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Journal of Function Spaces and Applications
Volumeย 2012ย (2012), Article IDย 815475, 18 pages
Research Article

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

1Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradoviฤ‡a 4, 21000 Novi Sad, Serbia
2Faculty of Technology, University of Novi Sad, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia
3Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, 9000 Gent, Belgium

Received 14 November 2011; Accepted 27 May 2012

Academic Editor: Hans G.ย Feichtinger

Copyright ยฉ 2012 Stevan Pilipoviฤ‡ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces ๐’ฎ0(โ„๐‘›) and ๐’ฎ(โ„๐‘›+1). We then introduce and study a new class of weighted Hรถlder-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 Hรถlder-Zygmund spaces via the wavelet transform [1โ€“3]. It is well known [1, 4โ€“6] that the wavelet transforms of elements of the classical Zygmund space ๐ถ๐›ผโˆ—(โ„๐‘›) satisfy the size estimate |๐’ฒ๐œ“๐‘“(๐‘ฅ,๐‘ฆ)|โ‰ค๐ถ๐‘ฆ๐›ผ, which, plus a side condition, essentially characterizes the space itself. We will replace the regularity measurement ๐‘ฆ๐›ผ by weights from the interesting class of regularly varying functions [7, 8]. Familiar functions such as ๐‘ฆ๐›ผ,๐‘ฆ๐›ผ|log๐‘ฆ|๐›ฝ,๐‘ฆ๐›ผ|log|log๐‘ฆ||๐›ฝ,โ€ฆ, 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 ๐’ฎ0(โ„๐‘›) and ๐’ฎ(โ„๐‘›+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 ๐ถ๐›ผโˆ—(โ„๐‘›), our analysis provides a new approach to the study of Hรถlder-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 ๐ถ๐›ผ,๐ฟ(โ„๐‘›) and ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›), the ๐ฟ-Hรถlder and ๐ฟ-Zygmund spaces of exponent ๐›ผ 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 Hรถlder-Zygmund spaces. In fact, as explained in Section 5, they satisfy the following useful inclusion relations: ๐ถ๐›ฝ(โ„๐‘›)โŠ‚๐ถ๐›ผ,๐ฟ(โ„๐‘›)โŠ‚๐ถ๐›พ(โ„๐‘›๐ถ),when0<๐›พ<๐›ผ<๐›ฝ,๐›ฝโˆ—(โ„๐‘›)โŠ‚๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›)โŠ‚๐ถ๐›พโˆ—(โ„๐‘›),if๐›พ<๐›ผ<๐›ฝ.(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 ๐ฟ(๐‘ฆ)=|log๐‘ฆ|๐›ฝ has been extensively studied (see, e.g., [1, page 276]). Our analysis will treat more general weights, specifically, the important case when ๐ฟ 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 Hรถlder-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 Hรถlder-Zygmund spaces in Section 5.

2. Notation and Notions

We denote by โ„๐‘›+1=โ„๐‘›ร—โ„+ the upper half space. If ๐‘ฅโˆˆโ„๐‘› and ๐‘šโˆˆโ„•๐‘›, then |๐‘ฅ| denotes the euclidean norm, ๐‘ฅ๐‘š=๐‘ฅ๐‘š11โ‹ฏ๐‘ฅ๐‘š๐‘›๐‘›,โ€‰โ€‰๐œ•๐‘š=๐œ•๐‘š๐‘ฅ=๐œ•๐‘š1๐‘ฅ1โ‹ฏ๐œ•๐‘š๐‘›๐‘ฅ๐‘›,โ€‰โ€‰๐‘š!=๐‘š1!๐‘š2!โ‹ฏ๐‘š๐‘›! and |๐‘š|=๐‘š1+โ‹ฏ+๐‘š๐‘›. If the ๐‘—th coordinate of ๐‘š is one and the others vanish, we then write ๐œ•๐‘—=๐œ•๐‘š๐‘ฅ. The set ๐ต(0,๐‘Ÿ) is the euclidean ball in โ„๐‘› of radius ๐‘Ÿ. In the sequel, we use ๐ถ and ๐ถโ€ฒ 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 ๐’ฎ(โ„๐‘›). We will fix constants in the Fourier transform as โˆซ๎๐œ‘(๐œ‰)=โ„๐‘›๐œ‘(๐‘ก)๐‘’โˆ’๐‘–๐œ‰โ‹…๐‘กd๐‘ก. The moments of ๐œ‘โˆˆ๐’ฎ(โ„๐‘›) are denoted by ๐œ‡๐‘šโˆซ(๐œ‘)=โ„๐‘›๐‘ก๐‘š๐œ‘(๐‘ก)d๐‘ก, ๐‘šโˆˆโ„•๐‘›.

Following [1], the space of highly time-frequency localized functions ๐’ฎ0(โ„๐‘›) is defined as ๐’ฎ0(โ„๐‘›)={๐œ‘โˆˆ๐’ฎ(โ„๐‘›)โˆถ๐œ‡๐‘š(๐œ‘)=0,forall๐‘šโˆˆโ„•๐‘›}; it is provided with the relative topology inhered from ๐’ฎ(โ„๐‘›). In [1], the topology of ๐’ฎ0(โ„๐‘›) is introduced in an apparently different way; however, both approaches are equivalent in view of the open mapping theorem. Observe that ๐’ฎ0(โ„๐‘›) is a closed subspace of ๐’ฎ(โ„๐‘›) and that ๐œ‘โˆˆ๐’ฎ0(โ„๐‘›) if and only if ๐œ•๐‘š๎๐œ‘(0)=0, for all ๐‘šโˆˆโ„•๐‘›. It is important to point out that ๐’ฎ0(โ„๐‘›) is also well known in the literature as the Lizorkin space of test functions (cf. [13]).

The space ๐’ฎ(โ„๐‘›+1) of highly localized functions on the half space [1] consists of those ฮฆโˆˆ๐ถโˆž(โ„๐‘›+1) for which ๐œŒ0๐‘™,๐‘˜,๐œˆ,๐‘š(ฮฆ)=sup(๐‘ฅ,๐‘ฆ)โˆˆโ„๐‘›+1๎‚ต๐‘ฆ๐‘™+1๐‘ฆ๐‘™๎‚ถ๎€ท1+|๐‘ฅ|2๎€ธ๐‘˜/2||๐œ•๐œˆ๐‘ฆ๐œ•๐‘š๐‘ฅ||ฮฆ(๐‘ฅ,๐‘ฆ)<โˆž,(2.1) for all ๐‘™,๐‘˜,๐œˆโˆˆโ„•, and ๐‘šโˆˆโ„•๐‘›. The canonical topology on ๐’ฎ(โ„๐‘›+1) is induced by this family of seminorms [1]. For later use, we will denote by ๐œŒ๐‘˜,๐‘š the corresponding seminorms in ๐’ฎ(โ„๐‘›), namely, ๐œŒ๐‘˜,๐‘š(๐œ‘)=sup๐‘กโˆˆโ„๐‘›๎€ท1+|๐‘ก|2๎€ธ๐‘˜/2||๐œ•๐‘š||๐œ‘(๐‘ก),๐‘˜โˆˆโ„•,๐‘šโˆˆโ„•๐‘›.(2.2)

The corresponding duals of these three spaces are ๐’ฎ๎…ž(โ„๐‘›), ๐’ฎ๎…ž0(โ„๐‘›), and ๐’ฎ๎…ž(โ„๐‘›+1). They are, respectively, the spaces of tempered distributions, Lizorkin distributions, and distributions of slow growth on โ„๐‘›+1. Since the elements of ๐’ฎ0(โ„๐‘›) are orthogonal to every polynomial, ๐’ฎ๎…ž0(โ„๐‘›) can be canonically identified with the quotient space of ๐’ฎ๎…ž(โ„๐‘›) modulo polynomials.

Finally, we will also make use of spaces of vector-valued tempered distributions [14, 15]. If ๐‘‹ is a locally convex topological vector space [15], then the space of ๐‘‹-valued tempered distributions is ๐’ฎ๎…ž(โ„๐‘›,๐‘‹)=๐ฟ๐‘(๐’ฎ(โ„๐‘›),๐‘‹), namely, the space of continuous linear mappings from ๐’ฎ(โ„๐‘›) into ๐‘‹.

2.2. Wavelet Transform

In this paper a wavelet simply means a function ๐œ“โˆˆ๐’ฎ(โ„๐‘›) that satisfies ๐œ‡0โˆซ(๐œ“)=โ„๐‘›๐œ“(๐‘ก)d๐‘ก=0.

The wavelet transform of ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) with respect to the wavelet ๐œ“โˆˆ๐’ฎ(โ„๐‘›) is defined as ๐’ฒ๐œ“๎ƒก1๐‘“(๐‘ฅ,๐‘ฆ)=๐‘“(๐‘ก),๐‘ฆ๐‘›๐œ“๎‚ต๐‘กโˆ’๐‘ฅ๐‘ฆ=1๎‚ถ๎ƒข๐‘ฆ๐‘›๎€œโ„๐‘›๐‘“(๐‘ก)๐œ“๎‚ต๐‘กโˆ’๐‘ฅ๐‘ฆ๎‚ถd๐‘ก,(2.3) where (๐‘ฅ,๐‘ฆ)โˆˆโ„๐‘›+1. The very last integral formula is a formal notation which makes sense when ๐‘“ is a function of tempered growth. Notice that the wavelet transform is also well defined via (2.3) for ๐‘“โˆˆ๐’ฎ๎…ž0(โ„๐‘›) if the wavelet ๐œ“โˆˆ๐’ฎ0(โ„๐‘›). The wavelet transform can be defined exactly in the same way for vector-valued distributions.

