Abstract

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space ̇𝐵𝑠𝑝,𝑞 in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ̇𝐵𝑠𝑝,𝑞 with 1𝑝,𝑞< and 𝑠.

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 {𝜓𝑗,𝑘}𝑗,𝑘𝐿2(𝑛) (consisting of sufficiently regular wavelets with vanishing moments) and 𝑓𝐿2(𝑛), the expansion𝑓=𝑗,𝑘𝑓,𝜓𝑗,𝑘𝜓𝑗,𝑘(1.1) converges not only in 𝐿2, but also in any other Besov space norm ̇𝐵𝑠𝑝,𝑞, as soon as 𝑓 is contained in that space. Furthermore, the latter condition can be read off the decay behaviour of the wavelet coefficients {𝑓,𝜓𝑗,𝑘}𝑗,𝑘 associated to 𝑓 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 [48], 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 [1618].

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 𝐺. 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 𝜙-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 ̇𝑏𝑠𝑝,𝑞. The chief result is Theorem 5.4, establishing that the wavelet coefficient sequence of ̇𝐵𝑓𝑠𝑝,𝑞 lies in ̇𝑏𝑠𝑝,𝑞. 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 Γ, the discrete wavelet system {𝜓𝑗,𝛾}𝑗,𝛾Γ obtained by shifts from 𝛾 and dilations by powers of 2 is a universal Banach frame for all Besov spaces. In other words, the wavelet system allows the decomposition𝑓=𝑗,𝛾𝑟𝑗,𝛾𝜓𝑗,𝛾(1.2) converging unconditionally in ̇𝐵𝑠𝑝,𝑞 whenever ̇𝐵𝑓𝑠𝑝,𝑞, with coefficients {𝑟𝑗,𝛾}𝑗,𝛾̇𝑏𝑠𝑝,𝑞 depending linearly and boundedly on 𝑓, and satisfying the norm equivalence𝑟𝑗,𝛾𝑗,𝛾̇𝑏𝑠𝑝,𝑞𝑓̇𝐵𝑠𝑝,𝑞.(1.3)

2. Preliminaries and Notation

Following the terminology in [19], we call a Lie group 𝐺 stratified if it is connected and simply connected, and its Lie algebra 𝔤 decomposes as a direct sum 𝔤=𝑉1𝑉𝑚, with [𝑉1,𝑉𝑘]=𝑉𝑘+1 for 1𝑘<𝑚 and [𝑉1,𝑉𝑚]={0}. Then 𝔤 is nilpotent of step 𝑚 and generated as a Lie algebra by 𝑉1. Euclidean spaces 𝑛 and the Heisenberg group 𝑛 are examples of stratified Lie groups.

If 𝐺 is stratified, its Lie algebra admits a canonical (natural) family of dilations, namely,𝛿𝑟𝑋1+𝑋2++𝑋𝑚=𝑟𝑋1+𝑟2𝑋2++𝑟𝑚𝑋𝑚𝑋𝑗𝑉𝑗(𝑟>0),(2.1) which are Lie algebra automorphisms. We identify 𝐺 with 𝔤 through the exponential map. Hence 𝐺 is a Lie group with underlying manifold 𝑛, for some 𝑛, and the group product provided by the Campbell-Baker-Hausdorff formula. The dilations are then also group automorphisms of 𝐺. Instead of writing 𝛿𝑎(𝑥) for 𝑥𝐺 and 𝑎>0, we simply use 𝑎𝑥, whenever a confusion with the Lie group product is excluded. After choosing a basis of 𝔤 obtained as a union of bases of the 𝑉𝑖, and a possible change of coordinates, one therefore has for 𝑥𝐺 and 𝑎>0 that𝑎𝑎𝑥=𝑑1𝑥1,,𝑎𝑑𝑛𝑥𝑛,(2.2) for integers 𝑑1𝑑𝑛, according to 𝑥𝑖𝑉𝑑𝑖.

Under our identification of 𝐺 with 𝔤, polynomials on 𝐺 are polynomials on 𝔤 (with respect to any linear coordinate system on the latter). Polynomials on 𝐺 are written as𝑝dim(𝐺)𝑖=1𝑥𝑖𝑌𝑖=𝐼𝑐𝐼𝑥𝐼,(2.3) where 𝑐𝐼 are the coefficients with respect to a suitable basis 𝑌1,𝑌2,, and 𝑥𝐼=𝑥𝐼11𝑥𝐼22,,𝑥𝐼𝑛𝑛 the monomials associated to the multi-indices 𝐼{1,,𝑛}. For a multi-index 𝐼, define𝑑(𝐼)=𝑛𝑖=1𝐼𝑖𝑛(𝑖),𝑛(𝑖)=𝑗for𝑌𝑖𝑉𝑗.(2.4) A polynomial of the type (2.3) is called of homogeneous degree 𝑘 if 𝑑(𝐼)𝑘 holds, for all multiindices 𝐼 with 𝑐𝐼0. We write 𝒫𝑘 for the space of polynomials of homogeneous degree 𝑘.

We let 𝒮(𝐺) denote the space of Schwartz functions on 𝐺. By definition,𝒮(𝐺)=𝒮(𝔤). Let 𝒮(𝐺) and 𝒮(𝐺)/𝒫 denote the space of distributions and distributions modulo polynomials on 𝐺, respectively. The duality between the spaces is denoted by the map (,)𝒮(𝐺)×𝒮(𝐺). Most of the time, however, we will work with the sesquilinear version 𝑓,𝑔=(𝑓,𝑔), for 𝑓𝒮(𝐺) and 𝑔𝒮(𝐺).

Left Haar measure on 𝐺 is induced by Lebesgue measure on its Lie algebra, and it is also right-invariant. The number 𝑄=𝑚1𝑗(dim𝑉𝑗) will be called the homogeneous dimension of 𝐺. (For instance, for 𝐺=𝑛 and 𝑛 we have 𝑄=𝑛 and 𝑄=2𝑛+2, respectively.) For any function 𝜙 on 𝐺 and 𝑎>0, the 𝐿1-normalized dilation of 𝜙 is defined by𝐷𝑎𝜙(𝑥)=𝑎𝑄𝜙(𝑎𝑥).(2.5) Observe that this action preserves the 𝐿1-norm, that is, 𝐷𝑎𝜙1=𝜙. We fix a homogeneous quasi-norm || on 𝐺 which is smooth away from 0 with, |𝑎𝑥|=𝑎|𝑥| for all 𝑥𝐺, 𝑎0, |𝑥1|=|𝑥| for all 𝑥𝐺, with |𝑥|>0 if 𝑥0, and fulfilling a triangle inequality |𝑥𝑦|𝐶(|𝑥|+|𝑦|), with constant 𝐶>0. Confer [19] for the construction of homogeneous norms, as well as further properties.

Moreover, by [19, Proposition  1.15], for any 𝑟>0, there is a finite 𝐶𝑟>0 such that |𝑥|>𝑅|𝑥|𝑄𝑟𝑑𝑥=𝐶𝑟𝑅𝑟 for all 𝑅>0.

Our conventions for left-invariant operators on 𝐺 are as follows. We let 𝑌1,,𝑌𝑛 denote a basis of 𝔤, obtained as a union of bases of the 𝑉𝑖. In particular, 𝑌1,,𝑌𝑙, for 𝑙=dim(𝑉1), is a basis of 𝑉1. Elements of the Lie algebra are identified in the usual manner with left-invariant differential operators on 𝐺. Given a multi-index 𝐼𝑛0, we write 𝑌𝐼 for 𝑌𝐼11𝑌𝐼𝑛𝑛. A convenient characterization of Schwartz functions in terms of left-invariant operators states that 𝑓𝒮(𝐺) if and only if, for all 𝑁, |𝑓|𝑁<, where||𝑓||𝑁=sup||𝐼||𝑁,𝑥𝐺(1+|𝑥|)𝑁||𝑌𝐼||.𝑓(𝑥)(2.6) In addition, the norms ||𝑁 induce the topology of 𝒮(𝐺) (see [19]).

The sub-Laplacian operator on 𝐺 can be viewed as the analog of the Laplacian operator on 𝑛 defined by 𝐿=𝑛𝑖=1𝜕2/𝜕𝑥2𝑘. Using the above conventions for the choice of basis 𝑌1,,𝑌𝑛 and 𝑙=dim(𝑉1), the sub-Laplacian is defined as 𝐿=𝑙𝑖=1𝑌2𝑖. 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 𝑇 on 𝐺 is called homogenous of degree 𝑙 if 𝑇(𝑓𝛿𝑎)=𝑎𝑙(𝑇𝑓)𝛿𝑎 for any 𝑓 on 𝐺. By choice of the 𝑌𝑖 for 𝑖𝑙, these operators are homogeneous of degree 1; it follows that 𝐿 is homogenous of degree 2, and 𝐿𝑘 is homogenous of degree 2𝑘. Furthermore, any operator of the form 𝑌𝐼 is homogeneous of degree 𝑑(𝐼).

When restricted to 𝐶𝑐, 𝐿 is formally self-adjoint: for any 𝑓,𝑔𝐶𝑐(𝐺), 𝐿𝑓,𝑔=𝑓,𝐿𝑔. (For more see [15].) Its closure has domain 𝒟={𝑢𝐿2(𝐺)𝐿𝑢𝐿2(𝐺)}, where we take 𝐿𝑢 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 𝐿𝐶𝑐; we denote this extension also by the symbol 𝐿.

Suppose that 𝐿 has spectral resolution𝐿=0𝜆𝑑𝑃𝜆,(2.7) where 𝑑𝑃𝜆 is the projection measure. For a bounded Borel function 𝑓 on [0,), the operator𝑓(𝐿)=0𝑓(𝜆)𝑑𝑃𝜆(2.8) is a bounded integral operator on 𝐿2(𝐺) with a convolution distribution kernel in 𝐿2(𝐺) denoted by 𝑓, and𝑓(𝐿)𝜂=𝜂𝑓𝜂𝒮(𝐺).(2.9) An important fact to be used later on is that for rapidly decaying smooth functions, 𝑓𝒮(+), the kernel associated to 𝑓(𝐿) is a Schwartz function. For a function 𝑓 on 𝐺 we define 𝑓(𝑥)=𝑓(𝑥1) and 𝑓=𝑓. For 𝑓𝐿2(𝐺)𝐿1(𝐺), the adjoint of the convolution operator 𝑔𝑔𝑓 is provided by 𝑔𝑔𝑓.

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 𝑢 as𝑢=𝑗𝑢𝜓𝑗𝜓𝑗,(3.1)

where 𝜓𝑗 is a dilated copy of a suitably chosen Schwartz function 𝜓. In the Euclidean setting, it is customary to construct 𝜓 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 𝐿, see Lemma 3.7. However, this approach raises the question to what extent the construction depends on the choice of 𝐿. It turns out that the precise choice of sub-Laplacian obtained from a basis of 𝑉1 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 𝑁. A function 𝑓𝐺 has polynomial decay order 𝑁 if there exists a constant 𝐶>0 such that, for all 𝑥𝐺, ||||𝑓(𝑥)𝐶(1+|𝑥|)𝑁.(3.2)𝑓 has vanishing moments of order 𝑁, if one has 𝑝𝒫𝑁1𝐺𝑓(𝑥)𝑝(𝑥)𝑑𝑥=0,(3.3) with absolute convergence of the integral.

Under our identification of 𝐺 with 𝔤, the inversion map 𝑥𝑥1 is identical to the additive inversion map. That is, 𝑥1=𝑥, and it follows that ̃𝑝𝒫𝑁 for all 𝑝𝒫𝑁. Thus, if 𝑓 has vanishing moments of order 𝑁, then for all 𝑝𝒫𝑁1𝐺𝑓(𝑥)𝑝(𝑥)𝑑𝑥=𝐺𝑓(𝑥)̃𝑝(𝑥)𝑑𝑥=0,(3.4) that is, 𝑓 has vanishing moments of order 𝑁 as well.

