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Journal of Function Spaces and Applications
Volumeย 2012ย (2012), Article IDย 523586, 41 pages
http://dx.doi.org/10.1155/2012/523586
Research Article

Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

1Lehrstuhl A fรผr Mathematik, RWTH Aachen University, D-52056 Aachen, Germany
2Department of Mathematics and Computer Sciences, City University of New York (CUNY), Queensborough College, 222-05 56th Avenue Bayside, NY 11364, USA

Received 17 January 2012; Accepted 29 January 2012

Academic Editor: Hans G.ย Feichtinger

Copyright ยฉ 2012 Hartmut Fรผhr and Azita Mayeli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ฬ‡๐ต๐‘ ๐‘,๐‘ž 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 ฬ‡๐ต๐‘ ๐‘,๐‘ž with 1โ‰ค๐‘,๐‘ž<โˆž and ๐‘ โˆˆโ„.

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 {๐œ“๐‘—,๐‘˜}๐‘—,๐‘˜โŠ‚๐ฟ2(โ„๐‘›) (consisting of sufficiently regular wavelets with vanishing moments) and ๐‘“โˆˆ๐ฟ2(โ„๐‘›), the expansion๎“๐‘“=๐‘—,๐‘˜๎ซ๐‘“,๐œ“๐‘—,๐‘˜๎ฌ๐œ“๐‘—,๐‘˜(1.1) converges not only in โ€–โ‹…โ€–๐ฟ2, but also in any other Besov space norm โ€–โ‹…โ€–ฬ‡๐ต๐‘ ๐‘,๐‘ž, as soon as ๐‘“ is contained in that space. Furthermore, the latter condition can be read off the decay behaviour of the wavelet coefficients {โŸจ๐‘“,๐œ“๐‘—,๐‘˜โŸฉ}๐‘—,๐‘˜ associated to ๐‘“ 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 [4โ€“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 Follandโ€™s paper [10].

The first wavelet systems on stratified Lie groups (fulfilling certain technical assumptions) were constructed by Lemariรฉ [11], by suitably adapting concepts from spline theory. Lemariรฉ also indicated that the wavelet systems constructed by his approach were indeed unconditional bases of Sakaโ€™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 Lemariรฉโ€™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 [16โ€“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 ๐บ. 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 ๐œ™-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 Calderรณn decomposition. For this purpose, we introduce the coefficient space ฬ‡๐‘๐‘ ๐‘,๐‘ž. The chief result is Theorem 5.4, establishing that the wavelet coefficient sequence of ฬ‡๐ต๐‘“โˆˆ๐‘ ๐‘,๐‘ž lies in ฬ‡๐‘๐‘ ๐‘,๐‘ž. 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 ฮ“, the discrete wavelet system {๐œ“๐‘—,๐›พ}๐‘—โˆˆโ„ค,๐›พโˆˆฮ“ obtained by shifts from ๐›พ and dilations by powers of 2 is a universal Banach frame for all Besov spaces. In other words, the wavelet system allows the decomposition๎“๐‘“=๐‘—,๐›พ๐‘Ÿ๐‘—,๐›พ๐œ“๐‘—,๐›พ(1.2) converging unconditionally in ฬ‡๐ต๐‘ ๐‘,๐‘ž whenever ฬ‡๐ต๐‘“โˆˆ๐‘ ๐‘,๐‘ž, with coefficients {๐‘Ÿ๐‘—,๐›พ}๐‘—,๐›พโˆˆฬ‡๐‘๐‘ ๐‘,๐‘ž depending linearly and boundedly on ๐‘“, and satisfying the norm equivalenceโ€–โ€–๎€ฝ๐‘Ÿ๐‘—,๐›พ๎€พ๐‘—,๐›พโ€–โ€–ฬ‡๐‘๐‘ ๐‘,๐‘žโ‰โ€–๐‘“โ€–ฬ‡๐ต๐‘ ๐‘,๐‘ž.(1.3)

2. Preliminaries and Notation

Following the terminology in [19], we call a Lie group ๐บ stratified if it is connected and simply connected, and its Lie algebra ๐”ค decomposes as a direct sum ๐”ค=๐‘‰1โŠ•โ‹ฏโŠ•๐‘‰๐‘š, with [๐‘‰1,๐‘‰๐‘˜]=๐‘‰๐‘˜+1 for 1โ‰ค๐‘˜<๐‘š and [๐‘‰1,๐‘‰๐‘š]={0}. Then ๐”ค is nilpotent of step ๐‘š and generated as a Lie algebra by ๐‘‰1. Euclidean spaces โ„๐‘› and the Heisenberg group โ„๐‘› are examples of stratified Lie groups.

If ๐บ is stratified, its Lie algebra admits a canonical (natural) family of dilations, namely,๐›ฟ๐‘Ÿ๎€ท๐‘‹1+๐‘‹2+โ‹ฏ+๐‘‹๐‘š๎€ธ=๐‘Ÿ๐‘‹1+๐‘Ÿ2๐‘‹2+โ‹ฏ+๐‘Ÿ๐‘š๐‘‹๐‘š๎€ท๐‘‹๐‘—โˆˆ๐‘‰๐‘—๎€ธ(๐‘Ÿ>0),(2.1) which are Lie algebra automorphisms. We identify ๐บ with ๐”ค through the exponential map. Hence ๐บ is a Lie group with underlying manifold โ„๐‘›, for some ๐‘›, and the group product provided by the Campbell-Baker-Hausdorff formula. The dilations are then also group automorphisms of ๐บ. Instead of writing ๐›ฟ๐‘Ž(๐‘ฅ) for ๐‘ฅโˆˆ๐บ and ๐‘Ž>0, we simply use ๐‘Ž๐‘ฅ, whenever a confusion with the Lie group product is excluded. After choosing a basis of ๐”ค obtained as a union of bases of the ๐‘‰๐‘–, and a possible change of coordinates, one therefore has for ๐‘ฅโˆˆ๐บ and ๐‘Ž>0 that๎€ท๐‘Ž๐‘Ž๐‘ฅ=๐‘‘1๐‘ฅ1,โ€ฆ,๐‘Ž๐‘‘๐‘›๐‘ฅ๐‘›๎€ธ,(2.2) for integers ๐‘‘1โ‰คโ‹ฏโ‰ค๐‘‘๐‘›, according to ๐‘ฅ๐‘–โˆˆ๐‘‰๐‘‘๐‘–.

Under our identification of ๐บ with ๐”ค, polynomials on ๐บ are polynomials on ๐”ค (with respect to any linear coordinate system on the latter). Polynomials on ๐บ are written as๐‘๎ƒฉdim(๐บ)๎“๐‘–=1๐‘ฅ๐‘–๐‘Œ๐‘–๎ƒช=๎“๐ผ๐‘๐ผ๐‘ฅ๐ผ,(2.3) where ๐‘๐ผโˆˆโ„‚ are the coefficients with respect to a suitable basis ๐‘Œ1,๐‘Œ2,โ€ฆ, and ๐‘ฅ๐ผ=๐‘ฅ๐ผ11๐‘ฅ๐ผ22,โ€ฆ,๐‘ฅ๐ผ๐‘›๐‘› the monomials associated to the multi-indices ๐ผโˆˆโ„•{1,โ€ฆ,๐‘›}. For a multi-index ๐ผ, define๐‘‘(๐ผ)=๐‘›๎“๐‘–=1๐ผ๐‘–๐‘›(๐‘–),๐‘›(๐‘–)=๐‘—for๐‘Œ๐‘–โˆˆ๐‘‰๐‘—.(2.4) A polynomial of the type (2.3) is called of homogeneous degree ๐‘˜ if ๐‘‘(๐ผ)โ‰ค๐‘˜ holds, for all multiindices ๐ผ with ๐‘๐ผโ‰ 0. We write ๐’ซ๐‘˜ for the space of polynomials of homogeneous degree ๐‘˜.

We let ๐’ฎ(๐บ) denote the space of Schwartz functions on ๐บ. By definition,๐’ฎ(๐บ)=๐’ฎ(๐”ค). Let ๐’ฎ๎…ž(๐บ) and ๐’ฎ๎…ž(๐บ)/๐’ซ denote the space of distributions and distributions modulo polynomials on ๐บ, respectively. The duality between the spaces is denoted by the map (โ‹…,โ‹…)โˆถ๐’ฎ๎…ž(๐บ)ร—๐’ฎ(๐บ)โ†’โ„‚. Most of the time, however, we will work with the sesquilinear version โŸจ๐‘“,๐‘”โŸฉ=(๐‘“,๐‘”), for ๐‘“โˆˆ๐’ฎ๎…ž(๐บ) and ๐‘”โˆˆ๐’ฎ(๐บ).

Left Haar measure on ๐บ is induced by Lebesgue measure on its Lie algebra, and it is also right-invariant. The number โˆ‘๐‘„=๐‘š1๐‘—(dim๐‘‰๐‘—) will be called the homogeneous dimension of ๐บ. (For instance, for ๐บ=โ„๐‘› and โ„๐‘› we have ๐‘„=๐‘› and ๐‘„=2๐‘›+2, respectively.) For any function ๐œ™ on ๐บ and ๐‘Ž>0, the ๐ฟ1-normalized dilation of ๐œ™ is defined by๐ท๐‘Ž๐œ™(๐‘ฅ)=๐‘Ž๐‘„๐œ™(๐‘Ž๐‘ฅ).(2.5) Observe that this action preserves the ๐ฟ1-norm, that is, โ€–๐ท๐‘Ž๐œ™โ€–1=โ€–๐œ™โ€–. We fix a homogeneous quasi-norm |โ‹…| on ๐บ which is smooth away from 0 with, |๐‘Ž๐‘ฅ|=๐‘Ž|๐‘ฅ| for all ๐‘ฅโˆˆ๐บ, ๐‘Žโ‰ฅ0, |๐‘ฅโˆ’1|=|๐‘ฅ| for all ๐‘ฅโˆˆ๐บ, with |๐‘ฅ|>0 if ๐‘ฅโ‰ 0, and fulfilling a triangle inequality |๐‘ฅ๐‘ฆ|โ‰ค๐ถ(|๐‘ฅ|+|๐‘ฆ|), with constant ๐ถ>0. Confer [19] for the construction of homogeneous norms, as well as further properties.

Moreover, by [19, Propositionโ€‰โ€‰1.15], for any ๐‘Ÿ>0, there is a finite ๐ถ๐‘Ÿ>0 such that โˆซ|๐‘ฅ|>๐‘…|๐‘ฅ|โˆ’๐‘„โˆ’๐‘Ÿ๐‘‘๐‘ฅ=๐ถ๐‘Ÿ๐‘…โˆ’๐‘Ÿ for all ๐‘…>0.

Our conventions for left-invariant operators on ๐บ are as follows. We let ๐‘Œ1,โ€ฆ,๐‘Œ๐‘› denote a basis of ๐”ค, obtained as a union of bases of the ๐‘‰๐‘–. In particular, ๐‘Œ1,โ€ฆ,๐‘Œ๐‘™, for ๐‘™=dim(๐‘‰1), is a basis of ๐‘‰1. Elements of the Lie algebra are identified in the usual manner with left-invariant differential operators on ๐บ. Given a multi-index ๐ผโˆˆโ„•๐‘›0, we write ๐‘Œ๐ผ for ๐‘Œ๐ผ11โˆ˜โ‹ฏโˆ˜๐‘Œ๐ผ๐‘›๐‘›. A convenient characterization of Schwartz functions in terms of left-invariant operators states that ๐‘“โˆˆ๐’ฎ(๐บ) if and only if, for all ๐‘โˆˆโ„•, |๐‘“|๐‘<โˆž, where||๐‘“||๐‘=sup||๐ผ||โ‰ค๐‘,๐‘ฅโˆˆ๐บ(1+|๐‘ฅ|)๐‘||๐‘Œ๐ผ||.๐‘“(๐‘ฅ)(2.6) In addition, the norms |โ‹…|๐‘ induce the topology of ๐’ฎ(๐บ) (see [19]).

