Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Abstract

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

1. Introduction

In recent years there were several efforts of extending Besov and Triebel-Lizorkin spaces from Euclidean spaces to other domains and non-isotropic settings. In particular, Han et al. [1] developed a theory of these function spaces on spaces of homogenous type with the additional reverse doubling property. That setting is quite general and includes for example Lie groups of polynomial growth. However, the high level of generality imposes restrictions on the possible values of the parameters of the function spaces.

For the purpose of studying subelliptic regularity, Folland [2] introduced fractional Sobolev spaces and Lipschitz spaces on stratified Lie groups. Later, Folland and Stein [3] established the theory of Hardy spaces on general homogeneous groups. Besov spaces on stratified Lie groups were first introduced by Saka [4], by means of the heat semigroup associated to the sub-Laplacian. Recently, Führ and Mayeli [5] introduced homogeneous Besov spaces on stratified Lie groups in terms of Littlewood-Paley-type decomposition and established wavelet characterization of them. However, the integrability parameter 𝑝 and the summability parameter 𝑞 of the function spaces studied in both [4, 5] are restricted to be no less than 1. Moreover, systematic treatment of Triebel-Lizorkin spaces on stratified Lie groups can not be found in the literature, to our best knowledge.

The purpose of this paper is to introduce and study homogeneous Triebel-Lizorkin spaces with full range of parameters on stratified Lie groups. Motivated by [5], we define these function spaces via Littlewood-Paley-type decomposition. We find that a helpful way to treat the case that either the integrability parameter 𝑝 or the summability parameter 𝑞 is less than 1 is to take the Peetre type maximal function into consideration. With the help of the almost orthogonality estimate on stratified Lie groups (see Lemma 2), we show that our definition of homogeneous Triebel-Lizorkin spaces is independent of the choice of the Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Thus, these function spaces reflect of properties of the group, not of the sub-Laplacian used for the construction of the decomposition.

Singular integral theory is a powerful tool for the study of partial differential equations. The 𝐿^𝑝-boundedness of convolution operators with homogeneous distribution kernels on Lie groups endowed with suitable homogeneous structure was proved by Knapp and Stein [6] (for 𝑝=2) and Korányi and Vági [7] (for 1<𝑝<∞). In Section 4 of this paper, we prove the boundedness on homogeneous Triebel-Lizorkin spaces of a class of convolution type singular integral operators on stratified Lie groups, which includes convolution operators with homogeneous distribution kernels.

This paper is organized as follows. After reviewing some basic notions concerning stratified Lie groups and their associated sub-Laplaicans in Section 2, in Section 3 we introduce homogeneous Triebel-Lizorkin spaces 𝐹̇^𝛼_𝑝,𝑞(𝐺) on stratified Lie groups, and give some basic properties of them. In Section 4 we show the 𝐹̇^𝛼_𝑝,𝑞(𝐺)-boundedness of a class of convolution singular integral operators. Throughout this paper the letter 𝐶 will denote a positive constant which is independent of the main variables involved but whose value may differ from line to line. The notation 𝑎≲𝑏 or 𝑏≳𝑎 for some variable quantities 𝑎 and 𝑏 means that 𝑎≤𝐶𝑏 for some constant 𝐶>0; 𝑎∼𝑏 stands for 𝑎≲𝑏≲𝑎. We agree that the set ℕ of natural numbers contains 0.

2. Preliminaries

In this section we briefly review the basic notions concerning stratified Lie groups and their associated sub-Laplacians. For more details we refer the reader to the monograph by Folland and Stein [3]. A Lie group 𝐺 is called a stratified Lie group if it is connected and simply connected, and its Lie algebra 𝔤 may be decomposed as a direct sum 𝔤=𝑉₁⊕⋯⊕𝑉_𝑚, with [𝑉₁,𝑉_𝑘]=𝑉_𝑘+1 for 1≤𝑘≤𝑚−1 and [𝑉₁,𝑉_𝑚]=0. Such a group 𝐺 is clearly nilpotent, and thus it may be identified with 𝔤 (as a manifold) via the exponential map exp∶𝔤→𝐺. Examples of stratified Lie groups include Euclidean spaces ℝ^𝑛 and the Heisenberg group ℍ^𝑛.

The algebra 𝔤 is equipped with a family of dilations {𝛿_𝑡∶𝑡>0} which are the algebra automorphisms defined by 𝛿_𝑡^𝑚_𝑗=1𝑋_𝑗=^𝑚_𝑗=1𝑡^𝑗𝑋_𝑗𝑋_𝑗∈𝑉_𝑗. (1) Under our identification of 𝐺 with 𝔤, 𝛿_𝑡 may also be viewed as a map 𝐺→𝐺. We generally write 𝑡𝑥 instead of 𝛿_𝑡(𝑥), for 𝑥∈𝐺. We shall denote by Δ=^𝑚_𝑗=1𝑗dim𝑉_𝑗 (2) the homogeneous dimension of 𝐺.

A homogeneous norm on G is a continuous function 𝑥↦|𝑥| from 𝐺 to [0,∞) smooth away from 0 (the group identity), vanishing only at 0, and satisfying |𝑥⁻¹|=|𝑥| and |𝑡𝑥|=𝑡|𝑥| for all 𝑥∈𝐺 and 𝑡>0. Homogeneous norms on 𝐺 always exist and any two of them are equivalent. We assume 𝐺 is provided with a fixed homogeneous norm. It satisfies a triangle inequality: there exists a constant 𝛾≥1 such that |𝑥𝑦|≤𝛾(|𝑥|+|𝑦|) for all 𝑥,𝑦∈𝐺. If 𝑥∈𝐺 and 𝑟>0 we define the ball of radius 𝑟 about 𝑥 by 𝐵(𝑥,𝑟)={𝑦∈𝐺∶|𝑦⁻¹𝑥|<𝑟}. The Lebesgue measure on 𝔤 induces a bi-invariant Haar measure on 𝐺. As done in [3], we fix the normalization of Haar measure by requiring that the measure of 𝐵(0,1) be 1. We shall denote the measure of any measurable 𝐸⊂𝐺 by |𝐸|. Clearly we have |𝛿_𝑡(𝐸)|=𝑡^Δ|𝐸|. All integrals on 𝐺 are with respect to (the normalization of) Haar measure. Convolution is defined by 𝑓∗𝑔(𝑥)=𝑓𝑦𝑔𝑦⁻¹𝑥𝑑𝑦=𝑓𝑥𝑦⁻¹𝑔𝑦𝑑𝑦. (3)

We consider 𝔤 as the Lie algebra of all left-invariant vector fields on 𝐺, and fix a basis 𝑋₁,…,𝑋_𝑛 of 𝔤, obtained as a union of bases of the 𝑉_𝑗. In particular, 𝑋₁,…,𝑋_𝜈, with 𝜈=dim(𝑉₁), is a basis of 𝑉₁. We denote by 𝑌₁,…,𝑌_𝑛 the corresponding basis for right-invariant vector fields, that is, 𝑌_𝑗𝑓(𝑥)=𝑑𝑓𝑑𝑡exp𝑡𝑋_𝑗𝑥∣_𝑡=0. (4) If 𝐼=(𝑖₁,…,𝑖_𝑛)∈ℕ^𝑛 is a multi-index we set 𝑋^𝐼=𝑋^𝑖1₁⋯𝑋^𝑖𝑛_𝑛 and 𝑌^𝐼=𝑌^𝑖1₁⋯𝑌^𝑖𝑛_𝑛. Moreover, we set |𝐼|=^𝑛_𝑘=1𝑖_𝑘,𝑑(𝐼)=^𝑛_𝑘=1𝑑_𝑘𝑖_𝑘, (5) where the integers 𝑑₁≤⋯≤𝑑_𝑛 are given according to that 𝑋_𝑘∈𝑉_𝑑𝑘. Then 𝑋^𝐼 (resp., 𝑌^𝐼) is a left-invariant (resp., right-invariant) differential operator, homogeneous of degree 𝑑(𝐼), with respect to the dilations 𝛿_𝑡, 𝑡>0.

A complex-valued function 𝑃 on 𝐺 is called a polynomial on 𝐺 if 𝑃∘exp is a polynomial on 𝔤. Let 𝜉₁,…,𝜉_𝑛 be the basis for the linear forms on 𝔤 dual to the basis 𝑋₁,…,𝑋_𝑛 for 𝔤, and set 𝜂_𝑗=𝜉_𝑗∘exp⁻¹. From our definition of polynomials on 𝐺, 𝜂₁,…,𝜂_𝑛 are generators of the algebra of polynomials on 𝐺. Thus, every polynomial on 𝐺 can be written uniquely as 𝑃=_𝐼𝑎_𝐼𝜂^𝐼,𝑎_𝐼∈ℂ, (6) where all but finitely many of the coefficients vanish, and 𝜂^𝐼=𝜂^𝑖1⋯𝜂^𝑖𝑛. A polynomial of the type (6) is called of homogeneous degree 𝐿, where 𝐿∈ℕ, if 𝑑(𝐼)≤𝐿 holds for all multi-indices 𝐼 with 𝑎_𝐼≠0. We let 𝒫 denote the space of all polynomials on 𝐺, and let 𝒫_𝐿 denote the space of polynomials on 𝐺 of homogeneous degree 𝐿. Note that 𝒫_𝐿 is invariant under left and right translations (see [3, Proposition 1.25]). A function 𝑓∶𝐺→ℂ is said to have vanishing moments of order 𝐿, if ∀𝑃∈𝒫_𝐿∶_𝐺𝑓(𝑥)𝑃(𝑥)𝑑𝑥=0, (7) with the absolute convergence of the integral.

The Schwartz class on 𝐺 is defined by 𝒮(𝐺)=𝑓∈𝐶^∞(𝐺)𝜕∶𝑃^|𝐼|𝑓𝜕𝜂^𝑖1⋯𝜕𝜂^𝑖𝑛∈𝐿^∞(𝐺),∀𝐼∈ℕ^𝑛,∀𝑃∈𝒫; (8) that is, 𝑓∈𝒮(𝐺) if and only if 𝑓∘exp∈𝒮(𝔤). As is indicated in [3, p. 35], 𝒮(𝐺) is a Fréchet space and several different choices of families of norms induce the same topology on 𝒮(𝐺). In this paper, for our purpose we use the family {‖⋅‖_(𝐿,𝑁)∶𝐿,𝑁∈ℕ} of norms given by 𝜙_(𝐿,𝑁)=sup_{𝑑(𝐼)≤𝐿,𝑥∈𝐺}(1+|𝑥|)^𝑁𝑋^𝐼𝜙(𝑥)+𝑋^𝐼𝜙(𝑥). (9) Here and in what follows, we use the notation convention 𝜙(𝑥)=𝜙(𝑥⁻¹) for any function 𝜙∶𝐺→ℂ. The dual space 𝒮^′(𝐺) of 𝒮(𝐺) is the space of tempered distributions on 𝐺. If 𝑓∈𝒮^′(𝐺) and 𝜙∈𝒮(𝐺) we shall denote the evaluation of 𝑓 on 𝜙 by ⟨𝑓,𝜙⟩.

Using the above conventions for the choice of the basis 𝑋₁,…,𝑋_𝑛, and 𝜈=dim(𝑉₁), the sub-Laplacian is defined by ℒ=−∑^𝜈_𝑗=1𝑋²_𝑗. When restricted to smooth functions with compact support, ℒ is essentially self-adjoint. Its closure has domain 𝒟={𝑢∈𝐿²(𝐺)∶ℒ𝑢∈𝐿²(𝐺)}, where ℒ𝑢 is taken in the sense of distributions. We denote this extension still by the symbol ℒ. By the spectral theorem, ℒ admits a spectral resolution ℒ=^∞₀𝜆𝑑𝐸(𝜆), (10) where 𝑑𝐸(𝜆) is the projection measure. If 𝑚 is a bounded Borel measurable function on [0,∞), the operator 𝑚(ℒ)=^∞₀𝑚(𝜆)𝑑𝐸(𝜆) (11) is bounded on 𝐿²(𝐺), and commutes with left translations. Thus, by the Schwartz kernel theorem, there exists a tempered distribution 𝑀 on 𝐺 such that 𝑚(ℒ)𝑓=𝑓∗𝑀,∀𝑓∈𝒮(𝐺). (12) Note that the point 𝜆=0 may be neglected in the spectral resolution, since the projection measure of {0} is zero (see [8, p. 76]). Consequently we should regard 𝑚 as functions on ℝ⁺≡(0,∞) rather than on [0,∞).

Let 𝒮(ℝ⁺) denote the space of restrictions to ℝ⁺ of functions in 𝒮(ℝ). An important fact proved by Hulanicki [9] is as in the following lemma.

Lemma 1. If 𝑚∈𝒮(ℝ⁺) then the distribution kernel 𝑀 of 𝑚(ℒ) is in 𝒮(𝐺).