2.3. Wavelet Synthesis Operator

Let ๐œ‚โˆˆ๐’ฎ0(โ„๐‘›). The wavelet synthesis transform of ฮฆโˆˆ๐’ฎ(โ„๐‘›+1) with respect to the wavelet ๐œ‚ is defined as โ„ณ๐œ‚๎€œฮฆ(๐‘ก)=โˆž0๎€œโ„๐‘›1ฮฆ(๐‘ฅ,๐‘ฆ)๐‘ฆ๐‘›๐œ‚๎‚ต๐‘กโˆ’๐‘ฅ๐‘ฆ๎‚ถd๐‘ฅd๐‘ฆ๐‘ฆ,๐‘กโˆˆโ„๐‘›.(2.4) Observe that the operator โ„ณ๐œ‚ may be extended to act on the space ๐’ฎ๎…ž(โ„๐‘›+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 ๐’ฎ(โ„๐‘›+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 ๐œ“โˆˆ๐’ฎ0(โ„๐‘›) admits a reconstruction wavelet ๐œ‚โˆˆ๐’ฎ0(โ„๐‘›). More precisely, it means that the constant ๐‘๐œ“,๐œ‚=๐‘๐œ“,๐œ‚๎€œ(๐œ”)=โˆž0๎๐œ“(๐‘Ÿ๐œ”)ฬ‚๐œ‚(๐‘Ÿ๐œ”)d๐‘Ÿ๐‘Ÿ,๐œ”โˆˆ๐•Š๐‘›โˆ’1(2.5) is different from zero and independent of the direction ๐œ”. Then, a straightforward calculation [1] shows that Id๐’ฎ0(โ„๐‘›)=1๐‘๐œ“,๐œ‚โ„ณ๐œ‚๐’ฒ๐œ“.(2.6) It is worth pointing out that (2.6) is also valid [1, 10] when ๐’ฒ๐œ“ and โ„ณ๐œ‚ act on the spaces ๐’ฎ๎…ž0(โ„๐‘›) and ๐’ฎ๎…ž(โ„๐‘›+1), respectively.

Furthermore, it is very important to emphasize that a wavelet ๐œ“ admits a reconstruction wavelet ๐œ‚ if and only if it is nondegenerate in the sense of the following definition [10].

Definition 2.1. A test function ๐œ‘โˆˆ๐’ฎ(โ„๐‘›) is said to be nondegenerate if for any ๐œ”โˆˆ๐•Š๐‘›โˆ’1 the function of one variable ๐‘…๐œ”(๐‘Ÿ)=๎๐œ‘(๐‘Ÿ๐œ”)โˆˆ๐ถโˆž[0,โˆž) is not identically zero, that is, supp๐‘…๐œ”โ‰ โˆ…, for each ๐œ”โˆˆ๐•Š๐‘›โˆ’1.

3. The Wavelet Transform of Test Functions

The wavelet and wavelet synthesis transforms induce the following bilinear mappings: ๐’ฒโˆถ(๐œ“,๐œ‘)โŸผ๐’ฒ๐œ“๐œ‘,โ„ณโˆถ(๐œ‚,ฮฆ)โŸผโ„ณ๐œ‚ฮฆ.(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)๐’ฒโˆถ๐’ฎ0(โ„๐‘›)ร—๐’ฎ0(โ„๐‘›)โ†’๐’ฎ(โ„๐‘›+1), (ii)โ„ณโˆถ๐’ฎ0(โ„๐‘›)ร—๐’ฎ(โ„๐‘›+1)โ†’๐’ฎ0(โ„๐‘›)are continuous.