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

Lemma 3.2. Let 𝑁,𝑘 be arbitrary.
(a)Let 𝑓𝐶𝑘, such that 𝑌𝐼(𝑓) is of decay order 𝑁, for all 𝐼 with 𝑑(𝐼)𝑘. Let 𝑔 have vanishing moments of order 𝑘 and decay order 𝑁+𝑘+𝑄+1. Then there exists a constant, depending only on the decay of 𝑌𝐼(𝑓) and 𝑔, such that ||𝐷𝑥𝐺0<𝑡<1𝑔𝑡𝑓(||𝑥)𝐶𝑡𝑘+𝑄(1+|𝑡𝑥|)𝑁.(3.5) In particular, if 𝑝>𝑄/𝑁, 𝐷𝑥𝐺0<𝑡<1𝑔𝑡𝑓𝑝𝐶𝑡𝑘+𝑄(11/𝑝).(3.6)(b)Now suppose that 𝑔𝐶𝑘, with 𝑌𝐼(̃𝑔) of decay order 𝑁 for all 𝐼 with 𝑑(𝐼)𝑘. Let 𝑓 have vanishing moments of order 𝑘 and decay order 𝑁+𝑘+𝑄+1. Then there exists a constant, depending only on the decay of 𝑓 and 𝑌𝐼(̃𝑔), such that ||𝐷𝑥𝐺1<𝑡<𝑔𝑡𝑓(||𝑥)𝐶𝑡𝑘(1+|𝑥|)𝑁.(3.7) In particular, if 𝑝>𝑄/𝑁, 𝐷𝑥𝐺1<𝑡<𝑔𝑡𝑓𝑝𝐶𝑡𝑘.(3.8)

Proof. First, let us prove (a). Let 0<𝑡<1. For 𝑥𝐺, let 𝑃𝑘𝑥,𝐷𝑡𝑓 denote the left Taylor polynomial of 𝐷𝑡𝑓 with homogeneous degree 𝑘1, see [19, Definition  1.44]. By that result, ||||𝐷𝑡𝑓𝑦1𝑥𝑃𝑘𝑥,𝐷𝑡𝑓||||(𝑦)𝐶𝑘||𝑦||𝑘sup|𝑧|𝑏𝑘||𝑦||,𝑑(𝐼)=𝑘|||𝑌𝐼𝐷𝑡𝑓|||,(𝑥𝑧)(3.9) with suitable positive constants 𝐶𝑘 and 𝑏. We next use the homogeneity properties of the partial derivatives [19, page 21], together with the decay condition on 𝑌𝐼𝑓 to estimate for 𝐼 with 𝑑(𝐼)=𝑘sup|𝑧|𝑏𝑘||𝑦|||||𝑌𝐼𝐷𝑡𝑓|||(𝑥𝑧)=𝑡𝑘sup|𝑧|𝑏𝑘||𝑦|||||𝐷𝑡𝑌𝐼𝑓|||(𝑥𝑧)=𝑡𝑘+𝑄sup|𝑧|𝑏𝑘||𝑦|||||𝑌𝐼𝑓|||(𝑡(𝑥𝑧))𝑡𝑘+𝑄sup|𝑧|𝑏𝑘||𝑦||𝐶𝑓||||1+𝑡(𝑥𝑧)𝑁𝑡𝑘+𝑄sup|𝑧|𝑏𝑘||𝑦||𝐶𝑓(1+|𝑡𝑥|)𝑁(1+|𝑡𝑧|)𝑁𝑡𝑘+𝑄(1+𝑏)𝑘𝑁𝐶𝑓(1+|𝑡𝑥|)𝑁||𝑦||1+𝑁,(3.10) where the penultimate inequality used [19, 1.10], and the final estimate used |𝑡𝑦|=𝑡|𝑦||𝑦|. Thus, ||||𝐷𝑡𝑓𝑦1𝑥𝑃𝑘𝑥,𝐷𝑡𝑓||||𝐶(𝑦)𝑘𝑡𝑘+𝑄||𝑦||1+𝑁+𝑘(1+|𝑡𝑥|)𝑁.(3.11) Next, using vanishing moments of 𝑔, ||𝑔𝐷𝑡𝑓||(𝑥)𝐺||||||||𝐷𝑔(𝑦)𝑡𝑓𝑦1𝑥𝑃𝑘𝑥,𝐷𝑡𝑓||||𝐶(𝑦)𝑑𝑦𝑘(1+|𝑡𝑥|)𝑁𝑡𝑘+𝑄𝐺||𝑔||||𝑦||(𝑦)1+𝑁+𝑘𝐶𝑑𝑦𝑘(1+|𝑡𝑥|)𝑁𝑡𝑘+𝑄𝐺𝐶𝑔||𝑦||1+𝑄1𝑑𝑦,(3.12) and the integral is finite by [19, 1.15]. This proves (3.5), and (3.6) follows by 𝑔𝐷𝑡𝑓𝑝𝐶𝑡𝑘+𝑄𝐺(1+|𝑡𝑥|)𝑁𝑝𝑑𝑥1/𝑝𝐶𝑡𝑘+𝑄𝑄/𝑝,(3.13) using 𝑁𝑝>𝑄.
For part (b), we first observe that 𝑔𝐷𝑡𝑓(𝑥)=𝑡𝑄𝑓𝐷𝑡1̃𝑔(𝑡𝑥).(3.14) Our assumptions on 𝑓,𝑔 allow to invoke part (a) with 𝑓̃𝑔, replacing 𝑓,𝑔, and (3.7) follows immediately. (3.8) is obtained from this by straightforward integration.

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

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

Proof. By definition, 𝒵(𝐺) is the intersection of kernels of a family of tempered distributions, hence a closed subspace. For 𝑝𝒫 and 𝑓𝒵(𝐺), one has by unimodularity of 𝐺 that 𝑝,𝑓=̃𝑝,𝑓=0, since ̃𝑝 is a polynomial. But then, for any 𝑔𝒮(𝐺) and 𝑓𝒵(𝐺), one has for all polynomials 𝑝 on 𝐺 that 𝑓𝑔𝑓,𝑝=𝑔,𝑝=𝑔,0=0,(3.15) since 𝑓𝒵(𝐺) implies 𝑝𝑓=0 (translation on 𝐺 is polynomial). Thus 𝑔𝑓𝒵(𝐺). All further properties of 𝒵(𝐺) follow from the corresponding statements concerning 𝒵(𝑛). For identification of 𝒵(𝑛) with the quotient space 𝒮(𝑛)/𝒫, we first observe that a tempered distribution 𝜑 vanishes on 𝒵(𝑛) if and only if its (Euclidean) Fourier transform is supported in {0}, which is well known to be the case if and only if 𝜑 is a polynomial. Using this observation, we map 𝑢𝒵(𝑛) to ̃𝑢+𝒫, where ̃𝑢 is a continuous extension of 𝑢 to all of 𝒮(𝑛); such an extension exists by the Hahn-Banach theorem. The map is well defined because the difference between two extensions of 𝑢 annihilates 𝒵(𝑛) and hence is a polynomial. Linearity follows from well-definedness. Furthermore, the inverse of the mapping is clearly obtained by assigning 𝑤+𝒫 to the restriction 𝑤𝒵(𝐺).

In the following, we will usually not explicitly distinguish between 𝑢𝒮(𝐺) and its equivalence class modulo polynomials, and we will occasionally write 𝑢𝒮(𝐺)/𝒫. The topology of 𝒮(𝐺)/𝒫 is just the topology of pointwise convergence on the elements of 𝒵(𝐺). For any net (𝑢𝑗+𝒫)𝑗𝐼, 𝑢𝑗+𝒫𝑢+𝒫 holds if and only if 𝑢𝑗,𝜑𝑢,𝜑, for all 𝜑𝒵(𝐺). We next study convolution on 𝒮(𝐺)/𝒫.

Lemma 3.4. For every 𝜓𝒮(𝐺), the map 𝑢𝑢𝜓 is a well-defined and continuous operator 𝒮(𝐺)/𝒫𝒮(𝐺)/𝒫. If 𝜓𝒵(𝐺), the associated convolution operator is a well-defined and continuous operator 𝒮(𝐺)/𝒫𝒮(𝐺).

Proof. Note that 𝒫𝒮(𝐺)𝒫. Hence 𝑢𝑢𝜓 induces a well-defined canonical map 𝒮(𝐺)/𝒫𝒮(𝐺)/𝒫. Furthermore, 𝑢𝑢𝜓 is continuous on 𝒮(𝐺), as a consequence of [19, Proposition  1.47]. Therefore, for any net 𝑢𝑗𝑢 and any 𝜑𝒵(𝐺), the fact that 𝜑𝜓𝒵(𝐺) allows to write 𝑢𝑗=𝑢𝜓,𝜑𝑗,𝜑𝜓𝑢,𝜑𝜓=𝑢𝜓,𝜑,(3.16) showing 𝑢𝑗𝜓𝑢𝜓 in 𝒮(𝐺)/𝒫.
For 𝜓𝒵(𝐺), the fact that 𝒫𝜓={0} makes the mapping 𝑢𝑢𝜓𝒮(𝐺) well-defined modulo polynomials. The continuity statement is proved by (3.16), with assumptions on 𝜓 and 𝜑 switched.

The definition of homogeneous Besov spaces requires taking 𝐿𝑝-norms of elements of 𝒮(𝐺)/𝒫. The following remark clarifies this.

Remark 3.5. Throughout this paper, we use the canonical embedding 𝐿𝑝(𝐺)𝒮(𝐺). For 𝑝<, this gives rise to an embedding 𝐿𝑝(𝐺)𝒮(𝐺)/𝒫, using that 𝒫𝐿𝑝(𝐺)={0}. Consequently, given 𝑢𝒮(𝐺)/𝒫, we let 𝑢𝑝=𝑢+𝑞𝑝whenever𝑢+𝑞𝐿𝑝(𝐺),forsuitable𝑞𝒫(3.17) assigning the value otherwise. Here the fact that 𝒫𝐿𝑝(𝐺)={0} guarantees that the decomposition is unique, and thus (3.17) well-defined.
By contrast, can only be defined on 𝒮(𝐺), if we assign the value to 𝑢𝒮(𝐺)𝐿(𝐺).
Note that with these definitions, the Hausdorff-Young inequality 𝑢𝑓𝑝𝑢𝑝𝑓1 remains valid for all 𝑓𝒮(𝐺), and all 𝑢𝒮(𝐺)/𝒫 (for 𝑝<), respectively, 𝑢𝒮(𝐺) (for 𝑝=). For 𝑝=, this is clear. For 𝑝<, note that if 𝑢+𝑞𝐿𝑝(𝐺), then (𝑢+𝑞)𝜓=𝑢𝜓+𝑞𝜓𝐿𝑝(𝐺) with 𝑞𝜓𝒫.

We now introduce a general Littlewood-Paley-type decomposition. For this purpose we define for 𝜓𝒮(𝐺),𝜓𝑗=𝐷2𝑗𝜓.(3.18)

Definition 3.6. A function 𝜓𝒮(𝐺) is called LP-admissible if for all 𝑔𝒵(𝐺), 𝑔=lim𝑁||𝑗||𝑁𝑔𝜓𝑗𝜓𝑗(3.19) holds, with convergence in the Schwartz space topology. Duality entails the convergence 𝑢=lim𝑁||𝑗||𝑁𝑢𝜓𝑗𝜓𝑗(3.20) for all 𝑢𝒮(𝐺)/𝒫.

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

Lemma 3.7. Let 𝜙 be a function in 𝐶 with support in [0,4] such that 0𝜙1 and 𝜙1 on [0,1/4]. Take 𝜓(𝜉)=𝜙(22𝜉)𝜙(𝜉). Thus, 𝜓𝐶𝑐(+), with support in the interval [1/4,4], and 1=𝑗||2𝜓2𝑗𝜉||2𝑎.𝑒.(3.21) Pick a sub-Laplacian 𝐿, and let 𝜓 denote the convolution kernel associated to the bounded left-invariant operator 𝜓(𝐿). Then 𝜓 is LP-admissible, with 𝜓𝒵(𝐺).

Proof. Let us first comment on the properties of 𝜓 that are immediate from the construction via spectral calculus: 𝜓𝒮(𝐺) follows from [20] and vanishing moments by [15, Proposition  1].
Now let 𝑔𝒵(𝐺). First note that 2-homogeneity of 𝐿 implies that the convolution kernel associated to 𝜓(22𝑗)(𝐿) coincides with 𝜓𝑗. Then, by the spectral theorem and (3.21), 𝑔=𝑗2𝜓2𝑗(𝐿)2𝜓2𝑗(𝐿)𝑔=𝑗𝑔𝜓𝑗𝜓𝑗(3.22) holds in 𝐿2-norm.
For any positive integer 𝑁, ||𝑗||𝑁𝑔𝜓𝑗𝜓𝑗=𝑔𝐷2𝑁+1𝜙𝑔𝐷2𝑁𝜙,(3.23) where 𝜙𝒮(𝐺) is the convolution kernel of 𝜙(𝐿). Since 𝜙 is a Schwartz function, it follows by [19, Proposition (1.49)] that 𝑔𝐷2𝑁+1𝜙𝑐𝜙𝑔, for 𝑁, for all 𝑔𝒮(𝐺), with convergence in 𝒮(𝐺) and a suitable constant 𝑐𝜙.
We next show that 𝑔𝐷𝑡𝑓0 in 𝒮(𝐺), as 𝑡0, for any 𝑓𝒮(𝐺). Fix a multi-index 𝐼 and 𝑁,𝑘 with 𝑘𝑁. Then left-invariance and homogeneity of 𝑌𝐼 yield ||𝑌𝐼𝑔𝐷𝑡𝑓||(𝑥)=𝑡𝑑(𝐼)||𝑔𝐷𝑡𝑌𝐼𝑓||(𝑥)𝐶𝑓,𝑔𝑡𝑘+𝑄+𝑑(𝐼)(1+|𝑡𝑥|)𝑁𝐶𝑓,𝑔𝑡𝑘+𝑄+𝑑(𝐼)𝑁(1+|𝑥|)𝑁.(3.24) Here the first inequality is an application of (3.5); the constant 𝐶𝑓,𝑔 can be estimated in terms of |𝑓|𝑀,|𝑔|𝑀, for 𝑀 sufficiently large. But this proves 𝑔𝐷𝑡𝑓0 in the Schwartz topology.
Summarizing, |𝑗|𝑁𝑔𝜓𝑗𝜓𝑗𝑐𝜙𝑔 in 𝒮(𝐺), and in addition by (3.22), |𝑗|𝑁𝑔𝜓𝑗𝜓𝑗𝑔 in 𝐿2, whence 𝑐𝜙=1 follows.