Moreover, from the proof of [10, Corollary 1] we see that if 𝑚 is a function in 𝒮(ℝ⁺) which vanishes identically near the origin, then 𝑀 is a Schwartz function with all moments vanishing.

In the sequel, if not other specified, we will generally use Greak alphabets with hats to denote functions in 𝒮(ℝ⁺), and use Greek alphabets without hats to denote the associated distribution kernels; for example, for 𝜙∈𝒮(ℝ⁺) we shall denote by 𝜙 the distribution kernel of the operator 𝜙(ℒ), where ℒ is a sub-Laplacian fixed in the context.

3. Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

For any function ℎ on 𝐺 and 𝑡>0, we define the 𝐿¹-normalized dilation of ℎ by 𝐷_𝑡ℎ(𝑥)=𝑡^Δℎ(𝑡𝑥). (13) Before we introduce the homogeneous Triebel-Lizorkin spaces on stratified Lie groups, we prove the following basic estimate, which is a generalization of [11, Lemma B.1] and will be frequently used throughout this paper.

Lemma 2. Let 𝐿,𝑁∈ℕ with 𝑁≥𝐿+Δ+2. Suppose both 𝜙,𝜓∈𝒮(𝐺) have vanishing moments of order 𝐿. Then there exists a constant 𝐶>0 such that for all 𝑗,ℓ∈ℤ and all 𝑥∈𝐺, 𝐷₂^𝑗𝜙∗𝐷₂^ℓ𝜓(𝑥𝜙)≤𝐶_{(𝐿+1,𝑁)}𝜓_{(𝐿+1,𝑁)}2^{−|𝑗−ℓ|𝐿}2^{(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ|𝑥|^𝑁, (14) where 𝑗∧ℓ∶=min{𝑗,ℓ}.

Proof. Using dilations and the facts (𝐷₂^𝑗𝜙)∗(𝐷₂^ℓ𝜓)(𝑥)=(𝐷₂^ℓ𝜓)∗(𝐷₂^𝑗𝜙)(𝑥⁻¹) and ‖𝜙‖_(𝐿,𝑁)=‖𝜙‖_(𝐿,𝑁) (see (9)), we may assume ℓ≥𝑗=0. To proceed we follow the idea in the proof of [11, Lemma B.1]. Let 𝐷₁={𝑦∶|𝑦|<1}, 𝐷₂={𝑦∶|𝑦|≥1and|𝑥𝑦⁻¹|≤|𝑥|/2𝛾} and 𝐷₃={𝑦∶|𝑦|≥1and|𝑥𝑦⁻¹|>|𝑥|/2𝛾}. Let 𝑦↦𝑃_𝑥,𝜙(𝑦) be the left Taylor polynomial of 𝜙 at 𝑥 of homogeneous degree 𝐿 (see [3, pp. 26-27]). Then using vanishing moments of 𝜓𝜙∗𝐷₂^ℓ𝜓(𝑥≤𝜙)𝑥𝑦⁻¹−𝑃_𝑥,𝜙𝑦⁻¹𝐷₂^ℓ𝜓𝑦≡𝑑𝑦_𝐷1+_𝐷2+_𝐷3. (15)
For 𝑦∈𝐷₁, the stratified Taylor formula (cf. [3, Corollary 1.44]) yields that, with 𝑏 a suitable positive constant, 𝜙𝑥𝑦⁻¹−𝑃_𝑥,𝜙𝑦⁻¹≲𝑦^𝐿+1sup_{|𝑧|≤𝑏𝐿+1𝑦𝑑(𝐼)=𝐿+1}𝑋^𝐼𝜙≲𝜙(𝑥𝑧)_{(𝐿+1,𝑁)}sup_{|𝑧|≤𝑏𝐿+1}𝑦^𝐿+1(1+|𝑥𝑧|)^𝑁≲𝜙_{(𝐿+1,𝑁)}𝑦^𝐿+1(1+|𝑥|)^𝑁, (16) since 2𝑏^𝐿+1(1+|𝑥𝑧|)≥2𝑏^𝐿+1+|𝑥𝑧|≥2𝑏^𝐿+1+|𝑥|/𝛾−|𝑧|≥𝑏^𝐿+1+|𝑥|/𝛾≳(1+|𝑥|)/𝛾 if |𝑧|≤𝑏^𝐿+1. Hence we have _𝐷1≲𝜙_{(𝐿+1,𝑁)}𝜓_(0,𝑁)2^ℓΔ(1+|𝑥|)^𝑁𝐷₁𝑦^𝐿+11+2^ℓ𝑦^𝑁=𝜙𝑑𝑦_{(𝐿+1,𝑁)}𝜓_(0,𝑁)2^{−ℓ(𝐿+1)}(1+|𝑥|)^𝑁𝐺2^ℓ𝑦^𝐿+11+2^ℓ𝑦^𝑁𝑑2^ℓ𝑦≲𝜙_{(𝐿+1,𝑁)}𝜓_(0,𝑁)2^{−ℓ(𝐿+1)}(1+|𝑥|)^𝑁, (17) where for the last inequality we used [3, Corollary 1.17] and that 𝑁−𝐿−1≥Δ+1.
For 𝑦∈𝐷₂, we have |𝑦|≥|𝑥|/𝛾−|𝑥𝑦⁻¹|≥|𝑥|/𝛾−|𝑥|/2𝛾=|𝑥|/2𝛾. On the other hand, |𝑦|≤𝛾(|𝑥|+|𝑥𝑦⁻¹|)≤𝛾(|𝑥|+|𝑥|/2𝛾)=(𝛾+(1/2))|𝑥|. Thus, we have 1+2^ℓ𝑦≥2^ℓ𝑦≥2^ℓ1+𝑦2≥2^ℓ(1+|𝑥|),𝑦4𝛾_𝐿≤1𝛾+2^𝐿|𝑥|^𝐿. (18) Also, we note that by [12, Proposition 20.3.14] the left Taylor polynomial 𝑃_𝑥,𝜙 is of the form 𝑃_𝑥,𝜙𝑦=𝜙(𝑥)+^𝐿ℎ_{ℎ=1𝑘=11≤𝑖1,…,𝑖𝑘𝑑≤𝑛𝑖1+⋯+𝑑𝑖𝑘=ℎ}𝜂_𝑖1𝑦…𝜂_𝑖𝑘𝑦𝑋𝑘!^𝑖1⋯𝑋_𝑖𝑘𝜙(𝑥), (19) where the integers 𝑑₁≤⋯≤𝑑_𝑛 are given according to that 𝑋_𝑗∈𝑉_𝑑𝑗. From these remarks, it follows that _𝐷2≲𝜙_(𝐿,𝑁)𝜓_(0,𝑁)×_𝐷211+𝑥𝑦^−1𝑁+_{0≤ℎ≤𝐿}𝑦^ℎ(1+|𝑥|)^𝑁×2^ℓΔ1+2^ℓ𝑦^𝑁≲𝜙𝑑𝑦_(𝐿,𝑁)𝜓_{(0,𝑁)𝐷2}11+𝑥𝑦^−1𝑁+|𝑥|^𝐿(1+|𝑥|)^𝑁×2^ℓΔ2^ℓ(1+|𝑥|)^𝑁=𝜙𝑑𝑦_(𝐿,𝑁)𝜓_(0,𝑁)2^{−ℓ(𝑁−Δ)}(1+|𝑥|)^𝑁×_𝐺11+𝑥𝑦^−1𝑁+|𝑥|𝑑𝑦_𝐿(1+|𝑥|)^{𝑁|𝑥𝑦−1|≤|𝑥|/2𝛾}≲𝜙𝑑𝑦_(𝐿,𝑁)𝜓_(0,𝑁)2^{−ℓ(𝑁−Δ)}(1+|𝑥|)^𝑁×_𝐺11+𝑦^𝑁+|𝑥|𝑑𝑦_𝐿(1+|𝑥|)^{𝑁|𝑦|≤𝛾(|𝑥|+|𝑥|/2𝛾)}≲𝜙𝑑𝑦_(𝐿,𝑁)𝜓_(0,𝑁)2^{−ℓ(𝑁−Δ)}(1+|𝑥|)^𝑁|𝑥|1+^𝐿+Δ(1+|𝑥|)^𝑁≲𝜙_(𝐿,𝑁)𝜓_(0,𝑁)2^−ℓ𝐿(1+|𝑥|)^𝑁, (20) where we used that |{𝑦∶|𝑦|≤𝛾(|𝑥|+|𝑥|/2𝛾)}|∼|𝑥|^Δ and 𝑁−Δ>𝐿.
For 𝑦∈𝐷₃ we have |𝑥𝑦⁻¹|>|𝑥|/2𝛾, and, hence _𝐷3≲𝜙_(𝐿,𝑁)𝜓_(0,𝑁)×_𝐷311+𝑥𝑦^−1𝑁+_{0≤ℎ≤𝐿}𝑦^ℎ(1+|𝑥|)^𝑁×2^ℓΔ1+2^ℓ𝑦^𝑁≲𝜙𝑑𝑦_(𝐿,𝑁)𝜓_(0,𝑁)2^ℓΔ(1+|𝑥|)^{𝑁|𝑦|≥1}𝑦^𝐿1+2^ℓ𝑦^𝑁≤𝜙𝑑𝑦_(𝐿,𝑁)𝜓_(0,𝑁)2^−ℓ𝐿(1+|𝑥|)^𝑁𝐺2^ℓ𝑦^𝐿1+2^ℓ𝑦^𝑁𝑑2^ℓ𝑦≲𝜙_(𝐿,𝑁)𝜓_(0,𝑁)2^−ℓ𝐿(1+|𝑥|)^𝑁, (21) where for the last inequality we used [3, Corollary 1.17] and 𝑁−𝐿>Δ+1.
Combining the above estimates, we arrive at 𝜙∗𝐷₂^ℓ𝜓(𝑥)≤𝜙_{(𝐿+1,𝑁)}𝜓_{(𝐿+1,𝑁)}2^−ℓ𝐿1(1+|𝑥|)^𝑁. (22) This is exactly what we need.

Let 𝒵(𝐺) denote the space of Schwartz functions with all moments vanishing. We then consider 𝒵(𝐺) as a subspace of 𝒮(𝐺), including the topology. It is shown in [5, Lemma 3.3] that 𝒵(𝐺) is a closed subspace of 𝒮(𝐺), and the topology dual 𝒵^′(𝐺) of 𝒵(𝐺) can be canonically identified with the factor space 𝒮^′(𝐺)/𝒫.

We now have the following Calderón type reproducing formula.

Lemma 3. Suppose ℒ is a sub-Laplacian on 𝐺, and 𝜙∈𝒮(ℝ⁺) is a function with compact support, vanishing identically near the origin, and satisfying _𝑗∈ℤ𝜙2^−2𝑗𝜆=1,∀𝜆∈ℝ⁺. (23) Then for all 𝑔∈𝒵(𝐺), it holds that 𝑔=lim_{𝑘→∞𝑗≤𝑘}𝑔∗𝐷₂^𝑗𝜙, (24) with convergence in 𝒵(𝐺). Duality entails that, for all 𝑢∈𝒮^′(𝐺)/𝒫, 𝑢=lim_{𝑘→∞𝑗≤𝑘}𝑢∗𝐷₂^𝑗𝜙, (25) and the convergence is in 𝒮^′(𝐺)/𝒫.

Proof. First note that the 2-homogeneity of ℒ implies that the distribution kernel of 𝜙(2^−2𝑗ℒ) coincides with 𝐷₂^𝑗𝜙. Let 𝐼∈ℕ^𝑛 and 𝑁∈ℕ be arbitrarily chosen. Then take 𝐿∈ℕ such that 𝐿≥𝑁+𝑑(𝐼)+1. Since both 𝑔 and 𝜙 are Schwartz functions with all moments vanishing, it follows by Lemma 2 that (1+|𝑥|)^𝑁𝑋^𝐼𝑔∗𝐷₂^𝑗𝜙(𝑥)=2^{𝑑(𝐼)𝑗}(1+|𝑥|)^𝑁𝑔∗𝐷₂^𝑗𝑋^𝐼𝜙(𝑥)≤𝐶2^{𝑑(𝐼)𝑗}2^{−|𝑗|𝐿}2^(𝑗∧0)Δ1+2^(𝑗∧0)|𝑥|^{−(𝐿+Δ+2)}(1+|𝑥|)^𝑁≤𝐶2^{𝑑(𝐼)𝑗}2^{−|𝑗|𝐿}2^(𝑗∧0)Δ1+2^(𝑗∧0)|𝑥|^−𝑁(1+|𝑥|)^𝑁≤𝐶2^{−|𝑗|(𝐿−𝑁−𝑑(𝐼))}, (26) where the constant 𝐶 is a suitable multiple of ‖𝑔‖_{(𝐿+1,𝐿+Δ+2)}‖𝑋^𝐼𝜙‖_{(𝐿+1,𝐿+Δ+2)}. This implies that ∑_𝑗∈ℤ(1+|𝑥|)^𝑁|𝑋^𝐼[𝑔∗(𝐷₂^𝑗𝜙)](𝑥)| converges uniformly in 𝑥, for every 𝐼∈ℕ^𝑛 and every 𝑁∈ℕ. Consequently there exists ℎ∈𝒮(𝐺) such that ∑_{|𝑗|≤𝑘}𝑔∗(𝐷₂^𝑗𝜙) converges in the topology of 𝒮(𝐺) to ℎ, as 𝑘→∞. On the other hand, by (23) and the spectral theorem (cf. [13, Theorem VII.2]), 𝑔=_𝑗∈ℤ𝜙2^−2𝑗ℒ𝑔=_𝑗∈ℤ𝑔∗𝐷₂^𝑗𝜙 (27) holds in 𝐿²-norm. Therefore, ℎ=𝑔, which completes the proof.