The sub-Laplacian operator on ๐บ can be viewed as the analog of the Laplacian operator on โ„๐‘› defined by โˆ‘๐ฟ=โˆ’๐‘›๐‘–=1๐œ•2/๐œ•๐‘ฅ2๐‘˜. Using the above conventions for the choice of basis ๐‘Œ1,โ€ฆ,๐‘Œ๐‘› and ๐‘™=dim(๐‘‰1), the sub-Laplacian is defined as โˆ‘๐ฟ=โˆ’๐‘™๐‘–=1๐‘Œ2๐‘–. 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 ๐‘‡ on ๐บ is called homogenous of degree ๐‘™ if ๐‘‡(๐‘“โˆ˜๐›ฟ๐‘Ž)=๐‘Ž๐‘™(๐‘‡๐‘“)โˆ˜๐›ฟ๐‘Ž for any ๐‘“ on ๐บ. By choice of the ๐‘Œ๐‘– for ๐‘–โ‰ค๐‘™, these operators are homogeneous of degree 1; it follows that ๐ฟ is homogenous of degree 2, and ๐ฟ๐‘˜ is homogenous of degree 2๐‘˜. Furthermore, any operator of the form ๐‘Œ๐ผ is homogeneous of degree ๐‘‘(๐ผ).

When restricted to ๐ถโˆž๐‘, ๐ฟ is formally self-adjoint: for any ๐‘“,๐‘”โˆˆ๐ถโˆž๐‘(๐บ), โŸจ๐ฟ๐‘“,๐‘”โŸฉ=โŸจ๐‘“,๐ฟ๐‘”โŸฉ. (For more see [15].) Its closure has domain ๐’Ÿ={๐‘ขโˆˆ๐ฟ2(๐บ)โˆถ๐ฟ๐‘ขโˆˆ๐ฟ2(๐บ)}, where we take ๐ฟ๐‘ข 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 ๐ฟโˆฃ๐ถโˆž๐‘; we denote this extension also by the symbol ๐ฟ.

Suppose that ๐ฟ has spectral resolution๎€œ๐ฟ=โˆž0๐œ†๐‘‘๐‘ƒ๐œ†,(2.7) where ๐‘‘๐‘ƒ๐œ† is the projection measure. For a bounded Borel function ๎๐‘“ on [0,โˆž), the operator๎๎€œ๐‘“(๐ฟ)=โˆž0๎๐‘“(๐œ†)๐‘‘๐‘ƒ๐œ†(2.8) is a bounded integral operator on ๐ฟ2(๐บ) with a convolution distribution kernel in ๐ฟ2(๐บ) denoted by ๐‘“, and๎๐‘“(๐ฟ)๐œ‚=๐œ‚โˆ—๐‘“โˆ€๐œ‚โˆˆ๐’ฎ(๐บ).(2.9) An important fact to be used later on is that for rapidly decaying smooth functions, ๐‘“โˆˆ๐’ฎ(โ„+), the kernel associated to ๎๐‘“(๐ฟ) is a Schwartz function. For a function ๐‘“ on ๐บ we define ๎‚๐‘“(๐‘ฅ)=๐‘“(๐‘ฅโˆ’1) and ๐‘“โˆ—=๎‚๐‘“. For ๐‘“โˆˆ๐ฟ2(๐บ)โˆฉ๐ฟ1(๐บ), the adjoint of the convolution operator ๐‘”โ†ฆ๐‘”โˆ—๐‘“ is provided by ๐‘”โ†ฆ๐‘”โˆ—๐‘“โˆ—.

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 ๐‘ข as๎“๐‘ข=๐‘—โˆˆโ„ค๐‘ขโˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—,(3.1)

where ๐œ“๐‘— is a dilated copy of a suitably chosen Schwartz function ๐œ“. In the Euclidean setting, it is customary to construct ๐œ“ 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 ๐ฟ, see Lemma 3.7. However, this approach raises the question to what extent the construction depends on the choice of ๐ฟ. It turns out that the precise choice of sub-Laplacian obtained from a basis of ๐‘‰1 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 ๐‘โˆˆโ„•. A function ๐‘“โˆถ๐บโ†’โ„‚ has polynomial decay order ๐‘ if there exists a constant ๐ถ>0 such that, for all ๐‘ฅโˆˆ๐บ, ||||๐‘“(๐‘ฅ)โ‰ค๐ถ(1+|๐‘ฅ|)โˆ’๐‘.(3.2)๐‘“ has vanishing moments of order ๐‘, if one has โˆ€๐‘โˆˆ๐’ซ๐‘โˆ’1โˆถ๎€œ๐บ๐‘“(๐‘ฅ)๐‘(๐‘ฅ)๐‘‘๐‘ฅ=0,(3.3) with absolute convergence of the integral.

Under our identification of ๐บ with ๐”ค, the inversion map ๐‘ฅโ†ฆ๐‘ฅโˆ’1 is identical to the additive inversion map. That is, ๐‘ฅโˆ’1=โˆ’๐‘ฅ, and it follows that ฬƒ๐‘โˆˆ๐’ซ๐‘ for all ๐‘โˆˆ๐’ซ๐‘. Thus, if ๐‘“ has vanishing moments of order ๐‘, then for all ๐‘โˆˆ๐’ซ๐‘โˆ’1๎€œ๐บ๎‚๎€œ๐‘“(๐‘ฅ)๐‘(๐‘ฅ)๐‘‘๐‘ฅ=๐บ๐‘“(๐‘ฅ)ฬƒ๐‘(๐‘ฅ)๐‘‘๐‘ฅ=0,(3.4) that is, ๎‚๐‘“ has vanishing moments of order ๐‘ as well.

Vanishing moments are central to most estimates in wavelet analysis, by the following principle: in a convolution product of the type ๐‘”โˆ—๐ท๐‘ก๐‘“, vanishing moments of one factor together with smoothness of the other result in decay. Later on, we will apply the lemma to Schwartz functions ๐‘“,๐‘”, where only the vanishing moment assumptions are nontrivial. The more general version given here is included for reference.

Lemma 3.2. Let ๐‘,๐‘˜โˆˆโ„• be arbitrary.
(a)Let ๐‘“โˆˆ๐ถ๐‘˜, such that ๐‘Œ๐ผ(๐‘“) is of decay order ๐‘, for all ๐ผ with ๐‘‘(๐ผ)โ‰ค๐‘˜. Let ๐‘” have vanishing moments of order ๐‘˜ and decay order ๐‘+๐‘˜+๐‘„+1. Then there exists a constant, depending only on the decay of ๐‘Œ๐ผ(๐‘“) and ๐‘”, such that ||๎€ท๐ทโˆ€๐‘ฅโˆˆ๐บโˆ€0<๐‘ก<1โˆถ๐‘”โˆ—๐‘ก๐‘“๎€ธ(||๐‘ฅ)โ‰ค๐ถ๐‘ก๐‘˜+๐‘„(1+|๐‘ก๐‘ฅ|)โˆ’๐‘.(3.5) In particular, if ๐‘>๐‘„/๐‘, โ€–โ€–๎€ท๐ทโˆ€๐‘ฅโˆˆ๐บโˆ€0<๐‘ก<1โˆถ๐‘”โˆ—๐‘ก๐‘“๎€ธโ€–โ€–๐‘โ‰ค๐ถโ€ฒ๐‘ก๐‘˜+๐‘„(1โˆ’1/๐‘).(3.6)(b)Now suppose that ๐‘”โˆˆ๐ถ๐‘˜, with ๐‘Œ๐ผ(ฬƒ๐‘”) of decay order ๐‘ for all ๐ผ with ๐‘‘(๐ผ)โ‰ค๐‘˜. Let ๐‘“ have vanishing moments of order ๐‘˜ and decay order ๐‘+๐‘˜+๐‘„+1. Then there exists a constant, depending only on the decay of ๐‘“ and ๐‘Œ๐ผ(ฬƒ๐‘”), such that ||๎€ท๐ทโˆ€๐‘ฅโˆˆ๐บโˆ€1<๐‘ก<โˆžโˆถ๐‘”โˆ—๐‘ก๐‘“๎€ธ(||๐‘ฅ)โ‰ค๐ถ๐‘กโˆ’๐‘˜(1+|๐‘ฅ|)โˆ’๐‘.(3.7) In particular, if ๐‘>๐‘„/๐‘, โ€–โ€–๎€ท๐ทโˆ€๐‘ฅโˆˆ๐บโˆ€1<๐‘ก<โˆžโˆถ๐‘”โˆ—๐‘ก๐‘“๎€ธโ€–โ€–๐‘โ‰ค๐ถโ€ฒ๐‘กโˆ’๐‘˜.(3.8)

Proof. First, let us prove (a). Let 0<๐‘ก<1. For ๐‘ฅโˆˆ๐บ, let ๐‘ƒ๐‘˜๐‘ฅ,๐ท๐‘ก๎‚๐‘“ denote the left Taylor polynomial of ๐ท๐‘ก๎‚๐‘“ with homogeneous degree ๐‘˜โˆ’1, see [19, Definitionโ€‰โ€‰1.44]. By that result, ||||๐ท๐‘ก๐‘“๎€ท๐‘ฆโˆ’1๐‘ฅ๎€ธโˆ’๐‘ƒ๐‘˜๐‘ฅ,๐ท๐‘ก๎‚๐‘“||||(๐‘ฆ)โ‰ค๐ถ๐‘˜||๐‘ฆ||๐‘˜sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ||,๐‘‘(๐ผ)=๐‘˜|||๐‘Œ๐ผ๎‚€๐ท๐‘ก๎‚๐‘“๎‚|||,(๐‘ฅ๐‘ง)(3.9) with suitable positive constants ๐ถ๐‘˜ and ๐‘. We next use the homogeneity properties of the partial derivatives [19, page 21], together with the decay condition on ๐‘Œ๐ผ๐‘“ to estimate for ๐ผ with ๐‘‘(๐ผ)=๐‘˜sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ|||||๐‘Œ๐ผ๎‚€๐ท๐‘ก๎‚๐‘“๎‚|||(๐‘ฅ๐‘ง)=๐‘ก๐‘˜sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ|||||๐ท๐‘ก๎‚€๐‘Œ๐ผ๎‚๐‘“๎‚|||(๐‘ฅ๐‘ง)=๐‘ก๐‘˜+๐‘„sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ|||||๎‚€๐‘Œ๐ผ๎‚๐‘“๎‚|||(๐‘ก(๐‘ฅโ‹…๐‘ง))โ‰ค๐‘ก๐‘˜+๐‘„sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ||๐ถ๐‘“๎€ท||||๎€ธ1+๐‘ก(๐‘ฅโ‹…๐‘ง)โˆ’๐‘โ‰ค๐‘ก๐‘˜+๐‘„sup|๐‘ง|โ‰ค๐‘๐‘˜||๐‘ฆ||๐ถ๐‘“(1+|๐‘ก๐‘ฅ|)โˆ’๐‘(1+|๐‘ก๐‘ง|)๐‘โ‰ค๐‘ก๐‘˜+๐‘„(1+๐‘)๐‘˜๐‘๐ถ๐‘“(1+|๐‘ก๐‘ฅ|)โˆ’๐‘๎€ท||๐‘ฆ||๎€ธ1+๐‘,(3.10) where the penultimate inequality used [19, 1.10], and the final estimate used |๐‘ก๐‘ฆ|=๐‘ก|๐‘ฆ|โ‰ค|๐‘ฆ|. Thus, ||||๐ท๐‘ก๐‘“๎€ท๐‘ฆโˆ’1๐‘ฅ๎€ธโˆ’๐‘ƒ๐‘˜๐‘ฅ,๐ท๐‘ก๎‚๐‘“||||โ‰ค๎‚๐ถ(๐‘ฆ)๐‘˜๐‘ก๐‘˜+๐‘„๎€ท||๐‘ฆ||๎€ธ1+๐‘+๐‘˜(1+|๐‘ก๐‘ฅ|)โˆ’๐‘.(3.11) Next, using vanishing moments of ๐‘”, ||๎€ท๐‘”โˆ—๐ท๐‘ก๐‘“๎€ธ||โ‰ค๎€œ(๐‘ฅ)๐บ||||||||๐ท๐‘”(๐‘ฆ)๐‘ก๐‘“๎€ท๐‘ฆโˆ’1๐‘ฅ๎€ธโˆ’๐‘ƒ๐‘˜๐‘ฅ,๐ท๐‘ก๎‚๐‘“||||โ‰ค๎‚๐ถ(๐‘ฆ)๐‘‘๐‘ฆ๐‘˜(1+|๐‘ก๐‘ฅ|)โˆ’๐‘๐‘ก๐‘˜+๐‘„๎€œ๐บ||๐‘”||๎€ท||๐‘ฆ||๎€ธ(๐‘ฆ)1+๐‘+๐‘˜โ‰ค๎‚๐ถ๐‘‘๐‘ฆ๐‘˜(1+|๐‘ก๐‘ฅ|)โˆ’๐‘๐‘ก๐‘˜+๐‘„๎€œ๐บ๐ถ๐‘”๎€ท||๐‘ฆ||๎€ธ1+โˆ’๐‘„โˆ’1๐‘‘๐‘ฆ,(3.12) and the integral is finite by [19, 1.15]. This proves (3.5), and (3.6) follows by โ€–โ€–๐‘”โˆ—๐ท๐‘ก๐‘“โ€–โ€–๐‘โ‰ค๐ถโ€ฒ๐‘ก๐‘˜+๐‘„๎‚ต๎€œ๐บ(1+|๐‘ก๐‘ฅ|)โˆ’๐‘๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘โ‰ค๐ถ๎…ž๎…ž๐‘ก๐‘˜+๐‘„โˆ’๐‘„/๐‘,(3.13) using ๐‘๐‘>๐‘„.
For part (b), we first observe that ๎€ท๐‘”โˆ—๐ท๐‘ก๐‘“๎€ธ(๐‘ฅ)=๐‘ก๐‘„๎‚€๎‚๐‘“โˆ—๐ท๐‘กโˆ’1๎‚ฬƒ๐‘”(๐‘กโ‹…๐‘ฅ).(3.14) Our assumptions on ๐‘“,๐‘” allow to invoke part (a) with ๎‚๐‘“ฬƒ๐‘”, replacing ๐‘“,๐‘”, and (3.7) follows immediately. (3.8) is obtained from this by straightforward integration.