Note that an LP-admissible function 𝜓 as constructed in Lemma 3.7 fulfills the convenient relation||||𝑗,𝑙𝑗𝑙>1𝜓𝑗𝜓𝑙=0,(3.25) which follows from [𝜓(22𝑗)(𝐿)][𝜓(22𝑙)(𝐿)]=0.

Remark 3.8. By spectral calculus, we find that 𝜓=𝐿𝑘𝑔𝑘, with 𝑔𝑘𝒵(𝐺). In particular, the decomposition 𝑓=lim𝑁|𝑗|𝑁𝑓𝜓𝑗𝐷2𝑗𝐿𝑘𝑔𝑘=lim𝑁𝐿𝑘||𝑗||𝑁𝑓𝜓𝑗2𝑘𝑗𝐷2𝑗𝑔𝑘(3.26) shows that 𝐿𝑘(𝒵(𝐺))𝒵(𝐺) is dense.

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

Definition 3.9. Let 𝜓𝒵(𝐺) be LP-admissible, let 1𝑝, 1𝑞, and 𝑠. The homogeneous Besov space associated to 𝜓 is defined aṡ𝐵𝑠,𝜓𝑝,𝑞=𝑢𝒮2(𝐺)/𝒫𝑗𝑠𝑢𝜓𝑗𝑝𝑗𝑞(),(3.27) with associated norm 𝑢̇𝐵𝑠,𝜓𝑝,𝑞=2𝑗𝑠𝑢𝜓𝑗𝑝𝑗𝑞().(3.28)

Remark 3.10. The definition relies on the conventions regarding 𝐿𝑝-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 𝜓1,𝜓2𝒵(𝐺) be LP-admissible. Let 𝑠 and  1𝑝,𝑞. Then, ̇𝐵𝑠,𝜓1𝑝,𝑞=̇𝐵𝑠,𝜓2𝑝,𝑞, with equivalent norms.

Proof. It is sufficient to prove the norm equivalence, and here symmetry with respect to 𝜓1 and 𝜓2 immediately reduces the proof to showing, for a suitable constant 𝐶>0, 𝑢𝒮(𝐺)/𝒫𝑢̇𝐵1𝑠,𝜓𝑝,𝑞𝐶𝑢̇𝐵2𝑠,𝜓𝑝,𝑞,(3.29) in the extended sense that the left-hand side is finite whenever the right-hand side is. Hence assume that ̇𝐵𝑢𝑠,𝜓2𝑝,𝑞; otherwise, there is nothing to show. In the following, let 𝜓𝑖,𝑗=𝐷2𝑗𝜓𝑖 (𝑖=1,2).
By LP-admissibility of 𝜓2, 𝑢=lim𝑁||𝑗||𝑁𝑢𝜓2,𝑗𝜓2,𝑗,(3.30) with convergence in 𝒮(𝐺)/𝒫. Accordingly, 𝑢𝜓1,=lim𝑁||𝑗||𝑁𝑢𝜓2,𝑗𝜓2,𝑗𝜓1,,(3.31) where the convergence on the right-hand side holds in 𝒮(𝐺), by Lemma 3.4. We next show that the right-hand side also converges in 𝐿𝑝. For this purpose, we observe that 𝜓2,𝑗𝜓1,1=𝐷2𝑗𝜓2𝐷2𝑗𝜓111=𝜓2𝐷2𝑗𝜓11𝐶2|𝑗|𝑘,(3.32) where 𝑘>𝑠 is a fixed integer. For 𝑗0, this follows directly from (3.8), using 𝜓1,𝜓2𝒮(𝐺), and vanishing moments of 𝜓1, whereas for 𝑗<0, the vanishing moments of 𝜓2 allow to apply (3.6).
Using Young’s inequality, we estimate with 𝐶 from above that 𝑗𝑢𝜓2,𝑗𝜓2,𝑗𝜓1,𝑝𝑗𝑢𝜓2,𝑗𝑝𝜓2,𝑗𝜓1,1𝐶𝑢𝜓2,𝑗𝑝2|𝑗|𝑘(3.33)𝐶𝑗2𝑗𝑠𝑢𝜓2,𝑗𝑝2|𝑗|𝑘𝑗𝑠.(3.34) Next observe that 2|𝑗|𝑘𝑗𝑠=2𝑠2|𝑗|(𝑘+𝑠)2𝑗|𝑗|(𝑘𝑠)𝑗<𝑙2𝑠2|𝑗|(𝑘|𝑠|).(3.35) By assumption, the sequence (2𝑗𝑠𝑢𝜓𝑗,2𝑝)𝑗 is in 𝑞, in particular, bounded. Therefore, 𝑘|𝑠|>0 yields that (3.34) converges. But then the right-hand side of (3.31) converges unconditionally with respect to 𝑝. This limit coincides with the 𝒮(𝐺)/𝒫-limit 𝑢𝜓1, (which because of 𝜓1,𝒵(𝐺) is even a 𝒮(𝐺)-limit), yielding 𝑢𝜓1,𝐿𝑝(𝐺), with 2𝑠𝑢𝜓1,𝑝2𝑠𝑗𝑢𝜓2,𝑗𝜓2,𝑗𝜓1,𝑝𝐶32𝑠𝑗2𝑗𝑠𝑢𝜓2,𝑗𝑝2|𝑗|(𝑘|𝑠|).(3.36) Now an application of Young’s inequality for convolution over , again using 𝑘|𝑠|>0, provides (3.29).

As a consequence, we write ̇𝐵𝑠𝑝,𝑞=̇𝐵𝑠,𝜓𝑝,𝑞, for any LP-admissible 𝜓𝒵(𝐺). These spaces coincide with the homogeneous Besov spaces for the Heisenberg group in [9], and with the usual definitions in the case 𝐺=𝑛.

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 1𝑝,𝑞 and all 𝑠, one has continuous inclusion maps ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞𝑆(𝐺)/𝒫, as well as ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞, where the latter denotes the dual of ̇𝐵𝑠𝑝,𝑞. For 𝑝,𝑞<, ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞 is dense.

Proof. We pick 𝜓 as in Lemma 3.7 and define Δ𝑗𝑔=𝑔𝜓𝑗 for 𝑔𝒮(𝐺). For the inclusion ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞, note that (3.6) and (3.8) allow to estimate for all 𝑔𝒵(𝐺) and 𝑘 that Δ𝑗𝑔𝑝𝐶𝑘2|𝑗|𝑘.(3.37) Here the constant 𝐶𝑘 is a suitable multiple of |𝑔|𝑀, for 𝑀=𝑀(𝑘) sufficiently large. But this implies that ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞 continuously.
For the other embedding, repeated applications of Hölder’s inequality yield the estimate ||||=|||||𝑓,𝑔𝑗𝑓,𝑔𝜓𝑗𝜓𝑗|||||𝑗||𝑓𝜓𝑗,𝑔𝜓𝑗||𝑗𝑓𝜓𝑗𝑝𝑔𝜓𝑗𝑝=𝑗2𝑗𝑠𝑓𝜓𝑗𝑝2𝑗𝑠𝑓𝜓𝑗𝑝𝑓𝑠𝑝,𝑞𝑔𝑠𝑝,𝑞(3.38) valid for all ̇𝐵𝑓𝒵(𝐺)𝑠𝑝,𝑞 and ̇𝐵𝑔𝑠𝑝,𝑞. Here 𝑝,𝑞 are the conjugate exponents of 𝑝,𝑞, respectively. But this estimate implies continuity of the embeddings ̇𝐵𝑠𝑝,𝑞𝒮(𝐺)/𝒫 and ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞.
For the density statement, let ̇𝐵𝑢𝑠𝑝,𝑞, and 𝜖>0. For convenience, we pick 𝜓 according to Lemma 3.7. Since 𝑞<, there exists 𝑁 such that ||𝑗||>𝑁12𝑗𝑠𝑞Δ𝑗𝑢𝑞𝑝<𝜖.(3.39) Next define 𝐾𝑁=||𝑗||𝑁𝜓𝑗𝜓𝑗=𝐷2𝑁+1𝜙𝐷2𝑁𝜙.(3.40)
Let 𝑤=𝑢𝐾𝑁. By assumption on 𝑢 and Young’s inequality, 𝑤𝐿𝑝(𝐺), and since 𝑝<, there exists 𝑔𝒮(𝐺) with 𝑤𝑔𝑝<𝜖1/𝑞. Let 𝑓=𝑔𝐾𝑁, then 𝑓𝒵(𝐺), and for 𝑗, Δ𝑗(𝑢𝑓)𝑝=(𝑢𝑓)𝜓𝑗𝑝𝑢𝜓𝑗𝑢𝐾𝑁𝜓𝑗𝑝+𝑤𝜓𝑗𝑔𝐾𝑁𝜓𝑗𝑝.(3.41) For |𝑗|𝑁1, the construction of 𝜓𝑗 and 𝐾𝑁 implies that 𝐾𝑁𝜓𝑗=𝜓𝑗, whereas for |𝑗|>𝑁+1, one has 𝐾𝑁𝜓𝑗=0. As a consequence, one finds for |𝑗|<𝑁1Δ𝑗(𝑢𝑓)𝑝𝑤𝑔𝑝𝜓𝑗1=𝑤𝑔𝑝𝜓1<𝜖1/𝑞𝜓1,(3.42) and for|𝑗|>𝑁+1Δ𝑗(𝑢𝑓)𝑝𝑢𝜓𝑗𝑝<𝜖1/𝑞.(3.43) For ||𝑗|𝑁|1, one finds Δ𝑗(𝑢𝑓)𝑝𝐶𝜖1/𝑞(3.44) with some constant 𝐶>0 depending only on 𝜓. For instance, for 𝑗=𝑁, Δ𝑗(𝑢𝑓)𝑝𝑢𝜓𝑁𝜓𝑢𝑁1𝜓𝑁1+𝜓𝑁𝜓𝑁𝜓𝑁𝑝+𝑤𝜓𝑁𝜓𝑔𝑁1𝜓𝑁1+𝜓𝑁𝜓𝑁𝜓𝑁𝑝.(3.45) A straight forward application of triangle and Young’s inequality yields 𝑢𝜓𝑁𝜓𝑢𝑁1𝜓𝑁1+𝜓𝑁𝜓𝑁𝜓𝑁𝑝𝑢𝜓𝑁𝑝1+2𝜓𝜓1<𝜖1/𝑞1+2𝜓𝜓1.(3.46) Similar considerations applied to 𝑤=𝑢𝐾𝑁 yield 𝑤𝜓𝑁𝜓𝑔𝑁1𝜓𝑁1+𝜓𝑁𝜓𝑁𝜓𝑁𝑝2𝑢𝜓𝑁𝑝𝜓𝜓1+2𝑔𝜓𝑁𝑝𝜓𝜓12𝜖1/𝑞𝜓𝜓1+2𝑤𝜓𝑁𝑝+(𝑤𝑔)𝜓𝑁𝑝𝜓𝜓14𝜓𝜓1+𝜓𝜓1𝜓1𝜖1/𝑞.(3.47) Now summation over 𝑗 yields 𝑢𝑓̇𝐵𝑠𝑝,𝑞𝐶𝜖,(3.48) as desired.

Remark 3.13. Let 𝜓be as in Lemma 3.7. As a byproduct of the proof, we note that the space 𝒟=𝑓𝐾𝑁𝑓𝒮(𝐺),𝑁(3.49) is dense in 𝒵(𝐺) as well as ̇𝐵𝑠𝑝,𝑞, if 𝑝,𝑞<. In 𝒟, the decomposition 𝑔=𝑗𝑔𝜓𝑗𝜓𝑗(3.50) holds with finitely many nonzero terms.

We next extend the Littlewood-Paley decomposition to the elements of the Besov space. For simplicity, we prove the result only for certain LP-admissible functions.

Proposition 3.14. Let 1𝑝,𝑞<, and let 𝜓𝒵(𝐺) be an LP-admissible vector constructed via Lemma 3.7. Then the decomposition (3.19) converges for all ̇𝐵𝑔𝑠𝑝,𝑞 in the Besov space norm.

Proof. Consider the operators Σ𝑁̇𝐵𝑠𝑝,𝑞̇𝐵𝑠𝑝,𝑞, Σ𝑁𝑔=||𝑗||𝑁𝑔𝜓𝑗𝜓𝑗.(3.51) By suitably adapting the arguments proving the density statement of Lemma 3.12, it is easy to see that the family of operators (Σ𝑁)𝑁 is bounded in the operator norm. As noted in Remark 3.13, the Σ𝑁 strongly converges to the identity operator on a dense subspace. But then boundedness of the family implies strong convergence everywhere.

A further class of spaces for which the decomposition converges is 𝐿𝑝.

Proposition 3.15. Let 1<𝑝<, and let 𝜓𝒵(𝐺) be an LP-admissible vector constructed via Lemma 3.7. Then the decomposition (3.19) converges with respect to 𝑝, for all 𝑔𝐿𝑝(𝐺).