Proof. Continuity of the Wavelet Mapping. We will prove that for arbitrary ๐‘™,๐‘˜,๐œˆโˆˆโ„•,๐‘šโˆˆโ„•๐‘›, there exist ๐‘โˆˆโ„• and ๐ถ>0 such that ๐œŒ0๐‘™,๐‘˜,๐œˆ,๐‘š๎€ท๐’ฒ๐œ“๐œ‘๎€ธ๎“โ‰ค๐ถ๐‘–โ€ฒ,|๐‘—โ€ฒ|,๐‘–,|๐‘—|โ‰ค๐‘๐œŒ๐‘–โ€ฒ,๐‘—โ€ฒ(๐œ“)๐œŒ๐‘–,๐‘—(๐œ‘).(3.2) We begin by making some reductions. Observe that, for constants ๐‘๐‘— which do not depend on ๐œ‘ and ๐œ“, ๐œ•๐œˆ๐‘ฆ๐œ•๐‘š๐‘ฅ๐’ฒ๐œ“๐œ‘(๐‘ฅ,๐‘ฆ)=๐œ•๐œˆ๐‘ฆ๐œ•๐‘š๐‘ฅ๎€œโ„๐‘›๐œ‘(๐‘ฆ๐‘ก+๐‘ฅ)=๎“๐œ“(๐‘ก)d๐‘ก|๐‘—|โ‰ค๐œˆ๐‘๐‘—๎€œโ„๐‘›๐œ•๐‘š+๐‘—๐œ‘(๐‘ฆ๐‘ก+๐‘ฅ)๐‘ก๐‘—=๎“๐œ“(๐‘ก)d๐‘ก|๐‘—|โ‰ค๐œˆ๐‘๐‘—๐’ฒ๐œ“๐‘—๐œ‘๐‘š+๐‘—(๐‘ฅ,๐‘ฆ),(3.3) where ๐œ“๐‘—(๐‘ก)=๐‘ก๐‘—๐œ“(๐‘ก)โˆˆ๐’ฎ0(โ„๐‘›) and ๐œ‘๐‘š+๐‘—=๐œ•๐‘š+๐‘—๐œ‘โˆˆ๐’ฎ0(โ„๐‘›). It is therefore enough to show (3.2) for ๐œˆ=0 and ๐‘š=0. Next, we may assume that ๐‘˜ is even. We then have, for constants ๐‘๐‘Ÿ,๐‘  independent of ๐œ“ and ๐œ‘, ๎€ท1+|๐‘ฅ|2๎€ธ๐‘˜/2๐’ฒ๐œ“1๐œ‘(๐‘ฅ,๐‘ฆ)=(2๐œ‹)๐‘›๎‚ฌ(1โˆ’ฮ”๐œ‰)๐‘˜/2๐‘’๐‘–๐œ‰โ‹…๐‘ฅ,๎๐œ‘(๐œ‰)๎‚ญ=1๎๐œ“(๐‘ฆ๐œ‰)(2๐œ‹)๐‘›๎“|๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜๐‘๐‘Ÿ,๐‘ ๐‘ฆ|๐‘ |๎€œโ„๐‘›๐‘’๐‘–๐œ‰โ‹…๐‘ฅ๐œ•๐‘Ÿ๎๐œ‘(๐œ‰)๐œ•๐‘ ๐œ‰=1๎๐œ“(๐‘ฆ๐œ‰)d๐œ‰(2๐œ‹)๐‘›๎“|๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜๐‘๐‘Ÿ,๐‘ ๐‘ฆ|๐‘ |๐’ฒ๐œ“๐‘ ๐œ‘๐‘Ÿ(๐‘ฅ,๐‘ฆ),(3.4) where ๐œ‘๐‘Ÿ(๐‘ก)=(โˆ’๐‘–๐‘ก)๐‘Ÿ๐œ‘(๐‘ก) and ๐œ“๐‘ (๐‘ก)=(๐‘–๐‘ก)๐‘ ๐œ“(๐‘ก). Thus, it clearly suffices to establish (3.2) for ๐‘˜=๐œˆ=0 and ๐‘š=0. We may also assume that ๐‘™โ‰ฅ๐‘›.
We first estimate the term ๐‘ฆ๐‘™|๐’ฒ๐œ“๐œ‘(๐‘ฅ,๐‘ฆ)|. Since ๐œ•๐‘—๎๐œ‘(0)=0 for every ๐‘—โˆˆโ„•๐‘›, we can apply the Taylor formula to obtain ๎“๎๐œ‘(๐œ‰)=||๐‘—||=๐‘™โˆ’๐‘›๐œ•๐‘—๎€ท๐‘ง๎๐œ‘๐œ‰๎€ธ๐œ‰๐‘—!๐‘—,forsome๐‘ง๐œ‰[].inthelinesegment0,๐œ‰(3.5) Hence, ๐‘ฆ๐‘™||๐’ฒ๐œ“||=๐‘ฆ๐œ‘(๐‘ฅ,๐‘ฆ)๐‘™(2๐œ‹)๐‘›|||๎‚ฌ๐‘’๐‘–๐œ‰โ‹…๐‘ฅ๎๐œ‘(๐œ‰),๎‚ญ|||โ‰ค1๎๐œ“(๐‘ฆ๐œ‰)(2๐œ‹)๐‘›๎€œโ„๐‘›||||๐‘ฆ๎๐œ‘(๐œ‰)๐‘™||||โ‰ค1๎๐œ“(๐‘ฆ๐œ‰)d๐œ‰(2๐œ‹)๐‘›๎“||๐‘—||=๐‘™โˆ’๐‘›1๎€œ๐‘—!โ„๐‘›||๐œ•๐‘—๎€ท๐‘ง๎๐œ‘๐œ‰๎€ธ||๐‘ฆ๐‘›||(๐‘ฆ๐œ‰)๐‘—||โ‰ค1๎๐œ“(๐‘ฆ๐œ‰)d๐œ‰(2๐œ‹)๐‘›๎“||๐‘—||=๐‘™โˆ’๐‘›๐œŒ0,๐‘—๎€ท๎€ธ๎๐œ‘๎€œ๐‘—!โ„๐‘›||๐œ‰๐‘—||โ‰ค1๎๐œ“(๐œ‰)d๐œ‰(2๐œ‹)๐‘›๎“||๐‘—||=๐‘™โˆ’๐‘›๐œŒ0,๐‘—๎€ท๎€ธ๐œŒ๎๐œ‘๐‘™+๐‘›,0๎€ท๎€ธ๎๐œ“๎€œ๐‘—!โ„๐‘›d๐œ‰๎‚€||๐œ‰||1+2๎‚๐‘›โ‰ค๐ถ๐œŒ๐‘™+๐‘›,0๎“(๐œ‘)||๐‘—||โ‰ค2๐‘™+2๐‘›๐œŒ2๐‘›,๐‘—(๐œ“),โˆ€(๐‘ฅ,๐‘ฆ)โˆˆโ„๐‘›+1.(3.6)
It remains to estimate ๐‘ฆโˆ’๐‘™|๐’ฒ๐œ“๐œ‘(๐‘ฅ,๐‘ฆ)|. We will now use the fact that all the moments of ๐œ“ vanish. If we apply the Taylor formula, we have, for some ๐‘ง๐‘ก=๐‘ง(๐‘ก,๐‘ฅ,๐‘ฆ) in the line segment [๐‘ฅ,๐‘ฆ๐‘ก], 1๐‘ฆ๐‘™||๐’ฒ๐œ“||=1๐œ‘(๐‘ฅ,๐‘ฆ)๐‘ฆ๐‘™||||๎€œโ„๐‘›๐œ‘(๐‘ฆ๐‘ก+๐‘ฅ)๐œ“||||=1(๐‘ก)d๐‘ก๐‘ฆ๐‘™|||||๎€œโ„๐‘›๎ƒฉ๎“๐œ“(๐‘ก)|๐‘—|<๐‘™๐œ•๐‘—๐œ‘(๐‘ฅ)๐‘—!(๐‘ฆ๐‘ก)๐‘—+๎“|๐‘—|=๐‘™๐œ•๐‘—๐œ‘๎€ท๐‘ง๐‘ก๎€ธ๐‘—!(๐‘ฆ๐‘ก)๐‘—๎ƒช|||||โ‰ค๎“d๐‘ก||๐‘—||=๐‘™๐œŒ0,๐‘—(๐œ‘)๎€œ๐‘—!โ„๐‘›๎€ท1+|๐‘ก|2๎€ธ๐‘™/2||||๐œ“(๐‘ก)d๐‘กโ‰ค๐ถ๐œŒ๐‘™+2๐‘›,0๎“(๐œ“)||๐‘—||=๐‘™๐œŒ0,๐‘—(๐œ‘).(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 ๐‘˜โˆˆโ„• and ๐œ…โˆˆโ„•๐‘› there exist ๐‘โˆˆโ„• and ๐ถ>0 such that ๐œŒ๐‘˜,๐œ…๎€ทโ„ณ๐œ‚ฮฆ๎€ธ๎“โ‰ค๐ถ๐‘˜1,๐‘˜2||๐‘—||,๐‘™,|๐‘š|โ‰ค๐‘๐‘–,โ‰ค๐‘๐œŒ๐‘–,๐‘—(๐œ‚)๐œŒ0๐‘˜1,๐‘˜2,๐‘™,๐‘š(ฮฆ).(3.8) Since ๐œ•๐œ…๐‘กโ„ณ๐œ‚ฮฆ=โ„ณ๐œ‚๐œ•๐œ…๐‘ฅฮฆ, it is enough to prove (3.8) for ๐œ…=0. We denote below ๎ฮฆ the partial Fourier transform of ฮฆ with respect to the space coordinate, that is, ๎โˆซฮฆ(๐œ‰,๐‘ฆ)=โ„๐‘›ฮฆ(๐‘ฅ,๐‘ฆ)๐‘’โˆ’๐‘–๐œ‰โ‹…๐‘ฅd๐‘ฅ. We may assume that ๐‘˜ is even. We then have (1+|๐‘ก|2)๐‘˜/2||โ„ณ๐œ‚||ฮฆ(๐‘ก)=(1+|๐‘ก|2)๐‘˜/2||||๎€œโ„+๎€œโ„๐‘›1ฮฆ(๐‘กโˆ’๐‘ฅ,๐‘ฆ)๐‘ฆ๐‘›๐œ‚๎‚ต๐‘ฅ๐‘ฆ๎‚ถd๐‘ฅd๐‘ฆ๐‘ฆ||||=(1+|๐‘ก|2)๐‘˜/2(2๐œ‹)๐‘›|||||๎€œโ„+๎€œโ„๐‘›(1โˆ’ฮ”๐œ‰)๐‘˜/2๐‘’๐‘–๐‘ก๐œ‰(1+|๐‘ก|2)๐‘˜/2๎ฮฆ(๐œ‰,๐‘ฆ)ฬ‚๐œ‚(๐‘ฆ๐œ‰)d๐œ‰d๐‘ฆ๐‘ฆ|||||โ‰ค1(2๐œ‹)๐‘›๎€œโ„+๎€œโ„๐‘›|||(1โˆ’ฮ”๐œ‰)๐‘˜/2๎‚€๎ฮฆ๎‚|||(๐œ‰,๐‘ฆ)ฬ‚๐œ‚(๐‘ฆ๐œ‰)d๐œ‰d๐‘ฆ๐‘ฆโ‰ค1(2๐œ‹)๐‘›๎“|๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜๐‘๐‘Ÿ,๐‘ ๎€œโ„+๎€œโ„๐‘›||๐œ•๐‘Ÿ๐œ‰๎||๐‘ฆฮฆ(๐œ‰,๐‘ฆ)|๐‘ |โˆ’1||๐œ•๐‘ ||โ‰ค1ฬ‚๐œ‚(๐‘ฆ๐œ‰)d๐œ‰d๐‘ฆ(2๐œ‹)๐‘›๎“|๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜๐‘๐‘Ÿ,๐‘ ๐œŒ0,๐‘ ๎€œ(ฬ‚๐œ‚)โ„+๎€œโ„๐‘›๐‘ฆ|๐‘ |โˆ’1||๐œ•๐‘Ÿ๐œ‰๎ฮฆ||(๐œ‰,๐‘ฆ)d๐œ‰d๐‘ฆโ‰ค๐ถ๎…ž๎“|๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜๐œŒ0,๐‘ ๎‚€๐œŒ(ฬ‚๐œ‚)0|๐‘ |โˆ’1,2๐‘›,0,๐‘Ÿ๎‚€๎ฮฆ๎‚+๐œŒ0|๐‘ |+1,2๐‘›,0,๐‘Ÿ๎‚€๎ฮฆ๎€œ๎‚๎‚โ„+๎€œโ„๐‘›d๐œ‰(1+|๐œ‰|2)๐‘›d๐‘ฆ1+๐‘ฆ2๎“โ‰ค๐ถ||๐‘—|||๐‘Ÿ|+|๐‘ |โ‰ค๐‘˜โ‰ค2๐‘›๐œŒ|๐‘ |+2๐‘›,0(๎‚€๐œŒ๐œ‚)0|๐‘ |โˆ’1,|๐‘Ÿ|+2๐‘›,0,๐‘—(ฮฆ)+๐œŒ0|๐‘ |+1,|๐‘Ÿ|+2๐‘›,0,๐‘—(๎‚.ฮฆ)(3.9) This completes the proof.

Remark 3.2. It follows from the proof of the continuity of โ„ณ that we can extend the bilinear mapping โ„ณโˆถ(๐œ‚,ฮฆ)โ†ฆโ„ณ๐œ‚ฮฆ to act on โ„ณโˆถ๐’ฎ(โ„๐‘›๎€ทโ„)ร—๐’ฎ๐‘›+1๎€ธโŸถ๐’ฎ(โ„๐‘›),(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 Hรถlder-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 โ€œdegenerates.โ€ We measure in the next definition how big that ball is.

Definition 4.1. Let ๐œ“โˆˆ๐’ฎ(โ„๐‘›) be a nondegenerate wavelet. Its index of nondegenerateness is the (finite) number ๎€ฝ๐œ=inf๐‘Ÿโˆˆโ„+โˆถsupp๐‘…๐œ”โˆฉ[]0,๐‘Ÿโ‰ โˆ…,โˆ€๐œ”โˆˆ๐•Š๐‘›โˆ’1๎€พ,(4.1) where ๐‘…๐œ” are the functions of one variable ๐‘…๐œ”(๐‘Ÿ)=๎๐œ“(๐‘Ÿ๐œ”).