We let ๐’ต(๐บ) denote the space of Schwartz functions with all moments vanishing. We next consider properties of ๐’ต(๐บ) as a subspace of ๐’ฎ(๐บ) with the relative topology.

Lemma 3.3. ๐’ต(๐บ) is a closed subspace (in particular complete) of ๐’ฎ(๐บ), with ๐’ฎ(๐บ)โˆ—๐’ต(๐บ)โŠ‚๐’ต(๐บ), as well as ๎‚๐‘“โˆˆ๐’ต(๐บ) for all ๐‘“โˆˆ๐’ต(๐บ). The topological dual of ๐’ต(๐บ), ๐’ต๎…ž(๐บ), can be canonically identified with the factor space ๐’ฎ๎…ž(๐บ)/๐’ซ.

Proof. By definition, ๐’ต(๐บ) is the intersection of kernels of a family of tempered distributions, hence a closed subspace. For ๐‘โˆˆ๐’ซ and ๐‘“โˆˆ๐’ต(๐บ), one has by unimodularity of ๐บ that ๎‚โŸจ๐‘,๐‘“โŸฉ=โŸจฬƒ๐‘,๐‘“โŸฉ=0, since ฬƒ๐‘ is a polynomial. But then, for any ๐‘”โˆˆ๐’ฎ(๐บ) and ๐‘“โˆˆ๐’ต(๐บ), one has for all polynomials ๐‘ on ๐บ that ๎‚ฌ๎‚๐‘“๎‚ญโŸจ๐‘”โˆ—๐‘“,๐‘โŸฉ=๐‘”,๐‘โˆ—=โŸจ๐‘”,0โŸฉ=0,(3.15) since ๐‘“โˆˆ๐’ต(๐บ) implies ๎‚๐‘โˆ—๐‘“=0 (translation on ๐บ is polynomial). Thus ๐‘”โˆ—๐‘“โˆˆ๐’ต(๐บ). All further properties of ๐’ต(๐บ) follow from the corresponding statements concerning ๐’ต(โ„๐‘›). For identification of ๐’ต๎…ž(โ„๐‘›) with the quotient space ๐’ฎ(โ„๐‘›)โ€ฒ/๐’ซ, we first observe that a tempered distribution ๐œ‘ vanishes on ๐’ต(โ„๐‘›) if and only if its (Euclidean) Fourier transform is supported in {0}, which is well known to be the case if and only if ๐œ‘ is a polynomial. Using this observation, we map ๐‘ขโˆˆ๐’ต๎…ž(โ„๐‘›) to ฬƒ๐‘ข+๐’ซ, where ฬƒ๐‘ข is a continuous extension of ๐‘ข to all of ๐’ฎ(โ„๐‘›); such an extension exists by the Hahn-Banach theorem. The map is well defined because the difference between two extensions of ๐‘ข annihilates ๐’ต(โ„๐‘›) and hence is a polynomial. Linearity follows from well-definedness. Furthermore, the inverse of the mapping is clearly obtained by assigning ๐‘ค+๐’ซ to the restriction ๐‘คโˆฃ๐’ต(๐บ).

In the following, we will usually not explicitly distinguish between ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ) and its equivalence class modulo polynomials, and we will occasionally write ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ. The topology of ๐’ฎ๎…ž(๐บ)/๐’ซ is just the topology of pointwise convergence on the elements of ๐’ต(๐บ). For any net (๐‘ข๐‘—+๐’ซ)๐‘—โˆˆ๐ผ, ๐‘ข๐‘—+๐’ซโ†’๐‘ข+๐’ซ holds if and only if โŸจ๐‘ข๐‘—,๐œ‘โŸฉโ†’โŸจ๐‘ข,๐œ‘โŸฉ, for all ๐œ‘โˆˆ๐’ต(๐บ). We next study convolution on ๐’ฎ๎…ž(๐บ)/๐’ซ.

Lemma 3.4. For every ๐œ“โˆˆ๐’ฎ(๐บ), the map ๐‘ขโ†ฆ๐‘ขโˆ—๐œ“ is a well-defined and continuous operator ๐’ฎ๎…ž(๐บ)/๐’ซโ†’๐’ฎ๎…ž(๐บ)/๐’ซ. If ๐œ“โˆˆ๐’ต(๐บ), the associated convolution operator is a well-defined and continuous operator ๐’ฎ๎…ž(๐บ)/๐’ซโ†’๐’ฎ๎…ž(๐บ).

Proof. Note that ๐’ซโˆ—๐’ฎ(๐บ)โŠ‚๐’ซ. Hence ๐‘ขโ†ฆ๐‘ขโˆ—๐œ“ induces a well-defined canonical map ๐’ฎ๎…ž(๐บ)/๐’ซโ†’๐’ฎ๎…ž(๐บ)/๐’ซ. Furthermore, ๐‘ขโ†ฆ๐‘ขโˆ—๐œ“ is continuous on ๐’ฎ๎…ž(๐บ), as a consequence of [19, Propositionโ€‰โ€‰1.47]. Therefore, for any net ๐‘ข๐‘—โ†’๐‘ข and any ๐œ‘โˆˆ๐’ต(๐บ), the fact that ๐œ‘โˆ—๐œ“โˆ—โˆˆ๐’ต(๐บ) allows to write ๎ซ๐‘ข๐‘—๎ฌ=๎ซ๐‘ขโˆ—๐œ“,๐œ‘๐‘—,๐œ‘โˆ—๐œ“โˆ—๎ฌโŸถโŸจ๐‘ข,๐œ‘โˆ—๐œ“โˆ—โŸฉ=โŸจ๐‘ขโˆ—๐œ“,๐œ‘โŸฉ,(3.16) showing ๐‘ข๐‘—โˆ—๐œ“โ†’๐‘ขโˆ—๐œ“ in ๐’ฎ๎…ž(๐บ)/๐’ซ.
For ๐œ“โˆˆ๐’ต(๐บ), the fact that ๐’ซโˆ—๐œ“={0} makes the mapping ๐‘ขโ†ฆ๐‘ขโˆ—๐œ“โˆˆ๐’ฎ๎…ž(๐บ) well-defined modulo polynomials. The continuity statement is proved by (3.16), with assumptions on ๐œ“ and ๐œ‘ switched.

The definition of homogeneous Besov spaces requires taking ๐ฟ๐‘-norms of elements of ๐’ฎ๎…ž(๐บ)/๐’ซ. The following remark clarifies this.

Remark 3.5. Throughout this paper, we use the canonical embedding ๐ฟ๐‘(๐บ)โŠ‚๐’ฎ๎…ž(๐บ). For ๐‘<โˆž, this gives rise to an embedding ๐ฟ๐‘(๐บ)โŠ‚๐’ฎ๎…ž(๐บ)/๐’ซ, using that ๐’ซโˆฉ๐ฟ๐‘(๐บ)={0}. Consequently, given ๐‘ขโˆˆ๐’ฎโ€ฒ(๐บ)/๐’ซ, we let โ€–๐‘ขโ€–๐‘=โ€–๐‘ข+๐‘žโ€–๐‘whenever๐‘ข+๐‘žโˆˆ๐ฟ๐‘(๐บ),forsuitable๐‘žโˆˆ๐’ซ(3.17) assigning the value โˆž otherwise. Here the fact that ๐’ซโˆฉ๐ฟ๐‘(๐บ)={0} guarantees that the decomposition is unique, and thus (3.17) well-defined.
By contrast, โ€–โ‹…โ€–โˆž can only be defined on ๐’ฎ๎…ž(๐บ), if we assign the value โˆž to ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)โงต๐ฟโˆž(๐บ).
Note that with these definitions, the Hausdorff-Young inequality โ€–๐‘ขโˆ—๐‘“โ€–๐‘โ‰คโ€–๐‘ขโ€–๐‘โ€–๐‘“โ€–1 remains valid for all ๐‘“โˆˆ๐’ฎ(๐บ), and all ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ (for ๐‘<โˆž), respectively, ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ) (for ๐‘=โˆž). For ๐‘=โˆž, this is clear. For ๐‘<โˆž, note that if ๐‘ข+๐‘žโˆˆ๐ฟ๐‘(๐บ), then (๐‘ข+๐‘ž)โˆ—๐œ“=๐‘ขโˆ—๐œ“+๐‘žโˆ—๐œ“โˆˆ๐ฟ๐‘(๐บ) with ๐‘žโˆ—๐œ“โˆˆ๐’ซ.

We now introduce a general Littlewood-Paley-type decomposition. For this purpose we define for ๐œ“โˆˆ๐’ฎ(๐บ),๐œ“๐‘—=๐ท2๐‘—๐œ“.(3.18)

Definition 3.6. A function ๐œ“โˆˆ๐’ฎ(๐บ) is called LP-admissible if for all ๐‘”โˆˆ๐’ต(๐บ), ๐‘”=lim๐‘โ†’โˆž๎“||๐‘—||โ‰ค๐‘๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—(3.19) holds, with convergence in the Schwartz space topology. Duality entails the convergence ๐‘ข=lim๐‘โ†’โˆž๎“||๐‘—||โ‰ค๐‘๐‘ขโˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—(3.20) for all ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ.

