New classes of weighted H\"older-Zygmund spaces and the wavelet transform

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces $ \mathcal{S}_0(\mathbb{R}^n) $ and $ \mathcal{S}(\mathbb{H}^{n+1})$. We then introduce and study a new class of weighted H\"older-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.


Introduction
The purpose of this article is two folded. The main one is to define and analyze a new class of weighted Hölder-Zygmund spaces via the wavelet transform [6,8,11]. It is well known [5,6,7,18] that the wavelet transforms of elements of the classical Zymund space C α * (R n ) satisfy the size estimate |W ψ f (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 [1,17]. Familiar functions such as y α , y α |log y| β , y α |log |log y|| β , ..., are regularly varying.
The continuity of the wavelet transform and its left-inverse on test function spaces [4] plays a very important role when analyzing many function and distribution spaces [6], such as the ones introduced in this article. Our second aim is to provide a new proof of the continuity theorem, originally obtained in [4], for these transforms on the function spaces S 0 (R n ) and S(H n+1 ). Our approach to the proof is completely elementary and substancially simplifies the much longer original proof from [4] (see also [6,Chap. 1]).
The definition of our weighted Zygmund spaces is based on the useful concept of (generalized) Littlewood-Paley pairs, introduced in Subsection 4.1, which generalizes the familiar notion of (continuous) Littlewood-Paley decomposition of the unity [7]. 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 [13,23]. Even in the classical case C α * , our analysis provides a new approach to the study of Hölder-Zygmund spaces. It is then very likely that this kind of arguments may also be applied to study other types of smooth spaces, such as Besov type spaces.
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 shall 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 in Section 5.

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 its euclidean norm, . . m n ! and |m| = m 1 + · · · + m n . If the j-th coordinate of m is one and the others vanish, we then write ∂ j = ∂ m x . 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 [15] Schwartz space of rapidly decreasing smooth test functions is denoted by S(R n ). We will fix constants in the Fourier transform asφ(u) = R n ϕ(t)e −iu·t dt. The moments of ϕ ∈ S(R n ) are denoted by µ m (ϕ) = R n t m ϕ(t)dt, m ∈ N n .
Following [6], the space of highly time-frequency localized functions S 0 (R n ) is defined by S 0 (R n ) = {ϕ ∈ S(R n ) : µ m (ϕ) = 0, ∀m ∈ N n } , it is provided with the relative topology inhered from S(R n ). In [6], the topology of S 0 (R n ) is introduced in an apparently different way; however, both approaches are equivalent in view of the open mapping theorem. Observe that S 0 (R n ) is a closed subspace of S(R n ) and that ϕ ∈ S 0 (R n ) if and only if ∂ mφ (0) = 0 for all m ∈ N n . It is important to point out that S 0 (R n ) is also well known in the literature as the Lizorkin space of test functions (cf. [14]).
The space S(H n+1 ) of highly localized functions on the half-space [6] consists of those Φ ∈ C ∞ (H n+1 ) for which 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 [6]. For later use, we shall denote by ρ k,m the corresponding seminorms in S(R n ), namely, 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 S 0 (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 shall also make use of spaces of vector-valued tempered distributions [16,19]. If X is a locally convex topological vector spaces [19], 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.
The wavelet transform of f ∈ S ′ (R n ) with respect to the wavelet ψ ∈ S(R n ) is defined as 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 (2.1) for f ∈ S ′ 0 (R n ) if the wavelet ψ ∈ S 0 (R n ). The wavelet transform can be defined exactly in the same way for vector-valued distributions.
2.3. Wavelet synthesis operator. Let η ∈ S 0 (R n ). The wavelet synthesis transform of Φ ∈ S(H n+1 ) with respect to the wavelet η is defined as Observe that the operator M η may be extended to act on the space S ′ (H n+1 ) via duality arguments, see [6] for details (cf. [13] 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 ∈ S 0 (R n ) admits a reconstruction wavelet η ∈ S 0 (R n ). More precisely, it means that the constant is different from zero and independent of the direction ω. Then, a straightforward calculation [6] shows that It is worth pointing out that (2.3) is also valid [6,13] 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 non-degenerate in the sense of the following definition [13]:
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 [4,6]. Our proof is elementary and significantly simpler than the one given in [6].
Theorem 3.1. The two bilinear mappings 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 We begin by making some reductions. Observe that, for constants c j which do not depend on ϕ and ψ, where ψ j (t) = t j ψ(t) ∈ S 0 (R n ) and ϕ m+j = ∂ m+j ϕ ∈ S 0 (R n ). It is therefore enough to show (3.1) 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, where ϕ r (t) = (−it) r ϕ(t) and ψ s (t) = (it) s ψ(t). Thus, it clearly suffices to establish (3.1) for k = ν = 0 and m = 0. We may also assume that l ≥ n.
We first estimate the term y l |W ψ ϕ(x, y)|. Since ∂ jφ (0) = 0 for every j ∈ N n , we can apply the Taylor formula to obtain Hence, It remains to estimate y −l |W ψ ϕ(x, y)|. We shall now use the fact that all the moments of ψ vanish. If we apply the Taylor formula, we have, for some 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 2) for κ = 0. We denote belowΦ the partial Fourier transform of Φ with respect to the space coordinate, i.e.,Φ(ξ, y) = R n Φ(x, y)e −iξ·x dx. We may assume that k is even. We then have, This completes the proof.
and it is still continuous.