If we only know values of ๐’ฒ๐œ“๐‘“(๐‘ฅ,๐‘ฆ) at scale ๐‘ฆ<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 ๐œ™ (cf. [16]).

Definition 4.2. Let ๐›ผโˆˆโ„, ๐œ™โˆˆ๐’ฎ(โ„๐‘›). Let ๐œ“โˆˆ๐’ฎ(โ„๐‘›) be a nondegenerate wavelet with the index of nondegenerateness ๐œ. The pair (๐œ™,๐œ“) is said to be a Littlewood-Paley pair (LP-pair) of order ๐›ผ if ๎๐œ™(๐œ‰)โ‰ 0 for |๐œ‰|โ‰ค๐œ and ๐œ‡๐‘š(๐œ“)=0 for all multi-index ๐‘šโˆˆโ„•๐‘› with |๐‘š|โ‰ค[๐›ผ].

Example 4.3. Let ๐œ™โˆˆ๐’ฎ(โ„๐‘›) be a radial function such that ๎๐œ™ is nonnegative, ๎๐œ™(๐œ‰)=1 for |๐œ‰|<1/2 and ๎๐œ™(๐œ‰)=0 for |๐œ‰|>1. Set ๎๐œ“(๐œ‰)=โˆ’๐œ‰โ‹…โˆ‡๎๐œ‘(๐œ‰). The pair (๐œ™,๐œ“) is then clearly an LP-pair of order โˆž. 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 ๐ถ๐›ผโˆ—(โ„๐‘›) (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 (๐œ™,๐œ“) be an LP-pair, the wavelet ๐œ“ having index of nondegenerateness ๐œ and ๐‘Ÿ>๐œ being a number such that ๎๐œ™(๐œ‰)โ‰ 0 for |๐œ‰|<๐‘Ÿ. Pick any ๐œŽ in between ๐œ and ๐‘Ÿ. If ๐œ‚โˆˆ๐’ฎ0(โ„๐‘›) is a reconstruction wavelet for ๐œ“ whose Fourier transform has support in ๐ต(0,๐œŽ) and ๐œ‘โˆˆ๐’Ÿ(โ„๐‘›) is such that ๐œ‘(๐œ‰)=1 for ๐œ‰โˆˆ๐ต(0,๐œŽ) and supp๐œ‘โŠ‚๐ต(0,๐‘Ÿ), then, for all ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) and ๐œƒโˆˆ๐’ฎ(โ„๐‘›)๎€œโŸจ๐‘“,๐œƒโŸฉ=โ„๐‘›(๐‘“โˆ—๐œ™)(๐‘ก)๐œƒ11(๐‘ก)d๐‘ก+๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐’ฒ๐œ“๐‘“(๐‘ฅ,๐‘ฆ)๐’ฒ๐œ‚๐œƒ2(๐‘ฅ,๐‘ฆ)d๐‘ฅd๐‘ฆ๐‘ฆ,(4.2) where ฬ‚๐œƒ1ฬ‚๎(๐œ‰)=๐œƒ(๐œ‰)๐œ‘(๐œ‰)/๐œ™(โˆ’๐œ‰) and ฬ‚๐œƒ2ฬ‚(๐œ‰)=๐œƒ(๐œ‰)(1โˆ’๐œ‘(๐œ‰)).

Proof. Observe that โŸจ๐‘“โˆ—๐œ™,๐œƒ1โŸฉ=(2๐œ‹)โˆ’๐‘›โŸจ๎๐‘“ฬ‚๐œƒ(๐œ‰),(โˆ’๐œ‰)๐œ‘(โˆ’๐œ‰)โŸฉ.(4.3) It is therefore enough to assume that ๐œƒ1=0 so that ๐œƒ=๐œƒ2. Our assumption over ๐œ‚ is that ๐œ‚โˆˆ๐’ฎ0(โ„๐‘›), suppฬ‚๐œ‚โŠ‚๐ต(0,๐œŽ), and ๐‘๐œ“,๐œ‚=๎€œโˆž0๎๐œ“(๐‘Ÿ๐œ”)ฬ‚๐œ‚(๐‘Ÿ๐œ”)dr๐‘Ÿโ‰ 0(4.4) does not depend on the direction ๐œ”. We remark that such a reconstruction wavelet can always be found (see the proof of [10, Theorem 7.7]). Therefore, ๐’ฒ๐œ‚๐œƒ(๐‘ฅ,๐‘ฆ)=0 for all (๐‘ฅ,๐‘ฆ)โˆˆโ„๐‘›ร—(1,โˆž). Exactly as in [1, page 66], the usual calculation shows that 1๐œƒ(๐‘ก)=๐‘๐œ“,๐œ‚โ„ณ๐œ“๎€ท๐’ฒ๐œ‚๐œƒ๎€ธ1(๐‘ก)=๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐œ“๎‚ต๐‘กโˆ’๐‘ฅ๐‘ฆ๎‚ถ๐’ฒ๐œ‚๐œƒ(๐‘ฅ,๐‘ฆ)d๐‘ฅd๐‘ฆ๐‘ฆ.(4.5) Furthermore, since ๐’ฒ๐œ‚๐œƒโˆˆ๐’ฎ(โ„๐‘›+1) (cf. Theorem 3.1), the last integral can be expressed as the limit in ๐’ฎ(โ„๐‘›) of Riemann sums. That justifies the exchange of dual pairing and integral in ๎ƒก1โŸจ๐‘“,๐œƒโŸฉ=๐‘“(๐‘ก),๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐œ“๎‚ต๐‘กโˆ’๐‘ฅ๐‘ฆ๎‚ถ๐’ฒ๐œ‚๐œƒ(๐‘ฅ,๐‘ฆ)d๐‘ฅd๐‘ฆ๐‘ฆ๎ƒข=1๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐’ฒ๐œ“๐‘“(๐‘ฅ,๐‘ฆ)๐’ฒ๐œ‚๐œƒ(๐‘ฅ,๐‘ฆ)d๐‘ฅd๐‘ฆ๐‘ฆ.(4.6)

4.2. Slowly Varying Functions

The weights in our weighted versions of Hรถlder-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 ๐ฟ is called slowly varying (at the origin) if it is asymptotically invariant under rescaling, that is, lim๐œ€โ†’0+๐ฟ(๐‘Ž๐œ€)๐ฟ(๐œ€)=1,foreach๐‘Ž>0.(4.7) Familiar functions such as 1, |log๐œ€|๐›ฝ, |log|log๐œ€||๐›ฝ, โ€ฆ, are slowly varying. Regularly varying functions are then those that can be written as ๐œ€๐›ผ๐ฟ(๐œ€), where ๐ฟ is slowly varying and ๐›ผโˆˆโ„.

4.3. Weak Asymptotics

We will use some notions from the theory of asymptotics of generalized functions [10, 17โ€“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 ๐ธ be a Banach space with norm โ€–โ€– and let ๐ฟ be slowly varying. For ๐Ÿโˆˆ๐’ฎโ€ฒ(โ„๐‘›,๐ธ), we write ๐Ÿ๎€ท๐‘ฅ0๎€ธ+๐œ€๐‘ก=๐‘‚(๐œ€๐›ผ๐ฟ(๐œ€))as๐œ€โŸถ0+in๐’ฎ๎…ž(โ„๐‘›,๐ธ),(4.8) if the order growth relation holds after evaluation at each test function, that is, โ€–โ€–๎ซ๐Ÿ๎€ท๐‘ฅ0๎€ธ๎ฌโ€–โ€–+๐œ€๐‘ก,๐œ‘(๐‘ก)โ‰ค๐ถ๐œ‘๐œ€๐›ผ๐ฟ(๐œ€),0<๐œ€โ‰ค1,(4.9) for each test function ๐œ‘โˆˆ๐’ฎ(โ„๐‘›). 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 ลojasiewicz in [21, 22] (see also [23โ€“25]). The vector-valued distribution ๐Ÿโˆˆ๐’ฎ๎…ž(โ„๐‘›,๐ธ) is said to have a value ๐ฏโˆˆ๐ธ at the point ๐‘ฅ0โˆˆโ„๐‘› if lim๐œ€โ†’0+๐Ÿ(๐‘ฅ0+๐œ€๐‘ก)=๐ฏ, distributionally, that is, for each ๐œ‘โˆˆ๐’ฎ(โ„๐‘›)lim๐œ€โ†’0+๎ƒก1๐Ÿ(๐‘ก),๐œ€๐‘›๐œ‘๎‚ต๐‘กโˆ’๐‘ฅ0๐œ€๎€œ๎‚ถ๎ƒข=๐ฏโ„๐‘›๐œ‘(๐‘ก)d๐‘กโˆˆ๐ธ.(4.10) In such a case, we simply write ๐Ÿ(๐‘ฅ0)=๐ฏ, distributionally.

4.4. Pointwise Weak Hรถlder Space

An important tool in Section 5 will be the concept of pointwise weak Hรถlder 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 ๐ธ a Banach space, ๐ฟ is a slowly varying function at the origin.

