1. Introduction

To a large extent, the success of wavelets in applications can be attributed to the realization that wavelet bases are universal unconditional bases for a large class of smoothness spaces, including all homogeneous Besov spaces. Given a wavelet orthonormal basis (consisting of sufficiently regular wavelets with vanishing moments) and , the expansion converges not only in , but also in any other Besov space norm , as soon as is contained in that space. Furthermore, the latter condition can be read off the decay behaviour of the wavelet coefficients associated to in a straightforward manner.

This observation provided important background and heuristics for many wavelet-based methods in applications such as denoising and data compression, but it was also of considerable theoretical interest, for example, for the study of operators. In this paper we provide similar results for simply connected stratified Lie groups. To our knowledge, studies of Besov spaces in this context have been largely restricted to the inhomogeneous cases. The definition of inhomogeneous Besov spaces on stratified Lie groups was introduced independently by Saka [1], and in a somewhat more general setting by Pesenson [2, 3]. Since then, the study of Besov spaces on Lie groups remained restricted to the inhomogeneous cases [4–8], with the notable exception of [9] which studied homogeneous Besov spaces on the Heisenberg group. A further highly influential source for the study of function spaces associated to the sub-Laplacian is Folland’s paper [10].

The first wavelet systems on stratified Lie groups (fulfilling certain technical assumptions) were constructed by Lemarié [11], by suitably adapting concepts from spline theory. Lemarié also indicated that the wavelet systems constructed by his approach were indeed unconditional bases of Saka’s inhomogeneous Besov spaces. Note that an adaptation, say, of the arguments in [12] for a proof of such a characterization requires a sampling theory for bandlimited functions on stratified groups, which was established only a few years ago by Pesenson [13]; see also [14].

More recent constructions of both continuous and discrete wavelet systems were based on the spectral theory of the sub-Laplacian [15]. Given the central role of the sub-Laplacian both in [8, 15], and in view of Lemarié’s remarks, it seemed quite natural to expect a wavelet characterization of homogeneous Besov spaces, and it is the aim of this paper to work out the necessary details. New results in this direction were recently published in [16–18].

The paper is structured as follows. After reviewing the basic notions concerning stratified Lie groups and their associated sub-Laplacians in Section 2, in Section 3 we introduce a Littlewood-Paley-type decomposition of functions and tempered discributions on . It is customary to employ the spectral calculus of a suitable sub-Laplacian for the definition of such decompositions, see, for example, [8], and this approach is also used here (Lemma 3.7). However, this raises the issue of consistency: the spaces should reflect properties of the group, not of the sub-Laplacian used for the construction of the decomposition. Using a somewhat more general notion than the -functions in [12] allows to establish that different choices of sub-Laplacian result in the same scale of Besov spaces (Theorem 3.11). In Section 4, we derive a characterization of Besov spaces in terms of continuous wavelet transform, with a wide variety of wavelets to choose from (Theorem 4.4). As a special case one obtains a characterization of homogeneous Besov spaces in terms of the heat semigroup. (See the remarks before Theorem 4.4.)

In Section 5, we study discrete characterizations of Besov spaces obtained by sampling the Calderón decomposition. For this purpose, we introduce the coefficient space . The chief result is Theorem 5.4, establishing that the wavelet coefficient sequence of lies in . Section 5 introduces our most important tool to bridge the gap between continuous and discrete decompositions, namely, oscillation estimates.

We then proceed to study wavelet synthesis and frame properties of the wavelet system. Our main result in this respect is that for all sufficiently dense regular sampling sets , the discrete wavelet system obtained by shifts from and dilations by powers of 2 is a universal Banach frame for all Besov spaces. In other words, the wavelet system allows the decomposition converging unconditionally in whenever , with coefficients depending linearly and boundedly on , and satisfying the norm equivalence

2. Preliminaries and Notation

Following the terminology in [19], we call a Lie group stratified if it is connected and simply connected, and its Lie algebra decomposes as a direct sum , with for and . Then is nilpotent of step and generated as a Lie algebra by . Euclidean spaces and the Heisenberg group are examples of stratified Lie groups.