Let 𝒜 denote the class of all functions 𝜙 in 𝒮(ℝ⁺) satisfying supp𝜙⊂2⁻²,2²,𝜙(𝜆3)≥𝐶>0for𝜆∈5²,53². (28)

Definition 4. Let 𝛼∈ℝ, 0<𝑝<∞ and 0<𝑞≤∞. Let ℒ be a sub-Laplacian on 𝐺 and 𝜙∈𝒜. We define 𝐹̇^𝛼_𝑝,𝑞(ℒ,𝜙) as the space of all 𝑓∈𝒮^′(𝐺)/𝒫 such that 𝑓_{̇𝐹𝛼𝑝,𝑞(ℒ,}𝜙⁾≡_𝑗∈ℤ2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙^𝑞1/𝑞_𝐿𝑝<∞, (29) with the usual modification for 𝑞=∞.

We then introduce the Peetre type maximal functions: Given 𝑓∈𝒮^′(𝐺), 𝜙∈𝒮(ℝ⁺), ℒ a sub-Laplacian, and 𝑎,𝑡>0, we define 𝑀^∗_𝑎,𝑡𝑓,ℒ,𝜙(𝑥)=sup_𝑦∈𝐺𝑓∗𝐷_𝑡−1𝜙(𝑥)1+𝑡⁻¹𝑦⁻¹𝑥^𝑎𝑀^∗∗_𝑎,𝑡𝑓,ℒ,𝜙(𝑥)=sup_{𝑦∈𝐺1≤𝑘≤𝜈}𝑋_𝑘𝑓∗𝐷_𝑡−1𝜙(𝑥)1+𝑡⁻¹𝑦⁻¹𝑥^𝑎. (30)

Lemma 5. Suppose ℒ is a sub-Laplacian and 𝜙∈𝒜. Then for every 𝑎>0 there is a constant 𝐶>0 such that for all 𝑓∈𝒮^′(𝐺)/𝒫, all 𝑗∈ℤ, and all 𝑥∈𝐺, 𝑀^∗∗_{𝑎,2−𝑗}𝑓,ℒ,𝜙(𝑥)≤𝐶2^𝑗𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ,𝜙(𝑥). (31)

Proof. Because of (28) it is possible to find a function 𝜓∈𝒮(ℝ⁺) supported in [2⁻²,2²] such that ∑_𝑗∈ℤ𝜙(2^−2𝑗𝜆)𝜓(2^−2𝑗𝜆)=1 for 𝜆∈ℝ⁺. Set 𝜁̂(𝜆)=∑¹_𝑗=−1𝜙(2^−2𝑗𝜆)𝜓(2^−2𝑗𝜆), 𝜆∈ℝ⁺. Then 𝜙(𝜆)=𝜙(𝜆)𝜁̂(𝜆) for all 𝜆∈ℝ⁺. Consequently, for 𝑗∈ℤ and 1≤𝑘≤𝜈, 𝑋_𝑘𝑓∗𝐷₂^𝑗𝜙𝑦=𝑋_𝑘𝑓∗𝐷₂^𝑗𝜙∗𝐷₂^𝑗𝜁𝑦≤𝑓∗𝐷₂^𝑗𝜙(𝑧)𝑋_𝑘𝐷₂^𝑗𝜁𝑧⁻¹𝑦𝑑𝑧=2^𝑗(Δ+1)𝑓∗𝐷₂^𝑗𝜙(𝑧)𝑋_𝑘𝜁2^𝑗𝑧⁻¹𝑦𝑑𝑧≲2^𝑗𝑀^∗_{𝑎,2−𝑗}𝜙(𝑥)×2𝑓,ℒ,_𝑗Δ1+2^𝑗𝑧⁻¹𝑥^𝑎1+2^𝑗𝑧⁻¹𝑦^{−𝑎−(Δ+1)}𝑑𝑧≲2^𝑗𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ,𝜙(𝑥)1+2^𝑗𝑦⁻¹𝑥^𝑎, (32) where for the last inequality we used [3, Corollary 1.17] and that (1+2^𝑗|𝑧⁻¹𝑥|)^𝑎(1+2^𝑗|𝑧⁻¹𝑦|)^−𝑎≲(1+2^𝑗|𝑦⁻¹𝑥|)^𝑎. Dividing both sides of the above estimate by (1+2^𝑗|𝑦⁻¹𝑥|)^𝑎, and then taking the supremum over 𝑦∈𝐺 and 1≤𝑘≤𝜈, we obtain the desired estimate.

Lemma 6. Suppose ℒ is a sub-Laplacian, 𝜙∈𝒜, and 𝑟>0. Then there exists a constant 𝐶 such that for all 𝑓∈𝒮^′(𝐺)/𝒫, all 𝑗∈ℤ, and all 𝑥∈𝐺, 𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ,𝜙(𝑥)≤𝐶𝑀𝑓∗𝐷₂^𝑗𝜙^𝑟(𝑥)^1/𝑟, (33) where 𝑎=Δ/𝑟, and 𝑀 is the Hardy-Littlewood maximal operator on 𝐺.

Proof. Let 𝑔∈𝐶^∞(𝐺) and 𝑦₀∈𝐺. The stratified mean value theorem (cf. [3, Theorem 1.41]) gives that for every 𝛿>0 and every 𝑦∈𝐺 with |𝑦⁻¹𝑦₀|<𝛿, 𝑔𝑦−𝑔𝑦₀≤𝐶𝑦₀⁻¹𝑦sup_{|𝑧|𝑦≤𝑏0−1𝑦1≤𝑘≤𝜈}𝑋_𝑘𝑔𝑦₀𝑧≤𝐶𝛿sup_{|𝑧|≤𝑏𝛿1≤𝑘≤𝜈}𝑋_𝑘𝑔𝑦₀𝑧, (34) where 𝑏 is a suitable positive constant. Hence we have 𝑔𝑦₀≤𝐶𝛿sup_{𝑦−1𝑦0≤𝑏𝛿1≤𝑘≤𝜈}𝑋_𝑘𝑔𝑦+𝛿^−𝑎_{|𝑦−1𝑦0|≤𝛿}𝑔𝑦^𝑟𝑑𝑦^1/𝑟. (35) Putting 𝑔=𝑓∗(𝐷₂^𝑗𝜙), dividing both sides by (1+2^𝑗|𝑦₀⁻¹𝑥|)^𝑎, and using Lemma 5, we have 𝑓∗𝐷₂^𝑗𝜙𝑦₀1+2^𝑗𝑦₀⁻¹𝑥^𝑎≤𝐶𝛿sup_{𝑦−1𝑦0≤𝑏𝛿1≤𝑘≤𝜈}𝑋_𝑘𝑓∗𝐷₂^𝑗𝜙𝑦1+2^𝑗𝑦⁻¹𝑥^𝑎1+2^𝑗𝑦⁻¹𝑥^𝑎1+2^𝑗𝑦₀⁻¹𝑥^𝑎+𝛿^−𝑎1+2^𝑗𝑦₀⁻¹𝑥^{𝑎|𝑦−1𝑦}₀^|≤𝛿𝑓∗𝐷₂^𝑗𝜙𝑦^𝑟𝑑𝑦^1/𝑟≤𝐶𝛿sup_{𝑦−1𝑦0≤𝑏𝛿}1+2^𝑗𝑦⁻¹𝑦₀^𝑎𝑀^∗∗_{𝑎,2−𝑗}𝜙(𝑥)+𝛿𝑓,ℒ,_−𝑎1+2^𝑗𝑦₀⁻¹𝑥^𝑎×_{|𝑦−1𝑥|≤𝛾(𝛿+|𝑦0−1𝑥|)}𝑓∗𝐷₂^𝑗𝜙𝑦^𝑟𝑑𝑦^1/𝑟≤𝐶𝛿1+2^𝑗𝑏𝛿^𝑎2^𝑗𝑀^∗_{𝑎,2−𝑗}𝜙𝑥)+𝛿𝑓,ℒ,(_−𝑎𝛾^𝑎𝛿+𝑦₀⁻¹𝑥^𝑎1+2^𝑗𝑦₀⁻¹𝑥^𝑎𝑀𝑓∗𝐷₂^𝑗𝜙^𝑟(𝑥)^1/𝑟≤𝐶𝜀(1+𝑏𝜀)^𝑎𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ,𝜙(𝑥)+𝛾^𝑎1+𝜀^−1𝑎𝑀𝑓∗𝐷₂^𝑗𝜙^𝑟(𝑥)^1/𝑟, (36) where we have set 𝛿=2^−𝑗𝜀 and used that (1+2^𝑗𝜀⁻¹|𝑦₀⁻¹𝑥|)/(1+2^𝑗|𝑦₀⁻¹𝑥|)≲1+𝜀⁻¹. Finally, taking 𝜀 sufficiently small (such that 𝐶𝜀(1+𝑏𝜀)^𝑎<1/2), and taking the supremum over 𝑦₀∈𝐺, we get the desired estimate.

Theorem 7. Suppose ℒ⁽¹⁾,ℒ⁽²⁾ are any two sub-Laplacians on 𝐺, and 𝜙⁽¹⁾,𝜙⁽²⁾ are any two functions in 𝒜. Then, for 𝛼∈ℝ, 0<𝑝<∞ and 0<𝑞≤∞, we have the (quasi-)norm equivalence 𝑓_{̇𝐹𝛼𝑝,𝑞(ℒ(1),}𝜙⁽¹⁾₎∼𝑓_{̇𝐹𝛼𝑝,𝑞(ℒ(2),}𝜙⁽²⁾₎,𝑓∈𝒮^′(𝐺)/𝒫. (37)

Proof. First we note that if 𝑎>Δ/min{𝑝,𝑞} then we have _𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ^(𝑖),𝜙^{(𝑖)𝑞1/𝑞}_𝐿𝑝∼_𝑗∈ℤ2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙^{(𝑖)𝑞1/𝑞}_𝐿𝑝 (38) for 𝑖=1,2, where 𝜙^(𝑖) denotes the distribution kernel of 𝜙^(𝑖)(ℒ^(𝑖)). Indeed, the direction “≳” of (38) is obvious, and the other direction follows by Lemma 6 and the Fefferman-Stein vector-valued maximal inequality on spaces of homogeneous type (see, e.g., [14]). Thus, to prove (37), it suffices to show that _𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙^{(1)𝑞1/𝑞}_𝐿𝑝∼_𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽²⁾,𝜙^{(2)𝑞1/𝑞}_𝐿𝑝. (39)
To this end, let 𝜓⁽¹⁾ be a function in 𝒮(ℝ⁺) with support in [2⁻²,2²] such that ∑_𝑗∈ℤ𝜙⁽¹⁾(2^−2𝑗𝜆)𝜓⁽¹⁾(2^−2𝑗𝜆)=1 for 𝜆∈ℝ⁺. For 𝑓∈𝒮^′(𝐺)/𝒫, by Lemma 3 we have 𝑓=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙⁽¹⁾∗𝐷₂^𝑗𝜓⁽¹⁾ (40) with convergence in 𝒮^′(𝐺)/𝒫. Here 𝜓⁽¹⁾ is the distribution kernel of 𝜓⁽¹⁾(ℒ⁽¹⁾). Hence, since 𝜙⁽²⁾∈𝒵(𝐺), we have the pointwise representation 𝑓∗𝐷₂^ℓ𝜙⁽²⁾𝑦=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙⁽¹⁾∗𝐷₂^𝑗𝜓⁽¹⁾∗𝐷₂^ℓ𝜙⁽²⁾𝑦,∀𝑦∈𝐺. (41) It follows that 𝑓∗𝐷₂^ℓ𝜙⁽²⁾𝑦≤_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙⁽¹⁾(𝑧×𝐷_2𝑗)𝜓⁽¹⁾∗𝐷₂^ℓ𝜙⁽²⁾𝑧⁻¹𝑦𝑑𝑧≤_𝑗∈ℤ𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾𝑦×1+2^𝑗|𝑧|^𝑎𝐷₂^𝑗𝜓⁽¹⁾∗𝐷₂^ℓ𝜙⁽²⁾(𝑧)𝑑𝑧≡_𝑗∈ℤ𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾𝑦𝐼_𝑗,ℓ. (42) Since both 𝜓⁽¹⁾ and 𝜙⁽²⁾ are Schwartz functions with all moments vanishing, we can use Lemma 2 to estimate that, with 𝐿 sufficiently large, 𝐼_𝑗,ℓ≤𝐶1+2^𝑗|𝑧|^𝑎2^{−|𝑗−ℓ|𝐿}2^{(𝑗∧ℓ)Δ}×1+2^𝑗∧ℓ|𝑧|^{−(𝐿+Δ+2)}𝑑𝑧≤𝐶1+2^𝑗|𝑧|^𝑎2^{−|𝑗−ℓ|𝐿}2^{(𝑗∧ℓ)Δ}×1+2^𝑗∧ℓ|𝑧|^{−(𝑎+Δ+1)}𝑑𝑧≤𝐶2^{−|𝑗−ℓ|(𝐿−𝑎)}2^{(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ|𝑧|^−(Δ+1)𝑑𝑧≤𝐶2^{−|𝑗−ℓ|(𝐿−𝑎)}. (43) Here, the constant 𝐶 is a suitable multiple of ‖𝜓⁽¹⁾‖_{(𝐿+1,𝐿+Δ+2)}‖𝜙⁽²⁾‖_{(𝐿+1,𝐿+Δ+2)}. On the other hand we observe that 𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾𝑦≤𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾(𝑥)1+2^𝑗𝑦⁻¹𝑥^𝑎≲𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾(𝑥)×1+2^ℓ𝑦⁻¹𝑥^𝑎max1,2^{(𝑗−ℓ)𝑎}. (44) Putting these estimates into (42), multiplying both sides by 2^ℓ𝛼, dividing both sides by (1+2^ℓ|𝑦⁻¹𝑥|)^𝑎 and then taking the supremum over 𝑦∈𝐺, we obtain 2^ℓ𝛼𝑀^∗_𝑎,2−ℓ𝑓,ℒ⁽²⁾,𝜙⁽²⁾(𝑥)≤_𝑗∈ℤ2^{−|𝑗−ℓ|(𝐿−2𝑎−|𝛼|)}2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,ℒ⁽¹⁾,𝜙⁽¹⁾(𝑥). (45) In view of [15, Lemma 2], taking 𝐿>2𝑎+|𝛼| in the above inequality yields the direction “≲” of (39). By symmetricity, (39) holds, and the proof is complete.