Proof. Let the operator family (Σ𝑁)𝑁 be defined as in the previous proof. Then, Σ𝑁𝑓=𝑔𝐷2𝑁+1𝜙𝑔𝐷2𝑁𝜙, and Young’s inequality implies that the sequence of operators is norm-bounded. It therefore suffices to prove the desired convergence on the dense subspace 𝒮(𝐺). By [19, Proposition  1.49],𝑔𝐷2𝑁+1𝜙𝑐𝜙𝑔. Furthermore, for 𝑁, 𝑔𝐷2𝑁𝜙(𝑥)=2𝑁𝑄𝐺2𝑔(𝑦)𝜙𝑁𝑦1𝑥=𝑑𝑦𝐺𝑔2𝑁𝑦𝜙𝑦12𝑁𝑥𝑑𝑦=2𝑁𝑄𝐷2𝑁2𝑔𝜙𝑁𝑥,(3.52) and thus 𝑔𝐷2𝑁𝑄𝜙𝑝=2𝑁𝑄𝐺||𝐷2𝑁2𝑔𝜙𝑁𝑥||𝑝𝑑𝑥1/𝑝=2𝑁𝑄+𝑁𝑄/𝑝𝐷2𝑁𝑔𝜙𝑝.(3.53) Again by [19, Proposition  1.49],(𝐷2𝑁𝑔𝜙)𝑐𝑔𝜙, in particular, 2𝑁𝑄+𝑁𝑄/𝑝𝐷2𝑁𝑔𝜙𝑝0as𝑁.(3.54) Hence, Σ𝑁𝑔𝑐𝜙𝑔, and the case 𝑝=2 yields 𝑐𝜙=1.

Theorem 3.16. ̇𝐵𝑠𝑝,𝑞 is a Banach space.

Proof. Completeness is the only issue here. Again, we pick 𝜓𝒵(𝐺) as an LP-admissible vector via Lemma 3.7. Suppose that {𝑢𝑛}𝑛̇𝐵𝑠𝑝,𝑞 is a Cauchy sequence. As a consequence, one has in particular, for all 𝑗, that {𝑢𝑛𝜓𝑗}𝑛𝐿𝑝(𝐺) is a Cauchy sequence, hence 𝑢𝑛𝜓𝑗𝑣𝑗, for a suitable 𝑣𝑗𝐿𝑝(𝐺). Furthermore, the Cauchy property of {𝑢𝑛}𝑛̇𝐵𝑠𝑝,𝑞 implies that 2𝑗𝑠𝑢𝑛𝜓𝑗𝑝𝑗𝑛𝑞()(3.55) is a Cauchy sequence. On the other hand, the sequence converges pointwise to {2𝑗𝑠𝑣𝑗𝑝}𝑗, whence 𝑗2𝑗𝑠𝑞𝑣𝑗𝑞𝑝<.(3.56) We define 𝑢=lim𝑀||𝑗||𝑀𝑣𝑗𝜓𝑗.(3.57) Now, using (3.56) and ̇𝐵𝒵(𝐺)𝑠𝑝,𝑞, where 𝑝,𝑞 are the conjugate exponents of 𝑝,𝑞, respectively, a straightforward calculation as in the proof of Lemma 3.12 shows that the sum defining 𝑢 converges in 𝒮(𝐺)/𝒫. Furthermore, (3.56) and (3.25) easily imply that ̇𝐵𝑢𝑠𝑝,𝑞. Finally, for the proof of 𝑢𝑛𝑢, we employ (3.25) together with the equality 𝜓𝑗=|𝑙𝑗|1𝜓𝑙𝜓𝑙𝜓𝑗, to show that 𝑢𝑛𝑢𝜓𝑗𝑝=𝑢𝑛𝜓𝑗||||𝑙𝑗1𝑣𝑙𝜓𝑙𝜓𝑗𝑝||||𝑙𝑗1𝑢𝑛𝜓𝑙𝑣𝑙𝜓𝑙𝜓𝑗𝑝||||𝑙𝑗1𝑢𝑛𝜓𝑙𝑣𝑙𝑝𝜓𝑙𝜓𝑗10,as𝑛.(3.58) Summarizing, the sequence {{2𝑗𝑠(𝑢𝑛𝑢)𝜓𝑗𝑝}𝑗}𝑛𝑞() is a Cauchy sequence, converging pointwise to 0. But then 𝑢𝑛𝑢̇𝐵𝑠𝑝,𝑞0 follows.

4. Characterization via Continuous Wavelet Transform

The following definition can be viewed as a continuous-scale analog of LP-admissibility.

Definition 4.1. 𝜓𝒮(𝐺) is called 𝒵-admissible, if for all 𝑓𝒵(𝐺), 𝑓=lim𝜖0,𝐴𝐴𝜖𝑓𝐷𝑎𝜓𝜓𝑑𝑎𝑎(4.1) holds with convergence in the Schwartz topology.

The next theorem reveals a large class of 𝒵-admissible wavelets. In fact, all the wavelets studied in [15] are also 𝒵-admissible in the sense considered here. Its proof is an adaptation of the argument showing [15, Theorem  1].

We let𝑆+=𝑓𝐶(0,)𝑘0,𝑓(𝑘)decreasesrapidly,lim𝜉𝑓(𝑘)(𝜉)exists.(4.2)

Theorem 4.2. Let 𝒮(+), and let 𝜓 be the distribution kernel associated to the operator 𝐿(𝐿). Then 𝜓 is 𝒵-admissible up to normalization.

Proof. The main idea of the proof is to write, for 𝑓𝒵(𝐺), 𝐴𝜖𝑓𝐷𝑎𝜓𝜓𝑑𝑎𝑎=𝑓𝐴𝜖𝐷𝑎𝜓𝜓𝑑𝑎𝑎=𝑓𝐷𝐴𝑔𝑓𝐷𝜖𝑔,(4.3) with suitable 𝑔𝒮(𝐺). Once this is established, 𝑓𝐷𝐴𝑔𝑐𝑔𝑓 for 𝐴 follows by [19, Proposition (1.49)], with convergence in the Schwartz topology. Moreover, 𝑓𝒵(𝐺) entails that 𝑓𝐷𝜖𝑔0 in the Schwartz topology: given any 𝑁>0 and 𝐼𝑛0 with associated left-invariant differential operator 𝑌𝐼, we can employ (3.5) to estimate sup𝑥𝐺(1+|𝑥|)𝑁||𝑌𝐼𝑓𝐷𝜖𝑔||(𝑥)=sup𝑥𝐺(1+|𝑥|)𝑁𝜖𝑄+𝑑(𝐼)||𝑓𝐷𝜖𝑌𝐼𝑔||(𝑥)𝐶sup𝑥𝐺(1+|𝑥|)𝑁𝜖𝑄+𝑑(𝐼)+𝑘(1+|𝜖𝑥|)𝑀𝐶sup𝑥𝐺(1+|𝑥|)𝑁𝑀𝜖𝑄+𝑑(𝐼)+𝑘𝑀,(4.4) which converges to zero for 𝜖0, as soon as 𝑀𝑁 and 𝑘>𝑀𝑄𝑑(𝐼). But this implies 𝑓𝐷𝜖𝑔0 in 𝒮(𝐺), by [19].
Thus it remains to construct 𝑔. To this end, define 1̂𝑔(𝜉)=2𝜉𝑎|||𝑎2|||2𝑑𝑎,(4.5) which is clearly in 𝒮(+), and let 𝑔 denote the associated convolution kernel of ̂𝑔(𝐿). By the definition, 𝑔𝒮(𝐺). Let 𝜑1,𝜑2 be in 𝒮(𝐺), and let 𝑑𝜆𝜑1,𝜑2 denote the scalar-valued Borel measure associated to 𝜑1,𝜑2 by the spectral measure. Then, by spectral calculus and the invariance properties of 𝑑𝑎/𝑎, 𝐴𝜖𝜑1𝐷𝑎𝜓𝑓𝜓𝑑𝑎𝑎,𝜑2=0𝐴𝜖𝑎2𝜉2|||𝑎2𝜉|||2𝑑𝑎𝑎𝑑𝜆𝜑1,𝜑2=1(𝜉)20𝐴2𝜉𝜖2𝜉𝑎|||𝑎2𝜉|||2𝑑𝑎𝑑𝜆𝜑1,𝜑2(=𝜉)0𝐴̂𝑔2𝜉𝜖̂𝑔2𝜉𝑑𝜆𝜑1,𝜑2=𝜑(𝜉)1𝐷𝐴𝑔𝐷𝜖𝑔,𝜑2,(4.6) as desired.

Hence, by [15, Corollary  1] we have the following.

Corollary 4.3. (a) There exist 𝒵-admissible 𝜓𝒵(𝐺).
(b) There exist 𝒵-admissible 𝜓𝐶𝑐(𝐺) with vanishing moments of arbitrary finite order.

Given a tempered distribution 𝑢𝒮(𝐺)/𝒫 and a 𝒵(𝐺)-admissible function 𝜓, the continuous wavelet transform of 𝑢 is the family (𝑢𝐷𝑎𝜓)𝑎>0 of convolution products. We will now prove a characterization of Besov spaces in terms of the continuous wavelet transform.

Another popular candidate for defining scales of Besov spaces is the heat semigroup; see for example, [1] for the inhomogeneous case on stratified groups, or rather [21] for the general treatment. In our setting, the heat semigroup associated to the sub-Laplacian is given by right convolution with 𝑡(𝑥)=𝐷𝑡(𝑥), where is the kernel of (𝐿) with (𝜉)=𝑒𝜉. Theorem 4.2 implies that 𝜓=𝐿𝑘 is 𝒵-admissible; it can be viewed as an analog of the well-known Mexican Hat wavelet. (For general stratified Lie groups, this class of wavelets was studied for the first time in [15].) The wavelet transform of 𝑓𝒮(𝐺) associated to 𝜓 is then very closely related to the 𝑘-fold time derivative of the solution to the heat equation with initial condition 𝑓. By choice of ,𝑢(𝑥,𝑡)=𝑓𝐷𝑡(𝑥)(4.7) denotes the solution of the heat equation associated to 𝐿, with initial condition 𝑓. A formal calculation using left invariance of 𝐿 then yields𝜕𝑘𝑡𝑢=𝐿𝑘𝑓𝐷𝑡=𝑓𝐿𝑘𝐷𝑡=𝑡2𝑘𝑓𝐷𝑡𝜓.(4.8) Thus the following theorem also implies a characterization of Besov spaces in terms of the heat semigroup.

Theorem 4.4. Let 𝜓𝒮(𝐺) be 𝒵-admissible, with vanishing moments of order 𝑘. Then, for all 𝑠 with |𝑠|<𝑘, and all 1𝑝<, 1𝑞, the following norm equivalence holds: 𝑢𝑆(𝐺)/𝑝𝑢̇𝐵𝑠𝑝,𝑞𝑎𝑎𝑠𝑢𝐷𝑎𝜓𝑝𝐿𝑞(+;𝑑𝑎/𝑎).(4.9) Here the norm equivalence is understood in the extended sense that one side is finite if and only if the other side is. If 𝜓𝒵(𝐺), the equivalence is also valid for the case 𝑝=.