If is stratified, its Lie algebra admits a canonical (natural) family of dilations, namely, which are Lie algebra automorphisms. We identify with through the exponential map. Hence is a Lie group with underlying manifold , for some , and the group product provided by the Campbell-Baker-Hausdorff formula. The dilations are then also group automorphisms of . Instead of writing for and , we simply use , whenever a confusion with the Lie group product is excluded. After choosing a basis of obtained as a union of bases of the , and a possible change of coordinates, one therefore has for and that for integers , according to .

Under our identification of with , polynomials on are polynomials on (with respect to any linear coordinate system on the latter). Polynomials on are written as where are the coefficients with respect to a suitable basis , and the monomials associated to the multi-indices . For a multi-index , define A polynomial of the type (2.3) is called of homogeneous degree if holds, for all multiindices with . We write for the space of polynomials of homogeneous degree .

We let denote the space of Schwartz functions on . By definition,. Let and denote the space of distributions and distributions modulo polynomials on , respectively. The duality between the spaces is denoted by the map . Most of the time, however, we will work with the sesquilinear version , for and .

Left Haar measure on is induced by Lebesgue measure on its Lie algebra, and it is also right-invariant. The number will be called the homogeneous dimension of . (For instance, for and we have and , respectively.) For any function on and , the -normalized dilation of is defined by Observe that this action preserves the -norm, that is, . We fix a homogeneous quasi-norm on which is smooth away from 0 with, for all , , for all , with if , and fulfilling a triangle inequality , with constant . Confer [19] for the construction of homogeneous norms, as well as further properties.

Moreover, by [19, Proposition  1.15], for any , there is a finite such that for all .

Our conventions for left-invariant operators on are as follows. We let denote a basis of , obtained as a union of bases of the . In particular, , for , is a basis of . Elements of the Lie algebra are identified in the usual manner with left-invariant differential operators on . Given a multi-index , we write for . A convenient characterization of Schwartz functions in terms of left-invariant operators states that if and only if, for all , , where In addition, the norms induce the topology of (see [19]).

The sub-Laplacian operator on can be viewed as the analog of the Laplacian operator on defined by . Using the above conventions for the choice of basis and , the sub-Laplacian is defined as . Note that a less restrictive notion of sub-Laplacians can also be found in the literature (e.g., any sum of squares of Lie algebra generators); we stress that the results in this paper crucially rely on the definition presented here. A linear differential operator on is called homogenous of degree if for any on . By choice of the for , these operators are homogeneous of degree 1; it follows that is homogenous of degree 2, and is homogenous of degree . Furthermore, any operator of the form is homogeneous of degree .

When restricted to , is formally self-adjoint: for any , . (For more see [15].) Its closure has domain , where we take in the sense of distributions. From this fact it quickly follows that this closure is self-adjoint and is in fact the unique self-adjoint extension of ; we denote this extension also by the symbol .

Suppose that has spectral resolution where is the projection measure. For a bounded Borel function on , the operator is a bounded integral operator on with a convolution distribution kernel in denoted by , and An important fact to be used later on is that for rapidly decaying smooth functions, , the kernel associated to is a Schwartz function. For a function on we define and . For , the adjoint of the convolution operator is provided by .

3. Homogeneous Besov Spaces on Stratified Lie Groups

In this section we define homogeneous Besov spaces on stratified Lie groups via Littlewood-Paley decompositions of distributions as

where is a dilated copy of a suitably chosen Schwartz function . In the Euclidean setting, it is customary to construct by picking a dyadic partition of unity on the Fourier transform side and applying Fourier inversion. The standard way of transferring this construction to stratified Lie groups consists in replacing the Fourier transform by the spectral decomposition of a sub-Laplacian , see Lemma 3.7. However, this approach raises the question to what extent the construction depends on the choice of . It turns out that the precise choice of sub-Laplacian obtained from a basis of is irrelevant. In order to prove this, we study Littlewood-Paley decompositions in somewhat different terms. The right setting for the study of such decompositions is the space of tempered distributions modulo polynomials, and the easiest approach to this convergence is via duality to a suitable space of Schwartz functions.

