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We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces

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 [

The continuity of the wavelet transform and its left inverse on test function spaces [

The definition of our weighted Zygmund spaces is based on the useful concept of (generalized) Littlewood-Paley pairs, introduced in Section

Our new classes of spaces

The paper is organized as follows. We review in Section

We denote by

The well-known [

Following [

The space

The corresponding duals of these three spaces are

Finally, we will also make use of spaces of vector-valued tempered distributions [

In this paper a wavelet simply means a function

The wavelet transform of

Let

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

Furthermore, it is very important to emphasize that a wavelet

A test function

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

The two bilinear mappings

We first estimate the term

It remains to estimate

It follows from the proof of the continuity of

Our next task is to define and study the properties of a new class of weighted Hölder-Zygmund spaces. We postpone that for Section

In our definition of weighted Zygmund spaces, we will employ a generalized Littlewood-Paley pair [

Let us start by introducing the index of nondegenerateness of wavelets, as defined in [

Let

If we only know values of

Let

Let

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

Let

Observe that

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 [

We will use some notions from the theory of asymptotics of generalized functions [

One can also use these ideas to study exact pointwise scaling asymptotic properties of distributions (cf. [

An important tool in Section

For a given

The pointwise weak Hölder space

Let

It is worth mentioning that the elements of

Throughout this section, we assume that

We now introduce weighted Hölder spaces with respect to

The space

We now proceed to define the weighted Zygmund space

Observe that we clearly have

The definition of

In view of Proposition

Let

The estimate (

We obtain the following useful properties.

The following properties hold:

if

the mapping

(i) It is enough to consider

(ii) If

(iii) Since

We can also use Proposition

The space

Let

We have arrived to the main and last result of this section. It provides the

Let

If

If

In addition, (

Observe that the

Suppose that

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).