Proof. The strategy consists in adapting the proof of Theorem 3.11 to the setting where one summation over scales is replaced by integration. This time, however, we have to deal with both directions of the norm equivalence. In the following estimates, the symbol 𝐶 denotes a constant that may change from line to line, but in a way that is independent of 𝑢𝑆(𝐺).
Let us first assume that 𝑎𝑠𝑞𝑢𝐷𝑎𝜓𝑞𝑝𝑑𝑎𝑎<,(4.10) for 𝑢𝒮(𝐺)/𝒫, 1𝑝,𝑞, for a 𝒵-admissible function 𝜓𝑆(𝐺) with 𝑘𝜓>|𝑠| vanishing moments (𝜓𝒵(𝐺), if 𝑝=). Let 𝜑𝒵(𝐺) be LP-admissible. Then, for all 𝑗, 𝑢𝜑𝑗=lim𝜖0,𝐴𝐴𝜖𝑢𝐷𝑎𝜓𝐷𝑎𝜓𝜑𝑗𝑑𝑎𝑎(4.11) holds in 𝒮(𝐺), by Lemma 3.4.
We next prove that the right-hand side of (4.11) converges in 𝐿𝑝. For this purpose, introduce 𝑐𝑗=0𝑢𝐷𝑎𝜓𝐷𝑎𝜓𝜑𝑗𝑝𝑑𝑎𝑎.(4.12) We estimate 𝑐𝑗0𝑢𝐷𝑎𝜓𝑝𝐷𝑎𝜓𝜑𝑗1𝑑𝑎𝑎=21𝑢𝐷𝑎2𝜓𝑝𝐷𝑎2𝜓𝜑𝑗1𝑑𝑎𝑎(4.13)21𝑢𝐷𝑎2𝜓𝑝𝐷𝑎2𝜓𝜑𝑗1𝑞𝑑𝑎𝑎1/𝑞log(2)1/𝑞,(4.14) where we used that 𝑑𝑎/𝑎 is scaling invariant. Note that the last inequality is Hölder’s inequality for 𝑞<. In this case, taking 𝑞th powers and summing over 𝑗 yields 𝑗2𝑗𝑠𝑞𝑐𝑞𝑗𝐶21𝑗2𝑗𝑠𝑞𝑢𝐷𝑎2𝜓𝑝𝐷𝑎2𝜓𝜑𝑗1𝑞𝑑𝑎𝑎.(4.15) Using vanishing moments and Schwartz properties of 𝜓 and 𝜑, we can now employ (3.6) and (3.8) to obtain 𝐷𝑎2𝜓𝜑𝑗1𝐶2|𝑗|𝑘,(4.16) with a constant independent of 𝑎[1,2]. But then, since 𝑘>|𝑠|, we may proceed just as in the proof of Theorem 3.11 to estimate the integrand in (4.15) via 𝑗2𝑗𝑠𝑞𝑢𝐷𝑎2𝜓𝑝𝐷𝑎2𝜓𝜑𝑗1𝑞𝐶2𝑠𝑞𝑢𝐷𝑎2𝜓𝑞𝑝.(4.17) Summarizing, we obtain 𝑗2𝑗𝑠𝑞𝑐𝑞𝑗𝐶212𝑠𝑞𝑢𝐷𝑎2𝜓𝑞𝑝𝑑𝑎𝑎𝐶0𝑎𝑠𝑞𝑢𝐷𝑎2𝜓𝑞𝑝𝑑𝑎𝑎<.(4.18) In particular, 𝑐𝑗<. But then the right-hand side of (4.11) converges to 𝑢𝜑𝑗 in 𝐿𝑝. The Minkowski inequality for integrals yields 𝑢𝜑𝑗𝑝𝑐𝑗, and thus 𝑢𝑞̇𝐵𝑠𝑝,𝑞𝐶0𝑎𝑠𝑞𝑢𝐷𝑎2𝜓𝑞𝑝𝑑𝑎𝑎,(4.19) as desired. In the case 𝑞=, (4.16) yields that sup𝑗2𝑗𝑠𝑢𝐷𝑎2𝜓𝑝𝐷𝑎2𝜓𝜑𝑗1𝐶sup2𝑠𝑢𝐷𝑎2𝜓𝑞𝑝.(4.20) Thus, by (4.13), sup𝑗2𝑗𝑠𝑐𝑗𝐶21sup2𝑠𝑢𝐷𝑎2𝜓𝑝𝑑𝑎𝑎𝐶esssup𝑎𝑎𝑠𝑢𝐷𝑎𝜓𝑝.(4.21) The remainder of the argument is the same as for the case 𝑞<.
Next assume ̇𝐵𝑢𝑠𝑝,𝑞. Then, for all 𝑎[1,2] and , 𝑢𝐷𝑎2𝜓=𝑗𝑢𝜑𝑗𝜑𝑗𝐷𝑎2𝜓,(4.22) with convergence in 𝒮(𝐺)/𝒫; for 𝜓𝒵(𝐺), convergence holds even in 𝒮(𝐺). As before, 𝑗𝑢𝜑𝑗𝜑𝑗𝐷𝑎2𝜓𝑝𝑗𝑢𝜑𝑗𝑝𝜑𝑗𝐷𝑎2𝜓1.(4.23) Again, we have 𝜑𝑗𝐷𝑎2𝜓12|𝑗|𝑘 with a constant independent of 𝑎. Hence, one concludes in the same fashion as in the proof of Theorem 3.11 that, for all 𝑎[1,2], 2𝑠𝑢𝐷𝑎2𝜓𝑝𝑞2𝐶𝑗𝑠𝑢𝜑𝑗𝑝𝑗𝑞,(4.24) again with a constant independent of 𝑎. In the case 𝑞=, this finishes the proof immediately, and for 𝑞<, we integrate the 𝑞th power over 𝑎[1,2] and sum over to obtain the desired inequality.

Remark 4.5. Clearly, the proof of Theorem 4.4 can be adapted to consider discrete Littlewood-Paley decompositions based on integer powers of any 𝑎>1 instead of 𝑎=2. Thus consistently replacing powers of 2 in Definitions 3.6 and 3.9 by powers of 𝑎>1 results in the same scale of Besov spaces.

As an application of the characterization via continuous wavelet transforms, we exhibit certain of the homogeneous Besov spaces as homogeneous Sobolev spaces, and we investigate the mapping properties of sub-Laplacians between Besov spaces of different smoothness exponents.

Lemma 4.6. ̇𝐵02,2=𝐿2(𝐺), with equivalent norms.

Proof. Pick 𝜓 by Lemma 3.7. Then spectral calculus implies that for all 𝑓𝒵(𝐺)𝑓2̇𝐵02,2=𝑗𝑓𝜓𝑗22=𝑓22.(4.25) Since 𝒵(𝐺) is dense in both spaces, and both spaces are complete, it follows that ̇𝐵02,2=𝐿2(𝐺).

The next lemma investigates the mapping properties of sub-Laplacians between Besov spaces of different smoothness exponents. Its proof is greatly facilitated by the characterization via continuous wavelet transforms.

Lemma 4.7. Let 𝐿 denote a sub-Laplacian. For all 𝑢𝒮(𝐺)/𝒫, 1𝑝,𝑞<, 𝑠 and 𝑘0, 𝐿𝑘𝑢̇𝐵𝑠2𝑘𝑝,𝑞𝑢̇𝐵𝑠𝑝,𝑞,(4.26) in the extended sense that one side is infinite if and only if the other side is. In particular, 𝐿𝑘̇𝐵𝑠𝑝,𝑞̇𝐵𝑠2𝑘𝑝,𝑞 is a bijection, and it makes sense to extend the definition to negative 𝑘. Thus, for all 𝑘, 𝐿𝑘̇𝐵𝑠𝑝,𝑞̇𝐵𝑠2𝑘𝑝,𝑞(4.27) is a topological isomorphism of Banach spaces.

Proof. Pick a nonzero real-valued 𝒮(+), an integer 𝑚>|𝑠| and let 𝜓 denote the distribution kernel of 𝐿𝑚(𝐿). Hence 𝜓 is admissible by Theorem 4.2, with vanishing moments of order 2𝑚 and 𝜓=𝜓. On 𝐿2(𝐺), the convolution operator 𝑢𝑢𝐷𝑎𝜓 can be written as Ψ𝑎(𝐿) with a suitable function Ψ𝑎. For 𝑢𝒵(𝐺)𝐿2(𝐺), spectral calculus implies 𝐿𝑘𝑢𝐷𝑎𝜓𝑝=Ψ𝑎(𝐿)𝐿𝑘(𝑢)𝑝=𝐿𝑘Ψ𝑎=𝐿(𝐿)(𝑢)𝑘𝑢𝐷𝑎𝜓𝑝=𝑢𝐿𝑘𝐷𝑎𝜓𝑝=𝑎2𝑘𝑢𝐷𝑎𝐿𝑘𝜓𝑝,(4.28) where we employed left invariance to pull 𝐿𝑘 past 𝑢 in the convolution. Note that up to normalization, 𝐿𝑘𝜓 is admissible with vanishing moments of order 2𝑚+2𝑘>|𝑠2𝑘|. Thus, applying Theorem 4.4, we obtain 𝐿𝑘𝑢̇𝐵𝑠2𝑘𝑝,𝑞𝑎𝑎𝑠2𝑘𝐿𝑘𝑢𝐷𝑎𝜓𝑝𝐿𝑞(+;𝑑𝑎/𝑎)=𝑎𝑎𝑠𝑢𝐷𝑎𝐿𝑘𝜓𝑝𝐿𝑞(+;𝑑𝑎/𝑎)𝑢̇𝐵𝑠𝑝,𝑞.(4.29) Now assume that 𝐿𝑘̇𝐵𝑢𝑠2𝑘𝑝,𝑞. Then, combining the density statements from Lemma 3.12 and Remark 3.8, we obtain a sequence {𝑢𝑛}𝑛𝒵(𝐺) with 𝐿𝑘𝑢𝑛𝐿𝑘𝑢 in ̇𝐵𝑠2𝑘𝑝,𝑞; thus also with convergence in 𝒮(𝐺)/𝒫. The norm equivalence and completeness of ̇𝐵𝑠𝑝,𝑞 yield that 𝑢𝑛̇𝐵𝑣𝑠𝑝,𝑞, for suitable ̇𝐵𝑣𝑠𝑝,𝑞. Again, this implies convergence in 𝒮(𝐺)/𝒫. Since 𝐿𝑘 is continuous on that space, it follows that 𝐿𝑘𝑢𝑛𝐿𝑘𝑣, establishing that 𝐿𝑘𝑣=𝐿𝑘𝑢. Since any distribution annihilated by 𝐿𝑘 is a polynomial, this finally yields ̇𝐵𝑢=𝑣𝑠𝑝,𝑞, and 𝑢̇𝐵𝑠𝑝,𝑞𝐿𝑘𝑢̇𝐵𝑠2𝑘𝑝,𝑞 follows by taking limits. A similar but simpler argument establishes the norm equivalence under the assumption that ̇𝐵𝑢𝑠𝑝,𝑞.

This observation shows that we can regard certain Besov spaces as homogeneous Sobolev spaces, or, more generally, as generalizations of Riesz potential spaces.

Corollary 4.8. For all 𝑘: 𝐵2𝑘2,2={𝑓𝒮(𝐺)/𝒫𝐿𝑘𝑓𝐿2(𝐺)}.

As a further corollary, we obtain the following interesting result relating two sub-Laplacians 𝐿1 and 𝐿2. For all 𝑘, the operator𝐿𝑘1𝐿2𝑘𝐿2(𝐺)𝐿2(𝐺)(4.30)

is densely defined and has a bounded extension with bounded inverse. More general analogues involving more than two sub-Laplacians are also easily formulated. For the Euclidean case, this is easily derived using the Fourier transform, which can be viewed as a joint spectral decomposition of commuting operators. In the general, nonabelian case however, this tool is not readily available, and we are not aware of a direct proof of this observation, nor of a previous source containing it.

5. Characterization of Besov Spaces by Discrete Wavelet Systems

We next show that the Littlewood-Paley characterization of ̇𝐵𝑠𝑝,𝑞 can be discretized by sampling the convolution products 𝑓𝜓𝑗 over a given discrete set Γ𝐺. This is equivalent to the study of the analysis operator associated to a discrete wavelet system {𝜓𝑗,𝛾}𝑗,𝛾Γ, defined by𝜓𝑗,𝛾(𝑥)=𝐷2𝑗𝑇𝛾𝜓(𝑥)=2𝑗𝑄𝜓𝛾12𝑗𝑥.(5.1) Throughout the rest of the paper, we assume that the wavelet 𝜓𝒵(𝐺) has been chosen according to Lemma 3.7 and 𝜓=𝜓.

We first define the discrete coefficient spaces which will be instrumental in the characterization of the Besov spaces.

Definition 5.1. Fix a discrete set Γ𝐺. For a family {𝑐𝑗,𝛾}𝑗,𝛾Γ of complex numbers, we define 𝑐𝑗,𝛾𝑗,𝛾Γ̇𝑏𝑠𝑝,𝑞=𝑗𝛾Γ2𝑗(𝑠𝑄/𝑝)||𝑐𝑗,𝛾||𝑝𝑞/𝑝1/𝑞.(5.2) The coefficient space ̇𝑏𝑠𝑝,𝑞(Γ) associated to ̇𝐵𝑠𝑝,𝑞 and Γ is then defined as ̇𝑏𝑠𝑝,𝑞𝑐(Γ)=𝑗,𝛾𝑗,𝛾Γ𝑐𝑗,𝛾𝑗,𝛾Γ̇𝑏𝑠𝑝,𝑞<.(5.3) We simply write ̇𝑏𝑠𝑝,𝑞 if Γ is understood from the context.

We define the analysis operator 𝐴𝜓 associated to the function 𝜓 and Γ, assigning each 𝑢𝒮(𝐺)/𝑃 the family of coefficients 𝐴𝜓(𝑢)={𝑢,𝜓𝑗,𝛾}𝑗,𝛾. Note that the analysis operator is implicitly assumed to refer to the same set Γ that is used in the definition of ̇𝑏𝑠𝑝,𝑞.

We next formulate properties of the sampling sets we intend to use in the following. We will focus on regular sampling, as specified in the next definition. Most of the results are obtainable for less regular sampling sets, at the cost of more intricate notation.

Definition 5.2. A subset Γ𝐺 is called regular sampling set, if there exists a relatively compact Borel neighborhood 𝑊𝐺 of the identity element of 𝐺 satisfying 𝛾Γ𝛾𝑊=𝐺 (up to a set of measure zero) as well as |𝛾𝑊𝛼𝑊|=0, for all distinct 𝛾,𝛼Γ. Such a set 𝑊 is called a Γ- tile. A regular sampling set Γ is called 𝑈-dense, for 𝑈𝐺, if there exists a Γ-tile 𝑊𝑈.

Note that the definition of 𝑈-dense used here is somewhat more restrictive than, for example, in [14]. A particular class of regular sampling sets is provided by lattices, that is, cocompact discrete subgroups Γ𝐺. Here, Γ-tiles are systems of representatives mod Γ. However, not every stratified Lie group admits a lattice. By contrast, there always exist sufficiently dense regular sampling sets, as the following result shows.

Lemma 5.3. For every neighborhood 𝑈 of the identity, there exists a 𝑈-dense regular sampling set.

Proof. By [14, Lemma  5.10], there exists Γ𝐺 and a relatively compact 𝑊 with nonempty open interior, such that 𝛾𝑊 tiles 𝐺 (up to sets of measure zero). Then 𝑉=𝑊𝑥01 is a Γ-tile, for some point 𝑥0 in the interior of 𝑊. Finally, choosing 𝑏>0 sufficiently small ensures that 𝑏𝑉𝑈, and 𝑏𝑉 is a 𝑏Γ-tile.