Definition 3.1. Let . A function has polynomial decay order if there exists a constant such that, for all , has vanishing moments of order , if one has with absolute convergence of the integral.

Under our identification of with , the inversion map is identical to the additive inversion map. That is, , and it follows that for all . Thus, if has vanishing moments of order , then for all that is, has vanishing moments of order as well.

Vanishing moments are central to most estimates in wavelet analysis, by the following principle: in a convolution product of the type , vanishing moments of one factor together with smoothness of the other result in decay. Later on, we will apply the lemma to Schwartz functions , where only the vanishing moment assumptions are nontrivial. The more general version given here is included for reference.

Lemma 3.2. Let be arbitrary.
(a)Let , such that is of decay order , for all with . Let have vanishing moments of order and decay order . Then there exists a constant, depending only on the decay of and , such that In particular, if , (b)Now suppose that , with of decay order for all with . Let have vanishing moments of order and decay order . Then there exists a constant, depending only on the decay of and , such that In particular, if ,

Proof. First, let us prove (a). Let . For , let denote the left Taylor polynomial of with homogeneous degree , see [19, Definition  1.44]. By that result, with suitable positive constants and . We next use the homogeneity properties of the partial derivatives [19, page 21], together with the decay condition on to estimate for with where the penultimate inequality used [19, 1.10], and the final estimate used . Thus, Next, using vanishing moments of , and the integral is finite by [19, 1.15]. This proves (3.5), and (3.6) follows by using .
For part (b), we first observe that Our assumptions on allow to invoke part (a) with replacing , and (3.7) follows immediately. (3.8) is obtained from this by straightforward integration.

We let denote the space of Schwartz functions with all moments vanishing. We next consider properties of as a subspace of with the relative topology.

Lemma 3.3. is a closed subspace (in particular complete) of , with , as well as for all . The topological dual of , , can be canonically identified with the factor space .

Proof. By definition, is the intersection of kernels of a family of tempered distributions, hence a closed subspace. For and , one has by unimodularity of that , since is a polynomial. But then, for any and , one has for all polynomials on that since implies (translation on is polynomial). Thus . All further properties of follow from the corresponding statements concerning . For identification of with the quotient space , we first observe that a tempered distribution vanishes on if and only if its (Euclidean) Fourier transform is supported in , which is well known to be the case if and only if is a polynomial. Using this observation, we map to , where is a continuous extension of to all of ; such an extension exists by the Hahn-Banach theorem. The map is well defined because the difference between two extensions of annihilates and hence is a polynomial. Linearity follows from well-definedness. Furthermore, the inverse of the mapping is clearly obtained by assigning to the restriction .

In the following, we will usually not explicitly distinguish between and its equivalence class modulo polynomials, and we will occasionally write . The topology of is just the topology of pointwise convergence on the elements of . For any net , holds if and only if , for all . We next study convolution on .

Lemma 3.4. For every , the map is a well-defined and continuous operator . If , the associated convolution operator is a well-defined and continuous operator .

Proof. Note that . Hence induces a well-defined canonical map . Furthermore, is continuous on , as a consequence of [19, Proposition  1.47]. Therefore, for any net and any , the fact that allows to write showing in .
For , the fact that makes the mapping well-defined modulo polynomials. The continuity statement is proved by (3.16), with assumptions on and switched.

The definition of homogeneous Besov spaces requires taking -norms of elements of . The following remark clarifies this.

Remark 3.5. Throughout this paper, we use the canonical embedding . For , this gives rise to an embedding , using that . Consequently, given , we let assigning the value otherwise. Here the fact that guarantees that the decomposition is unique, and thus (3.17) well-defined.
By contrast, can only be defined on , if we assign the value to .
Note that with these definitions, the Hausdorff-Young inequality remains valid for all , and all (for ), respectively, (for ). For , this is clear. For , note that if , then with .

We now introduce a general Littlewood-Paley-type decomposition. For this purpose we define for ,

Definition 3.6. A function is called LP-admissible if for all , holds, with convergence in the Schwartz space topology. Duality entails the convergence for all .

The following lemma yields the chief construction of LP-admissible functions.