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

Lemma 3.7. Let ๎๐œ™ be a function in ๐ถโˆž with support in [0,4] such that ๎0โ‰ค๐œ™โ‰ค1 and ๎๐œ™โ‰ก1 on [0,1/4]. Take ๎”๎๐œ“(๐œ‰)=๎๐œ™(2โˆ’2๎๐œ‰)โˆ’๐œ™(๐œ‰). Thus, ๎๐œ“โˆˆ๐ถโˆž๐‘(โ„+), with support in the interval [1/4,4], and ๎“1=๐‘—โˆˆโ„ค||๎€ท2๎๐œ“2๐‘—๐œ‰๎€ธ||2๐‘Ž.๐‘’.(3.21) Pick a sub-Laplacian ๐ฟ, and let ๐œ“ denote the convolution kernel associated to the bounded left-invariant operator ๎๐œ“(๐ฟ). Then ๐œ“ is LP-admissible, with ๐œ“โˆˆ๐’ต(๐บ).

Proof. Let us first comment on the properties of ๐œ“ that are immediate from the construction via spectral calculus: ๐œ“โˆˆ๐’ฎ(๐บ) follows from [20] and vanishing moments by [15, Propositionโ€‰โ€‰1].
Now let ๐‘”โˆˆ๐’ต(๐บ). First note that 2-homogeneity of ๐ฟ implies that the convolution kernel associated to ๎๐œ“(2โˆ’2๐‘—โ‹…)(๐ฟ) coincides with ๐œ“๐‘—. Then, by the spectral theorem and (3.21), ๎“๐‘”=๐‘—โˆˆโ„ค๎€บ๎€ท2๎๐œ“โˆ’2๐‘—โ‹…๎€ธ(๎€ป๐ฟ)โˆ—โˆ˜๎€บ๎€ท2๎๐œ“โˆ’2๐‘—โ‹…๎€ธ(๎€ป๎“๐ฟ)๐‘”=๐‘—โˆˆโ„ค๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—(3.22) holds in ๐ฟ2-norm.
For any positive integer ๐‘, ๎“||๐‘—||โ‰ค๐‘๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—=๐‘”โˆ—๐ท2๐‘+1๐œ™โˆ’๐‘”โˆ—๐ท2โˆ’๐‘๐œ™,(3.23) where ๐œ™โˆˆ๐’ฎ(๐บ) is the convolution kernel of ๎๐œ™(๐ฟ). Since ๐œ™ is a Schwartz function, it follows by [19, Proposition (1.49)] that ๐‘”โˆ—๐ท2๐‘+1๐œ™โ†’๐‘๐œ™๐‘”, for ๐‘โ†’โˆž, for all ๐‘”โˆˆ๐’ฎ(๐บ), with convergence in ๐’ฎ(๐บ) and a suitable constant ๐‘๐œ™.
We next show that ๐‘”โˆ—๐ท๐‘ก๐‘“โ†’0 in ๐’ฎ(๐บ), as ๐‘กโ†’0, for any ๐‘“โˆˆ๐’ฎ(๐บ). Fix a multi-index ๐ผ and ๐‘,๐‘˜โˆˆโ„• with ๐‘˜โ‰ฅ๐‘. Then left-invariance and homogeneity of ๐‘Œ๐ผ yield ||๐‘Œ๐ผ๎€ท๐‘”โˆ—๐ท๐‘ก๐‘“๎€ธ||(๐‘ฅ)=๐‘ก๐‘‘(๐ผ)||๐‘”โˆ—๐ท๐‘ก๎€ท๐‘Œ๐ผ๐‘“๎€ธ||(๐‘ฅ)โ‰ค๐ถ๐‘“,๐‘”๐‘ก๐‘˜+๐‘„+๐‘‘(๐ผ)(1+|๐‘ก๐‘ฅ|)โˆ’๐‘โ‰ค๐ถ๐‘“,๐‘”๐‘ก๐‘˜+๐‘„+๐‘‘(๐ผ)โˆ’๐‘(1+|๐‘ฅ|)โˆ’๐‘.(3.24) Here the first inequality is an application of (3.5); the constant ๐ถ๐‘“,๐‘” can be estimated in terms of |๐‘“|๐‘€,|๐‘”|๐‘€, for ๐‘€ sufficiently large. But this proves ๐‘”โˆ—๐ท๐‘ก๐‘“โ†’0 in the Schwartz topology.
Summarizing, โˆ‘|๐‘—|โ‰ค๐‘๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—โ†’๐‘๐œ™๐‘” in ๐’ฎ(๐บ), and in addition by (3.22), โˆ‘|๐‘—|โ‰ค๐‘๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—โ†’๐‘” in ๐ฟ2, whence ๐‘๐œ™=1 follows.

Note that an LP-admissible function ๐œ“ as constructed in Lemma 3.7 fulfills the convenient relation||||โˆ€๐‘—,๐‘™โˆˆโ„คโˆถ๐‘—โˆ’๐‘™>1โŸน๐œ“โˆ—๐‘—โˆ—๐œ“๐‘™=0,(3.25) which follows from [๎๐œ“(2โˆ’2๐‘—โ‹…)(๐ฟ)]โˆ˜[๎๐œ“(2โˆ’2๐‘™โ‹…)(๐ฟ)]=0.

Remark 3.8. By spectral calculus, we find that ๐œ“=๐ฟ๐‘˜๐‘”๐‘˜, with ๐‘”๐‘˜โˆˆ๐’ต(๐บ). In particular, the decomposition ๐‘“=lim๐‘โ†’โˆž๎“|๐‘—|โ‰ค๐‘๐‘“โˆ—๐œ“โˆ—๐‘—โˆ—๐ท2๐‘—๐ฟ๐‘˜๎€ท๐‘”๐‘˜๎€ธ=lim๐‘โ†’โˆž๐ฟ๐‘˜โŽ›โŽœโŽœโŽ๎“||๐‘—||โ‰ค๐‘๐‘“โˆ—๐œ“โˆ—๐‘—โˆ—2โˆ’๐‘˜๐‘—๐ท2๐‘—๐‘”๐‘˜โŽžโŽŸโŽŸโŽ (3.26) shows that ๐ฟ๐‘˜(๐’ต(๐บ))โŠ‚๐’ต(๐บ) is dense.

We now associate a scale of homogeneous Besov spaces to the function ๐œ“.

Definition 3.9. Let ๐œ“โˆˆ๐’ต(๐บ) be LP-admissible, let 1โ‰ค๐‘โ‰คโˆž, 1โ‰ค๐‘žโ‰คโˆž, and ๐‘ โˆˆโ„. The homogeneous Besov space associated to ๐œ“ is defined asฬ‡๐ต๐‘ ,๐œ“๐‘,๐‘ž=๎‚ป๐‘ขโˆˆ๐’ฎ๎…ž๎‚†2(๐บ)/๐’ซโˆถ๐‘—๐‘ โ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘๎‚‡๐‘—โˆˆโ„คโˆˆโ„“๐‘ž๎‚ผ(โ„ค),(3.27) with associated norm โ€–๐‘ขโ€–ฬ‡๐ต๐‘ ,๐œ“๐‘,๐‘ž=โ€–โ€–โ€–๎‚†2๐‘—๐‘ โ€–๐‘ขโˆ—๐œ“โˆ—๐‘—โ€–๐‘๎‚‡๐‘—โˆˆโ„คโ€–โ€–โ€–โ„“๐‘ž(โ„ค).(3.28)

Remark 3.10. The definition relies on the conventions regarding ๐ฟ๐‘-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 ๐œ“1,๐œ“2โˆˆ๐’ต(๐บ) be LP-admissible. Let ๐‘ โˆˆโ„ andโ€‰ 1โ‰ค๐‘,๐‘žโ‰คโˆž. Then, ฬ‡๐ต๐‘ ,๐œ“1๐‘,๐‘ž=ฬ‡๐ต๐‘ ,๐œ“2๐‘,๐‘ž, with equivalent norms.

Proof. It is sufficient to prove the norm equivalence, and here symmetry with respect to ๐œ“1 and ๐œ“2 immediately reduces the proof to showing, for a suitable constant ๐ถ>0, โˆ€๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซโˆถโ€–๐‘ขโ€–ฬ‡๐ต1๐‘ ,๐œ“๐‘,๐‘žโ‰ค๐ถโ€–๐‘ขโ€–ฬ‡๐ต2๐‘ ,๐œ“๐‘,๐‘ž,(3.29) in the extended sense that the left-hand side is finite whenever the right-hand side is. Hence assume that ฬ‡๐ต๐‘ขโˆˆ๐‘ ,๐œ“2๐‘,๐‘ž; otherwise, there is nothing to show. In the following, let ๐œ“๐‘–,๐‘—=๐ท2๐‘—๐œ“๐‘– (๐‘–=1,2).
By LP-admissibility of ๐œ“2, ๐‘ข=lim๐‘โ†’โˆž๎“||๐‘—||โ‰ค๐‘๐‘ขโˆ—๐œ“โˆ—2,๐‘—โˆ—๐œ“2,๐‘—,(3.30) with convergence in ๐’ฎโ€ฒ(๐บ)/๐’ซ. Accordingly, ๐‘ขโˆ—๐œ“โˆ—1,โ„“=lim๐‘โ†’โˆž๎“||๐‘—||โ‰ค๐‘๐‘ขโˆ—๐œ“โˆ—2,๐‘—โˆ—๐œ“2,๐‘—โˆ—๐œ“โˆ—1,โ„“,(3.31) where the convergence on the right-hand side holds in ๐’ฎ๎…ž(๐บ), by Lemma 3.4. We next show that the right-hand side also converges in ๐ฟ๐‘. For this purpose, we observe that โ€–โ€–๐œ“2,๐‘—โˆ—๐œ“โˆ—1,โ„“โ€–โ€–1=โ€–โ€–๐ท2๐‘—๎€ท๐œ“2โˆ—๐ท2โ„“โˆ’๐‘—๐œ“11โˆ—๎€ธโ€–โ€–1=โ€–โ€–๐œ“2โˆ—๐ท2โ„“โˆ’๐‘—๐œ“1โˆ—โ€–โ€–1โ‰ค๐ถ2โˆ’|โ„“โˆ’๐‘—|๐‘˜,(3.32) where ๐‘˜>๐‘  is a fixed integer. For โ„“โˆ’๐‘—โ‰ฅ0, this follows directly from (3.8), using ๐œ“1,๐œ“2โˆˆ๐’ฎ(๐บ), and vanishing moments of ๐œ“1, whereas for โ„“โˆ’๐‘—<0, the vanishing moments of ๐œ“2 allow to apply (3.6).
Using Youngโ€™s inequality, we estimate with ๐ถ from above that ๎“๐‘—โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—2,๐‘—โˆ—๐œ“2,๐‘—โˆ—๐œ“โˆ—1,โ„“โ€–โ€–๐‘โ‰ค๎“๐‘—โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—2,๐‘—โ€–โ€–๐‘โ€–โ€–๐œ“2,๐‘—โˆ—๐œ“โˆ—1,โ„“โ€–โ€–1โ€–โ€–โ‰ค๐ถ๐‘ขโˆ—๐œ“โˆ—2,๐‘—โ€–โ€–๐‘2โˆ’|๐‘—โˆ’โ„“|๐‘˜๎“(3.33)โ‰ค๐ถ๐‘—โˆˆโ„ค2๐‘—๐‘ โ€–โ€–๐‘ขโˆ—๐œ“โˆ—2,๐‘—โ€–โ€–๐‘2โˆ’|๐‘—โˆ’โ„“|๐‘˜โˆ’๐‘—๐‘ .(3.34) Next observe that 2โˆ’|๐‘—โˆ’โ„“|๐‘˜โˆ’๐‘—๐‘ =2โˆ’โ„“๐‘ โ‹…๎‚ป2โˆ’|๐‘—โˆ’โ„“|(๐‘˜+๐‘ )2๐‘—โ‰ฅโ„“โˆ’|๐‘—โˆ’โ„“|(๐‘˜โˆ’๐‘ )๐‘—<๐‘™โ‰ค2โˆ’โ„“๐‘ 2โˆ’|๐‘—โˆ’โ„“|(๐‘˜โˆ’|๐‘ |).(3.35) By assumption, the sequence (2๐‘—๐‘ โ€–๐‘ขโˆ—๐œ“โˆ—๐‘—,2โ€–๐‘)๐‘—โˆˆโ„ค is in โ„“๐‘ž, in particular, bounded. Therefore, ๐‘˜โˆ’|๐‘ |>0 yields that (3.34) converges. But then the right-hand side of (3.31) converges unconditionally with respect to โ€–โ‹…โ€–๐‘. This limit coincides with the ๐’ฎ๎…ž(๐บ)/๐’ซ-limit ๐‘ขโˆ—๐œ“โˆ—1,โ„“ (which because of ๐œ“โˆ—1,โ„“โˆˆ๐’ต(๐บ) is even a ๐’ฎ๎…ž(๐บ)-limit), yielding ๐‘ขโˆ—๐œ“โˆ—1,โ„“โˆˆ๐ฟ๐‘(๐บ), with 2โ„“๐‘ โ€–โ€–๐‘ขโˆ—๐œ“โˆ—1,โ„“โ€–โ€–๐‘โ‰ค2โ„“๐‘ ๎“๐‘—โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—2,๐‘—โˆ—๐œ“2,๐‘—โˆ—๐œ“โˆ—1,โ„“โ€–โ€–๐‘โ‰ค๐ถ32โ„“๐‘ ๎“๐‘—โˆˆโ„ค2๐‘—๐‘ โ€–โ€–๐‘ขโˆ—๐œ“โˆ—2,๐‘—โ€–โ€–๐‘2โˆ’|๐‘—โˆ’โ„“|(๐‘˜โˆ’|๐‘ |).(3.36) Now an application of Youngโ€™s inequality for convolution over โ„ค, again using ๐‘˜โˆ’|๐‘ |>0, provides (3.29).