Remark 8. From Theorem 7 we see that the space 𝐹̇^𝛼_𝑝,𝑞(ℒ,𝜙) is actually independent of the choice of ℒ and 𝜙. Thus, in what follows we don't specify the choice of ℒ and 𝜙 and write 𝐹̇^𝛼_𝑝,𝑞(𝐺) instead of 𝐹̇^𝛼_𝑝,𝑞(ℒ,𝜙). Henceforth we shall fix any sub-Laplacian ℒ. Moreover, for the sake of briefness, we will write 𝑀^∗_𝑎,𝑡(𝑓,𝜙)(𝑥) instead of 𝑀^∗_𝑎,𝑡(𝑓,ℒ,𝜙)(𝑥).

Proposition 9. For 𝛼∈ℝ, 0<𝑝<∞ and 0<𝑞≤∞, one has the continuous inclusion maps 𝒵(𝐺)↪𝐹̇^𝛼_𝑝,𝑞(𝐺)↪𝒮^′(𝐺)/𝒫.

Proof. Let 𝑔∈𝒵(𝐺) and 𝜙∈𝒜. Choose 𝑁≥(Δ+1)/𝑝 and 𝐿≥𝑁+|𝛼|+1. Since both 𝑔 and 𝜙 are Schwartz functions with all moments vanishing, it follows by Lemma 2 that 𝑔∗𝐷₂^𝑗𝜙(𝑥𝑔)≤𝐶_{(𝐿+1,𝐿+Δ+2)}2^{−|𝑗|𝐿}2^(𝑗∧0)Δ1+2^𝑗∧0|𝑥|^{−(𝐿+Δ+2)}≤𝐶𝑔_{(𝐿+1,𝐿+Δ+2)}2^{−|𝑗|𝐿}2^(𝑗∧0)Δ1+2^𝑗∧0|𝑥|^−𝑁≤𝐶𝑔_{(𝐿+1,𝐿+Δ+2)}2^{−|𝑗|(𝐿−𝑁)}(1+|𝑥|)^−𝑁 (46) for all 𝑗∈ℤ and all 𝑥∈𝐺, where the constant 𝐶 is a suitable multiple of ‖𝜙‖_{(𝐿+1,𝐿+Δ+2)}. This together with [3, Corollary 1.17] give that 𝑔_{̇𝐹𝛼𝑝,𝑞(𝐺)}≲𝑔_{(𝐿+1,𝐿+Δ+2)𝑗∈ℤ}2^{−|𝑗|(𝐿−𝑁−|𝛼|)𝑞1/𝑞}≲𝑔_{(𝐿+1,𝐿+Δ+2)}, (47) which implies that 𝒵(𝐺)↪𝐹̇^𝑠_𝑝,𝑞(𝐺) continuously.
Now we show the other embedding. Let 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺) and take any 𝜓∈𝒵(𝐺). Let 𝜙∈𝒜, and then let 𝜁̂∈𝒮(ℝ⁺) be a function with support in [2⁻²,2²] such that ∑_𝑗∈ℤ𝜙(2^−2𝑗𝜆)𝜁̂(2^−2𝑗𝜆)=1 for 𝜆∈ℝ⁺. Then by Lemma 3 we have 𝑓,𝜓=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙∗𝐷₂^𝑗𝜁,𝜓≤_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙,𝜓∗𝐷₂^𝑗𝜁̃≤_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜙_𝐿∞𝜓∗𝐷₂^𝑗𝜁̃_𝐿1. (48) To proceed we claim that 𝑓∗𝐷₂^𝑗𝜙_𝐿∞≤𝐶2^𝑗Δ/𝑝𝑓∗𝐷₂^𝑗𝜙_𝐿𝑝 (49) for all 0<𝑝<∞. Assuming the claim for a moment, it follows from (48) that 𝑓,𝜓≲sup_𝑗∈ℤ2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙_𝐿𝑝×_𝑗∈ℤ2^{𝑗[(Δ/𝑝)−𝛼]}𝜓∗𝐷₂^𝑗𝜁̃_𝐿1. (50) It is easy to see that sup_𝑗∈ℤ2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙_𝐿𝑝≲𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}. (51) To estimate the sum in (50), we note that if we choose 𝑁≥Δ+1 and 𝐿≥𝑁+(Δ/𝑝)+|𝛼|+1 then similarly to (46) we have 𝜓∗𝐷₂^𝑗𝜁̃(𝑥)≤𝐶𝜓_{(𝐿+1,𝐿+Δ+2)}2^{−|𝑗|(𝐿−𝑁)}(1+|𝑥|)^−𝑁, (52) where the constant 𝐶 is a suitable multiple of ‖𝜁̃‖_{(𝐿+1,𝐿+Δ+2)}. From this it follows that _𝑗∈ℤ2^{𝑗(Δ/𝑝−𝛼)}𝜓∗𝐷₂^𝑗𝜁̃_𝐿1≲_𝑗∈ℤ2^{−|𝑗|[𝐿−𝑁−(Δ/𝑝)−|𝛼|]}𝜓_{(𝐿+1,𝐿+Δ+2)}≲𝜓_{(𝐿+1,𝐿+Δ+2)}. (53) Therefore, 𝑓,𝜓≲𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}𝜓_{(𝐿+1,𝐿+Δ+2)}. (54) This implies that 𝐹̇^𝛼_𝑝,𝑞(𝐺)↪𝒮^′(𝐺)/𝒫 continuously.
We are left with showing the claim. Indeed, if 𝑟>0 is fixed then by Lemma 6 we have, for all 𝑥∈𝐺, 𝑓∗𝐷₂^𝑗𝜙(𝑥)∼inf_{𝑦∈𝐵𝑥,2−𝑗}𝑓∗𝐷₂^𝑗𝜙(𝑥)1+2^𝑗𝑥⁻¹𝑦^Δ/𝑟≤inf_{𝑦∈𝐵𝑥,2−𝑗}𝑀^∗_{Δ/𝑟,2−𝑗}𝑓,𝜙𝑦≲inf_{𝑦∈𝐵𝑥,2−𝑗}𝑀𝑓∗𝐷₂^𝑗𝜙^𝑟(𝑦)^1/𝑟. (55) Taking 𝑟<𝑝 in the above estimate and using Hardy-Littlewood maximal inequality, we have 𝑓∗𝐷₂^𝑗𝜙(𝑥)^𝑝≲1𝐵𝑥,2^−𝑗_𝐵(𝑥,2^−𝑗₎𝑀𝑓∗𝐷₂^𝑗𝜙^𝑟𝑦^𝑝/𝑟𝑑𝑦≤2^𝑗Δ𝑓∗𝐷₂^𝑗𝜙𝑦^𝑝𝑑𝑦. (56) Since 𝑥 is arbitrary, the claim follows.

Since all the necessary tools are developed in the above arguments, the following proposition can be proved in the same manner as its Euclidean counterpart; see, for example, the proof of [16, Theorem 2.3.3].

Proposition 10. For 𝛼∈ℝ, 0<𝑝<∞ and 0<𝑞≤∞, 𝐹̇^𝛼_𝑝,𝑞(𝐺) is a quasi-Banach space.

Let us introduce a class of functions. We say that 𝑓∈ℛ(𝐺), if there exists 𝜙∈𝒮(ℝ⁺) whose support is compact and which vanishes identically near the origin, and 𝑔∈𝒮(𝐺), such that 𝑓=𝜙(ℒ)𝑔. Clearly ℛ(𝐺)⊂𝒵(𝐺).

Lemma 11. Let 𝛼∈ℝ and 0<𝑝,𝑞<∞. Then ℛ(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺). In particular, 𝒵(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺).

Proof. Take any 𝑢∈𝐹̇^𝛼_𝑝,𝑞(𝐺) and any 𝜀>0. In the appendix we show that 𝐹̇^𝛼_𝑝,𝑞(𝐺) admits smooth atomic decomposition. By the smooth atomic decomposition, we see that 𝐶^∞₀(𝐺)∩𝐹̇^𝛼_𝑝,𝑞(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺), for 𝛼∈ℝ and 0<𝑝,𝑞<∞. Thus, we can find 𝑔∈𝐶^∞₀(𝐺)∩𝐹̇^𝛼_𝑝,𝑞(𝐺) such that ‖𝑔−𝑢‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}<𝜀/2. On the other hand, the argument in Step 5 of the proof of [16, Theorem 2.3.3] shows that there exists a sufficiently large 𝑁∈ℕ such that ‖𝑔−∑^𝑁_𝑗=−𝑁𝑔∗(𝐷₂^𝑗𝜙)‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}<𝜀/2. Now we put ℎ=^𝑁_𝑗=−𝑁𝜙2^−2𝑗ℒ𝑔=^𝑁_𝑗=−𝑁𝑔∗𝐷₂^𝑗𝜙. (57) Then ℎ∈ℛ(𝐺), and we have ‖ℎ−𝑢‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}≲ℎ−𝑔_{̇𝐹𝛼𝑝,𝑞(𝐺)}+𝑔−𝑢_{̇𝐹𝛼𝑝,𝑞(𝐺)}<𝜀. (58) This proves the claimed statement.

We next consider lifting property of 𝐹̇^𝛼_𝑝,𝑞(𝐺). For 𝜎∈ℝ, the power ℒ^𝜎 is naturally given by ℒ^𝜎𝑓=^∞₀𝜆^𝜎𝑑𝐸(𝜆)𝑓,𝑓∈Domℒ^𝜎⊂𝐿²(𝐺). (59)

Remark 12. By [17, Theorem 13.24], we have ℛ(𝐺)⊂Dom(ℒ^𝜎) for all 𝜎∈ℝ. As a consequence, Dom(ℒ^𝜎)∩𝐹̇^𝛼_𝑝,𝑞(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺), for all 𝛼,𝜎∈ℝ and 0<𝑝,𝑞<∞.
We now have the lifting property of 𝐹̇^𝛼_𝑝,𝑞(𝐺).

Theorem 13. Let 𝛼,𝜎∈ℝ and 0<𝑝, 𝑞<∞.(i)The operator ℒ^𝜎, initially defined on Dom(ℒ^𝜎)∩𝐹̇^𝛼_𝑝,𝑞(𝐺), extends to a continuous operator from 𝐹̇^𝛼_𝑝,𝑞(𝐺) to 𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺).(ii)Let 𝑇_𝜎∶𝐹̇^𝛼_𝑝,𝑞(𝐺)→𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) denote the continuous extension of ℒ^𝜎. Then 𝑇_𝜎 is an isomorphism, and ‖𝑇_𝜎𝑓‖_̇𝐹^𝛼−2𝜎_{𝑝,𝑞(𝐺)} is an equivalent quasi-norm of 𝐹̇^𝛼_𝑝,𝑞(𝐺).