Lemma 3.7. Let be a function in with support in such that and on . Take . Thus, , with support in the interval , and Pick a sub-Laplacian , and let denote the convolution kernel associated to the bounded left-invariant operator . Then is LP-admissible, with .

Proof. Let us first comment on the properties of that are immediate from the construction via spectral calculus: follows from [20] and vanishing moments by [15, Proposition  1].
Now let . First note that 2-homogeneity of implies that the convolution kernel associated to coincides with . Then, by the spectral theorem and (3.21), holds in -norm.
For any positive integer , where is the convolution kernel of . Since is a Schwartz function, it follows by [19, Proposition (1.49)] that , for , for all , with convergence in and a suitable constant .
We next show that in , as , for any . Fix a multi-index and with . Then left-invariance and homogeneity of yield Here the first inequality is an application of (3.5); the constant can be estimated in terms of , for sufficiently large. But this proves in the Schwartz topology.
Summarizing, in , and in addition by (3.22), in , whence follows.

Note that an LP-admissible function as constructed in Lemma 3.7 fulfills the convenient relation which follows from .

Remark 3.8. By spectral calculus, we find that , with . In particular, the decomposition shows that is dense.

We now associate a scale of homogeneous Besov spaces to the function .

Definition 3.9. Let be LP-admissible, let , , and . The homogeneous Besov space associated to is defined as with associated norm

Remark 3.10. The definition relies on the conventions regarding -norms of distributions (modulo polynomials), as outlined in Remark 3.5. Definiteness of the Besov norm holds because of (3.20).

The combination of Lemma 3.7 with Definition 3.9 shows that we cover the homogeneous Besov spaces defined in the usual manner via the spectral calculus of sub-Laplacians. Hence the following theorem implies in particular that different sub-Laplacians yield the same homogeneous Besov spaces (at least within the range of sub-Laplacians that we consider).

Theorem 3.11. Let be LP-admissible. Let and  . Then, , with equivalent norms.

Proof. It is sufficient to prove the norm equivalence, and here symmetry with respect to and immediately reduces the proof to showing, for a suitable constant , in the extended sense that the left-hand side is finite whenever the right-hand side is. Hence assume that ; otherwise, there is nothing to show. In the following, let ().
By LP-admissibility of , with convergence in . Accordingly, where the convergence on the right-hand side holds in , by Lemma 3.4. We next show that the right-hand side also converges in . For this purpose, we observe that where is a fixed integer. For , this follows directly from (3.8), using , and vanishing moments of , whereas for , the vanishing moments of allow to apply (3.6).
Using Young’s inequality, we estimate with from above that Next observe that By assumption, the sequence is in , in particular, bounded. Therefore, yields that (3.34) converges. But then the right-hand side of (3.31) converges unconditionally with respect to . This limit coincides with the -limit (which because of is even a -limit), yielding , with Now an application of Young’s inequality for convolution over , again using , provides (3.29).

As a consequence, we write , for any LP-admissible . These spaces coincide with the homogeneous Besov spaces for the Heisenberg group in [9], and with the usual definitions in the case .

In the remainder of the section we note some functional-analytic properties of Besov spaces and Littlewood-Paley-decompositions for later use.

Lemma 3.12. For all and all , one has continuous inclusion maps , as well as , where the latter denotes the dual of . For , is dense.

Proof. We pick as in Lemma 3.7 and define for . For the inclusion , note that (3.6) and (3.8) allow to estimate for all and that Here the constant is a suitable multiple of , for sufficiently large. But this implies that continuously.
For the other embedding, repeated applications of Hölder’s inequality yield the estimate valid for all and . Here are the conjugate exponents of , respectively. But this estimate implies continuity of the embeddings and .
For the density statement, let , and . For convenience, we pick according to Lemma 3.7. Since , there exists such that Next define
Let . By assumption on and Young’s inequality, , and since , there exists with . Let , then , and for , For , the construction of and implies that , whereas for , one has . As a consequence, one finds for and for For , one finds with some constant depending only on . For instance, for , A straight forward application of triangle and Young’s inequality yields Similar considerations applied to yield Now summation over yields as desired.