As a consequence, we write ฬ‡๐ต๐‘ ๐‘,๐‘ž=ฬ‡๐ต๐‘ ,๐œ“๐‘,๐‘ž, for any LP-admissible ๐œ“โˆˆ๐’ต(๐บ). These spaces coincide with the homogeneous Besov spaces for the Heisenberg group in [9], and with the usual definitions in the case ๐บ=โ„๐‘›.

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 1โ‰ค๐‘,๐‘žโ‰คโˆž and all ๐‘ โˆˆโ„, one has continuous inclusion maps ฬ‡๐ต๐’ต(๐บ)โ†ช๐‘ ๐‘,๐‘žโ†ช๐‘†๎…ž(๐บ)/๐’ซ, as well as ฬ‡๐ต๐’ต(๐บ)โ†ช๐‘ โˆ—๐‘,๐‘ž, where the latter denotes the dual of ฬ‡๐ต๐‘ ๐‘,๐‘ž. For ๐‘,๐‘ž<โˆž, ฬ‡๐ต๐’ต(๐บ)โŠ‚๐‘ ๐‘,๐‘ž is dense.

Proof. We pick ๐œ“ as in Lemma 3.7 and define ฮ”๐‘—๐‘”=๐‘”โˆ—๐œ“โˆ—๐‘— for ๐‘”โˆˆ๐’ฎ๎…ž(๐บ). For the inclusion ฬ‡๐ต๐’ต(๐บ)โŠ‚๐‘ ๐‘,๐‘ž, note that (3.6) and (3.8) allow to estimate for all ๐‘”โˆˆ๐’ต(๐บ) and ๐‘˜โˆˆโ„• that โ€–โ€–ฮ”๐‘—๐‘”โ€–โ€–๐‘โ‰ค๐ถ๐‘˜2โˆ’|๐‘—|๐‘˜.(3.37) Here the constant ๐ถ๐‘˜ is a suitable multiple of |๐‘”|๐‘€, for ๐‘€=๐‘€(๐‘˜) sufficiently large. But this implies that ฬ‡๐ต๐’ต(๐บ)โŠ‚๐‘ ๐‘,๐‘ž continuously.
For the other embedding, repeated applications of Hรถlderโ€™s inequality yield the estimate ||||=|||||๎“โŸจ๐‘“,๐‘”โŸฉ๐‘—โˆˆโ„ค๎ซ๐‘“,๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—๎ฌ|||||โ‰ค๎“๐‘—โˆˆโ„ค||๎ซ๐‘“โˆ—๐œ“โˆ—๐‘—,๐‘”โˆ—๐œ“โˆ—๐‘—๎ฌ||โ‰ค๎“๐‘—โˆˆโ„คโ€–โ€–๐‘“โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘๎…žโ€–โ€–๐‘”โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘=๎“๐‘—โˆˆโ„ค๎‚€2โˆ’๐‘—๐‘ โ€–โ€–๐‘“โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘โ€ฒ2๎‚๎‚€๐‘—๐‘ โ€–โ€–๐‘“โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘๎‚โ‰คโ€–๐‘“โ€–โˆ’๐‘ ๐‘๎…ž,๐‘ž๎…žโ€–๐‘”โ€–๐‘ ๐‘,๐‘ž(3.38) valid for all ฬ‡๐ต๐‘“โˆˆ๐’ต(๐บ)โŠ‚โˆ’๐‘ ๐‘๎…ž,๐‘ž๎…ž and ฬ‡๐ต๐‘”โˆˆ๐‘ ๐‘,๐‘ž. Here ๐‘โ€ฒ,๐‘žโ€ฒ are the conjugate exponents of ๐‘,๐‘ž, respectively. But this estimate implies continuity of the embeddings ฬ‡๐ต๐‘ ๐‘,๐‘žโŠ‚๐’ฎ๎…ž(๐บ)/๐’ซ and ฬ‡๐ต๐’ต(๐บ)โŠ‚๐‘ โˆ—๐‘,๐‘ž.
For the density statement, let ฬ‡๐ต๐‘ขโˆˆ๐‘ ๐‘,๐‘ž, and ๐œ–>0. For convenience, we pick ๐œ“ according to Lemma 3.7. Since ๐‘ž<โˆž, there exists ๐‘โˆˆโ„• such that ๎“||๐‘—||>๐‘โˆ’12๐‘—๐‘ ๐‘žโ€–โ€–ฮ”๐‘—๐‘ขโ€–โ€–๐‘ž๐‘<๐œ–.(3.39) Next define ๐พ๐‘=๎“||๐‘—||โ‰ค๐‘๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—=๐ท2๐‘+1๐œ™โˆ’๐ท2โˆ’๐‘๐œ™.(3.40)
Let ๐‘ค=๐‘ขโˆ—๐พ๐‘. By assumption on ๐‘ข and Youngโ€™s inequality, ๐‘คโˆˆ๐ฟ๐‘(๐บ), and since ๐‘<โˆž, there exists ๐‘”โˆˆ๐’ฎ(๐บ) with โ€–๐‘คโˆ’๐‘”โ€–๐‘<๐œ–1/๐‘ž. Let ๐‘“=๐‘”โˆ—๐พ๐‘, then ๐‘“โˆˆ๐’ต(๐บ), and for ๐‘—โˆˆโ„ค, โ€–โ€–ฮ”๐‘—(โ€–โ€–๐‘ขโˆ’๐‘“)๐‘=โ€–โ€–(๐‘ขโˆ’๐‘“)โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘โ‰คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘—โˆ’๐‘ขโˆ—๐พ๐‘โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘+โ€–โ€–๐‘คโˆ—๐œ“โˆ—๐‘—โˆ’๐‘”โˆ—๐พ๐‘โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘.(3.41) For |๐‘—|โ‰ค๐‘โˆ’1, the construction of ๐œ“๐‘— and ๐พ๐‘ implies that ๐พ๐‘โˆ—๐œ“โˆ—๐‘—=๐œ“โˆ—๐‘—, whereas for |๐‘—|>๐‘+1, one has ๐พ๐‘โˆ—๐œ“โˆ—๐‘—=0. As a consequence, one finds for |๐‘—|<๐‘โˆ’1โ€–โ€–ฮ”๐‘—โ€–โ€–(๐‘ขโˆ’๐‘“)๐‘โ‰คโ€–๐‘คโˆ’๐‘”โ€–๐‘โ€–โ€–๐œ“โˆ—๐‘—โ€–โ€–1=โ€–๐‘คโˆ’๐‘”โ€–๐‘โ€–๐œ“โ€–1<๐œ–1/๐‘žโ€–๐œ“โ€–1,(3.42) and for|๐‘—|>๐‘+1โ€–โ€–ฮ”๐‘—โ€–โ€–(๐‘ขโˆ’๐‘“)๐‘โ‰คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘<๐œ–1/๐‘ž.(3.43) For ||๐‘—|โˆ’๐‘|โ‰ค1, one finds โ€–โ€–ฮ”๐‘—โ€–โ€–(๐‘ขโˆ’๐‘“)๐‘โ‰ค๐ถ๐œ–1/๐‘ž(3.44) with some constant ๐ถ>0 depending only on ๐œ“. For instance, for ๐‘—=๐‘, โ€–โ€–ฮ”๐‘—โ€–โ€–(๐‘ขโˆ’๐‘“)๐‘โ‰คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘๎€ท๐œ“โˆ’๐‘ขโˆ—โˆ—๐‘โˆ’1โˆ—๐œ“๐‘โˆ’1+๐œ“โˆ—๐‘โˆ—๐œ“๐‘๎€ธโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘+โ€–โ€–๐‘คโˆ—๐œ“โˆ—๐‘๎€ท๐œ“โˆ’๐‘”โˆ—โˆ—๐‘โˆ’1โˆ—๐œ“๐‘โˆ’1+๐œ“โˆ—๐‘โˆ—๐œ“๐‘๎€ธโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘.(3.45) A straight forward application of triangle and Youngโ€™s inequality yields โ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘๎€ท๐œ“โˆ’๐‘ขโˆ—โˆ—๐‘โˆ’1โˆ—๐œ“๐‘โˆ’1+๐œ“โˆ—๐‘โˆ—๐œ“๐‘๎€ธโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘โ‰คโ€–โ€–๐‘ขโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘๎€ท1+2โ€–๐œ“โˆ—โˆ—๐œ“โ€–1๎€ธ<๐œ–1/๐‘ž๎€ท1+2โ€–๐œ“โˆ—โˆ—๐œ“โ€–1๎€ธ.(3.46) Similar considerations applied to ๐‘ค=๐‘ขโˆ—๐พ๐‘ yield โ€–โ€–๐‘คโˆ—๐œ“โˆ—๐‘๎€ท๐œ“โˆ’๐‘”โˆ—โˆ—๐‘โˆ’1โˆ—๐œ“๐‘โˆ’1+๐œ“โˆ—๐‘โˆ—๐œ“๐‘๎€ธโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘โ€–โ€–โ‰ค2๐‘ขโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘โ€–๐œ“โˆ—โˆ—๐œ“โ€–1โ€–โ€–+2๐‘”โˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘โ€–๐œ“โˆ—โˆ—๐œ“โ€–1โ‰ค2๐œ–1/๐‘žโ€–๐œ“โˆ—โˆ—๐œ“โ€–1๎‚€โ€–โ€–+2๐‘คโˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘+โ€–โ€–(๐‘คโˆ’๐‘”)โˆ—๐œ“โˆ—๐‘โ€–โ€–๐‘๎‚โ€–๐œ“โˆ—โˆ—๐œ“โ€–1โ‰ค๎€ท4โ€–๐œ“โˆ—โˆ—๐œ“โ€–1+โ€–๐œ“โˆ—โˆ—๐œ“โ€–1โ€–๐œ“โ€–1๎€ธ๐œ–1/๐‘ž.(3.47) Now summation over ๐‘— yields โ€–๐‘ขโˆ’๐‘“โ€–ฬ‡๐ต๐‘ ๐‘,๐‘žโ‰ค๐ถโ€ฒ๐œ–,(3.48) as desired.

