑐𝜓,𝜂01/𝑦𝑛𝑎𝛼1𝐿(𝑎𝑦)||𝒲𝐿(𝑦)𝜂𝜃2,𝑦||(𝑏,𝑎)d𝑏d𝑎𝐶𝑓𝐶𝛼,𝐿0𝑛𝑎𝛼+𝑎𝛼2||𝒲𝜂𝜃2,𝑦(||𝑏,𝑎)d𝑏d𝑎.(5.13) Notice that {𝜃2,𝑦𝒮0(𝑛)𝜃𝔅,𝑦(0,1]} is a bounded set in 𝒮0(𝑛) because the derivatives of 𝜑 are supported in {𝜉𝜎|𝜉|𝑟}. Thus, due to the continuity of 𝒲𝜂 (cf. Theorem 3.1), {𝒲𝜂𝜃𝑦𝜃𝔅,𝑦(0,1]} is a bounded set in 𝒮(𝑛+1). This implies that the integrals involved in the very last estimate are uniformly bounded for 𝜃𝔅 and 𝑦(0,1]. Consequently, we obtain that sup0<𝑦1sup𝑥𝑛||𝐹2||(𝑥,𝑦)𝑦𝛼𝐿(𝑦)𝐶𝑓𝐶𝛼,𝐿.(5.14) Next, for 𝐹3, we have ||𝐹3||(𝑥,𝑦)𝑦𝛼𝐿(𝑦)𝐶𝑓𝐶𝛼,𝐿10𝑛𝑎𝛼3/2+𝑎𝛼1/2||𝒲𝜂𝜃3,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)𝜕𝑚𝐶𝛼,𝐿(𝑛)𝐶𝛼|𝑚|,𝐿(𝑛) is continuous, for any 𝑚𝑛,(ii)if 𝑓𝜙𝐿(𝑛) and 𝜕𝑗𝑓𝐶𝛼1,𝐿(𝑛) for 𝑗=1,,𝑛, then 𝑓𝐶𝛼,𝐿(𝑛),(iii)the mapping (1Δ)𝛽/2 is an isomorphism of the space 𝐶𝛼,𝐿(𝑛) onto 𝐶𝛼𝛽,𝐿(𝑛), for arbitrary 𝛼,𝛽.

Proof. (i) It is enough to consider 𝜕𝑗. We have that 𝜕𝑗𝑓𝜙=𝑓𝜕𝑗𝜙 and 𝒲𝜓𝜕𝑗𝑓(𝑥,𝑦)=𝑦1𝒲𝜕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 Δ𝑓𝐶𝛼2,𝐿(𝑛). In view of Proposition 5.1, it remains to observe that 1𝑦𝛼𝐿𝒲(𝑦)Δ𝜓1𝑓(𝑥,𝑦)=𝑦𝛼2𝒲𝐿(𝑦)𝜓(Δ𝑓)(𝑥,𝑦).(5.16)
(iii) Since (1Δ)𝛽/2 is the inverse of (1Δ)𝛽/2, it suffices to show that (1Δ)𝛽/2 maps continuously 𝐶𝛼,𝐿(𝑛) into 𝐶𝛼𝛽,𝐿(𝑛). Using (5.6) with 𝜃=(1Δ)𝛽/2𝜙, we obtain that (1Δ)𝛽/2𝑓𝜙𝐿𝐶𝑓𝐶𝛼,𝐿. We also have 𝒲𝜓(1Δ)𝛽/21𝑓(𝑥,𝑦)=(2𝜋)𝑛𝑦𝛽𝑓(𝜉),𝑒𝑖𝑥𝜉𝑦2+||||𝑦𝜉2𝛽/2=1𝜓(𝑦𝜉)𝑦𝛽𝒲𝜃𝑦𝑓(𝑥,𝑦),where𝜃𝑦=𝑦2Δ𝛽/2𝜓.(5.17) Finally, we can apply (5.7) because 𝔅={(𝑦2Δ)𝛽/2𝜓}𝑦(0,1] is a bounded set in 𝒮(𝑛) and 𝜇𝑚(𝜃𝑦)=0 for each multi-index |𝑚|[𝛼].