For a given ๐‘ฅ0โˆˆโ„๐‘› and ๐›ผโˆˆโ„, the pointwise weak Hรถlder space [10] ๐ถ๐‘ค๐›ผ,๐ฟ(๐‘ฅ0,๐ธ) consists of those distributions ๐Ÿโˆˆ๐’ฎ๎…ž(โ„๐‘›,๐ธ) for which there is an ๐ธ-valued polynomial ๐ of degree less than ๐›ผ such that (cf. Section 4.3) ๐Ÿ๎€ท๐‘ฅ0๎€ธ+๐œ€๐‘ก=๐(๐œ€๐‘ก)+๐‘‚(๐œ€๐›ผ๐ฟ(๐œ€))as๐œ€โŸถ0+in๐’ฎ๎…ž(โ„๐‘›,๐ธ).(4.11) Observe that if ๐›ผ<0, then the polynomial is irrelevant. In addition, when ๐›ผโ‰ฅ0, this polynomial is unique; in fact (4.11) readily implies that the ลojasiewicz point values ๐œ•๐‘š๐Ÿ(๐‘ฅ0) exist, distributionally, for |๐‘š|<๐›ผ and that ๐ is the โ€œTaylor polynomialโ€ ๎“๐(๐‘ก)=|๐‘š|<๐›ผ๐œ•๐‘š๐Ÿ๎€ท๐‘ฅ0๎€ธ๐‘ก๐‘š!๐‘š.(4.12)

The pointwise weak Hรถlder space ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) of second type is defined as follows: ๐Ÿโˆˆ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) if (4.9) is just assumed to hold for each ๐œ‘โˆˆ๐’ฎ(โ„๐‘›) satisfying the requirement ๐œ‡๐‘š(๐œ‘)=0 for all multi-index |๐‘š|โ‰ค๐›ผ. Naturally, the previous requirement is empty if ๐›ผ<0, thus, in such a case, ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ)=๐ถ๐‘ค๐›ผ,๐ฟ(๐‘ฅ0,๐ธ). One can also show that if ๐›ผโˆ‰โ„•, the equality between these two spaces remains true [10]. On the other hand, when ๐›ผโˆˆโ„•, we have the strict inclusion ๐ถ๐‘ค๐›ผ,๐ฟ(๐‘ฅ0,๐ธ)โŠŠ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ)(cf. comments below Theorem 4.5). The usefulness of ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) 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 ๐›ผโˆˆโ„ and let ๐œ“โˆˆ๐’ฎ(โ„๐‘›) be a nondegenerate wavelet with ๐œ‡๐‘š(๐œ“)=0 for |๐‘š|โ‰ค[๐›ผ]. Then, ๐Ÿโˆˆ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) if and only if there is ๐‘˜โˆˆโ„• such that limsup๐œ€โ†’0+sup|๐‘ฅ|2+๐‘ฆ2=1,๐‘ฆ>0๐‘ฆ๐‘˜๐œ€๐›ผโ€–โ€–๐’ฒ๐ฟ(๐œ€)๐œ“๐Ÿ๎€ท๐‘ฅ0๎€ธโ€–โ€–+๐œ€๐‘ฅ,๐œ€๐‘ฆ<โˆž.(4.13)

It is worth mentioning that the elements of ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) for ๐›ผ=๐‘โˆˆโ„• can be characterized by pointwise weak-asymptotic expansions. We have [10] that ๐Ÿโˆˆ๐ถ๐‘,๐ฟโˆ—,๐‘ค(๐‘ฅ0,๐ธ) if and only if it admits the following weak expansion: ๐Ÿ๎€ท๐‘ฅ0๎€ธ=๎“+๐œ€๐‘ก|๐‘š|<๐‘๐œ•๐‘š๐Ÿ๎€ท๐‘ฅ0๎€ธ๐‘š!(๐œ€๐‘ก)๐‘š+๐œ€๐‘๎“|๐‘š|=๐‘๐‘ก๐‘š๐œ๐‘š(๐œ€)+๐‘‚(๐œ€๐‘๐ฟ(๐œ€))in๐’ฎ๎…ž(โ„๐‘›,๐ธ),(4.14) where ๐œ•๐‘š๐Ÿ(๐‘ฅ0) are interpreted in the ลojasiewicz sense and the ๐œ๐‘šโˆถ(0,โˆž)โ†’๐ธ are continuous functions. Comparison of this weak expansion with (4.11) explains the difference between the two pointwise spaces when ๐›ผ=๐‘โˆˆโ„•.

5. New Class of Hรถlder-Zygmund Spaces

Throughout this section, we assume that ๐ฟ is a slowly varying function such that ๐ฟ and 1/๐ฟ are locally bounded on (0,1].

5.1. ๐ฟ-Hรถlder Spaces

We now introduce weighted Hรถlder spaces with respect to ๐ฟ. They were already defined and studied in [10]. Let ๐›ผโˆˆโ„+โงตโ„•. We say that a function ๐‘“ belongs to the space ๐ถ๐›ผ,๐ฟ(โ„๐‘›) if ๐‘“ has continuous derivatives up to order less than ๐›ผ and โ€–๐‘“โ€–๐ถ๐›ผ,๐ฟ๎“โˆถ=||๐‘—||โ‰ค[๐›ผ]sup๐‘กโˆˆโ„๐‘›||๐œ•๐‘—||+๎“๐‘“(๐‘ก)|๐‘š|=[๐›ผ]sup0<|๐‘กโˆ’๐‘ฅ|โ‰ค1||๐œ•๐‘š๐‘“(๐‘ก)โˆ’๐œ•๐‘š||๐‘“(๐‘ฅ)|๐‘กโˆ’๐‘ฅ|๐›ผโˆ’[๐›ผ]๐ฟ(|๐‘กโˆ’๐‘ฅ|)<โˆž.(5.1) When ๐›ผ=๐‘+1โˆˆโ„•, we replace the previous requirement by โ€–๐‘“โ€–๐ถ๐›ผ,๐ฟ๎“โˆถ=||๐‘—||โ‰ค๐‘sup๐‘กโˆˆโ„๐‘›||๐œ•๐‘—||+๎“๐‘“(๐‘ก)|๐‘š|=๐‘sup0<|๐‘กโˆ’๐‘ฅ|โ‰ค1||๐œ•๐‘š๐‘“(๐‘ก)โˆ’๐œ•๐‘š||๐‘“(๐‘ฅ)|๐‘กโˆ’๐‘ฅ|๐ฟ(|๐‘กโˆ’๐‘ฅ|)<โˆž.(5.2)

The space ๐ถ๐›ผ,๐ฟ(โ„๐‘›) is clearly a Banach space with the above norm. The conditions imposed over ๐ฟ ensure that ๐ถ๐›ผ,๐ฟ(โ„๐‘›) depends only on the behavior of ๐ฟ near 0, thus, it is invariant under dilations. When ๐ฟโ‰ก1, this space reduces to ๐ถ๐›ผ,๐ฟ(โ„๐‘›)=๐ถ๐›ผ(โ„๐‘›), the usual global (inhomogeneous) Hรถlder space [2, 3, 5]. Consequently, as in [10], we call ๐ถ๐›ผ,๐ฟ(โ„๐‘›) the global Hรถlder space with respect to ๐ฟ. Note that, because of the properties of ๐ฟ [7, 8], we have the following interesting inclusion relations: ๐ถ๐›ฝ(โ„๐‘›)โŠŠ๐ถ๐›ผ,๐ฟ(โ„๐‘›)โŠŠ๐ถ๐›พ(โ„๐‘›),whenever0<๐›พ<๐›ผ<๐›ฝ.(5.3)

5.2. ๐ฟ-Zygmund Spaces

We now proceed to define the weighted Zygmund space ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›). Let ๐›ผโˆˆโ„ and fix an LP-pair (๐œ™,๐œ“) of order ๐›ผ. A distribution ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) is said to belong to the ๐ฟ-Zygmund space ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) of exponent ๐›ผ if โ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟโˆถ=โ€–๐‘“โˆ—๐œ™โ€–๐ฟโˆž+sup๐‘ฅโˆˆโ„๐‘›sup0<๐‘ฆโ‰ค11๐‘ฆ๐›ผ๐ฟ||๐’ฒ(๐‘ฆ)๐œ“||๐‘“(๐‘ฅ,๐‘ฆ)<โˆž.(5.4)

Observe that we clearly have ๐ถ๐›ผ,๐ฟ(โ„๐‘›)โŠ†๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›), for ๐›ผ>0. We will show that if ๐›ผโˆˆโ„+โงตโ„•, we actually have the equality ๐ถ๐›ผ,๐ฟ(โ„๐‘›)=๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›). When ๐›ผ is a positive integer, we have in turn ๐ถ๐›ผ,๐ฟ(โ„๐‘›)โŠŠCโˆ—๐›ผ,๐ฟ(โ„๐‘›). As in the case of ๐ฟ-Hรถlder spaces, our ๐ฟ-Zygmund spaces refine the scale of classical Zygmund spaces; more precisely, we have again the following inclusions: ๐ถ๐›ฝโˆ—(โ„๐‘›)โŠŠ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›)โŠŠ๐ถ๐›พโˆ—(โ„๐‘›),whenever๐›พ<๐›ผ<๐›ฝ.(5.5) The definition of ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) 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 ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) 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 ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›). Therefore, when ๐ฟโ‰ก1, we recover the classical Zygmund space ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›)=๐ถ๐›ผโˆ—(โ„๐‘›) [5]. Proposition 5.1 follows at once from the following lemma.