Remark 3.13. Let ๐œ“be as in Lemma 3.7. As a byproduct of the proof, we note that the space ๎€ฝ๐’Ÿ=๐‘“โˆ—๐พ๐‘๎€พโˆถ๐‘“โˆˆ๐’ฎ(๐บ),๐‘โˆˆโ„•(3.49) is dense in ๐’ต(๐บ) as well as ฬ‡๐ต๐‘ ๐‘,๐‘ž, if ๐‘,๐‘ž<โˆž. In ๐’Ÿ, the decomposition ๎“๐‘”=๐‘—โˆˆโ„ค๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—(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 1โ‰ค๐‘,๐‘ž<โˆž, and let ๐œ“โˆˆ๐’ต(๐บ) be an LP-admissible vector constructed via Lemma 3.7. Then the decomposition (3.19) converges for all ฬ‡๐ต๐‘”โˆˆ๐‘ ๐‘,๐‘ž in the Besov space norm.

Proof. Consider the operators ฮฃ๐‘โˆถฬ‡๐ต๐‘ ๐‘,๐‘žโ†’ฬ‡๐ต๐‘ ๐‘,๐‘ž, ฮฃ๐‘๎“๐‘”=||๐‘—||โ‰ค๐‘๐‘”โˆ—๐œ“โˆ—๐‘—โˆ—๐œ“๐‘—.(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 (ฮฃ๐‘)๐‘โˆˆโ„• is bounded in the operator norm. As noted in Remark 3.13, the ฮฃ๐‘ 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 ๐ฟ๐‘.

Proposition 3.15. Let 1<๐‘<โˆž, and let ๐œ“โˆˆ๐’ต(๐บ) be an LP-admissible vector constructed via Lemma 3.7. Then the decomposition (3.19) converges with respect to โ€–โ‹…โ€–๐‘, for all ๐‘”โˆˆ๐ฟ๐‘(๐บ).

Proof. Let the operator family (ฮฃ๐‘)๐‘โˆˆโ„• be defined as in the previous proof. Then, ฮฃ๐‘๐‘“=๐‘”โˆ—๐ท2๐‘+1๐œ™โˆ’๐‘”โˆ—๐ท2โˆ’๐‘๐œ™, and Youngโ€™s inequality implies that the sequence of operators is norm-bounded. It therefore suffices to prove the desired convergence on the dense subspace ๐’ฎ(๐บ). By [19, Propositionโ€‰โ€‰1.49],๐‘”โˆ—๐ท2๐‘+1๐œ™โ†’๐‘๐œ™๐‘”. Furthermore, for ๐‘โˆˆโ„•, ๎€ท๐‘”โˆ—๐ท2โˆ’๐‘๐œ™๎€ธ(๐‘ฅ)=2โˆ’๐‘๐‘„๎€œ๐บ๎€ท2๐‘”(๐‘ฆ)๐œ™โˆ’๐‘๎€ท๐‘ฆโˆ’1๐‘ฅ=๎€œ๎€ธ๎€ธ๐‘‘๐‘ฆ๐บ๐‘”๎€ท2๐‘๐‘ฆ๎€ธ๐œ™๎€ท๐‘ฆโˆ’1โ‹…2โˆ’๐‘๐‘ฅ๎€ธ๐‘‘๐‘ฆ=2โˆ’๐‘๐‘„๎€ท๐ท2๐‘2๐‘”โˆ—๐œ™๎€ธ๎€ทโˆ’๐‘๐‘ฅ๎€ธ,(3.52) and thus โ€–โ€–๐‘”โˆ—๐ท2โˆ’๐‘๐‘„๐œ™โ€–โ€–๐‘=2โˆ’๐‘๐‘„๎‚ต๎€œ๐บ||๎€ท๐ท2๐‘2๐‘”โˆ—๐œ™๎€ธ๎€ทโˆ’๐‘๐‘ฅ๎€ธ||๐‘๎‚ถ๐‘‘๐‘ฅ1/๐‘=2โˆ’๐‘๐‘„+๐‘๐‘„/๐‘โ€–โ€–๐ท2๐‘โ€–โ€–๐‘”โˆ—๐œ™๐‘.(3.53) Again by [19, Propositionโ€‰โ€‰1.49],(๐ท2๐‘๐‘”โˆ—๐œ™)โ†’๐‘๐‘”๐œ™, in particular, 2โˆ’๐‘๐‘„+๐‘๐‘„/๐‘โ€–โ€–๐ท2๐‘โ€–โ€–๐‘”โˆ—๐œ™๐‘โŸถ0as๐‘โŸถโˆž.(3.54) Hence, ฮฃ๐‘๐‘”โ†’๐‘๐œ™๐‘”, and the case ๐‘=2 yields ๐‘๐œ™=1.

Theorem 3.16. ฬ‡๐ต๐‘ ๐‘,๐‘ž is a Banach space.

Proof. Completeness is the only issue here. Again, we pick ๐œ“โˆˆ๐’ต(๐บ) as an LP-admissible vector via Lemma 3.7. Suppose that {๐‘ข๐‘›}๐‘›โˆˆโ„•โŠ‚ฬ‡๐ต๐‘ ๐‘,๐‘ž is a Cauchy sequence. As a consequence, one has in particular, for all ๐‘—โˆˆโ„ค, that {๐‘ข๐‘›โˆ—๐œ“โˆ—๐‘—}๐‘›โˆˆโ„•โŠ‚๐ฟ๐‘(๐บ) is a Cauchy sequence, hence ๐‘ข๐‘›โˆ—๐œ“โˆ—๐‘—โ†’๐‘ฃ๐‘—, for a suitable ๐‘ฃ๐‘—โˆˆ๐ฟ๐‘(๐บ). Furthermore, the Cauchy property of {๐‘ข๐‘›}๐‘›โˆˆโ„•โŠ‚ฬ‡๐ต๐‘ ๐‘,๐‘ž implies that ๎‚ป๎‚†2๐‘—๐‘ โ€–โ€–๐‘ข๐‘›โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘๎‚‡๐‘—โˆˆโ„ค๎‚ผ๐‘›โˆˆโ„•โŠ‚โ„“๐‘ž(โ„ค)(3.55) is a Cauchy sequence. On the other hand, the sequence converges pointwise to {2๐‘—๐‘ โ€–๐‘ฃ๐‘—โ€–๐‘}๐‘—, whence ๎“๐‘—โˆˆโ„ค2๐‘—๐‘ ๐‘žโ€–โ€–๐‘ฃ๐‘—โ€–โ€–๐‘ž๐‘<โˆž.(3.56) We define ๐‘ข=lim๐‘€โ†’โˆž๎“||๐‘—||โ‰ค๐‘€๐‘ฃ๐‘—โˆ—๐œ“๐‘—.(3.57) Now, using (3.56) and ฬ‡๐ต๐’ต(๐บ)โŠ‚โˆ’๐‘ ๐‘๎…ž,๐‘ž๎…ž, where ๐‘โ€ฒ,๐‘žโ€ฒ are the conjugate exponents of ๐‘,๐‘ž, respectively, a straightforward calculation as in the proof of Lemma 3.12 shows that the sum defining ๐‘ข converges in ๐’ฎ๎…ž(๐บ)/๐’ซ. Furthermore, (3.56) and (3.25) easily imply that ฬ‡๐ต๐‘ขโˆˆ๐‘ ๐‘,๐‘ž. Finally, for the proof of ๐‘ข๐‘›โ†’๐‘ข, we employ (3.25) together with the equality ๐œ“โˆ—๐‘—=โˆ‘|๐‘™โˆ’๐‘—|โ‰ค1๐œ“โˆ—๐‘™โˆ—๐œ“๐‘™โˆ—๐œ“โˆ—๐‘—, to show that โ€–โ€–๎€ท๐‘ข๐‘›๎€ธโˆ’๐‘ขโˆ—๐œ“๐‘—โ€–โ€–๐‘=โ€–โ€–โ€–โ€–๐‘ข๐‘›โˆ—๐œ“๐‘—โˆ’๎“||||๐‘™โˆ’๐‘—โ‰ค1๐‘ฃ๐‘™โˆ—๐œ“๐‘™โˆ—๐œ“โˆ—๐‘—โ€–โ€–โ€–โ€–๐‘โ‰ค๎“||||๐‘™โˆ’๐‘—โ‰ค1โ€–โ€–๎€ท๐‘ข๐‘›โˆ—๐œ“โˆ—๐‘™โˆ’๐‘ฃ๐‘™๎€ธโˆ—๐œ“๐‘™โˆ—๐œ“โˆ—๐‘—โ€–โ€–๐‘โ‰ค๎“||||๐‘™โˆ’๐‘—โ‰ค1โ€–โ€–๐‘ข๐‘›โˆ—๐œ“โˆ—๐‘™โˆ’๐‘ฃ๐‘™โ€–โ€–๐‘โ€–โ€–๐œ“๐‘™โˆ—๐œ“โˆ—๐‘—โ€–โ€–1โŸถ0,as๐‘›โŸถโˆž.(3.58) Summarizing, the sequence {{2๐‘—๐‘ โ€–(๐‘ข๐‘›โˆ’๐‘ข)โˆ—๐œ“โˆ—๐‘—โ€–๐‘}๐‘—โˆˆโ„ค}๐‘›โˆˆโ„•โˆˆโ„“๐‘ž(โ„•) is a Cauchy sequence, converging pointwise to 0. But then โ€–๐‘ข๐‘›โˆ’๐‘ขโ€–ฬ‡๐ต๐‘ ๐‘,๐‘žโ†’0 follows.

4. Characterization via Continuous Wavelet Transform

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

Definition 4.1. ๐œ“โˆˆ๐’ฎ(๐บ) is called ๐’ต-admissible, if for all ๐‘“โˆˆ๐’ต(๐บ), ๐‘“=lim๐œ–โ†’0,๐ดโ†’โˆž๎€œ๐ด๐œ–๐‘“โˆ—๐ท๐‘Ž๎€ท๐œ“โˆ—๎€ธโˆ—๐œ“๐‘‘๐‘Ž๐‘Ž(4.1) holds with convergence in the Schwartz topology.

The next theorem reveals a large class of ๐’ต-admissible wavelets. In fact, all the wavelets studied in [15] are also ๐’ต-admissible in the sense considered here. Its proof is an adaptation of the argument showing [15, Theoremโ€‰โ€‰1].

We let๐‘†๎€ทโ„+๎€ธ=๎‚ป๐‘“โˆˆ๐ถโˆž(0,โˆž)โˆถโˆ€๐‘˜โˆˆโ„•0,๐‘“(๐‘˜)decreasesrapidly,lim๐œ‰๐‘“(๐‘˜)๎‚ผ(๐œ‰)exists.(4.2)

Theorem 4.2. Let ๎โ„Žโˆˆ๐’ฎ(โ„+), and let ๐œ“ be the distribution kernel associated to the operator ๐ฟ๎โ„Ž(๐ฟ). Then ๐œ“ is ๐’ต-admissible up to normalization.