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 shall employ a generalized Littlewood-Paley pair [20]. 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 non-degenerateness of wavelets, as defined in [13]. Even if a wavelet is non-degenerate, 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.
where R ω are the functions of one variable R ω (r) =ψ(rω).
If we only know values of W ψ f (x, y) at scale y < 1, then the wavelet transform can be blind when analyzing certain distributions (cf. [13,Sec. 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. [20]).
Example 4.3. Let φ ∈ S(R n ) be a radial function such thatφ is nonnegative,φ(ξ) = 1 for |ξ| < 1/2 andφ(ξ) = 0 for |ξ| > 1. Setψ(ξ) = −ξ · ∇φ(ξ). The pair (φ, ψ) is then clearly a 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., [7]). We pointed out above that LP-pairs enjoy of powerful reconstruction properties. Let us make this more precise.  Let (φ, ψ) be a LP-pair, the wavelet ψ having index of non-degenerateness τ and r > τ being a number such thatφ(ξ) = 0 for |ξ| < r. Pick any σ in between τ and r. If η ∈ S 0 (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 supp ϕ ⊂ B(0, r), then, for all f ∈ S ′ (R n ) and θ ∈ S(R n ) (4.1) Proof. Observe that It is therefore enough to assume that θ 1 = 0 so that θ = θ 2 . Our assumption over η is that η ∈ S 0 (R n ), suppη ⊂ B(0, σ) and does not depend on the direction ω. We remark that such a reconstruction wavelet can always be found (see the proof of [13,Theorem 7.7]). Therefore, Wηθ(x, y) = 0 for all (x, y) ∈ R n × (1, ∞). Exactly as in [6, p. 66], the usual calculation shows that Furthermore, since Wηθ ∈ S(H n+1 ) (cf. Theorem 3.1), the last integral can be express as the limit in S(R n ) of Riemann sums. That justifies the exchange of dual pairing and integral in 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 [1,17] 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, Familiar functions such as 1, |log ε| β , |log |log ε|| β , ..., are slowly varying. Regularly varying functions are then those that can be written as ε α L(ε), where L is slowly varying and α ∈ R.

4.3.
Weak-asymptotics. We shall use some notions from the theory of asymptotics of generalized functions [2,12,13,24]. The weak-asymptotics of distributions, also known as quasi-asymptotics, 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 if the order growth relation holds after evaluation at each test function, i.e., (4.3) f 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 [11]. One can also use these ideas to study exact pointwise scaling asymptotic properties of distributions (cf. [2,12,22,13]). We restrict our attention here to the important notion of the value of a distribution at a point, introduced and studied by Lojasiewicz in [9,10] (see also [3,21,25]). The vector-valued distribution f ∈ S ′ (R n , E) is said to have a value v ∈ E at the point x 0 ∈ R n if lim ε→0 + f (x 0 + εt) = v, distributionally, i.e., for each ϕ ∈ S(R n ) In such a case we simply write f (x 0 ) = 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 [13]. They are extended versions of Meyer's pointwise weak spaces from [11]. They are also close relatives of Bonny's two-microlocal spaces [8,11]. Again, we denote by E a Banach space, L is a slowly varying function at the origin. For a given x 0 ∈ R n and α ∈ R, the pointwise weak Hölder space [13] C α,L w (x 0 , E) consists of those distributions f ∈ S ′ (R n , E) for which there is an E-valued polynomial P such that (cf. Subsection 4.3) Observe that if α < 0, then the polynomial is irrelevant. In addition, when α ≥ 0, this polynomial is unique; in fact (4.4) readily implies that the Lojasiewicz point values ∂ m f (x 0 ) exist, distributionally, for |m| < α and that P is the "Taylor polynomial" The pointwise weak Hölder space C α,L * ,w (x 0 , E) of second type is defined as follows: 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 (x 0 , E) = C α,L w (x 0 , E). One can also show that if α / ∈ N, the equality between these two spaces remains true [13]. On the other hand, when α ∈ N, we have the strict inclusion C α,L w (x 0 , E) C α,L * ,w (x 0 , E) (cf. comments below). The usefulness of C α,L * ,w (x 0 , E) lies in the fact that it admits a precise wavelet characterization. The following theorem is shown in [13], it forms part of more general Tauberian-type results that will not be discussed here.
It is worth mentioning that the elements of C α,L * ,w (x 0 , E) for α = p ∈ N can be characterized by pointwise weak-asymptotic expansions. We have [13] that f ∈ C p,L * ,w (x 0 , E) if and only if it admits the weak-expansion where ∂ m f (x 0 ) are interpreted in the Lojasiewicz sense and the c m : (0, ∞) → E are continuous functions. Comparison of this weak-expansion with (4.4) explains the difference between the two pointwise spaces when α = p ∈ N.

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 introduce weighted Hölder spaces with respect to L. They were already defined and studied in [13]. 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 (5.1) When α = p + 1 ∈ N, we replace the previous requirement by 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 [7,8,11]. Consequently, as in [13], we call C α,L (R n ) the global Hölder space with respect to L. Note that, because of the properties of L [1, 17], we have the following interesting inclusion relations: whenever 0 < γ < α < β.

L-Zygmund spaces.
We now proceed to define the weighted Zygmund space C α,L * (R n ). Let α ∈ R and fix a LP-pair (φ, ψ) of order α.
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 inclusions: 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. In view of Proposition 5.1, we may employ a 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 ) [7]. Proposition 5.1 follows at once from the following lemma.

as required
We obtain the following useful properties.
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. Proof. Let η, ϕ, θ 1 , θ 2 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 We define the distribution f ∈ S ′ (R n ) whose action on test functions θ ∈ S(R n ) is given by Since the f j have the representation (4.1), we immediately see that f j → f in S ′ (R n ). Thus, pointwisely, and 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 shall use in its proof a technique based on the Tauberian theorem for pointwise weak regularity of vector-valued distributions, explained in Subsection 4.4. We denote below by C b (R n ) the Banach space of continuous and bounded functions.
(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 In addition, (5.5) produces a norm that is equivalent to (5.2).