Proof. (i) Let 𝜙∈𝒜. Set 𝜁̂(𝜆)=𝜆^𝜎 and 𝜓(𝜆)=𝜙(𝜆)𝜁̂(𝜆)=𝜙(𝜆)𝜆^𝜎. Clearly 𝜓 is also in 𝒜, and 2^2𝑗𝜎𝜓(2^−2𝑗𝜆)=𝜙(2^−2𝑗𝜆)𝜁̂(𝜆). By [17, Theorem 13.24], we have 𝜙(2^−2𝑗ℒ)∘𝜁̂(ℒ)⊂(𝜙(2^−2𝑗⋅)𝜁̂(⋅))(ℒ), and moreover Dom𝜙2^−2𝑗ℒ∘𝜁̂(ℒ)=Dom𝜁̂(ℒ)∩Dom𝜙2^−2𝑗⋅𝜁̂(⋅ℒℒ)()=Dom^𝜎∩𝐿²(𝐺)=Domℒ^𝜎. (60) Hence, for every 𝑓∈Dom(ℒ^𝜎), we have 2^2𝑗𝜎𝜓(2^−2𝑗ℒ)𝑓=𝜙(2^−2𝑗ℒ)ℒ^𝜎𝑓.
Now let 𝑓∈Dom(ℒ^𝜎)∩𝐹̇^𝑠_𝑝,𝑞(𝐺). By the above remarks, we have 2^2𝑗𝜎[𝑓∗(𝐷₂^𝑗𝜓)]=(ℒ^𝜎𝑓)∗(𝐷₂^𝑗𝜙). It follows that ℒ^𝜎𝑓_{̇𝐹𝛼−2𝜎𝑝,𝑞(𝐺)}∼_𝑗∈ℤ2^{𝑗(𝛼−2𝜎)}ℒ^𝜎𝑓∗𝐷₂^𝑗𝜙^𝑞1/𝑞_𝐿𝑝=_𝑗∈ℤ2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜓^𝑞1/𝑞_𝐿𝑝∼𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}. (61) Since Dom(ℒ^𝜎)∩𝐹̇^𝛼_𝑝,𝑞(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺) (see Remark 12), the mapping ℒ^𝜎∶Dom(ℒ^𝜎)∩𝐹̇^𝛼_𝑝,𝑞(𝐺)↦𝐹̇^𝛼_𝑝,𝑞(𝐺) extends to a continuous operator from 𝐹̇^𝛼_𝑝,𝑞(𝐺) to 𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺). We denote this extension by 𝑇_𝜎∶𝐹̇^𝛼_𝑝,𝑞(𝐺)↦𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺).
(ii) Let us first show that the mapping 𝑇_𝜎∶𝐹̇^𝛼_𝑝,𝑞(𝐺)→𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) is injective. Indeed, assume 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺) such that 𝑇_𝜎𝑓 is the zero element of 𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺). By Remark 12 we can find a sequence 𝑓_ℓ in Dom(ℒ^𝜎)∩𝐹̇^𝛼_𝑝,𝑞(𝐺) which converges in 𝐹̇^𝛼_𝑝,𝑞(𝐺) to 𝑓. Then applying (i) to ℒ^𝜎 yields that ℒ^𝜎𝑓_ℓ converges in 𝐹̇^𝑠−2𝜎_𝑝,𝑞(𝐺) to the zero element. Since ℒ^𝜎𝑓_ℓ∈Dom(ℒ^−𝜎)∩𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) and 𝑓_ℓ=ℒ^−𝜎(ℒ^𝜎𝑓_ℓ), applying (i) to the operator ℒ^−𝜎 we see that 𝑓_ℓ converges in 𝐹̇^𝛼_𝑝,𝑞(𝐺) to the zero element. Therefore, 𝑓 is the zero element in 𝐹̇^𝛼_𝑝,𝑞(𝐺). This proves that 𝑇_𝜎 is injective.
Next we show that 𝑇_𝜎∶𝐹̇^𝛼_𝑝,𝑞(𝐺)→𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) is surjective. Indeed, given 𝑓∈𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺), we let 𝑓_ℓ be a sequence in Dom(ℒ^−𝜎)∩𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) which converges in 𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) to 𝑓. Then from (i) we see that ℒ^−𝜎𝑓_ℓ converges in 𝐹̇^𝛼_𝑝,𝑞(𝐺). Denote this limit by 𝑔. We claim that 𝑇_𝜎𝑔=𝑓. Indeed, since 𝑓_ℓ=ℒ^𝜎(ℒ^−𝜎𝑓_ℓ), it follows from (i) that 𝑓_ℓ converges to 𝑇_𝜎𝑔 in 𝐹̇^𝛼_𝑝,𝑞(𝐺). Hence 𝑇_𝜎𝑔=𝑓 in 𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺). This proves that 𝑇_𝜎 is surjective.
The above arguments also show that both 𝑇_−𝜎∘𝑇_𝜎 and 𝑇_𝜎∘𝑇_−𝜎 are identity operators on 𝐹̇^𝛼_𝑝,𝑞(𝐺). Furthermore, by an easy density argument we see that (61) holds for all 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺), provided that ℒ^𝜎 in (61) is replaced by 𝑇_𝜎. Thus, 𝑇_𝜎∶𝐹̇^𝛼_𝑝,𝑞(𝐺)→𝐹̇^𝛼−2𝜎_𝑝,𝑞(𝐺) is an isomorphism, and ‖𝑇_𝜎𝑓‖_̇𝐹^𝛼−2𝜎_{𝑝,𝑞(𝐺)} is an equivalent quasi-norm of 𝐹̇^𝛼_𝑝,𝑞(𝐺).

Our next goal is to show the Lusin and Littlewood-Paley function characterizations of 𝐹̇^𝛼_𝑝,𝑞(𝐺). If 𝛼∈ℝ, 0<𝑞<∞, 𝜆>0, 𝑏∈ℝ, and 𝑢(𝑥,𝑘) is a function on 𝐺×ℤ, we define 𝑔^𝛼_𝑞(𝑢)(𝑥)=^∞_𝑘=−∞2^𝑘𝛼|𝑢(𝑥,𝑘)|^𝑞1/𝑞,𝐺^𝛼_𝜆,𝑞(𝑢)(𝑥)=^∞_{𝑘=−∞𝐺}2^𝑘𝛼𝑢𝑦,𝑘^𝑞×1+2^𝑘𝑦⁻¹𝑥^−𝜆𝑞2^𝑘Δ𝑑𝑦^1/𝑞,𝑆^𝛼_𝑏,𝑞(𝑢𝑥)=)(_{∞𝑘=−∞|𝑦−1𝑥|≤2𝑏−𝑘}2^𝑘𝛼𝑢𝑦,𝑘^𝑞2^{(𝑘−𝑏)Δ}𝑑𝑦^1/𝑞. (62) The following proposition shows that the spaces 𝐹̇^𝛼_𝑝,𝑞(𝐺) are characterized by Lusin and Littlewood-Paley functions.

Proposition 14. Let 𝛼∈ℝ, 0<𝑝, 𝑞<∞, 𝑏∈ℝ and 𝜆>max{Δ/𝑝,Δ/𝑞}. Let 𝑓∈𝒮^′(𝐺)/𝒫 and 𝜙∈𝒜. Put 𝑢(𝑥,𝑘)=𝑓∗(𝐷₂^𝑘𝜙)(𝑥), for 𝑥∈𝐺 and 𝑘∈ℤ. Then one has 𝑔^𝛼_𝑞(𝑢)_𝐿𝑝∼𝑆^𝛼_𝑏,𝑞(𝑢)_𝐿𝑝∼𝐺^𝛼_𝜆,𝑞(𝑢)_𝐿𝑝. (63)

Proof. Step  1. Show that if 𝜆>max{Δ/𝑝,Δ/𝑞} then 𝑔^𝛼_𝑞(𝑢)_𝐿𝑝≲𝐺^𝛼_𝜆,𝑞(𝑢)_𝐿𝑝≲𝑆^𝛼_𝑏,𝑞(𝑢)_𝐿𝑝. (64) Indeed, the proof of the first inequality in (64) is essentially the same as that of [18, Theorem 2.3]. To see the second inequality in (64), one only needs to examine the proofs of Theorems 1, 2 and 4 in [19, Chapter 4] and observe that, although the function 𝑢 considered in [19] is defined on the half space ℝ₊^𝑛+1=ℝ^𝑛×(0,∞), the arguments there can also be adapted to functions 𝑢 which are defined on 𝐺×ℤ.
Step  2. We prove that ‖𝑆^𝛼_𝑏,𝑞(𝑢)‖_𝐿^𝑝≲‖𝑔^𝛼_𝑞(𝑢)‖_𝐿^𝑝. First note that, by an argument similar to the proofs of [19, Theorems 1 and 2], it follows that ‖𝑆^𝛼_𝑏,𝑞(𝑢)‖_𝐿^𝑝∼‖𝑆^𝛼_0,𝑞(𝑢)‖ for every fixed 𝑏∈ℝ. Hence, in view of (38), it is enough to show that 𝑆^𝛼_0,𝑞(𝑢)_𝐿𝑝≲_𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝜆,2−𝑗}𝑓,𝜙^𝑞1/𝑞_𝐿𝑝. (65) But this is a consequence of the following elementary estimate: _{|𝑦−1𝑥|<2−𝑗}2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙𝑦^𝑞2^𝑗Δ𝑑𝑦≲sup_{𝑦∈𝐵𝑥,2−𝑗}2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙𝑦^𝑞1+2^𝑗𝑦⁻¹𝑥^𝜆𝑞≤sup_𝑦∈𝐺2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜙𝑦^𝑞1+2^𝑗𝑦⁻¹𝑥^𝜆𝑞=2^𝑗𝛼𝑀^∗_{𝜆,2−𝑗}𝑓,𝜙(𝑥)^𝑞. (66) The proof of Proposition 14 is thus complete.

Corollary 15. Let 𝐺 be a stratified Lie group. Then 𝐹̇⁰_𝑝,2(𝐺)=𝐻^𝑝(𝐺) with equivalent (quasi-)norms, for 0<𝑝<∞. Here 𝐻^𝑝(𝐺) are Hardy spaces on 𝐺.

Proof. In [3, Chapter 7], Folland and Stein proved the characterization of Hardy spaces 𝐻^𝑝(𝐺) by continuous version Lusin function, for 0<𝑝<∞. Note that the arguments in [3, Chapter 7] are still valid if we replace the continuous version Lusin function by discrete version one defined above; see also [20] for a treatment of discrete version Lusin funcion. This fact together with Proposition 14 yield the identification of 𝐹̇⁰_𝑝,2(𝐺) with 𝐻^𝑝(𝐺) for 0<𝑝<∞.

4. Convolution Singular Integral Operators on 𝐹̇^𝛼_𝑝,𝑞(𝐺)

In this section we study boundedness of convolution singular integral operators on homogeneous Triebel-Lizorkin spaces on stratified Lie groups. Motivated by [21, Section 5.3 in Chapter XIII], we introduce a class of singular convolution kernels as follows.

Definition 16. Let 𝑟 be a positive integer. A kernel of order 𝑟 is a distribution 𝐾∈𝒮^′(𝐺) with the following properties:
(i) 𝐾 coincides with a 𝐶^𝑟 function 𝐾(𝑥) away from the group identity 0 and enjoys the regularity condition: 𝑋^𝐼𝐾(𝑥)≤𝐶_𝐼|𝑥|^{−𝑄−𝑑(𝐼)},for|𝐼|≤𝑟,𝑥≠0. (67)
(ii) 𝐾 satisfies the cancellation condition: For all normalized bump function 𝜙 and all 𝑅>0, we have 𝐾,𝜙^𝑅≤𝐶, (68) where 𝜙^𝑅(𝑥)=𝜙(𝑅𝑥), and 𝐶 is a constant independent of 𝜙 and 𝑅. Here, by a normalized bump function we mean a function 𝜙 supported in 𝐵(0,1) and satisfying 𝑋^𝐼𝜙(𝑥)≤1,∀|𝐼|≤𝑁,∀𝑥∈𝐺, (69) for some fixed positive integer 𝑁.
The convolution operator 𝑇 with kernel of order 𝑟 is called a singular integral operator of order 𝑟.