Proof. The main idea of the proof is to write, for ๐‘“โˆˆ๐’ต(๐บ), ๎€œ๐ด๐œ–๐‘“โˆ—๐ท๐‘Ž๎€ท๐œ“โˆ—๎€ธโˆ—๐œ“๐‘‘๐‘Ž๐‘Ž๎€œ=๐‘“โˆ—๐ด๐œ–๐ท๐‘Ž๎€ท๐œ“โˆ—๎€ธโˆ—๐œ“๐‘‘๐‘Ž๐‘Ž=๐‘“โˆ—๐ท๐ด๐‘”โˆ’๐‘“โˆ—๐ท๐œ–๐‘”,(4.3) with suitable ๐‘”โˆˆ๐’ฎ(๐บ). Once this is established, ๐‘“โˆ—๐ท๐ด๐‘”โ†’๐‘๐‘”๐‘“ for ๐ดโ†’โˆž follows by [19, Proposition (1.49)], with convergence in the Schwartz topology. Moreover, ๐‘“โˆˆ๐’ต(๐บ) entails that ๐‘“โˆ—๐ท๐œ–๐‘”โ†’0 in the Schwartz topology: given any ๐‘>0 and ๐ผโˆˆโ„•๐‘›0 with associated left-invariant differential operator ๐‘Œ๐ผ, we can employ (3.5) to estimate sup๐‘ฅโˆˆ๐บ(1+|๐‘ฅ|)๐‘||๎€ท๐‘Œ๐ผ๐‘“โˆ—๐ท๐œ–๐‘”๎€ธ||(๐‘ฅ)=sup๐‘ฅโˆˆ๐บ(1+|๐‘ฅ|)๐‘๐œ–๐‘„+๐‘‘(๐ผ)||๐‘“โˆ—๐ท๐œ–๎€ท๐‘Œ๐ผ๐‘”๎€ธ||(๐‘ฅ)โ‰ค๐ถsup๐‘ฅโˆˆ๐บ(1+|๐‘ฅ|)๐‘๐œ–๐‘„+๐‘‘(๐ผ)+๐‘˜(1+|๐œ–๐‘ฅ|)โˆ’๐‘€โ‰ค๐ถsup๐‘ฅโˆˆ๐บ(1+|๐‘ฅ|)๐‘โˆ’๐‘€๐œ–๐‘„+๐‘‘(๐ผ)+๐‘˜โˆ’๐‘€,(4.4) which converges to zero for ๐œ–โ†’0, as soon as ๐‘€โ‰ฅ๐‘ and ๐‘˜>๐‘€โˆ’๐‘„โˆ’๐‘‘(๐ผ). But this implies ๐‘“โˆ—๐ท๐œ–๐‘”โ†’0 in ๐’ฎ(๐บ), by [19].
Thus it remains to construct ๐‘”. To this end, define 1ฬ‚๐‘”(๐œ‰)=โˆ’2๎€œโˆž๐œ‰๐‘Ž|||๎โ„Ž๎€ท๐‘Ž2๎€ธ|||2๐‘‘๐‘Ž,(4.5) which is clearly in ๐’ฎ(โ„+), and let ๐‘” denote the associated convolution kernel of ฬ‚๐‘”(๐ฟ). By the definition, ๐‘”โˆˆ๐’ฎ(๐บ). Let ๐œ‘1,๐œ‘2 be in ๐’ฎ(๐บ), and let ๐‘‘๐œ†๐œ‘1,๐œ‘2 denote the scalar-valued Borel measure associated to ๐œ‘1,๐œ‘2 by the spectral measure. Then, by spectral calculus and the invariance properties of ๐‘‘๐‘Ž/๐‘Ž, ๎ƒก๎€œ๐ด๐œ–๐œ‘1โˆ—๐ท๐‘Ž๎€ท๐œ“โˆ—๎€ธ๐‘“โˆ—๐œ“๐‘‘๐‘Ž๐‘Ž,๐œ‘2๎ƒข=๎€œโˆž0๎€œ๐ด๐œ–๎€ท๐‘Ž2๐œ‰๎€ธ2|||๎โ„Ž๎€ท๐‘Ž2๐œ‰๎€ธ|||2๐‘‘๐‘Ž๐‘Ž๐‘‘๐œ†๐œ‘1,๐œ‘2=1(๐œ‰)2๎€œโˆž0๎€œ๐ด2๐œ‰๐œ–2๐œ‰๐‘Ž|||๎โ„Ž๎€ท๐‘Ž2๐œ‰๎€ธ|||2๐‘‘๐‘Ž๐‘‘๐œ†๐œ‘1,๐œ‘2(=๎€œ๐œ‰)โˆž0๎€ท๐ดฬ‚๐‘”2๐œ‰๎€ธ๎€ท๐œ–โˆ’ฬ‚๐‘”2๐œ‰๎€ธ๐‘‘๐œ†๐œ‘1,๐œ‘2=๎ซ๐œ‘(๐œ‰)1โˆ—๎€ท๐ท๐ด๐‘”โˆ’๐ท๐œ–๐‘”๎€ธ,๐œ‘2๎ฌ,(4.6) as desired.

Hence, by [15, Corollaryโ€‰โ€‰1] we have the following.

Corollary 4.3. (a) There exist ๐’ต-admissible ๐œ“โˆˆ๐’ต(๐บ).
(b) There exist ๐’ต-admissible ๐œ“โˆˆ๐ถโˆž๐‘(๐บ) with vanishing moments of arbitrary finite order.

Given a tempered distribution ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ and a ๐’ต(๐บ)-admissible function ๐œ“, the continuous wavelet transform of ๐‘ข is the family (๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—)๐‘Ž>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 โ„Ž๐‘ก(๐‘ฅ)=๐ท๐‘กโ„Ž(๐‘ฅ), where โ„Ž is the kernel of ๎โ„Ž(๐ฟ) with ๎โ„Ž(๐œ‰)=๐‘’โˆ’๐œ‰. Theorem 4.2 implies that ๐œ“=๐ฟ๐‘˜โ„Ž is ๐’ต-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 ๐‘“โˆˆ๐’ฎ๎…ž(๐บ) associated to ๐œ“ is then very closely related to the ๐‘˜-fold time derivative of the solution to the heat equation with initial condition ๐‘“. By choice of โ„Ž,๐‘ข๎€ท(๐‘ฅ,๐‘ก)=๐‘“โˆ—๐ท๐‘กโ„Ž๎€ธ(๐‘ฅ)(4.7) denotes the solution of the heat equation associated to ๐ฟ, with initial condition ๐‘“. A formal calculation using left invariance of ๐ฟ then yields๐œ•๐‘˜๐‘ก๐‘ข=๐ฟ๐‘˜๎€ท๐‘“โˆ—๐ท๐‘กโ„Ž๎€ธ=๐‘“โˆ—๐ฟ๐‘˜๎€ท๐ท๐‘กโ„Ž๎€ธ=๐‘ก2๐‘˜๐‘“โˆ—๐ท๐‘ก๐œ“โˆ—.(4.8) Thus the following theorem also implies a characterization of Besov spaces in terms of the heat semigroup.

Theorem 4.4. Let ๐œ“โˆˆ๐’ฎ(๐บ) be ๐’ต-admissible, with vanishing moments of order ๐‘˜. Then, for all ๐‘ โˆˆโ„ with |๐‘ |<๐‘˜, and all 1โ‰ค๐‘<โˆž, 1โ‰ค๐‘žโ‰คโˆž, the following norm equivalence holds: โˆ€๐‘ขโˆˆ๐‘†๎…ž(๐บ)/๐‘โ€–๐‘ขโ€–ฬ‡๐ต๐‘ ๐‘,๐‘žโ‰โ€–โ€–๐‘ŽโŸผ๐‘Ž๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ฟ๐‘ž(โ„+;๐‘‘๐‘Ž/๐‘Ž).(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 ๐œ“โˆˆ๐’ต(๐บ), the equivalence is also valid for the case ๐‘=โˆž.

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 ๐ถ denotes a constant that may change from line to line, but in a way that is independent of ๐‘ขโˆˆ๐‘†๎…ž(๐บ).
Let us first assume that ๎€œโ„๐‘Ž๐‘ ๐‘žโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘ž๐‘๐‘‘๐‘Ž๐‘Ž<โˆž,(4.10) for ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ, 1โ‰ค๐‘,๐‘žโ‰คโˆž, for a ๐’ต-admissible function ๐œ“โˆˆ๐‘†(๐บ) with ๐‘˜๐œ“>|๐‘ | vanishing moments (๐œ“โˆˆ๐’ต(๐บ), if ๐‘=โˆž). Let ๐œ‘โˆˆ๐’ต(๐บ) be LP-admissible. Then, for all ๐‘—โˆˆโ„ค, ๐‘ขโˆ—๐œ‘โˆ—๐‘—=lim๐œ–โ†’0,๐ดโ†’โˆž๎€œ๐ด๐œ–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โˆ—๐ท๐‘Ž๐œ“โˆ—๐œ‘โˆ—๐‘—๐‘‘๐‘Ž๐‘Ž(4.11) holds in ๐’ฎ๎…ž(๐บ), by Lemma 3.4.
We next prove that the right-hand side of (4.11) converges in ๐ฟ๐‘. For this purpose, introduce ๐‘๐‘—=๎€œโˆž0โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โˆ—๐ท๐‘Ž๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–๐‘๐‘‘๐‘Ž๐‘Ž.(4.12) We estimate ๐‘๐‘—โ‰ค๎€œโˆž0โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๐‘‘๐‘Ž๐‘Ž=๎€œ21๎“โ„“โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๐‘‘๐‘Ž๐‘Žโ‰ค๎ƒฉ๎€œ(4.13)21๎ƒฉ๎“โ„“โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๎ƒช๐‘ž๐‘‘๐‘Ž๐‘Ž๎ƒช1/๐‘žlog(2)1/๐‘ž๎…ž,(4.14) where we used that ๐‘‘๐‘Ž/๐‘Ž is scaling invariant. Note that the last inequality is Hรถlderโ€™s inequality for ๐‘ž<โˆž. In this case, taking ๐‘žth powers and summing over ๐‘— yields ๎“๐‘—โˆˆโ„ค2๐‘—๐‘ ๐‘ž๐‘๐‘ž๐‘—๎€œโ‰ค๐ถ21๎“๐‘—โˆˆโ„ค2๐‘—๐‘ ๐‘ž๎ƒฉ๎“โ„“โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๎ƒช๐‘ž๐‘‘๐‘Ž๐‘Ž.(4.15) Using vanishing moments and Schwartz properties of ๐œ“ and ๐œ‘, we can now employ (3.6) and (3.8) to obtain โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1โ‰ค๐ถ2โˆ’|๐‘—โˆ’โ„“|๐‘˜,(4.16) with a constant independent of ๐‘Žโˆˆ[1,2]. But then, since ๐‘˜>|๐‘ |, we may proceed just as in the proof of Theorem 3.11 to estimate the integrand in (4.15) via ๎“๐‘—โˆˆโ„ค2๐‘—๐‘ ๐‘ž๎ƒฉ๎“โ„“โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๎ƒช๐‘ž๎“โ‰ค๐ถโ„“โˆˆโ„ค2โ„“๐‘ ๐‘žโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘ž๐‘.(4.17) Summarizing, we obtain ๎“๐‘—2๐‘—๐‘ ๐‘ž๐‘๐‘ž๐‘—๎€œโ‰ค๐ถ21๎“โ„“โˆˆโ„ค2โ„“๐‘ ๐‘žโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘ž๐‘๐‘‘๐‘Ž๐‘Ž๎€œโ‰ค๐ถโˆž0๐‘Ž๐‘ ๐‘žโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘ž๐‘๐‘‘๐‘Ž๐‘Ž<โˆž.(4.18) In particular, ๐‘๐‘—<โˆž. But then the right-hand side of (4.11) converges to ๐‘ขโˆ—๐œ‘โˆ—๐‘— in ๐ฟ๐‘. The Minkowski inequality for integrals yields โ€–๐‘ขโˆ—๐œ‘โˆ—๐‘—โ€–๐‘โ‰ค๐‘๐‘—, and thus โ€–๐‘ขโ€–๐‘žฬ‡๐ต๐‘ ๐‘,๐‘ž๎€œโ‰ค๐ถโˆž0๐‘Ž๐‘ ๐‘žโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘ž๐‘๐‘‘๐‘Ž๐‘Ž,(4.19) as desired. In the case ๐‘ž=โˆž, (4.16) yields that sup๐‘—2๐‘—๐‘ ๎ƒฉ๎“โ„“โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ท๐‘Ž2โ„“๐œ“โˆ—๐œ‘โˆ—๐‘—โ€–โ€–1๎ƒชโ‰ค๐ถsupโ„“2โ„“๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘ž๐‘.(4.20) Thus, by (4.13), sup๐‘—2๐‘—๐‘ ๐‘๐‘—๎€œโ‰ค๐ถ21supโ„“2โ„“๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘๐‘‘๐‘Ž๐‘Žโ‰ค๐ถesssup๐‘Ž๐‘Ž๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘.(4.21) The remainder of the argument is the same as for the case ๐‘ž<โˆž.
Next assume ฬ‡๐ต๐‘ขโˆˆ๐‘ ๐‘,๐‘ž. Then, for all ๐‘Žโˆˆ[1,2] and โ„“โˆˆโ„ค, ๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—=๎“๐‘—โˆˆโ„ค๐‘ขโˆ—๐œ‘โˆ—๐‘—โˆ—๐œ‘๐‘—โˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—,(4.22) with convergence in ๐’ฎ๎…ž(๐บ)/๐’ซ; for ๐œ“โˆˆ๐’ต(๐บ), convergence holds even in ๐’ฎ๎…ž(๐บ). As before, โ€–โ€–โ€–โ€–๎“๐‘—โˆˆโ„ค๐‘ขโˆ—๐œ‘โˆ—๐‘—โˆ—๐œ‘๐‘—โˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–โ€–โ€–๐‘โ‰ค๎“๐‘—โˆˆโ„คโ€–โ€–๐‘ขโˆ—๐œ‘โˆ—๐‘—โ€–โ€–๐‘โ€–โ€–๐œ‘๐‘—โˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–1.(4.23) Again, we have โ€–๐œ‘๐‘—โˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–1โชฏ2โˆ’|๐‘—โˆ’โ„“|๐‘˜ with a constant independent of ๐‘Ž. Hence, one concludes in the same fashion as in the proof of Theorem 3.11 that, for all ๐‘Žโˆˆ[1,2], โ€–โ€–โ€–๎‚€2โ„“๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž2โ„“๐œ“โˆ—โ€–โ€–๐‘๎‚โ„“โˆˆโ„คโ€–โ€–โ€–๐‘žโ€–โ€–โ€–๎‚€2โ‰ค๐ถ๐‘—๐‘ โ€–โ€–๐‘ขโˆ—๐œ‘โˆ—๐‘—โ€–โ€–๐‘๎‚๐‘—โˆˆโ„คโ€–โ€–โ€–๐‘ž,(4.24) again with a constant independent of ๐‘Ž. In the case ๐‘ž=โˆž, this finishes the proof immediately, and for ๐‘ž<โˆž, we integrate the ๐‘žth power over ๐‘Žโˆˆ[1,2] and sum over โ„“ 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 ๐‘Ž>1 instead of ๐‘Ž=2. Thus consistently replacing powers of 2 in Definitions 3.6 and 3.9 by powers of ๐‘Ž>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. ฬ‡๐ต02,2=๐ฟ2(๐บ), with equivalent norms.