Lemma 5.2. Let ๐‘“โˆˆ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›), then for every ๐œƒโˆˆ๐’ฎ(โ„๐‘›) there holds โ€–๐‘“โˆ—๐œƒโ€–๐ฟโˆžโ‰ค๐ถโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ,(5.6) where โ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ is given by (5.4). Furthermore, if ๐”…โŠ‚๐’ฎ(โ„๐‘›) is a bounded set such that ๐œ‡๐‘š(๐œƒ)=0 for all ๐œƒโˆˆ๐”… and all multi-index ๐‘šโ‰ค[๐›ผ], then sup๐‘ฅโˆˆโ„๐‘›sup0<๐‘ฆโ‰ค11๐‘ฆ๐›ผ๐ฟ||๐’ฒ(๐‘ฆ)๐œƒ||๐‘“(๐‘ฅ,๐‘ฆ)โ‰ค๐ถโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ,โˆ€๐œƒโˆˆ๐”….(5.7)

Proof. The estimate (5.6) easily follows from the representation (4.2) of ๐‘“ 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 ๐’ฒ๐œƒ๐‘“(๐‘ฅ,๐‘ฆ)=๐น1(๐‘ฅ,๐‘ฆ)+๐น2(๐‘ฅ,๐‘ฆ)+๐น3(๐‘ฅ,๐‘ฆ),(5.8) where ๐น1(1๐‘ฅ,๐‘ฆ)=(2๐œ‹)๐‘›๎€œโ„๐‘›๎€ท๎€ธ(๐‘“โˆ—๎๐œ‘๐‘ฅ+๐‘ฆ๐‘ก)๐น๐œƒ(๐‘ก)d๐‘ก,21(๐‘ฅ,๐‘ฆ)=๐‘๐œ“,๐œ‚๎€œ01/๐‘ฆ๎€œโ„๐‘›๐’ฒ๐œ“๐‘“(๐‘ฆ๐‘+๐‘ฅ,๐‘ฆ๐‘Ž)๐’ฒ๐œ‚๐œƒ2,๐‘ฆ(๐‘,๐‘Ž)d๐‘d๐‘Ž๐‘Ž,๐น31(๐‘ฅ,๐‘ฆ)=๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐’ฒ๐œ“๐‘“(๐‘ฆ๐‘+๐‘ฅ,๐‘ฆ๐‘Ž)๐’ฒ๐œ‚๐œƒ3,๐‘ฆ(๐‘,๐‘Ž)d๐‘d๐‘Ž๐‘Ž,(5.9) with ฬ‚๐œƒ2,๐‘ฆฬ‚(๐œ‰)=๐œƒ(๐œ‰)(1โˆ’๐œ‘(๐œ‰))(1โˆ’๐œ‘(๐œ‰/๐‘ฆ)) and ฬ‚๐œƒ3,๐‘ฆฬ‚(๐œ‰)=๐œƒ(๐œ‰)๐œ‘(๐œ‰)(1โˆ’๐œ‘(๐œ‰/๐‘ฆ)). To estimate ๐น1, we first observe that if ๐›ผ<0, then sup0<๐‘ฆโ‰ค1sup๐‘ฅโˆˆโ„๐‘›||๐น1||(๐‘ฅ,๐‘ฆ)๐‘ฆ๐›ผ||๐‘“||๐ฟ(๐‘ฆ)โ‰ค๐ถ๐ถโˆ—๐›ผ,๐ฟโ€–๐œƒโ€–๐ฟ1sup0<๐‘ฆโ‰ค11๐‘ฆ๐›ผ๐ฟ(๐‘ฆ)โ‰ค๐ถ๎…ž||๐‘“||๐ถโˆ—๐›ผ,๐ฟ,(5.10) because slowly varying functions satisfy the estimates ๐‘ฆ๐œ€<๐ถ๐ฟ(๐‘ฆ) for any exponent ๐œ€>0 [7, 8]. When ๐›ผโ‰ฅ0, we have that (๐‘“โˆ—๎๐œ‘) is a ๐ถโˆž-function with bounded derivatives of any order. Then, by the Taylor formula, (5.6) (with ๐œƒ=๐œ•๐‘š๎๐œ‘), and the assumption ๐œ‡๐‘š(๐œƒ)=0 for |๐‘š|โ‰ค[๐›ผ], we obtain sup0<๐‘ฆโ‰ค1sup๐‘ฅโˆˆโ„๐‘›||๐น1||(๐‘ฅ,๐‘ฆ)๐‘ฆ๐›ผ๐ฟ(๐‘ฆ)โ‰คsup0<๐‘ฆโ‰ค1๐ถโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ๐‘ฆ[๐›ผ]+1โˆ’๐›ผ๎€œ๐ฟ(๐‘ฆ)โ„๐‘›|๐‘ก|[๐›ผ]+1||||๐œƒ(๐‘ก)d๐‘กโ‰ค๐ถโ€ฒโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ.(5.11) We now bound ๐น2 and ๐น3. If ๐œ€>0, Potter's estimate [7, page 25] gives the existence of a constant ๐ถ=๐ถ๐œ€ such that ๐ฟ(๐‘Ž๐‘ฆ)๎‚€๐‘Ž๐ฟ(๐‘ฆ)<๐ถ๐œ€+1๐‘Ž๐œ€๎‚,โˆ€0<๐‘ฆ<1,๐‘Ž<1/๐‘ฆ.(5.12) Thus, ||๐น2||(๐‘ฅ,๐‘ฆ)๐‘ฆ๐›ผโ‰ค๐ฟ(๐‘ฆ)โ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ๐‘๐œ“,๐œ‚๎€œ01/๐‘ฆ๎€œโ„๐‘›๐‘Ž๐›ผโˆ’1๐ฟ(๐‘Ž๐‘ฆ)||๐’ฒ๐ฟ(๐‘ฆ)๐œ‚๐œƒ2,๐‘ฆ||(๐‘,๐‘Ž)d๐‘d๐‘Žโ‰ค๐ถ๎…žโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ๎€œโˆž0๎€œโ„๐‘›๎€ท๐‘Ž๐›ผ+๐‘Ž๐›ผโˆ’2๎€ธ||๐’ฒ๐œ‚๐œƒ2,๐‘ฆ(||๐‘,๐‘Ž)d๐‘d๐‘Ž.(5.13) Notice that {๐œƒ2,๐‘ฆโˆˆ๐’ฎ0(โ„๐‘›)โˆถ๐œƒโˆˆ๐”…,๐‘ฆโˆˆ(0,1]} is a bounded set in ๐’ฎ0(โ„๐‘›) because the derivatives of ๐œ‘ are supported in {๐œ‰โˆถ๐œŽโ‰ค|๐œ‰|โ‰ค๐‘Ÿ}. Thus, due to the continuity of ๐’ฒ๐œ‚ (cf. Theorem 3.1), {๐’ฒ๐œ‚๐œƒ๐‘ฆโˆถ๐œƒโˆˆ๐”…,๐‘ฆโˆˆ(0,1]} is a bounded set in ๐’ฎ(โ„๐‘›+1). This implies that the integrals involved in the very last estimate are uniformly bounded for ๐œƒโˆˆ๐”… and ๐‘ฆโˆˆ(0,1]. Consequently, we obtain that sup0<๐‘ฆโ‰ค1sup๐‘ฅโˆˆโ„๐‘›||๐น2||(๐‘ฅ,๐‘ฆ)๐‘ฆ๐›ผ๐ฟ(๐‘ฆ)โ‰ค๐ถโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ.(5.14) Next, for ๐น3, we have ||๐น3||(๐‘ฅ,๐‘ฆ)๐‘ฆ๐›ผ๐ฟ(๐‘ฆ)โ‰ค๐ถโ€ฒโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ๎€œ10๎€œโ„๐‘›๎€ท๐‘Ž๐›ผโˆ’3/2+๐‘Ž๐›ผโˆ’1/2๎€ธ||๐’ฒ๐œ‚๐œƒ3,๐‘ฆ||(๐‘,๐‘Ž)d๐‘d๐‘Ž.(5.15) As in the proof of Theorem 3.1, the above integrand can be uniformly estimated by ๐ถ(1+|๐‘|2)โˆ’๐‘›. This completes the proof of (5.7).

We obtain the following useful properties.

Corollary 5.3. The following properties hold: (i)๐œ•๐‘šโˆถ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›)โ†’๐ถโˆ—๐›ผโˆ’|๐‘š|,๐ฟ(โ„๐‘›) is continuous, for any ๐‘šโˆˆโ„•๐‘›,(ii)if ๐‘“โˆ—๐œ™โˆˆ๐ฟโˆž(โ„๐‘›) and ๐œ•๐‘—๐‘“โˆˆ๐ถโˆ—๐›ผโˆ’1,๐ฟ(โ„๐‘›) for ๐‘—=1,โ€ฆ,๐‘›, then ๐‘“โˆˆ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›),(iii)the mapping (1โˆ’ฮ”)๐›ฝ/2 is an isomorphism of the space ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) onto ๐ถโˆ—๐›ผโˆ’๐›ฝ,๐ฟ(โ„๐‘›), for arbitrary ๐›ผ,๐›ฝโˆˆโ„.