Remark 17. Using [3, Proposition 1.29], it is easy to verify that (67) is equivalent to the following condition: 𝑌^𝐼𝐾(𝑥)≤𝐶_𝐼|𝑥|^{−𝑄−𝑑(𝐼)},for|𝐼|≤𝑟,𝑥≠0. (70)

Examples of such kernels include the class of distributions which are homogeneous of degree −Δ (see Folland and Stein [3, p. 11] for definition) and agree with 𝐶^∞ functions away from 0. Indeed, assume 𝐾∈𝒮^′(𝐺) is such a distribution, then it is easy to verify that 𝐾 satisfies the regularity condition (i) in Definition 16; moreover, from [3, Proposition 6.13] we see that 𝐾 is a principle value distribution such that ∫_{𝜀<|𝑥|<𝐿}𝐾(𝑥)𝑑𝑥=0 for all 0<𝜀<𝐿<∞. Hence, for every normalized bump function 𝜙, by the homogeneity of 𝐾 we have 𝐾,𝜙^𝑅==𝐾,𝜙lim_{𝜀→0𝜀<|𝑥|<2}𝐾(𝑥𝜙(𝑥)(0≤)−𝜙)𝑑𝑥_|𝑥|<2|𝐾(𝑥)|𝜙(𝑥)−𝜙(0)𝑑𝑥. (71) Using stratified mean value theorem (cf. [3, Theorem 1.41]) and (67)–(69), it is easy to verify that the last integral converges absolutely and is bounded by a constant independent of 𝜙 and 𝑅. Hence 𝐾 satisfies the condition (ii) in Definition 16.

Now we state the main result of this section.

Theorem 18. Let 𝛼∈ℝ, 0<𝑝,𝑞<∞, and let 𝑟 be a positive integer such that 𝑟>Δ/min{𝑝,𝑞}+|𝛼|+2. Suppose 𝑇 is a singular integral operator of order 𝑟. Then 𝑇 extends to a bounded operator on 𝐹̇^𝛼_𝑝,𝑞(𝐺).

If 𝐾∈𝒮^′(𝐺) and 𝑡>0, we define 𝐷_𝑡𝐾 as the tempered distribution given by ⟨𝐷_𝑡𝐾,𝜙⟩=⟨𝐾,𝜙(𝑡⁻¹⋅)⟩ (∀𝜙∈𝒮(𝐺)). For the proof of Theorem 18, we will need the following lemma, in which 𝑏 is the positive constant as in [3, Corollary 1.44].

Lemma 19. Let 𝑟 be a positive integer. Suppose 𝐾 is a kernel of order 𝑟, and 𝜙 is a smooth function supported in 𝐵(0,1/4𝛾²𝑏^𝑟) and having vanishing moments of order 𝑟−1. Then, there exists a constant 𝐶>0 such that for all 𝑗∈ℤ, all 𝐼∈ℕ^𝑛 with |𝐼|≤𝑟, and all 𝑥∈𝐺max𝑋^𝐼𝜙∗𝐷₂^𝑗𝐾(𝑥),𝑌^𝐼𝜙∗𝐷₂^𝑗𝐾𝑥(|𝑥()≤𝐶1+|)^−Δ−𝑟. (72)
Moreover, 𝜙∗(𝐷₂^𝑗𝐾) have vanishing moments of the same order as 𝜙.

Proof. Recall that the convolution of 𝜙∈𝒮(𝐺) with 𝐾∈𝒮^′(𝐺) is defined by 𝜙∗𝐾(𝑥)∶=⟨𝐾,(^𝑥𝜙)^∼⟩, where ^𝑥𝜙 is the function given by ^𝑥𝜙(𝑧)=𝜙(𝑥𝑧), and as before 𝑓(𝑥)∶=𝑓(𝑥⁻¹) for any function 𝑓∶𝐺→ℂ. From [3, p. 38] we see that 𝜙∗(𝐷₂^𝑗𝐾) are 𝐶^∞ functions, 𝑗∈ℤ. We claim that for every 𝑥 with |𝑥|≤1/2𝛾, the function 𝑧↦(^𝑥𝜙)^∼(𝑧) is a normalized bump function multiplied with a constant independent of 𝑥. Indeed, using the quasi-triangle inequality satisfied by the homogeneous norm it is easy to verify that the function 𝑧↦(^𝑥𝜙)^∼(𝑧) is supported in 𝐵(0,1); moreover, since |𝑥|≤1/2𝛾 and since 𝑌^𝐼=_{|𝐽|≤|𝐼|𝑑(𝐽)(𝐼)≥𝑑}𝑃_𝐼,𝐽𝑋^𝐽, (73) where 𝑃_𝐼,𝐽 are polynomials of homogeneous degree 𝑑(𝐽)−𝑑(𝐼) (see [3, Proposition 1.29]), we have 𝑋^𝐼𝑥𝜙^∼𝑧=𝑌()_𝐼𝑥𝜙𝑧⁻¹≲_{|𝐽|≤|𝐼|𝑑(𝐽)(𝐼)≥𝑑}𝑃_𝐼,𝐽𝑧⁻¹𝑋^𝐽𝑥𝜙𝑧⁻¹=_{|𝐽|≤|𝐼|𝑑(𝐽)(𝐼)≥𝑑}𝑃_𝐼,𝐽𝑧⁻¹𝑋^𝐽𝜙𝑥𝑧⁻¹≤𝐶_𝐼. (74) Here 𝐶_𝐼 is a constant depending on 𝐼 but not on 𝑥. Hence the claim is true. The above argument also shows that, for every 𝑥 with |𝑥|≤1/2𝛾 and for every 𝐼∈ℕ^𝑛, [^𝑥(𝑌^𝐼𝜙)]^∼ is a normalized bump function multiplied with a constant 𝐶^′_𝐼 independent of 𝑥. Thus, by the condition (ii) in Definition 16, there exits a constant 𝐶>0 such that for all 𝑗∈ℤ, all 𝐼∈ℕ^𝑛 with |𝐼|≤𝑟, and all 𝑥 with |𝑥|≤1/2𝛾𝑌^𝐼𝜙∗𝐷₂^𝑗𝐾𝑥=𝑌()_𝐼𝜙∗𝐷₂^𝑗𝐾𝑥=𝐷()_2𝑗𝐾,^𝑥𝑌^𝐼𝜙^∼=𝐾,^𝑥𝑌^𝐼𝜙^∼2^−𝑗⋅≤𝐶. (75) From this and [3, Proposition 1.29], we also get that, for all 𝑗∈ℤ, all 𝐼∈ℕ^𝑛 with |𝐼|≤𝑟, and all 𝑥 with |𝑥|≤1/2𝛾𝑋^𝐼𝜙∗𝐷₂^𝑗𝐾(𝑥)≤𝐶_{𝐼′≤|𝐼|𝑑𝐼′(𝐼)≥𝑑}|𝑥|^{𝑑(𝐼′)−𝑑(𝐼)}𝑌^𝐼′𝜙∗𝐷₂^𝑗𝐾(𝑥)≤𝐶. (76)
Let now |𝑥|>1/2𝛾. Let 𝑦∈supp𝜙. Let 𝐼∈ℕ^𝑛 with |𝐼|≤𝑟, and denote by 𝑃_𝑥,𝑌^𝐼_{(𝐷2𝑗𝐾)} the right Taylor polynomial of 𝑌^𝐼(𝐷₂^𝑗𝐾) at 𝑥 of homogeneous degree 𝑟−|𝐼|−1 (see [3, pp. 26-27]). Then by the right-invariant version of [3, Corollary 1.44], we have 𝑋^𝐼𝐷₂^𝑗𝐾𝑦⁻¹𝑥−𝑃_𝑥,𝑋^𝐼_{(𝐷2𝑗𝐾)}𝑦⁻¹≤𝐶𝑦^{𝑟−|𝐼|}sup_{|𝑧|≤𝑏𝑟−|𝐼|𝑦𝑑(𝐽)|𝐼|=𝑟−}𝑌^𝐽𝑋^𝐼𝐷₂^𝐽𝐾(𝑧𝑥). (77) Observe that 𝐷₂^𝑗𝐾 satisfies (67) with the bound 𝐶 independent of 𝑗∈ℤ. Also note that for 𝑦∈supp𝜙 and |𝑧|≤𝑏^{𝑟−|𝐼|}|𝑦| we have 𝑧𝑥∈𝐺⧵{0}. Thus, for all 𝐽 with 𝑑(𝐽)=𝑟−|𝐼| and all 𝑧 with |𝑧|≤𝑏^{𝑟−|𝐼|}|𝑦|, by using (73) and (67) (with 𝐾 replaced by 𝐷₂^𝑗𝐾) we have 𝑌^𝐽𝑋^𝐼𝐷₂^𝑗𝐾(𝑧𝑥)≤_{𝐽′≤|𝐽|𝑑𝐽′(𝐽)≥𝑑}𝑃_𝐽,𝐽′(𝑧𝑥)𝑋^𝐽′+𝐼𝐷₂^𝑗𝐾(𝑧𝑥)≲_{𝐽′≤|𝐽|𝑑𝐽′(𝐽)≥𝑑}|𝑧𝑥|^{𝑑(𝐽′)−𝑑(𝐽)}|𝑧𝑥|^{−Δ−𝑑(𝐽′+𝐼)}≲|𝑧𝑥|^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}. (78) Here, for the second inequality we also used the observation that when 𝑑(𝐽)=𝑟−|𝐼| and |𝐽^′|≤|𝐽|, we have |𝐽^′+𝐼|≤|𝐽+𝐼|≤𝑑(𝐽)+|𝐼|=𝑟−|𝐼|+|𝐼|=𝑟. Inserting (78) into (77) we obtain 𝑋^𝐼𝐷₂^𝑗𝐾𝑦⁻¹𝑥−𝑃_𝑥,𝑋^𝐼_{(𝐷2𝑗𝐾)}𝑦⁻¹≤𝐶𝑦^{𝑟−|𝐼|}sup_{|𝑧|≤𝑏𝑟−|𝐼|𝑦}|𝑧𝑥|^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}. (79) Notice that for |𝑥|≥1/2𝛾, 𝑦∈supp𝜙 and |𝑧|≤𝑏^{𝑟−|𝐼|}|𝑦|, we have |𝑧𝑥|∼|𝑥|. Thus, by using the vanishing moments of 𝜙 and (79), we have 𝑋^𝐼𝜙∗𝐷₂^𝑗𝐾𝑥=𝜙𝑦𝑋()_𝐼𝐷₂^𝑗𝐾𝑦⁻¹𝑥𝜙𝑦𝑋𝑑𝑦≤^𝐼𝐷₂^𝑗𝐾𝑦⁻¹𝑥−𝑃_𝑥,𝑋^𝐼_{(𝐷2𝑗𝐾)}𝑦⁻¹𝑑𝑦≤𝐶sup_{|𝑧|≤𝑏𝑟−|𝐼|𝑦}|𝑧𝑥|^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}𝑦^{𝑟−|𝐼|}𝜙𝑦|𝑥|𝑑𝑦≤𝐶^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}𝑦^{𝑟−|𝐼|}𝜙𝑦|𝑥|𝑑𝑦≤𝐶^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}. (80) Combining (76) and (80), we see that, for all 𝑗∈ℤ, all 𝐼∈ℕ^𝑛 with |𝐼|<𝑟, and all 𝑥∈𝐺𝑋^𝐼𝜙∗𝐷₂^𝑗𝐾(𝑥)≤𝐶(1+|𝑥|)^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}. (81) From this and (73), we also get that, for all 𝑗∈ℤ, all 𝐼∈ℕ^𝑛 with |𝐼|<𝑟, and all 𝑥∈𝐺𝑌^𝐼𝜙∗𝐷₂^𝑗𝐾(𝑥)≤_{𝐼′≤|𝐼|𝑑𝐼′(𝐼)≥𝑑}𝑃_𝐼,𝐼′(𝑥)𝑋^𝐼′𝜙∗𝐷₂^𝑗𝐾(𝑥)≤𝐶_{𝐼′≤|𝐼|𝑑𝐼′(𝐼)≥𝑑}|𝑥|^{𝑑(𝐼′)−𝑑(𝐼)}(1+|𝑥|)^{−Δ−𝑟+|𝐼′|−𝑑(𝐼′)}≤𝐶(1+|𝑥|)^{−Δ−𝑟+|𝐼|−𝑑(𝐼)}. (82) Since |𝐼|≤𝑑(𝐼), (81) along with (82) yield (72).
It is straightforward to verify that 𝜙∗(𝐷₂^𝑗𝐾) have vanishing moments of the same order as 𝜙. The proof of Lemma 19 is therefore complete.

The proof of Theorem 18 also relies on the existence of smooth functions with compact support and having arbitrarily high order vanishing moments.

Lemma 20. Given any nonnegative integer 𝐿 and any positive number 𝛿, there exists a function 𝜁̂∈𝒮(ℝ⁺) with the following properties:(i)|𝜁̂(𝜆)|≥𝐶>0 for 𝜆∈[2^{−2(𝑘0+1)},2^{−2(𝑘0−1)}], with 𝑘₀ some (large) positive integer;(ii)𝜁 is a Schwartz function on 𝐺 having vanishing moments of order 𝐿;(iii)supp𝜁⊂𝐵(0,𝛿).