The chief result of this section is the following theorem which shows that the Besov norms can be expressed in terms of discrete coefficients. Note that the constants arising in the following norm equivalences may depend on the space, but the same sampling set is used simultaneously for all spaces.

Theorem 5.4. There exists a neighborhood 𝑈 of the identity, such that for all 𝑈-dense regular sampling sets Γ, and for all 𝑢𝒮(𝐺)/𝒫 and all 1𝑝,𝑞, the following implication holds: ̇𝐵𝑢𝑠𝑝,𝑞𝑢,𝜓𝑗,𝛾𝑗,𝛾Γ̇𝑏𝑠𝑝,𝑞(Γ).(5.4) Furthermore, the induced coefficient operator 𝐴𝜓̇𝐵𝑠𝑝,𝑞̇𝑏𝑠𝑝,𝑞 is a topological embedding. In other words, on ̇𝐵𝑠𝑝,𝑞 one has the norm equivalence 𝑢̇𝐵𝑠𝑝,𝑞𝑗𝛾2𝑗(𝑠𝑄/𝑝)||𝑢,𝜓𝑗,𝛾||𝑝𝑞/𝑝1/𝑞,(5.5) with constants depending on 𝑝,𝑞,𝑠, and Γ.

Remark 5.5. As a byproduct of the discussion in this section, we will obtain that the tightness of the frame estimates approaches 1, as the density of the sampling set increases. That is, the wavelet frames are asymptotically tight.

For the proof of Theorem 5.4, we need to introduce some notations. In the following, we write𝑋𝑗=𝑢𝜓𝑗𝑢𝒮,(𝐺)(5.6)

which is a space of smooth functions, as well as 𝑋𝑝𝑗=𝑋𝑗𝐿𝑝(𝐺). Furthermore, let Γ𝑗=2𝑗Γ, and denote by 𝑅Γ𝑗𝑋𝑗𝑔𝑔Γ𝑗 the restriction operator.

In order to prove Theorem 5.4, it is enough to prove the following sampling result for the spaces 𝑋𝑗; the rest of the argument consists in summing over 𝑗. In particular, note that the sampling set Γ is independent of 𝑝 and 𝑗, and the associated constants are independent of 𝑗.

Lemma 5.6. There exists a neighborhood 𝑈 of the identity, such that for all 𝑈-dense regular sampling sets Γ, the implication 𝑔𝑋𝑝𝑗𝑅Γ𝑗𝑔𝑝Γ𝑗,(5.7) holds. Furthermore, with suitable constants 0<𝑐(𝑝)𝐶(𝑝)< (for 1𝑝), the inequalities 𝑐(𝑝)𝑢𝜓𝑗𝑝𝛾Γ2𝑗𝑄||𝑢,𝜓𝑗,𝛾||𝑝1/𝑝𝐶(𝑝)𝑢𝜓𝑗𝑝(5.8) hold for all 𝑗 and all 𝑢𝑋𝑗.

Proof. Here we only show that the case 𝑗=0 implies the other cases; the rest will be established below. Hence assume (5.8) is known for 𝑗=0. Let 𝑔=𝑢𝜓𝑗𝑋𝑗. For arbitrary 𝑗, we have that 𝜓𝑗=2𝑗𝑄𝜓0𝛿2𝑗, and thus 𝑢𝜓𝑗=2𝑗𝑄𝜓𝑢𝛿2𝑗=𝑣𝑗𝜓𝛿2𝑗.(5.9) Here 𝑣𝑗=𝑢𝛿2𝑗, where the dilation action on distributions is defined in the usual manner by duality. The last equality follows from the fact that 𝛿2𝑗 is a group homomorphism. Recall that for any 𝑗 and 𝛾, 𝜓𝑗,𝛾(𝑥)=2𝑗𝑄𝜓(𝛾12𝑗𝑥), applying the case 𝑗=0, we obtain for 𝑝< that 𝛾Γ||𝑢,𝜓𝑗,𝛾||𝑝1/𝑝=𝛾Γ||𝑣𝑗,𝜓0,𝛾||𝑝1/𝑝𝑣𝐶(𝑝)𝑗𝜓0𝑝𝑣=𝐶(𝑝)𝑗𝜓0𝛿2𝑗𝛿2𝑗𝑝=𝐶(𝑝)𝑢𝜓𝑗𝛿2𝑗𝑝=𝐶(𝑝)2𝑗𝑄/𝑝𝑢𝜓𝑗𝑝,(5.10) which is the upper estimate for arbitrary 𝑗. The lower estimate and the case 𝑝= follow by similar calculations.

For the remainder of this section, we will therefore be concerned with the case 𝑗=0, which will be treated using ideas similar to the ones in [14], relying mainly on oscillation estimates. Given any function 𝑓 on 𝐺 and a set 𝑈𝐺, we define the oscillationosc𝑈(𝑓)(𝑥)=sup𝑦𝑈||𝑓(𝑥)𝑓𝑥𝑦1||.(5.11)

We can then formulate the following result.

Proposition 5.7. Let 𝑋0𝒮(𝐺) be a space of continuous functions. Suppose that there exists 𝐾𝒮(𝐺) such that, for all 𝑓𝑋0, 𝑓=𝑓𝐾 holds pointwise. Define 𝑋𝑝0=𝑋0𝐿𝑝(𝐺), for 1𝑝. Let 𝜖<1, and, 𝑈 be a neighborhood of the unit element fulfilling osc𝑈(𝐾)1𝜖. Then, for all 𝑈-dense regular sampling sets Γ, the following implication holds: 𝑓𝑋0𝑓𝑋𝑝0𝑓Γ𝑝(Γ).(5.12) The restriction map 𝑅Γ𝑓𝑓|Γ induces a topological embedding (𝑋𝑝0,𝑝)𝑙𝑝(Γ).
More precisely, for 𝑝<, 1||𝑊||1/𝑝(1𝜖)𝑓𝑝𝑅Γ𝑓𝑝1||𝑊||1/𝑝(1+𝜖)𝑓𝑝,𝑓𝑋𝑝0,(5.13) where 𝑊 denotes a Γ-tile, and (1𝜖)𝑓𝑅Γ𝑓(1+𝜖)𝑓,𝑓𝑋0.(5.14)

Proof. We introduce the auxiliary operator 𝑇𝑝(Γ)𝐿𝑝(𝐺) defined by 𝑇(𝑐)=𝛾Γ𝑐𝛾𝐿𝛾𝜒𝑊,(5.15) with 𝑐=(𝑐𝛾)𝛾Γ. Since the sets 𝛾𝑊 are pairwise disjoint, 𝑇 is a multiple of an isometry, 𝑇𝑐𝑝=|𝑊|1/𝑝𝑐𝑝. In particular, 𝑇 has a bounded inverse on its range, and 𝑇𝑐𝐿𝑝(𝐺) implies 𝑐𝑝(Γ) for any sequence 𝑐Γ.
The equation 𝑓=𝑓𝐾 implies the pointwise inequality osc𝑈||𝑓||(𝑓)osc𝑈(𝐾)(5.16) (see [14, page 185]). Now Young’s inequality provides for 𝑓𝑋𝑝: osc𝑈(𝑓)𝑝𝑓𝑝osc𝑈(𝐾)1𝜖𝑓𝑝.(5.17) Since the 𝛾𝑊’s are disjoint, we may then estimate, for all 𝑓𝑋𝑝, 𝑓𝑇𝑅Γ𝑓𝑝𝑝=𝛾Γ𝛾𝑊||||𝑓(𝑥)𝑓(𝛾)𝑝𝑑𝑥𝛾Γ𝛾𝑊||osc𝑈||(𝑓)(𝑥)𝑝=𝑑𝑥osc𝑈(𝑓)𝑝𝑝𝜖𝑝𝑓𝑝𝑝.(5.18) In particular, 𝑇𝑅Γ𝑓𝐿𝑝(𝐺), whence 𝑅Γ𝑓𝑝(Γ). In addition, we obtain the upper bound of the sampling inequality for 𝑓𝑋𝑝𝑅Γ𝑓𝑝=𝑇1𝑇𝑅Γ𝑓𝑝𝑇1𝑇𝑅Γ𝑓𝑝𝑇1𝑓𝑝+𝑓𝑇𝑅Γ𝑓𝑝𝑇1(1+𝜖)𝑓𝑝1||𝑊||1/𝑝(1+𝜖)𝑓𝑝.(5.19) The lower bound follows similarly by 𝑅Γ𝑓𝑝𝑇1𝑇𝑅Γ𝑓𝑝𝑇1𝑓𝑝𝑓𝑇𝑅Γ𝑓𝑝𝑇1(1𝜖)𝑓𝑝1||𝑈||1/𝑝(1𝜖)𝑓𝑝.(5.20) Thus (5.13) and (5.12) are shown, for 1𝑝<. For 𝑝=, we note that 𝑇=𝑇1=1. Furthermore, 𝑓𝑇𝑅𝛾𝑓sup𝛾esssup𝑥𝛾𝑊||||𝑓(𝑥)𝑓(𝛾)osc𝑈(𝑓).(5.21) Now the remainder of the proof is easily adapted from the case 𝑝<.

It remains to check the conditions of the proposition for𝑋0=𝑓=𝑢𝜓0𝒮𝑢(𝐺)𝑃.(5.22)

Lemma 5.8. There exists a Schwartz function 𝐾 acting as a reproducing kernel for 𝑋0, that is, 𝑓=𝑓𝐾 holds for all 𝑓𝑋𝑝0.

Proof. We pick a real-valued 𝐶𝑐-function 𝑘 on + that is identically 1 on the support of 𝜓0, and let 𝐾 be the associated distribution kernel to 𝑘(𝐿). Then 𝜓0=𝜓0𝐾, whence 𝑓=𝑓𝐾 follows, for all 𝑓𝑋0.

Lemma 5.9. Let 𝐾 be a Schwartz function. For every 𝜖>0, there exists a compact neighborhood 𝑈 of the unit element such that osc𝑈(𝐾)1<𝜖.

Proof. First observe that, by continuity, osc𝑈(𝐾)0 pointwise, as 𝑈 runs through a neighborhood base at the identity element. Thus by dominated convergence it suffices to prove osc𝑉(𝐾)1<, for some neighborhood 𝑉.
Let 𝑉={𝑥𝐺|𝑥|<1}. A straightforward application of the mean value theorem [19, Theorem  1.33] yields osc𝑉(𝐾)(𝑥)𝐶sup|𝑧|𝛽,1𝑖𝑛||𝑌𝑖||.𝐾(𝑥𝑧)(5.23) Here 𝐶 and 𝛽 are constants depending on 𝐺. The Sobolev estimate [22, (5.13)] for 𝑝=1 yields that for all 𝑧 with |𝑧|<𝛽||𝑌𝑖||𝐾(𝑥𝑧)𝐶𝑌𝑥𝑊||||𝑌𝐾(𝑦)𝑑𝑦,(5.24) where 𝑌 runs through all possible 𝑌𝐼 with 𝑑(𝐼)𝑄+1, including the identity operator corresponding to 𝐼=(0,,0). Furthermore, 𝑊={𝑥𝐺|𝑥|<𝛽}, and 𝐶>0 is a constant. Now integrating against Haar-measure (which is two-sided invariant) yields 𝐺osc𝑉(𝐾)(𝑥)𝑑𝑥𝐶𝑌𝐺𝑥𝑊||||𝑌𝐾(𝑦)𝑑𝑦𝑑𝑥=𝐶𝐶𝑌𝐺𝑊||||=||𝑊||𝑌𝐾(𝑥𝑦)𝑑𝑦𝑑𝑥𝐶𝐶𝑌𝐺||||𝑌𝐾(𝑥)𝑑𝑥,(5.25) and the last integral is finite because 𝐾 is a Schwartz function.

Now Lemma 5.6 is a direct consequence of Proposition 5.7 and Lemmas 5.8 and 5.9. Note that the tightness in Proposition 5.7 converges to 1, as 𝑈 runs through a neighborhood of the identity. This property is then inherited by the norm estimates in Theorem 5.4.

6. Banach Wavelet Frames for Besov Spaces

In Hilbert spaces a norm equivalence such as (5.5) would suffice to imply that the wavelet system is a frame, thus entailing a bounded reconstruction from the discrete coefficients. For Banach spaces one needs to use the extended definition of frames [23], that is, to show the invertibility of associated frame operator. In this section we will establish these statements for wavelet systems in Besov space. We retain the assumption that the wavelet 𝜓 was chosen according to Lemma 3.7.

We first prove that any linear combination of wavelet systems with coefficients in ̇𝑏𝑠𝑝,𝑞 converges unconditionally in ̇𝐵𝑠𝑝,𝑞, compare [12, Theorem  3.1]. We then show that for all sufficiently dense choices of the sampling set Γ, the wavelet system {2𝑗𝑄𝜓𝑗,𝛾} constitutes a Banach frame for ̇𝐵𝑠𝑝,𝑞.

Recall that the sampled convolution products studied in the previous sections can be read as scalar products𝑓𝜓𝑗2𝑗𝛾=𝑓,𝜓𝑗,𝛾,(6.1)

where 𝜓𝑗,𝛾(𝑥)=2𝑗𝑄𝜓(𝛾12𝑗𝑥) denotes the wavelet of scale 2𝑗 at position 2𝑗𝛾. In the following, the wavelet system is used for synthesis purposes, that is, we consider linear combinations of discrete wavelets. The next result can be viewed in parallel to synthesis results, for example, in [7]. It establishes synthesis for a large class of systems. Note in particular that the functions 𝑔𝑗,𝛾 need not be obtained by dilation and shifts from a single function 𝑔.