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

Proposition 5.4. The space 𝐶𝛼,𝐿(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 {𝑓𝑗}𝑗=0 is a Cauchy sequence in 𝐶𝛼,𝐿(𝑛). Then, there exist continuous functions 𝑔𝐿(𝑛) and 𝐺 defined on 𝑛×(0,1] such that 𝑓𝑗𝜙𝑔 in 𝐿(𝑛) and lim𝑗sup𝑦(0,1]sup𝑥𝑛1𝑦𝛼𝐿||𝒲(𝑦)𝜓𝑓𝑗||(𝑥,𝑦)𝐺(𝑥,𝑦)=0.(5.18) We define the distribution 𝑓𝒮(𝑛) whose action on test functions 𝜃𝒮(𝑛) is given by 𝑓,𝜃=𝑛𝑔(𝑡)𝜃11(𝑡)d𝑡+𝑐𝜓,𝜂10𝑛𝐺(𝑥,𝑦)𝒲𝜂𝜃2(𝑥,𝑦)d𝑥d𝑦𝑦.(5.19) Since the 𝑓𝑗 have the representation (4.2), we immediately see that 𝑓𝑗𝑓 in 𝒮(𝑛). Thus, pointwisely, (𝑓𝜙)(𝑡)=lim𝑗𝑓𝑗𝒲𝜙(𝑡)=𝑔(𝑡),𝜓𝑓(𝑥,𝑦)=lim𝑗𝒲𝜓𝑓𝑗(𝑥,𝑦)=𝐺(𝑥,𝑦).(5.20) This implies that lim𝑗𝑓𝑓𝑗𝐶𝛼,𝐿=0, and so 𝐶𝛼,𝐿(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 𝐶𝑏(𝑛) the Banach space of continuous and bounded functions.