Proof. From the appendix of [22] we see that there exists 𝜃∈𝒮(ℝ⁺) such that 𝜃(0)=1 and 𝜃 has compact support. Now let us define 𝜁̂(𝜆)=(𝑡⁻²𝜆)^𝑘𝜃(𝑡⁻²𝜆), 𝜆∈ℝ⁺. Here 𝑡>0 and 𝑘 is a nonnegative integer. Then 𝜁(𝑥)=𝐷_𝑡(ℒ^𝑘𝜃)(𝑥)=𝑡^Δ(ℒ^𝑘𝜃)(𝑡𝑥). Hence, if we take 𝑡,𝑘 sufficiently large, then (ii) and (iii) follow immediately. Moreover, since 𝜃(0)=1, it is easy to see that (i) is also satisfied, provided that 𝑘₀ is sufficiently large.

We are now ready to prove Theorem 18.

Proof of Theorem 18. Choose a function 𝜁̂∈𝒮(ℝ⁺) which satisfies conditions (i)–(iii) in Lemma 20 with 𝐿=𝑟−1 and 𝛿=1/4𝛾²𝑏^𝑟. The condition (i) guarantees the existence of a function 𝜓∈𝒮(ℝ⁺) with the following properties: supp𝜓⊂2^{−2(𝑘0+1)},2^{−2(𝑘0−1)},𝜓(𝜆)>0for𝜆∈2^{−2(𝑘0+1)},2^{−2(𝑘0−1)},_𝑗∈ℤ𝜓2^−2𝑗𝜆𝜁̂2^−2𝑗𝜆=1,∀𝜆∈ℝ⁺. (83) Note that 𝜓(2^−2𝑘0⋅)∈𝒜. For 𝑓∈𝒵(𝐺), by Lemma 3 we have 𝑓=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜓∗𝐷₂^𝑗𝜁 (84) with convergence in 𝒵(𝐺). Let 𝜙∈𝒜, and let 𝐾 be the convolution kernel of the operaotor 𝑇. Then we have the representation 𝑓∗𝐾∗𝐷₂^ℓ𝜙=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜓∗𝐷₂^𝑗𝜁∗𝐾∗𝐷₂^ℓ𝜙=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜓∗𝐷₂^𝑗𝜁∗𝐷₂^−𝑗𝐾∗𝐷₂^ℓ𝜙, (85) which holds pointwise and also in the sense of 𝒮^′(𝐺). Since (by Lemma 19) 𝜁∗(𝐷₂^−𝑗𝐾) satisfies the decay condition (72) (with the bound 𝐶 independent of 𝑗∈ℤ) and has vanishing moments of the same order as 𝜁, from the proof of Lemma 2 we see that 𝐷₂^𝑗𝜁∗𝐷₂^−𝑗𝐾∗𝐷₂^ℓ𝜙𝑦≤𝐶2^{−|𝑗−ℓ|(𝑟−2)}2^{(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ𝑦^𝑟+Δ. (86) This together with (85) gives that 𝑓∗𝐾∗𝐷₂^ℓ𝜙(𝑥≤_𝑗∈ℤ)𝑓∗𝐷₂^𝑗𝜓(𝑧×𝐷_2𝑗)𝜁∗𝐷₂^−𝑗𝐾∗𝐷₂^ℓ𝜙𝑧⁻¹𝑥≲𝑑𝑧_𝑗∈ℤ2^{−|𝑗−ℓ|(𝑟−2)}2^{(𝑗∧ℓ)Δ}𝑓∗𝐷₂^𝑗𝜓(𝑧)1+2^𝑗∧ℓ𝑧⁻¹𝑥^𝑟+Δ𝑑𝑧≲_𝑗∈ℤ2^{−|𝑗−ℓ|(𝑟−2−𝑎)}sup_𝑧∈𝐺𝑓∗𝐷₂^𝑗𝜓(𝑧)1+2^𝑗𝑧⁻¹𝑥^𝑎×2^{(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ𝑧⁻¹𝑥^{𝑟+Δ−𝑎}𝑑𝑧, (87) where for the last inequality we used that (1+2^𝑗∧ℓ|𝑧⁻¹𝑥|)^−𝑎≤2^{|𝑗−ℓ|𝑎}(1+2^𝑗|𝑧⁻¹𝑥|)^−𝑎. By the hypothesis we can choose 𝑎 such that 𝑎>Δ/min{𝑝,𝑞} and 𝑟−2−𝑎−|𝛼|>0. From [3, Corollary 1.17] we see that the last integral converges absolutely. Consequently, we obtain 2^ℓ𝛼𝑓∗𝐾∗𝐷₂^ℓ𝜙(𝑥≲_𝑗∈ℤ)2^{−|𝑗−ℓ|(𝑟−2−𝑎−|𝛼|)}2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,𝜓(𝑥). (88) Hence, it follows by [15, Lemma 2] that _ℓ∈ℤ2^ℓ𝛼𝑓∗𝐾∗𝐷₂^ℓ𝜙(𝑥)^𝑞1/𝑞_𝐿𝑝≲_𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝑎,2−𝑗}𝑓,𝜓^𝑞1/𝑞_𝐿𝑝. (89) This together with (38) imply that ‖𝑓∗𝐾‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}≲‖𝑓‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)} for all 𝑓∈𝒵(𝐺). Since 𝒵(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺), 𝑓↦𝑓∗𝐾 extends to an bounded operator on 𝐹̇^𝛼_𝑝,𝑞(𝐺). This completes the proof of Theorem 18.

Corollary 21. Let 𝛼∈ℝ, 0<𝑝,𝑞<∞, and let 𝑘 be a nonnegative integer. Then 𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}∼_{𝑑(𝐼)=𝑘}𝑋^𝐼𝑓_{̇𝐹𝛼−𝑘𝑝,𝑞(𝐺)}. (90)

Proof. Note that by the Poincaré-Birkhoff-Witt theorem (cf. [23, I.2.7]), the operators 𝑋^𝐼 form a basis of the algebra of the left-invariant differential operators on 𝐺. By this fact and the stratification of 𝐺, it suffices to show that 𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}∼^𝜈_𝑗=1𝑋_𝑗𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}. (91) To this end, we first note that when restricted to Schwartz functions, 𝑋_𝑗ℒ^−1/2 are convolution operators with distribution kernels homogeneous of degree −Δ and coincide with smooth functions in 𝐺⧵{0}. This follows from the fact that the operator ℒ^−1/2 is a convolution operator whose distribution kernel is homogeneous of degree −Δ+1 and coincides with a smooth function in 𝐺⧵{0} (see [2, Proposition 3.17]). Hence, by Theorem 18, 𝑋_𝑗ℒ^−1/2 extend to bounded operators on 𝐹̇^𝛼_𝑝,𝑞(𝐺). From this fact and the lifting property (Theorem 7), we deduce that 𝑋_𝑗𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}=𝑋_𝑗ℒ^−1/2ℒ^1/2𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}≲ℒ^1/2𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}∼𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}. (92) Hence ∑^𝜈_𝑗=1‖𝑋_𝑗𝑓‖_̇𝐹^𝛼−1_{𝑝,𝑞(𝐺)}≲‖𝑓‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}. To see the converse, we need to use [2, Lemma 4.12], which asserts that there exists tempered distributions 𝐾₁,…,𝐾_𝜈 homogeneous of degree −Δ+1 and coinciding with smooth functions in 𝐺⧵{0} such that 𝑓=∑^𝜈_𝑗=1(𝑋_𝑗𝑓)∗𝐾_𝑗 for all 𝑓∈𝒮(𝐺). By this result and Theorem 7, we have, at least for 𝑓∈ℛ(𝐺) (⊂Dom(ℒ^−1/2)), 𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}=ℒℒ^−1/2𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}=ℒ𝑓∗𝑅_{1̇𝐹𝛼−1𝑝,𝑞(𝐺)}=ℒ^𝜈_𝑗=1𝑋_𝑗𝑓∗𝐾_𝑗∗𝑅_{1̇𝐹𝛼−1𝑝,𝑞(𝐺)}=^𝜈_𝑗=1𝑋_𝑗𝑓∗ℒ𝐾_𝑗∗𝑅_{1̇𝐹𝛼−1𝑝,𝑞(𝐺)}, (93) where 𝑅₁ is the convolution kernel of the operator ℒ^−1/2. As is indicated in [2, p. 190], ℒ(𝐾_𝑗∗𝑅₁) are distributions homogeneous of degree −Δ and coincide with smooth functions away from 0. Thus it follows by Theorem 18 that 𝑋_𝑗𝑓∗ℒ𝐾_𝑗∗𝑅_{1̇𝐹𝛼−1𝑝,𝑞(𝐺)}≲𝑋_𝑗𝑓_{̇𝐹𝛼−1𝑝,𝑞(𝐺)}. (94) Inserting this into (93), we obtain ‖𝑓‖_̇𝐹^𝛼_{𝑝,𝑞(𝐺)}≲∑^𝜈_𝑗=1‖𝑋_𝑗𝑓‖_̇𝐹^𝛼−1_{𝑝,𝑞(𝐺)} for all 𝑓∈ℛ(𝐺). Since ℛ(𝐺) is dense in 𝐹̇^𝛼_𝑝,𝑞(𝐺), the latter inequality also holds for all 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺). This completes the proof.

Appendix

Smooth Atomic Decomposition of 𝐹̇^𝛼_𝑝,𝑞(𝐺)

In this appendix we show that homogeneous Triebel-Lizorkin spaces on stratified Lie groups admit smooth atomic decomposition. We follow the proof of [24, Theorem 6.6.3] with necessary modifications.

Equipped with Haar measure and the quasi-distance defined by the homogeneous norm, the group 𝐺 is a space of homogeneous type in the sense of Coifman and Weiss [25]. On such type of spaces, Christ [26] constructed a dyadic grid analogous to that of the Euclidean space as follows.

Lemma A.1. Let 𝐺 be a stratified Lie group. There exists a collection {𝑄^𝑘_𝛼∶𝑘∈ℤ,𝛼∈𝒜_𝑘} of open subsets of 𝐺, where 𝒜_𝑘 is some (possibly finite) index set, and constants 𝛿∈(0,1) and 𝐴₁,𝐴₂>0 such that(i)|𝐺⧵⋃_{𝛼∈𝒜𝑘}𝑄^𝑘_𝛼|=0 for each fixed 𝑘 and 𝑄^𝑘_𝛼∩𝑄^𝑘_𝛽=∅ if 𝛼≠𝛽;(ii)for any 𝛼,𝛽,𝑘,ℓ with ℓ≥𝑘, either 𝑄^ℓ_𝛽⊂𝑄^𝑘_𝛼 or 𝑄^ℓ_𝛽∩𝑄^𝑘_𝛼=∅;(iii)for each (𝑘,𝛼) and ℓ<𝑘, there exists a unique 𝛽 such that 𝑄^𝑘_𝛼⊂𝑄^ℓ_𝛽;(iv)diam(𝑄^𝑘_𝛼)≤𝐴₁𝛿^𝑘, where diam(𝑄^𝑘_𝛼)∶=sup{|𝑥⁻¹𝑦|∶𝑥,𝑦∈𝑄^𝑘_𝛼};(v)each 𝑄^𝑘_𝛼 contains some ball 𝐵(𝑧^𝑘_𝛼,𝐴₂𝛿^𝑘), where 𝑧^𝑘_𝛼∈𝐺.

The set 𝑄^𝑘_𝛼 can be thought of as a dyadic cube with diameter roughly 𝛿^𝑘 and centered at 𝑧^𝑘_𝛼. We denote by 𝒟 the family of all dyadic cubes on 𝐺. For 𝑘∈ℤ, we set 𝒟_𝑘={𝑄^𝑘_𝛼∶𝛼∈𝒜_𝑘}, so that 𝒟=⋃_𝑘∈ℤ𝒟_𝑘. For any dyadic cube 𝑄∈𝒟, we denote by 𝑧_𝑄 the “center” of 𝑄 and by 𝑘_𝑄 the unique integer 𝑘 such that 𝑄∈𝒟_𝑘.

Without loss of generality, in what follows we assume 𝛿=1/2. Otherwise we need to replace 2^𝑗 in Definition 4 by 𝛿^−𝑗, and also make some other necessary changes; see [27, pp. 96–98] for more details.

Definition A.2. Let 𝑄 be a dyadic cube and let 𝐿 be a nonnegative integer. A smooth function 𝑎_𝑄 on 𝐺 is called a smooth L-atom for 𝑄 if it satisfies(i)𝑎_𝑄 is supported in 𝐵(𝑧_𝑄,(𝛾(1+𝐴₁)/𝐴₂)2^−𝑘𝑄);(ii)∫𝑎_𝑄(𝑥)𝑃(𝑥)𝑑𝑥=0 for all 𝑃∈𝒫_𝐿;(iii)|𝑋^𝐼𝑎_𝑄(𝑥)|≤|𝑄|^{−(𝑑(𝐼)/Δ)−(1/2)} for all multi-indices 𝐼 with |𝐼|≤𝐿+1.