Theorem 6.1. Let Γ𝐺 be a regular sampling set. Let 1𝑝,𝑞<. (a)Suppose that one is given tempered distributions (𝑔𝑗,𝛾)𝑗,𝛾Γ satisfying the following decay conditions: for all 𝑁,𝜃, there exist constants 𝑐1,𝑐2 such that for all 𝑗,𝑙, 𝛾Γ, 𝑥𝐺: ||𝑔𝑗,𝛾𝜓𝑙||𝑐(𝑥)12𝑗𝑄2(𝑗𝑙)𝑁1+2𝑙||2𝑗𝛾1||𝑥(𝑄+1)𝑐for𝑙𝑗,22𝑗𝑄2(𝑙𝑗)𝜃1+2𝑗||2𝑗𝛾1||𝑥(𝑄+1)for𝑙𝑗,(6.2) Then for all {𝑐𝑗,𝛾}𝑗,𝛾Γ̇𝑏𝑠𝑝,𝑞(Γ), the sum 𝑓=𝑗,𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾(6.3) converges unconditionally in the Besov norm, with 𝑓̇𝐵𝑠𝑝,𝑞𝑐𝑗𝛾2𝑗(𝑠𝑄/𝑝)||𝑐𝑗,𝛾||𝑝𝑞/𝑝1/𝑞(6.4) for some constant 𝑐 independent of {𝑐𝑗,𝛾}𝑗,𝛾Γ. In other words, the synthesis operator ̇𝑏𝑠𝑝,𝑞̇𝐵(Γ)𝑠𝑝,𝑞 associated to the system (𝑔𝑗,𝛾)𝑗,𝛾 is bounded. (b)The synthesis result in (a) holds in particular for 𝑔𝑗,𝛾(𝑥)=𝜓𝑗,𝛾(𝑥)=2𝑗𝑄𝜓𝑗𝛾12𝑗𝑥.(6.5)

In order to motivate the following somewhat technical lemmas, let us give a short sketch of the proof strategy for the theorem. It suffices to show (6.4) for all finitely supported sequences; the rest follows by density arguments, using that ̇𝐵𝑠𝑝,𝑞 is a Banach space. Hence, given a finitely supported coefficient sequence {𝑐𝑗,𝛾} and 𝑓=𝑗,𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾, we need estimates for the 𝐿𝑝-norms of𝑓𝜓𝑙=𝑗,𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙.(6.6) These estimates are obtained by first looking at the summation over 𝛾, with 𝑗 fixed, and then summing over 𝑗. In both steps, we use the decay condition (6.2).

The following lemma shows that (6.2) is fulfilled for 𝑔𝑗,𝛾=𝜓𝑗,𝛾 and thus allows to conclude part (b) of Theorem 6.1.

Lemma 6.2. There exists a constant 𝐶>0 such that for any 𝑗,𝑙, 𝛾Γ, 𝑥𝐺, the following estimate holds: ||𝜓𝑗,𝛾𝜓𝑙||(𝑥)𝐶2𝑗𝑄1+2𝑗||2𝑗𝛾1||𝑥(𝑄+1)||||𝑙𝑗10otherwise.(6.7)

Proof. We first compute 𝜓𝑗,𝛾𝜓𝑙(𝑥)=𝐺2𝑗𝑄𝜓𝛾12𝑗𝑦2𝑙𝑄𝜓2𝑙𝑥1=𝑦𝑑𝑦𝐺𝜓𝛾12𝑦𝑙𝑄𝜓2𝑙𝑥12𝑗𝑦=𝑑𝑦𝜓(𝑦)2𝑗𝑄2𝑙𝑄𝜓2𝑙𝑗𝛾12𝑗𝑥1𝑦𝑑𝑦=2𝑗𝑄𝜓𝜓𝑙𝑗𝛾12𝑗𝑥.(6.8) In particular, (3.25) implies that the convolution vanishes if |𝑗𝑙|>1. For the other case, we observe that the convolution products 𝜓𝜓𝑙, for 𝑙{1,0,1} are Schwartz functions, hence ||𝜓𝑗,𝛾𝜓𝑙||(𝑥)𝐶2𝑗𝑄1+2𝑗||2𝑗𝛾1||𝑥𝑄1.(6.9)

For the convergence of the sums over Γ, we will need the Schur test for boundedness of infinite matrices on 𝑝-spaces.

Lemma 6.3. Let 1𝑝. Let Γ be some countable set, and let 𝐴=(𝑎𝜆,𝛾)𝜆,𝛾Γ denote a matrix of complex numbers. Assume that, for some finite constant 𝑀, sup𝛾𝜆Γ||𝑎𝜆,𝛾||𝑀,sup𝜆𝛾Γ||𝑎𝜆,𝛾||𝑀.(6.10) Then the operator 𝑇𝐴𝑥𝛾𝛾Γ𝛾Γ𝑎𝜆,𝛾𝑥𝛾𝜆Γ(6.11) is bounded on 𝑝(Γ), with operator norm 𝑀.

Lemma 6.4. Let 𝜂,𝑗, with 𝜂𝑗 and 𝑁𝑄+1. Let Γ𝐺 be separated. Then for any 𝑥𝐺, one has 𝛾2𝑗𝑄1+2𝜂||2𝑗𝛾1||𝑥𝑁𝐶2𝜂𝑄,(6.12) where the constant 𝐶 depends only on 𝑁 and Γ.

Proof. By assumption, there exists an open set 𝑊 such that 𝛾𝑊𝛾𝑊=, for 𝛾,𝛾Γ with 𝛾𝛾. In addition, we may assume 𝑊 is relatively compact. Then, 𝛾2𝑗𝑄1+2𝜂||2𝑗𝛾1||𝑥𝑁𝛾1||𝑊||2𝑗(𝛾𝑊)1+2𝜂||2𝑗𝛾1||𝑥𝑁𝑑𝑦.(6.13) For 𝑦2𝑗(𝛾𝑊), the triangle inequality of the quasi-norm yields 1+2𝜂||𝑦1𝑥||1+2𝜂𝐶||𝑦12𝑗𝛾||+||2𝑗𝛾1𝑥||1+𝐶2𝜂2𝑗||2diam(𝑊)+𝑗𝛾1𝑥||𝐶1+2𝜂||2𝑗𝛾1𝑥||,(6.14) with the last inequality due to 𝜂𝑗. Accordingly, 𝛾1||𝑊||2𝑗(𝛾𝑊)1+2𝜂||2𝑗𝛾1||𝑥𝑁𝑑𝑦𝐶𝛾2𝑗(𝛾𝑊)1+2𝜂||𝑦1||𝑥𝑁𝑑𝑦=𝐶2𝜂𝑄𝐺||𝑦||1+𝑁𝑑𝑦,(6.15) where the inequality used disjointness of the 𝛾𝑊. For 𝑁𝑄+1, the integral is finite.

The next lemma is an analog of [12, Lemma  3.4], which we will need for the proof of Theorem 6.1.

Lemma 6.5. Let 1𝑝 and 𝑗,𝜂 be fixed with 𝜂𝑗. Suppose that Γ𝐺 is a regular sampling set. For any 𝛾Γ, let 𝑓𝑗,𝛾 be a function on 𝐺. Assume that the 𝑓𝑗,𝛾 fulfill the decay estimate ||𝑓𝑥𝐺,𝜂,𝑗,𝛾Γ𝑗,𝛾||(𝑥)𝐶11+2𝜂||2𝑗𝛾1||𝑥(𝑄+1),(6.16) with a constant 𝐶1>0. Define 𝐹=𝛾Γ𝑐𝑗,𝛾𝑓𝑗,𝛾, where {𝑐𝑗,𝛾}𝛾𝑙𝑝(Γ). Then the series converges unconditionally in 𝐿𝑝, with 𝐹𝑝𝐶22(𝑗𝜂)𝑄2𝑗𝑄/𝑝𝑐𝑗,𝛾𝑝(Γ),(6.17) with a constant 𝐶2 independent of 𝑗,𝛾, 𝜂, and of the coefficient sequence.

Proof. To prove the assertion, let 𝑊 be a Γ-tile. Then, 𝐹𝑝𝑝=𝛼Γ2𝑗(𝛼𝑊)|||||𝛾𝑐𝑗,𝛾𝑓𝑗,𝛾|||||(𝑥)𝑝𝑑𝑥𝐶𝑝1𝛼Γ2𝑗(𝛼𝑊)|||||𝛾||𝑐𝑗,𝛾||1+2𝜂||2𝑗𝛾1||𝑥(𝑄+1)|||||𝑝𝑑𝑥.(6.18) On each integration patch 2𝑗(𝛼𝑊), the triangle inequality of the quasi-norm yields the estimate 1+2𝜂||2𝑗𝛾1𝛼||𝐶1+2𝜂||2𝑗𝛾1𝑥||,(6.19) compare the proof of Lemma 6.4, and thus the integrand can be estimated from above by the constant |||||𝛾||𝑐𝑗,𝛾||1+2𝜂||2𝑗𝛾1𝛼||(𝑄+1)|||||𝑝(6.20) whence 𝛼Γ2𝑗(𝛼𝑊)|||||𝛾||𝑐𝑗,𝛾||1+2𝜂||2𝑗𝛾1||𝑥(𝑄+1)|||||𝑝𝑑𝑥𝐶𝛼Γ2𝑗𝑄𝛾||𝑐𝑗,𝛾||1+2𝜂||2𝑗𝛾1𝛼||(𝑄+1)𝑝=𝐶𝛼Γ2𝑗𝑄𝛾||𝑐𝑗,𝛾||𝑎𝛼,𝛾𝑝.(6.21) Here 𝑎𝛼,𝛾=(1+2𝜂|2𝑗(𝛾1𝛼)|)(𝑄+1). Now Lemma 6.4 yields that the Schur test is fulfilled for the coefficients {𝑎𝛼,𝛾} with 𝑀=2𝑄(𝑗𝜂) (observe in particular that the right-hand side of the estimate above is independent of 𝑥), thus Lemma 6.3 yields 𝐹𝑝𝐶2𝑗𝑄/𝑝𝛼Γ𝛾||𝑐𝑗,𝛾||𝑎𝛼,𝛾𝑝1/𝑝𝐶22𝑗𝑄/𝑝2(𝑗𝜂)𝑄𝑐𝑗,𝛾𝛾𝑝,(6.22) as desired.

Proof of Theorem 6.1. We still need to prove part (a) of the theorem, and here it is sufficient to show the norm estimate for all finitely supported coefficient sequences {𝑐𝑗,𝛾}𝑗,𝛾. The full statement then follows by completeness of ̇𝐵𝑠𝑝,𝑞 and from the fact that the Kronecker-𝛿s are an unconditional basis of ̇𝑏𝑠𝑝,𝑞 (here we need 𝑝,𝑞<).
Repeated applications of the triangle inequality yield 𝑓̇𝐵𝑠𝑝,𝑞=2𝑙𝑠𝑗,𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝑙𝑞()2𝑙𝑠𝑙1𝑗=𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝑙𝑞()+2𝑙𝑠𝑗=𝑙𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝑙𝑞().(6.23) Pick 𝑁,𝜃 such that 𝑁>𝑄𝑠+1 and 𝜃>𝑠+1. Define 𝑑𝑗=2𝑗(𝑠𝑄/𝑝)𝑐𝑗,𝛾𝑝.(6.24) For 𝑗<𝑙, assumption (6.2) yields ||𝑔𝑗,𝛾𝜓𝑙||(𝑥)𝐶2𝑗𝑄2(𝑙𝑗)𝜃1+2𝑗||2𝑗𝛾1𝑥||𝑄1,(6.25) and thus, by Lemma 6.5, 𝛾Γ𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝐶2𝑗𝑄2(𝑙𝑗)𝜃2𝑗𝑄/𝑝𝑐𝑗,𝛾𝛾𝑝.(6.26) But then 2𝑙𝑠𝑙1𝑗=𝛾Γ𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝐶𝑙1𝑗=2(𝑙𝑗)(𝜃𝑠)2𝑗(𝑠𝑄/𝑝)𝑐𝑗,𝛾𝛾𝑝=𝐶𝑏𝑑𝑗𝑗(𝑙),(6.27) where denotes convolution over , and 𝑏(𝑗)=2𝑗(𝜃𝑠)𝜒(𝑗).(6.28) By choice of 𝜃, 𝑏1(), and Young’s inequality allows to conclude that 2𝑙𝑠𝑙1𝑗=𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝑙𝑞()𝐶𝑑𝑗𝑗𝑞=𝐶𝑐𝑗,𝛾𝑗,𝛾̇𝑏𝑠𝑝,𝑞.(6.29) For 𝑗𝑙, assumption (6.2) provides the estimate ||𝑔𝑗,𝛾𝜓𝑙||(𝑥)𝐶2𝑗𝑄2(𝑙𝑗)𝜃1+2𝑗||2𝑗𝛾1𝑥||𝑄1.(6.30) Here, Lemma 6.5 and straightforward calculation allow to conclude that 2𝑙𝑠𝑙1𝑗=𝛾Γ𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝐶̃𝑏𝑑𝑗𝑗(𝑙),(6.31) with ̃𝑏(𝑗)=2𝑗(𝑠+𝑁𝑄)𝜒(𝑗).(6.32) Hence, Young’s theorem applies again and yields 2𝑙𝑠𝑙1𝑗=𝛾𝑐𝑗,𝛾𝑔𝑗,𝛾𝜓𝑙𝑝𝑙𝑞()𝐶𝑐j,𝛾𝑗,𝛾̇𝑏𝑠𝑝,𝑞,(6.33) and we are done.