Theorem 5.5. Let 𝛼>0.(a)If 𝛼, then 𝐶𝛼,𝐿(𝑛)=𝐶𝛼,𝐿(𝑛). Moreover, the norms (5.4) and (5.1) are equivalent. (b)If 𝛼=𝑝+1, then 𝐶𝑝+1,𝐿(𝑛) consists of functions with continuous derivatives up to order 𝑝 such that |𝑚|𝑝𝜕𝑚𝑓𝐿+|𝑚|=𝑝sup𝑡𝑛0<||1||𝜕𝑚𝑓(𝑡+)+𝜕𝑚𝑓(𝑡)2𝜕𝑚||𝑓(𝑡)||||𝐿||||<.(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 𝑓𝐶𝛼,𝐿(𝑛). Consider the 𝐶𝑏(𝑛)-valued distribution 𝐟𝒮(𝑛𝑡,𝐶𝑏(𝑛𝜉)) given by 𝐟(𝑡)(𝜉)=𝑓(𝑡+𝜉), that is, the one whose action on test functions is given by ̌𝜃𝐟(𝑡),𝜃(𝑡)(𝜉)=𝑓(𝑡+𝜉),𝜃(𝑡)=𝑓(𝜉),𝜃𝒮(𝑛),𝜉𝑛.(5.22) It does take values in 𝐶𝑏(𝑛) 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 𝒲𝜓𝐟(𝜀𝑥,𝜀𝑦)𝐶𝑏(𝑛)𝐶𝜀𝛼𝐿(𝜀)𝑦𝛼1𝜀(0,1),(𝑥,𝑦)𝑛].×(0,1(5.23) Therefore, the Tauberian Theorem 4.5 yields 𝐟𝐶𝛼,𝐿,𝑤(0,𝐶𝑏(𝑛)). Now, the Łojasiewicz point values 𝜕𝑚𝐟(0)=𝑣𝑚𝐶𝑏(𝑛) exist, distributionally, for |𝑚|<𝛼. It explicitly means that for all 𝜃𝒮(𝑛)lim𝜀0+𝜕𝑚̌𝜃𝑓𝜀=lim𝜀0+𝜕𝑚𝐟(𝜀𝑡),𝜃(𝑡)=𝜇0(𝜃)𝑣𝑚in𝐶𝑏𝑛𝜉,(5.24) where ̌𝜃𝜀(𝑡)=𝜀𝑛𝜃(𝑡/𝜀). If we now take 𝜃 with 𝜇0(𝜃)=1, we then conclude that 𝜕𝑚𝑓=𝑣𝑚𝐶𝑏(𝑛) for each |𝑚|<𝛼. It remains in both cases to deal with the estimates for 𝜕𝑚𝑓; notice that 𝜕𝑚𝐟𝐶𝛼[𝛼],𝐿,𝑤(0,𝐶𝑏(𝑛)) when |𝑚|=[𝛼] and 𝜕𝑚𝐟𝐶1,𝐿,𝑤(0,𝐶𝑏(𝑛)) when |𝑚|=𝑝. We now divide the proof into two cases.
Case 𝛼. Fix a multi-index |𝑚|=[𝛼]. It suffices to show sup0<|𝑥𝑡|<1||𝜕𝑚𝑓(𝑥)𝜕𝑚||𝑓(𝑡)|𝑥𝑡|𝛼[𝛼]𝐿(|𝑥𝑡|)<.(5.25) We had already seen that 𝜕𝑚𝐟(𝑡)(𝜉)=𝜕𝑚𝑓(𝑡+𝜉)𝐶𝑤𝛼[𝛼],𝐿(0,𝐶𝑏(𝑛𝜉))=𝐶𝛼[𝛼],𝐿,𝑤(0,𝐶𝑏(𝑛𝜉)), that is, 𝜇0(𝜃)𝜕𝑚𝑓(𝜉)𝑛𝜕𝑚𝜀𝑓(𝜉+𝜀𝑡)𝜃(𝑡)d𝑡=𝑂𝛼[𝛼]𝐿(𝜀),𝜀0+,(5.26) in the space C𝑏(𝑛𝜉), for each 𝜃𝒮(𝑛). Hence, if 0<||1, we choose 𝜃 as before (𝜇0(𝜃)=1), and we use the fact that {𝜃𝜃(𝜔)|𝜔|=1} is compact in 𝒮(𝑛); we then have sup𝜉𝑛||𝜕𝑚𝑓(𝜉+)𝜕𝑚𝑓||(𝜉)2sup𝜉𝑛||||𝜕𝑚𝑓(𝜉)𝑛𝜕𝑚𝑓||||||||𝑡+𝜉𝜃(𝑡)d𝑡+sup𝜉𝑛||||𝑛𝜕𝑚𝑓||||𝑡||||𝜉+𝜃(𝑡)𝜃𝑡1||||||||d𝑡=𝑂𝛼[𝛼]𝐿||||,(5.27) and this completes the proof of (a).
Case 𝛼=𝑝+1. The proof is similar to that of (a). Fix now |𝑚|=𝑝. We now have 𝜕𝑚𝐟𝐶1,𝐿,𝑤(0,𝐶𝑏(𝑛)), which, as commented in Section 4.4, yields the distributional expansion 𝜕𝑚𝐟(𝜀𝑡)(𝜉)=𝜕𝑚𝑓(𝜉)+𝜀𝑛𝑗=1𝑡𝑗𝐜𝑗(𝜀,𝜉)+𝑂(𝜀𝐿(𝜀)),0<𝜀1,(5.28) in 𝒮(𝑛𝑡,𝐶𝑏(𝑛𝜉), where the 𝐜𝑗(𝜀,) are continuous 𝐶𝑏(𝑛𝜉)-valued functions in 𝜀. We apply (5.28) on a test function 𝜃𝒮(𝑛), with 𝜇0(𝜃)=1, and 𝑛𝑡𝑗𝜃(𝑡)d𝑡=0 for 𝑗=1,,𝑛, so we get 𝜕𝑚𝑓(𝜉)=𝑛𝜕𝑚𝑓||||𝑡||||𝐿||||||||𝜉+𝜃(𝑡)d𝑡+𝑂,0<1,(5.29) uniformly in 𝜉𝑛. Since {𝜃𝜔=𝜃(+𝜔)+𝜃(𝜔)2𝜃|𝜔|=1} is compact in 𝒮(𝑛) and 𝜇𝑚(𝜃𝜔)=0 for |𝑚|1, the relations (5.28) and (5.29) give sup𝜉𝑛||𝜕𝑚𝑓(𝜉+)+𝜕𝑚𝑓(𝜉)2𝜕𝑚||𝑓(𝜉)3sup𝜉𝑛||||𝜕𝑚𝑓(𝜉)𝑛𝜕𝑚𝑓||||||||+||||𝑡+𝜉𝜃(𝑡)d𝑡𝑛𝜕𝑚𝑓||||𝑡𝜃||||𝜉+𝑡+1||||+𝜃𝑡1||||||||𝐿||||||||2𝜃(𝑡)d𝑡=𝑂,0<1,(5.30) as claimed.


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).