Proof. (i) It is enough to consider ๐œ•๐‘—. We have that ๐œ•๐‘—๐‘“โˆ—๐œ™=๐‘“โˆ—๐œ•๐‘—๐œ™ and ๐’ฒ๐œ“๐œ•๐‘—๐‘“(๐‘ฅ,๐‘ฆ)=โˆ’๐‘ฆโˆ’1๐’ฒ๐œ•๐‘—๐œ“๐‘“(๐‘ฅ,๐‘ฆ). Thus, the result follows at once by applying (5.6) with ๐œƒ=๐œ•๐‘—๐œ™ and (5.7) with ๐œƒ=๐œ•๐‘—๐œ“.
(ii) If (๐œ™,๐œ“) is an LP-pair, so is (๐œ™,ฮ”๐œ“). Note that our assumption and (i) imply that ฮ”๐‘“โˆˆ๐ถโˆ—๐›ผโˆ’2,๐ฟ(โ„๐‘›). In view of Proposition 5.1, it remains to observe that 1๐‘ฆ๐›ผ๐ฟ๐’ฒ(๐‘ฆ)ฮ”๐œ“1๐‘“(๐‘ฅ,๐‘ฆ)=๐‘ฆ๐›ผโˆ’2๐’ฒ๐ฟ(๐‘ฆ)๐œ“(ฮ”๐‘“)(๐‘ฅ,๐‘ฆ).(5.16)
(iii) Since (1โˆ’ฮ”)โˆ’๐›ฝ/2 is the inverse of (1โˆ’ฮ”)๐›ฝ/2, it suffices to show that (1โˆ’ฮ”)๐›ฝ/2 maps continuously ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) into ๐ถโˆ—๐›ผโˆ’๐›ฝ,๐ฟ(โ„๐‘›). Using (5.6) with ๐œƒ=(1โˆ’ฮ”)๐›ฝ/2๐œ™, we obtain that โ€–(1โˆ’ฮ”)๐›ฝ/2๐‘“โˆ—๐œ™โ€–๐ฟโˆžโ‰ค๐ถโ€–๐‘“โ€–๐ถโˆ—๐›ผ,๐ฟ. We also have ๐’ฒ๐œ“(1โˆ’ฮ”)๐›ฝ/21๐‘“(๐‘ฅ,๐‘ฆ)=(2๐œ‹)๐‘›๐‘ฆ๐›ฝ๎ƒก๎๐‘“(๐œ‰),๐‘’๐‘–๐‘ฅโ‹…๐œ‰๎‚€๐‘ฆ2+||||๐‘ฆ๐œ‰2๎‚๐›ฝ/2๎ƒข=1๎๐œ“(๐‘ฆ๐œ‰)๐‘ฆ๐›ฝ๐’ฒ๐œƒ๐‘ฆ๐‘“(๐‘ฅ,๐‘ฆ),where๐œƒ๐‘ฆ=๎€ท๐‘ฆ2๎€ธโˆ’ฮ”๐›ฝ/2๐œ“.(5.17) Finally, we can apply (5.7) because ๐”…={(๐‘ฆ2โˆ’ฮ”)๐›ฝ/2๐œ“}๐‘ฆโˆˆ(0,1] is a bounded set in ๐’ฎ(โ„๐‘›) and ๐œ‡๐‘š(๐œƒ๐‘ฆ)=0 for each multi-index |๐‘š|โ‰ค[๐›ผ].

We can also use Proposition 4.4 to show that ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) is a Banach space, as stated in the following proposition.

Proposition 5.4. The space ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) is a Banach space when provided with the norm (5.4).

Proof. Let ๐œ‚,โ€‰โ€‰๐œ‘,โ€‰โ€‰๐œƒ1,โ€‰โ€‰๐œƒ2 be as in the statement of Proposition 4.4. Suppose that {๐‘“๐‘—}โˆž๐‘—=0 is a Cauchy sequence in ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›). Then, there exist continuous functions ๐‘”โˆˆ๐ฟโˆž(โ„๐‘›) and ๐บ defined on โ„๐‘›ร—(0,1] such that ๐‘“๐‘—โˆ—๐œ™โ†’๐‘” in ๐ฟโˆž(โ„๐‘›) and lim๐‘—โ†’โˆžsup๐‘ฆโˆˆ(0,1]sup๐‘ฅโˆˆโ„๐‘›1๐‘ฆ๐›ผ๐ฟ||๐’ฒ(๐‘ฆ)๐œ“๐‘“๐‘—||(๐‘ฅ,๐‘ฆ)โˆ’๐บ(๐‘ฅ,๐‘ฆ)=0.(5.18) We define the distribution ๐‘“โˆˆ๐’ฎ๎…ž(โ„๐‘›) whose action on test functions ๐œƒโˆˆ๐’ฎ(โ„๐‘›) is given by ๎€œโŸจ๐‘“,๐œƒโŸฉโˆถ=โ„๐‘›๐‘”(๐‘ก)๐œƒ11(๐‘ก)d๐‘ก+๐‘๐œ“,๐œ‚๎€œ10๎€œโ„๐‘›๐บ(๐‘ฅ,๐‘ฆ)๐’ฒ๐œ‚๐œƒ2(๐‘ฅ,๐‘ฆ)d๐‘ฅd๐‘ฆ๐‘ฆ.(5.19) Since the ๐‘“๐‘— have the representation (4.2), we immediately see that ๐‘“๐‘—โ†’๐‘“ in ๐’ฎ๎…ž(โ„๐‘›). Thus, pointwisely, (๐‘“โˆ—๐œ™)(๐‘ก)=lim๐‘—โ†’โˆž๎€ท๐‘“๐‘—๎€ธ๐’ฒโˆ—๐œ™(๐‘ก)=๐‘”(๐‘ก),๐œ“๐‘“(๐‘ฅ,๐‘ฆ)=lim๐‘—โ†’โˆž๐’ฒ๐œ“๐‘“๐‘—(๐‘ฅ,๐‘ฆ)=๐บ(๐‘ฅ,๐‘ฆ).(5.20) This implies that lim๐‘—โ†’โˆžโ€–๐‘“โˆ’๐‘“๐‘—โ€–๐ถโˆ—๐›ผ,๐ฟ=0, and so ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›) is complete.

We have arrived to the main and last result of this section. It provides the ๐ฟ-Hรถlderian characterization of the ๐ฟ-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 ๐ถ๐‘(โ„๐‘›) the Banach space of continuous and bounded functions.

Theorem 5.5. Let ๐›ผ>0.(a)If ๐›ผโˆ‰โ„•, then ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›)=๐ถ๐›ผ,๐ฟ(โ„๐‘›). Moreover, the norms (5.4) and (5.1) are equivalent. (b)If ๐›ผ=๐‘+1โˆˆโ„•, then ๐ถโˆ—๐‘+1,๐ฟ(โ„๐‘›) consists of functions with continuous derivatives up to order ๐‘ such that ๎“|๐‘š|โ‰ค๐‘โ€–๐œ•๐‘š๐‘“โ€–๐ฟโˆž+๎“|๐‘š|=๐‘sup๐‘กโˆˆโ„๐‘›0<|โ„Ž|โ‰ค1||๐œ•๐‘š๐‘“(๐‘ก+โ„Ž)+๐œ•๐‘š๐‘“(๐‘กโˆ’โ„Ž)โˆ’2๐œ•๐‘š||๐‘“(๐‘ก)||โ„Ž||๐ฟ๎€ท||โ„Ž||๎€ธ<โˆž.(5.21)
In addition, (5.21) produces a norm that is equivalent to (5.4).