We conclude this section by showing that wavelets provide a simultaneous Banach frame for ̇𝐵𝑠𝑝,𝑞, for all 1𝑝,𝑞< and 𝑠; see [24] for an introduction to Banach frames. In the following, we consider the frame operator associated to a regular sampling set Γ, given by𝑆𝜓,Γ(𝑓)=𝑗,𝛾Γ2𝑗𝑄𝑓,𝜓𝑗,𝛾𝜓𝑗,𝛾.(6.34) By Theorems 6.1 and 5.4, 𝑆𝜓,Γ̇𝐵𝑠𝑝,𝑞̇𝐵(𝐺)𝑠𝑝,𝑞(𝐺) is bounded, at least for sufficiently dense sampling sets Γ. Our aim is to show that, for all sufficiently dense regular sampling sets, the operator 𝑆𝜓,Γ is in fact invertible, showing that the wavelet system is a Banach frame for ̇𝐵𝑠𝑝,𝑞(𝐺). The following lemma contains the main technical ingredient for the proof. Once again, we will rely on oscillation estimates.

Lemma 6.6. Let 𝑓=𝑢𝜓𝑗, with 𝑢𝒮(𝐺)/𝒫, such that 𝑓L𝑝(𝐺), for some 1𝑝<. For 𝜖>0, there exists a neighborhood 𝑈 of the identity such that, for all 𝑈-dense regular sampling sets Γ𝐺 and all Γ-tiles 𝑊𝐺, one has 𝑓𝜓𝑗𝛾Γ||𝑊||2𝑗𝑄𝑢,𝜓𝑗,𝛾𝜓𝑗,𝛾𝑝𝜖𝑓𝜓𝑗𝑝.(6.35)

Proof. We first consider the case 𝑗=0. Let 𝑊 denote a Γ-tile. We define the auxiliary function =𝛾Γ𝑓(𝛾)𝐿𝛾𝜒𝛾𝑊=𝑇𝑅Γ𝑓,(6.36) using the notation of the proof of Proposition 5.7. By the triangle inequality, 𝑓𝜓0𝛾Γ||𝑊||𝑢,𝜓0,𝛾𝜓0,𝛾𝑝(𝑓)𝜓0𝑝+𝜓0𝛾Γ||𝑊||𝑢,𝜓0,𝛾𝜓0,𝛾𝑝.(6.37) Now Young’s inequality, together with the proof of Proposition 5.7, implies that for all sufficiently dense Γ, (𝑓)𝜓0𝑝𝑓𝑝𝜓01𝜖𝑓𝑝2.(6.38) For the second term in the right hand side of (6.37), we first observe that 𝑢,𝜓0,𝛾=𝑓(𝛾), and thus using the tiling 𝐺=𝛾Γ𝛾𝑊, |||||𝛾Γ||𝑊||𝑓(𝛾)𝜓0,𝛾𝜓0|||||=|||||(𝑦)𝛾Γ||𝑊||𝛾𝑓(𝛾)𝜓1𝑦𝛾Γ𝛾𝑊𝑓(𝛾)𝜓0𝑥1𝑦|||||=|||||𝑑𝑥𝛾Γ𝛾𝑊𝜓𝑓(𝛾)0𝛾1𝑦𝜓0𝑥1𝑦|||||𝑑𝑥𝛾Γ𝛾𝑊||||||𝜓𝑓(𝛾)0𝛾1𝑦𝜓0𝑥1𝑦||𝑑𝑥.(6.39) Since 𝑥𝛾𝑊 if and only if 𝑦1𝛾𝑦1𝑥𝑊1, it follows that ||𝜓0𝛾1𝑦𝜓0𝑥1𝑦||osc𝑊1𝜓0𝑦1𝑥,(6.40) thus we can continue the estimate by (6.39)𝛾Γ𝛾𝑊||||𝑓(𝛾)osc𝑊1𝜓0𝑦1𝑥=||||𝑑𝑥osc𝑊1𝜓0(𝑦),(6.41) leading to 𝜓0𝛾Γ||𝑊||𝑢,𝜓0,𝛾𝜓0,𝛾𝑝𝑝osc𝑊1𝜓01<𝜖𝑓𝑝2,(6.42) using 𝑝2𝑓𝑝 as well as osc𝑊1(𝜓0)1<𝜖/4, both valid for sufficiently dense Γ, by the proof of Proposition 5.7, and by Lemma 5.9, respectively.
Thus (6.35) is established for 𝑗=0. The statement for general 𝑗 now follows by dilation, similar to the proof of Lemma 5.6. We write 𝑓=𝑢𝜓𝑗=(𝑣𝑗𝜓0)𝛿2𝑗, where 𝑣𝑗=𝑢𝛿2𝑗. Hence, for 𝑔=𝛾Γ||𝑊||2𝑗𝑄𝑢,𝜓𝑗,𝛾𝜓𝑗,𝛾,(6.43) we obtain that 𝑓𝜓𝑗𝑔𝑝=𝑣𝑗𝜓0𝜓0𝛿2𝑗𝑔𝑝=𝑣𝑗𝜓0𝜓0𝑔𝛿2𝑗𝛿2𝑗𝑝=2𝑗𝑄/𝑝𝑣𝑗𝜓0𝜓0𝑔𝛿2𝑗𝑝.(6.44) Now 𝑔𝛿2𝑗=𝛾Γ||𝑊||2𝑗𝑄𝑢,𝜓𝑗,𝛾𝜓𝑗,𝛾𝛿2𝑗=𝛾Γ||𝑊||𝑢𝜓𝑗2𝑗𝛾𝜓0,𝛾=𝛾Γ||𝑊||𝑣𝑗𝜓0(𝛾)𝜓0,𝛾.(6.45) Thus, by the case 𝑗=0, 2𝑗𝑄/𝑝𝑣𝑗𝜓0𝜓0𝑔𝛿2𝑗𝑝=𝜖2𝑗𝑄/𝑝𝑣𝑗𝜓0𝜓0𝑝=𝜖𝑢𝜓𝑗𝑝,(6.46) as desired.

Now, invertibility of the frame operator is easily established. In fact, we can even show the existence of a dual frame and an atomic decomposition for our homogeneous Besov spaces. Note however that the notation of the following theorem is somewhat deceptive. The dual wavelet frame might depend on the space ̇𝐵𝑠𝑝,𝑞, whereas the well-known result for wavelet bases in the Euclidean setting allows to take 𝜓𝑗,𝑘=𝜓𝑗,𝑘, regardless of the Besov space under consideration.

Theorem 6.7 (Atomic decomposition). Let 1𝑝,𝑞<. There exists a neighborhood 𝑈 of the identity such that, for all 𝑈-dense regular sampling sets Γ𝐺, the frame operator 𝑆𝜓,Γ is an automorphism of ̇𝐵𝑠𝑝,𝑞(𝐺).
In this case, there exists a dual wavelet family {𝜓𝑗,𝛾}𝑗,𝛾̇𝐵𝑠𝑝,𝑞, such that for all ̇𝐵𝑓𝑠𝑝,𝑞(𝐺), one has 𝑓=𝑗,𝛾Γ2𝑗𝑄𝑓,𝜓𝑗,𝛾𝜓𝑗,𝛾,(6.47) and in addition 𝑓̇𝐵𝑠𝑝,𝑞𝑗𝛾Γ2𝑗(𝑠𝑄/𝑝)𝑝||𝑓,𝜓𝑗,𝛾||𝑝𝑞/𝑝1/𝑞.(6.48)

Proof. Fix 0<𝜖<1, and choose the neighborhood 𝑈 according to the previous lemma, with 𝜖 replaced by 𝜖0=𝜖(2𝑠𝑞+1+2𝑠𝑞)1/𝑞3(𝑞1)/𝑞𝜓01.(6.49)
Let Γ be a 𝑈-dense regular sampling set, and let 𝑊 denote a Γ-tile. Let 𝑓𝒟, where ̇𝐵𝒟𝑠𝑝,𝑞(𝐺) is the dense subspace of functions for which 𝑓=𝑗𝑓𝜓𝑗𝜓𝑗(6.50) holds with finitely many nonzero terms; see Remark 3.13. For 𝑙, we then obtain from (3.25) that ||𝑊||𝑆𝑓𝜓,Γ𝑓𝑝=||||𝑗𝑙1𝑓𝜓𝑗𝜓𝑗𝛾Γ2𝑗𝑄||𝑊||𝑓,𝜓𝑗,𝛾𝜓𝑗,𝛾𝜓𝑙𝑝||||𝑗𝑙1𝜖0𝑓𝜓𝑗𝑝𝜓01,(6.51) where the inequality used Lemma 6.6 and Young’s inequality. But then it follows that ||𝑊||𝑆𝑓𝜓,Γ𝑓𝑞̇𝐵𝑠𝑝,𝑞=𝑙2𝑙𝑠𝑞||𝑊||𝑆𝑓𝜓,Γ𝑓𝑞𝑝𝑙2𝑙𝑠𝑞||||𝑗𝑙1𝜖0𝑓𝜓𝑗𝑝𝜓01𝑞𝑙2𝑙𝑠𝑞||||𝑗𝑙13𝑞1𝜖𝑞0𝜓0𝑞1𝑓𝜓𝑗𝑞𝑝=𝑗2𝑗𝑠𝑞𝑓𝜓𝑗𝑞𝑝3𝑞1𝜖𝑞0(2𝑠𝑞+1+2𝑠𝑞)=𝜖𝑞𝑓𝑞̇𝐵𝑠𝑝,𝑞.(6.52) Since 𝑆𝜓,Γ is bounded, the estimate extends to all ̇𝐵𝑓𝑠𝑝,𝑞. Therefore, the operator 𝑆Γ,𝜓 is invertible by its Neumann series on ̇𝐵𝑠𝑝,𝑞, for all sufficiently dense quasi-lattices Γ. In this case, we define the dual wavelet frame by 𝑓,𝜓𝑗,𝛾=𝑆1𝜓,Γ(𝑓),𝜓𝑗,𝛾.(6.53)𝜓𝑗,𝛾̇𝐵𝑠𝑝,𝑞, since 𝑆1𝜓,Γ is bounded and 𝜓𝑗,𝛾̇𝐵𝑠𝑝,𝑞.
Let ̇𝐵𝑓𝑠𝑝,𝑞. By Theorem 5.4, 𝑆1𝜓,Γ̇𝐵(𝑓)𝑠𝑝,𝑞 implies {𝑓,𝜓𝑗,𝛾}𝑗,𝛾̇𝑏𝑠𝑝,𝑞. Theorem 6.1 then implies that 𝑓=𝑆𝜓,Γ𝑆1𝜓,Γ=(𝑓)𝑗,𝛾Γ𝑏2𝑗𝑄𝑆1𝜓,Γ(𝑓),𝜓𝑗,𝛾𝜓𝑗,𝛾=𝑗,𝛾Γ2𝑗𝑄𝑓,𝜓𝑗,𝛾𝜓𝑗,𝛾(6.54) with unconditional convergence in the Besov norm. Furthermore, Theorems 6.1 and 5.4 yield that 𝑓̇𝐵𝑠𝑝,𝑞𝑗𝛾2𝑗(𝑠𝑄/𝑝)𝑝||𝑓,𝜓𝑗,𝛾||𝑝𝑞/𝑝1/𝑞𝑆1𝜓,Γ(𝑓)̇𝐵𝑠𝑝,𝑞𝑓̇𝐵𝑠𝑝,𝑞,(6.55) up to constants depending on 𝑝,𝑞,𝑠, but not on 𝑓. This completes the proof.

Remark 6.8. We wish to stress that an appropriate choice of Γ provides a wavelet frame in ̇𝐵𝑠𝑝,𝑞, simultaneously valid for all 1𝑝,𝑞< and all 𝑠. As the discussion in Section 5 shows, the tightness of the oscillation estimates converges to one with increasing density of the quasi-lattices. As a consequence, the tightness of the wavelet frame in ̇𝐵𝑠𝑝,𝑞 converges to one also, at least when measured with respect to the Besov norm from Definition 3.9, applied to the same window 𝜓. However, the tightness will depend on 𝑝,𝑞, and 𝑠.

Remark 6.9. We expect to remove the restriction on 𝑝 and 𝑞 in our future work and prove the existence of (quasi)Banach frame for all homogeneous Besov spaces ̇𝐵𝑠𝑝,𝑞 with 0<𝑝,𝑞 and 𝑠.

Remark 6.10. Our treatment of discretization problems via oscillation estimates is heavily influenced by the work of Feichtinger and Gröchenig on atomic decomposition, in particular the papers [23, 25] on coorbit spaces. A direct application of these results to our problem is difficult, since the representations underlying our wavelet transforms are not irreducible if the group 𝐺 is noncommutative, whereas irreducibility is an underlying assumption in [23, 25]. However, the recent extensions of coorbit theory, most notably [26], provide a unified approach to our results (see [27]).

Acknowledgment

The authors thank the referees for useful comments and additional references.