Definition A.3. Let 𝛼∈ℝ and 0<𝑝,𝑞<∞. The sequence space 𝑓̇^𝛼_𝑝,𝑞 consists of all sequences {𝑠_𝑄}_𝑄∈𝒟 such that the function 𝑔^𝛼,𝑞𝑠_𝑄𝑄=_𝑄∈𝒟|𝑄|^{−(𝛼/Δ)−(1/2)}𝑠_𝑄𝜒_𝑄^𝑞1/𝑞 (A.1) is in 𝐿^𝑝(𝐺). For such a sequence 𝑠={𝑠_𝑄}_𝑄 we set ‖𝑠‖_̇𝑓^𝛼_𝑝,𝑞=𝑔^𝛼,𝑞(𝑠)_𝐿𝑝. (A.2)

The smooth atomic decomposition of homogeneous Triebel-Lizorkin spaces on stratified Lie groups can be stated as follows.

Theorem A.4. Let 𝛼∈ℝ, 0<𝑝,𝑞<∞, and let 𝐿 be a nonnegative integer satisfying 𝐿>[Δmax(1,1/𝑝,1/𝑞)−Δ−𝛼]+1. Then there is a constant 𝐶_{Δ,𝑝,𝑞,𝛼} such that for every sequence of smooth 𝐿-atoms {𝑎_𝑄}_𝑄∈𝒟 and every sequence of complex scalars {𝑠_𝑄}_𝑄∈𝒟 one has _𝑄∈𝒟𝑠_𝑄𝑎_{𝑄̇𝐹𝛼𝑝,𝑞(𝐺)}≤𝐶_{Δ,𝑝,𝑞,𝛼}𝑠_{𝑄𝑄∈𝒟̇𝑓𝛼𝑝,𝑞}. (A.3) Conversely, there is a constant 𝐶^′_{Δ,𝑝,𝑞,𝛼} such that given any distribution 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺) and any 𝐿≥0, there exists a sequence of smooth 𝐿-atoms {𝑎_𝑄}_𝑄∈𝒟 such that 𝑓=_𝑄∈𝒟𝑠_𝑄𝑎_𝑄, (A.4) where the sum converges in 𝒮^′(𝐺)/𝒫 and moreover 𝑠_{𝑄𝑄̇𝑓𝛼𝑝,𝑞}≤𝐶^′_{Δ,𝑝,𝑞,𝛼}𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}. (A.5)

Proof. Let 𝑄∈𝒟_ℓ for some ℓ∈ℤ, and let 𝑎_𝑄 be a smooth 𝐿-atom for 𝑄. We set 𝑎^′_𝑄(𝑥)=𝑎_𝑄(𝑧_𝑄𝑥). Then 𝑎^′_𝑄 is supported in 𝐵(0,(𝛾(1+𝐴₁)/𝐴₂)2^−ℓ); moreover (since 1+2^ℓ|𝑦|∶1 for 𝑦∈ supp 𝑎^′_𝑄), 𝑋^𝐼𝑎^′_𝑄𝑦≲2^−ℓΔ/22^{ℓ𝑑(𝐼)+ℓΔ}1+2^ℓ𝑦^𝑁,∀|𝐼|≤𝐿+1, (A.6) where 𝑁 can be chosen to be arbitrarily large. Set 𝑎_𝑄^′′(𝑥)=2^−ℓΔ𝑎^′_𝑄(2^−ℓ𝑥), that is, 𝑎^′_𝑄=𝐷₂^ℓ𝑎_𝑄^′′. Then the above inequality can be rewritten as 𝑋^𝐼𝑎_𝑄^′′𝑦≲2^−ℓΔ/211+𝑦^𝑁,∀|𝐼|≤𝐿+1. (A.7) Using this estimate, (73), and that supp𝑎_𝑄^′′⊂𝐵(0,(𝛾(1+𝐴₁))/𝐴₂), we also deduce that 𝑋^𝐼𝑎_𝑄^′′∼𝑦=𝑌^𝐼𝑎_𝑄^′′𝑦⁻¹≲_{|𝐽|≤|𝐼|𝑑(𝐽)(𝐼)≥𝑑}𝑦^{−1𝑑(𝐽)−𝑑(𝐼)}𝑋^𝐽𝑎_𝑄^′′𝑦⁻¹≲2^−ℓΔ/211+𝑦^𝑁,∀|𝐼|≤𝐿+1. (A.8) Thus, it follows from Lemma 2 that 𝑎^′_𝑄∗𝐷₂^𝑗𝜙(𝑥)=𝐷₂^ℓ𝑎_𝑄^′′∗𝐷₂^𝑗𝜙(𝑥)≲2^−ℓΔ/22^{−|𝑗−ℓ|𝐿}2^{−(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁, (A.9) where 𝜙∈𝒜, and 𝑁 can be taken to be arbitrarily large. Consequently, 𝑎_𝑄∗𝐷₂^𝑗𝜙(𝑥=𝑎_′𝑄)∗𝐷₂^𝑗𝜙𝑧_𝑄⁻¹𝑥≲2^−ℓΔ/22^{−|𝑗−ℓ|𝐿}2^{−(𝑗∧ℓ)Δ}1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁. (A.10)
Note that if 𝑟∈(0,1] and 𝑁>Δ/𝑟 then we have _{𝑄∈𝒟ℓ}𝑠_𝑄1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁≤_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^{𝑁𝑟1/𝑟}=_{𝐺𝑄∈𝒟ℓ}𝑠_𝑄^𝑟|𝑄|⁻¹𝜒_𝑄(𝑧)1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁𝑟𝑑𝑧^1/𝑟≲2^ℓΔ/𝑟_{|𝑥−1𝑧|≤(2𝛾𝐴1)2−(𝑗∧ℓ)𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁𝑟+𝑑𝑧_{∞𝑘=0(2𝛾𝐴1)2𝑘2−(𝑗∧ℓ)≤|𝑥−1𝑧|≤(2𝛾𝐴1)2𝑘+12−(𝑗∧ℓ)}×_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)1+2^𝑗∧ℓ𝑧_𝑄⁻¹𝑥^𝑁𝑟𝑑𝑧^1/𝑟≲2^ℓΔ/𝑟∫_{|𝑥−1𝑧|≤(2𝛾𝐴1)2−(𝑗∧ℓ)}∑_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)+𝑑𝑧_∞𝑘=01+𝐴₁2^{𝑘−𝑁𝑟}×_{(2𝛾𝐴1)2𝑘2−(𝑗∧ℓ)≤|𝑥−1𝑧|≤(2𝛾𝐴1)2𝑘+12−(𝑗∧ℓ)}×_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)𝑑𝑧^1/𝑟≲2^ℓΔ/𝑟2^{−(𝑗∧ℓ)Δ}𝐵𝑥,2𝛾𝐴₁2^{−(𝑗∧ℓ)𝐵(𝑥,2𝛾𝐴}₁^{2−(𝑗∧ℓ))}×_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)+𝑑𝑧_∞𝑘=01+𝐴₁2^{𝑘−𝑁𝑟}2^(𝑘+1)Δ2^{−(𝑗∧ℓ)Δ}𝐵𝑥,2𝛾𝐴₁2^𝑘+12^{−(𝑗∧ℓ)}×_{𝐵(𝑥,2𝛾𝐴12𝑘+12−(𝑗∧ℓ))}×_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑧)𝑑𝑧^1/𝑟≤2^ℓΔ/𝑟2^{−(𝑗∧ℓ)Δ}+^∞_𝑘=01+𝐴₁2^{𝑘−𝑁𝑟}2^(𝑘+1)Δ2^{−(𝑗∧ℓ)Δ1/𝑟}×𝑀∑_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑥)^1/𝑟≲2^ℓΔ/𝑟2^{−(𝑗∧ℓ)Δ/𝑟}𝑀_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑥)^1/𝑟=2^{max(ℓ−𝑗,0)Δ/𝑟}𝑀_{𝑄∈𝒟ℓ}𝑠_𝑄^𝑟𝜒_𝑄(𝑥)^1/𝑟. (A.11) Here we used the fact that for 𝑧∈𝑄 and |𝑥⁻¹𝑧|≥(2𝛾𝐴₁)2^𝑘2^{−(𝑗∧ℓ)} one has (1+2^𝑗∧ℓ|𝑧_𝑄⁻¹𝑥|)^𝑁𝑟≥(1+𝐴₁2^−𝑘)^𝑁𝑟, which can be easily verified by using Lemma 19 (iv) and the quasi-triangle inequality satisfied by the homogeneous norm. The above estimate and (A.10), along with the argument in [24, pp. 80-81], yield (A.3).
Now we show the converse statement of the theorem. By Lemma 20, there exists 𝜁̂∈𝒮(ℝ⁺) such that |𝜁̂(𝜆)|≥𝐶>0 for 𝜆∈[2^{−2(𝑘0+1)},2^{−2(𝑘0−1)}] for some (large) positive integer 𝑘₀, and that 𝜁 is supported in 𝐵(0,1) and has vanishing moments of order 𝐿. Then it is possible to find a function 𝜓∈𝒮(ℝ⁺) with the properties that supp𝜓⊂[2^{−2(𝑘0+1)},2^{−2(𝑘0−1)}], |𝜓(𝜆)|>0 for 𝜆∈(2^{−2(𝑘0+1)},2^{−2(𝑘0−1)}), and ∑_𝑗∈ℤ𝜓(2^−2𝑗𝜆)𝜁̂(2^−2𝑗𝜆)=1 for 𝜆∈ℝ⁺. Note that 𝜓(2^−2𝑘0⋅)∈𝒜. Given 𝑓∈𝐹̇^𝛼_𝑝,𝑞(𝐺), it follows from Lemma 3 that 𝑓=_𝑗∈ℤ𝑓∗𝐷₂^𝑗𝜓∗𝐷₂^𝑗𝜁 (A.12) with the convergence in 𝒮^′(𝐺)/𝒫. Let us decompose 𝑓 as 𝑓=_{𝑗∈ℤ𝑄∈𝒟𝑗𝑄}𝑓∗𝐷₂^𝑗𝜓𝑦𝐷₂^𝑗𝜁𝑦⁻¹𝑥𝑑𝑦=_{𝑗∈ℤ𝑄∈𝒟𝑗}𝑠_𝑄𝑎_𝑄, (A.13) where we have set, for 𝑄∈𝒟_𝑗, 𝑠_𝑄≡|𝑄|^1/2sup_𝑦∈𝑄𝑓∗𝐷₂^𝑗𝜓𝑦sup_{|𝐼|≤𝐿+1}𝑋^𝐼𝜁_𝐿1,𝑎_𝑄(𝑥)≡1𝑠_𝑄𝑄𝑓∗𝐷₂^𝑗𝜓𝑦𝐷₂^𝑗𝜁𝑦⁻¹𝑥𝑑𝑦. (A.14) It is straightforward to verify that 𝑎_𝑄 is supported in 𝐵(𝑧_𝑄,(𝛾(1+𝐴₁)/𝐴₂)2^−𝑗), and that 𝑎_𝑄 has vanishing moments of the same order as 𝜁. Moreover, for 𝑄∈𝒟_𝑗 we have 𝑋^𝐼𝑎_𝑄≤2^Δ𝑗2^{𝑑(𝐼)𝑗}𝑠_𝑄𝑄𝑓∗𝐷₂^𝑗𝜓𝑦𝑋^𝐼𝜁2^𝑗𝑦⁻¹𝑥≤2𝑑𝑦_{𝑑(𝐼)𝑗}𝑠_𝑄𝑋^𝐼𝜁_𝐿1sup_𝑦∈𝑄𝑓∗𝐷₂^𝑗𝜓𝑦≤|𝑄|^{−(𝑑(𝐼)/Δ)−(1/2)} (A.15) for all |𝐼|≤𝐿+1. Hence the function 𝑎_𝑄 a smooth 𝐿-atom for 𝑄. Choose any 𝜆>Δ/min{𝑝,𝑞}. We note that _{𝑄∈𝒟𝑗}|𝑄|^{−(𝛼/Δ)−(1/2)}𝑠_𝑄𝜒_𝑄(𝑥)^𝑞∼_{𝑄∈𝒟𝑗}2^𝑗𝛼sup_𝑦∈𝑄𝑓∗𝐷₂^𝑗𝜓𝑦𝜒_𝑄(𝑥)^𝑞≲sup_{𝑦∈𝐵𝑥,𝐴12−𝑗}2^𝑗𝛼𝑓∗𝐷₂^𝑗𝜓𝑦^𝑞≲2^𝑗𝛼𝑀^∗_{𝜆,2−𝑗}𝑓,𝜓(𝑥)^𝑞. (A.16) We thus obtain, using (38), 𝑠_{𝑄𝑄̇𝑓𝛼𝑝,𝑞}≲_𝑗∈ℤ2^𝑗𝛼𝑀^∗_{𝜆,2−𝑗}𝑓,𝜓^𝑞1/𝑞_𝐿𝑝≲𝑓_{̇𝐹𝛼𝑝,𝑞(𝐺)}. (A.17) This proves (A.5).

Acknowledgments

The author would like to thank Professor Hitoshi Arai for his patient guidance and constant support. He is also grateful to Professor Yoshihiro Sawano for his valuable comments.