Proof. Pick ๐œ“ by Lemma 3.7. Then spectral calculus implies that for all ๐‘“โˆˆ๐’ต(๐บ)โ€–๐‘“โ€–2ฬ‡๐ต02,2=๎“๐‘—โˆˆโ„คโ€–โ€–๐‘“โˆ—๐œ“โˆ—๐‘—โ€–โ€–22=โ€–๐‘“โ€–22.(4.25) Since ๐’ต(๐บ) is dense in both spaces, and both spaces are complete, it follows that ฬ‡๐ต02,2=๐ฟ2(๐บ).

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 ๐ฟ denote a sub-Laplacian. For all ๐‘ขโˆˆ๐’ฎ๎…ž(๐บ)/๐’ซ, 1โ‰ค๐‘,๐‘ž<โˆž, ๐‘ โˆˆโ„ and ๐‘˜โ‰ฅ0, โ€–โ€–๐ฟ๐‘˜๐‘ขโ€–โ€–ฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘žโ‰โ€–๐‘ขโ€–ฬ‡๐ต๐‘ ๐‘,๐‘ž,(4.26) in the extended sense that one side is infinite if and only if the other side is. In particular, ๐ฟ๐‘˜โˆถฬ‡๐ต๐‘ ๐‘,๐‘žโ†’ฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘ž is a bijection, and it makes sense to extend the definition to negative ๐‘˜. Thus, for all ๐‘˜โˆˆโ„ค, ๐ฟ๐‘˜โˆถฬ‡๐ต๐‘ ๐‘,๐‘žโŸถฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘ž(4.27) is a topological isomorphism of Banach spaces.

Proof. Pick a nonzero real-valued โ„Žโˆˆ๐’ฎ(โ„+), an integer ๐‘š>|๐‘ | and let ๐œ“ denote the distribution kernel of ๐ฟ๐‘š๎โ„Ž(๐ฟ). Hence ๐œ“ is admissible by Theorem 4.2, with vanishing moments of order 2๐‘š and ๐œ“โˆ—=๐œ“. On ๐ฟ2(๐บ), the convolution operator ๐‘ขโ†ฆ๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ— can be written as ๎ฮจ๐‘Ž(๐ฟ) with a suitable function ๎ฮจ๐‘Ž. For ๐‘ขโˆˆ๐’ต(๐บ)โŠ‚๐ฟ2(๐บ), spectral calculus implies โ€–โ€–๎€ท๐ฟ๐‘˜๐‘ข๎€ธโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘=โ€–โ€–๎‚€๎ฮจ๐‘Ž(๐ฟ)โˆ˜๐ฟ๐‘˜๎‚โ€–โ€–(๐‘ข)๐‘=โ€–โ€–๎‚€๐ฟ๐‘˜โˆ˜๎ฮจ๐‘Ž๎‚โ€–โ€–=โ€–โ€–๐ฟ(๐ฟ)(๐‘ข)๐‘˜๎€ท๐‘ขโˆ—๐ท๐‘Ž๐œ“โˆ—๎€ธโ€–โ€–๐‘=โ€–โ€–๐‘ขโˆ—๐ฟ๐‘˜๎€ท๐ท๐‘Ž๐œ“โˆ—๎€ธโ€–โ€–๐‘=๐‘Ž2๐‘˜โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๎€ท๐ฟ๐‘˜๐œ“๎€ธโˆ—โ€–โ€–๐‘,(4.28) where we employed left invariance to pull ๐ฟ๐‘˜ past ๐‘ข in the convolution. Note that up to normalization, ๐ฟ๐‘˜๐œ“ is admissible with vanishing moments of order 2๐‘š+2๐‘˜>|๐‘ โˆ’2๐‘˜|. Thus, applying Theorem 4.4, we obtain โ€–โ€–๐ฟ๐‘˜๐‘ขโ€–โ€–ฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘žโ‰โ€–โ€–๐‘ŽโŸผ๐‘Ž๐‘ โˆ’2๐‘˜โ€–โ€–๎€ท๐ฟ๐‘˜๐‘ข๎€ธโˆ—๐ท๐‘Ž๐œ“โˆ—โ€–โ€–๐‘โ€–โ€–๐ฟ๐‘ž(โ„+;๐‘‘๐‘Ž/๐‘Ž)=โ€–โ€–๐‘ŽโŸผ๐‘Ž๐‘ โ€–โ€–๐‘ขโˆ—๐ท๐‘Ž๎€ท๐ฟ๐‘˜๐œ“๎€ธโˆ—โ€–โ€–๐‘โ€–โ€–๐ฟ๐‘ž(โ„+;๐‘‘๐‘Ž/๐‘Ž)โ‰โ€–๐‘ขโ€–ฬ‡๐ต๐‘ ๐‘,๐‘ž.(4.29) Now assume that ๐ฟ๐‘˜ฬ‡๐ต๐‘ขโˆˆ๐‘ โˆ’2๐‘˜๐‘,๐‘ž. Then, combining the density statements from Lemma 3.12 and Remark 3.8, we obtain a sequence {๐‘ข๐‘›}๐‘›โˆˆโ„•โŠ‚๐’ต(๐บ) with ๐ฟ๐‘˜๐‘ข๐‘›โ†’๐ฟ๐‘˜๐‘ข in ฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘ž; thus also with convergence in ๐’ฎ๎…ž(๐บ)/๐’ซ. The norm equivalence and completeness of ฬ‡๐ต๐‘ ๐‘,๐‘ž yield that ๐‘ข๐‘›ฬ‡๐ตโ†’๐‘ฃโˆˆ๐‘ ๐‘,๐‘ž, for suitable ฬ‡๐ต๐‘ฃโˆˆ๐‘ ๐‘,๐‘ž. Again, this implies convergence in ๐’ฎ๎…ž(๐บ)/๐’ซ. Since ๐ฟ๐‘˜ is continuous on that space, it follows that ๐ฟ๐‘˜๐‘ข๐‘›โ†’๐ฟ๐‘˜๐‘ฃ, establishing that ๐ฟ๐‘˜๐‘ฃ=๐ฟ๐‘˜๐‘ข. Since any distribution annihilated by ๐ฟ๐‘˜ is a polynomial, this finally yields ฬ‡๐ต๐‘ข=๐‘ฃโˆˆ๐‘ ๐‘,๐‘ž, and โ€–๐‘ขโ€–ฬ‡๐ต๐‘ ๐‘,๐‘žโ‰โ€–๐ฟ๐‘˜๐‘ขโ€–ฬ‡๐ต๐‘ โˆ’2๐‘˜๐‘,๐‘ž follows by taking limits. A similar but simpler argument establishes the norm equivalence under the assumption that ฬ‡๐ต๐‘ขโˆˆ๐‘ ๐‘,๐‘ž.

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 ๐‘˜โˆˆโ„•: ๐ต2๐‘˜2,2={๐‘“โˆˆ๐’ฎ๎…ž(๐บ)/๐’ซโˆถ๐ฟ๐‘˜๐‘“โˆˆ๐ฟ2(๐บ)}.

As a further corollary, we obtain the following interesting result relating two sub-Laplacians ๐ฟ1 and ๐ฟ2. For all ๐‘˜โˆˆโ„ค, the operator๐ฟ๐‘˜1โˆ˜๐ฟ2โˆ’๐‘˜โˆถ๐ฟ2(๐บ)โŸถ๐ฟ2(๐บ)(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 ฬ‡๐ต๐‘ ๐‘,๐‘ž can be discretized by sampling the convolution products ๐‘“โˆ—๐œ“โˆ—๐‘— over a given discrete set ฮ“โŠ‚๐บ. This is equivalent to the study of the analysis operator associated to a discrete wavelet system {๐œ“๐‘—,๐›พ}๐‘—โˆˆโ„ค,๐›พโˆˆฮ“, defined by๐œ“๐‘—,๐›พ(๐‘ฅ)=๐ท2๐‘—๐‘‡๐›พ๐œ“(๐‘ฅ)=2๐‘—๐‘„๐œ“๎€ท๐›พโˆ’1โ‹…2๐‘—๐‘ฅ๎€ธ.(5.1) Throughout the rest of the paper, we assume that the wavelet ๐œ“โˆˆ๐’ต(๐บ) has been chosen according to Lemma 3.7 and ๐œ“โˆ—=๐œ“.

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 ฮ“โŠ‚๐บ. For a family {๐‘๐‘—,๐›พ}๐‘—โˆˆโ„ค,๐›พโˆˆฮ“ of complex numbers, we define โ€–โ€–๎€ฝ๐‘๐‘—,