Proof. Observe that the ๐ฟ-Hรถlderian 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 ๐‘“โˆˆ๐ถโˆ—๐›ผ,๐ฟ(โ„๐‘›). Consider the ๐ถ๐‘(โ„๐‘›)-valued distribution ๐Ÿโˆˆ๐’ฎ๎…ž(โ„๐‘›๐‘ก,๐ถ๐‘(โ„๐‘›๐œ‰)) given by ๐Ÿ(๐‘ก)(๐œ‰)โˆถ=๐‘“(๐‘ก+๐œ‰), that is, the one whose action on test functions is given by ๎€ทฬŒ๐œƒ๎€ธโŸจ๐Ÿ(๐‘ก),๐œƒ(๐‘ก)โŸฉ(๐œ‰)=โŸจ๐‘“(๐‘ก+๐œ‰),๐œƒ(๐‘ก)โŸฉ=๐‘“โˆ—(๐œ‰),๐œƒโˆˆ๐’ฎ(โ„๐‘›),๐œ‰โˆˆโ„๐‘›.(5.22) It does take values in ๐ถ๐‘(โ„๐‘›) because of (5.6). Clearly, ๐’ฒ๐œ“๐Ÿ(๐‘ฅ,๐‘ฆ)(๐œ‰)=๐’ฒ๐œ“๐‘“(๐‘ฅ+๐œ‰,๐‘ฆ). By (5.4) and Potter's estimate [7, page 25], we have that โ€–โ€–๐’ฒ๐œ“โ€–โ€–๐Ÿ(๐œ€๐‘ฅ,๐œ€๐‘ฆ)๐ถ๐‘(โ„๐‘›)โ‰ค๐ถ๐œ€๐›ผ๐ฟ(๐œ€)๐‘ฆ๐›ผโˆ’1โˆ€๐œ€โˆˆ(0,1),(๐‘ฅ,๐‘ฆ)โˆˆโ„๐‘›].ร—(0,1(5.23) Therefore, the Tauberian Theorem 4.5 yields ๐Ÿโˆˆ๐ถ๐›ผ,๐ฟโˆ—,๐‘ค(0,๐ถ๐‘(โ„๐‘›)). Now, the ลojasiewicz point values ๐œ•๐‘š๐Ÿ(0)=๐‘ฃ๐‘šโˆˆ๐ถ๐‘(โ„๐‘›) exist, distributionally, for |๐‘š|<๐›ผ. It explicitly means that for all ๐œƒโˆˆ๐’ฎ(โ„๐‘›)lim๐œ€โ†’0+๐œ•๐‘šฬŒ๐œƒ๐‘“โˆ—๐œ€=lim๐œ€โ†’0+โŸจ๐œ•๐‘š๐Ÿ(๐œ€๐‘ก),๐œƒ(๐‘ก)โŸฉ=๐œ‡0(๐œƒ)๐‘ฃ๐‘šin๐ถ๐‘๎‚€โ„๐‘›๐œ‰๎‚,(5.24) where ฬŒ๐œƒ๐œ€(๐‘ก)=๐œ€โˆ’๐‘›๐œƒ(โˆ’๐‘ก/๐œ€). If we now take ๐œƒ with ๐œ‡0(๐œƒ)=1, we then conclude that ๐œ•๐‘š๐‘“=๐‘ฃ๐‘šโˆˆ๐ถ๐‘(โ„๐‘›) for each |๐‘š|<๐›ผ. It remains in both cases to deal with the estimates for ๐œ•๐‘š๐‘“; notice that ๐œ•๐‘š๐Ÿโˆˆ๐ถ๐›ผโˆ’[๐›ผ],๐ฟโˆ—,๐‘ค(0,๐ถ๐‘(โ„๐‘›)) when |๐‘š|=[๐›ผ] and ๐œ•๐‘š๐Ÿโˆˆ๐ถ1,๐ฟโˆ—,๐‘ค(0,๐ถ๐‘(โ„๐‘›)) when |๐‘š|=๐‘. We now divide the proof into two cases.
Case ๐›ผโˆ‰โ„•. Fix a multi-index |๐‘š|=[๐›ผ]. It suffices to show sup0<|๐‘ฅโˆ’๐‘ก|<1||๐œ•๐‘š๐‘“(๐‘ฅ)โˆ’๐œ•๐‘š||๐‘“(๐‘ก)|๐‘ฅโˆ’๐‘ก|๐›ผโˆ’[๐›ผ]๐ฟ(|๐‘ฅโˆ’๐‘ก|)<โˆž.(5.25) We had already seen that ๐œ•๐‘š๐Ÿ(๐‘ก)(๐œ‰)=๐œ•๐‘š๐‘“(๐‘ก+๐œ‰)โˆˆ๐ถ๐‘ค๐›ผโˆ’[๐›ผ],๐ฟ(0,๐ถ๐‘(โ„๐‘›๐œ‰))=๐ถ๐›ผโˆ’[๐›ผ],๐ฟโˆ—,๐‘ค(0,๐ถ๐‘(โ„๐‘›๐œ‰)), that is, ๐œ‡0(๐œƒ)๐œ•๐‘š๎€œ๐‘“(๐œ‰)โˆ’โ„๐‘›๐œ•๐‘š๎€ท๐œ€๐‘“(๐œ‰+๐œ€๐‘ก)๐œƒ(๐‘ก)d๐‘ก=๐‘‚๐›ผโˆ’[๐›ผ]๎€ธ๐ฟ(๐œ€),๐œ€โŸถ0+,(5.26) in the space C๐‘(โ„๐‘›๐œ‰), for each ๐œƒโˆˆ๐’ฎ(โ„๐‘›). Hence, if 0<|โ„Ž|โ‰ค1, we choose ๐œƒ as before (๐œ‡0(๐œƒ)=1), and we use the fact that {๐œƒโˆ’๐œƒ(โ‹…โˆ’๐œ”)โˆถ|๐œ”|=1} is compact in ๐’ฎ(โ„๐‘›); we then have sup๐œ‰โˆˆโ„๐‘›||๐œ•๐‘š๐‘“(๐œ‰+โ„Ž)โˆ’๐œ•๐‘š๐‘“||(๐œ‰)โ‰ค2sup๐œ‰โˆˆโ„๐‘›||||๐œ•๐‘š๐‘“๎€œ(๐œ‰)โˆ’โ„๐‘›๐œ•๐‘š๐‘“๎€ท||โ„Ž||๎€ธ||||๐‘ก+๐œ‰๐œƒ(๐‘ก)d๐‘ก+sup๐œ‰โˆˆโ„๐‘›||||๎€œโ„๐‘›๐œ•๐‘š๐‘“๎€ท||โ„Ž||๐‘ก๎€ธ๎‚€๎‚€||โ„Ž||๐œ‰+๐œƒ(๐‘ก)โˆ’๐œƒ๐‘กโˆ’โˆ’1โ„Ž||||๎‚€||โ„Ž||๎‚๎‚d๐‘ก=๐‘‚๐›ผโˆ’[๐›ผ]๐ฟ๎€ท||โ„Ž||๎€ธ๎‚,(5.27) and this completes the proof of (a).
Case ๐›ผ=๐‘+1โˆˆโ„•. The proof is similar to that of (a). Fix now |๐‘š|=๐‘. We now have ๐œ•๐‘š๐Ÿโˆˆ๐ถ1,๐ฟโˆ—,๐‘ค(0,๐ถ๐‘(โ„๐‘›)), which, as commented in Section 4.4, yields the distributional expansion ๐œ•๐‘š๐Ÿ(๐œ€๐‘ก)(๐œ‰)=๐œ•๐‘š๐‘“(๐œ‰)+๐œ€๐‘›๎“๐‘—=1๐‘ก๐‘—๐œ๐‘—(๐œ€,๐œ‰)+๐‘‚(๐œ€๐ฟ(๐œ€)),0<๐œ€โ‰ค1,(5.28) in ๐’ฎ๎…ž(โ„๐‘›๐‘ก,๐ถ๐‘(โ„๐‘›๐œ‰), where the ๐œ๐‘—(๐œ€,โ‹…) are continuous ๐ถ๐‘(โ„๐‘›๐œ‰)-valued functions in ๐œ€. We apply (5.28) on a test function ๐œƒโˆˆ๐’ฎ(โ„๐‘›), with ๐œ‡0(๐œƒ)=1, and โˆซโ„๐‘›๐‘ก๐‘—๐œƒ(๐‘ก)d๐‘ก=0 for ๐‘—=1,โ€ฆ,๐‘›, so we get ๐œ•๐‘š๎€œ๐‘“(๐œ‰)=โ„๐‘›๐œ•๐‘š๐‘“๎€ท||โ„Ž||๐‘ก๎€ธ๎€ท||โ„Ž||๐ฟ๎€ท||โ„Ž||||โ„Ž||๐œ‰+๐œƒ(๐‘ก)d๐‘ก+๐‘‚๎€ธ๎€ธ,0<โ‰ค1,(5.29) uniformly in ๐œ‰โˆˆโ„๐‘›. Since {๐œƒ๐œ”โˆถ=๐œƒ(โ‹…+๐œ”)+๐œƒ(โ‹…โˆ’๐œ”)โˆ’2๐œƒโˆถ|๐œ”|=1} is compact in ๐’ฎ(โ„๐‘›) and ๐œ‡๐‘š(๐œƒ๐œ”)=0 for |๐‘š|โ‰ค1, the relations (5.28) and (5.29) give sup๐œ‰โˆˆโ„๐‘›||๐œ•๐‘š๐‘“(๐œ‰+โ„Ž)+๐œ•๐‘š๐‘“(๐œ‰โˆ’โ„Ž)โˆ’2๐œ•๐‘š||๐‘“(๐œ‰)โ‰ค3sup๐œ‰โˆˆโ„๐‘›||||๐œ•๐‘š๎€œ๐‘“(๐œ‰)โˆ’โ„๐‘›๐œ•๐‘š๐‘“๎€ท||โ„Ž||๎€ธ||||+||||๎€œ๐‘ก+๐œ‰๐œƒ(๐‘ก)d๐‘กโ„๐‘›๐œ•๐‘š๐‘“๎€ท||โ„Ž||๐‘ก๎€ธ๎‚€๐œƒ๎‚€||โ„Ž||๐œ‰+๐‘ก+โˆ’1โ„Ž๎‚๎‚€||โ„Ž||+๐œƒ๐‘กโˆ’โˆ’1โ„Ž๎‚๎‚||||๎€ท||โ„Ž||๐ฟ๎€ท||โ„Ž||||โ„Ž||โˆ’2๐œƒ(๐‘ก)d๐‘ก=๐‘‚๎€ธ๎€ธ,0<โ‰ค1,(5.30) as claimed.


S. Pilipoviฤ‡ acknowledges support by Project 174024 of the Serbian Ministry of Education and Sciences. D. Rakiฤ‡ 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).


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