# Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Authors: Guorong Hu
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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. TheLp-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 Section2, in Section 3 we introduce homogeneous Triebel-Lizorkin spaces F˙p,qα(G) on stratified Lie groups, and give some basic properties of them. In Section 4 we show the F˙p,qα(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⊕⋯⊕Vm, with [V1,Vk]=Vk+1 for 1≤k≤m-1 and [V1,Vm]=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 ℝn and the Heisenberg group ℍn.The algebra𝔤 is equipped with a family of dilations{δt:t>0} which are the algebra automorphisms defined by
(1)δt(∑j=1m‍Xj)=∑j=1m‍tjXj(Xj∈Vj).
Under our identification of G with 𝔤, δt may also be viewed as a map G→G. We generally write tx instead of δt(x), for x∈G. We shall denote by
(2)Δ=∑j=1m‍j[dim(Vj)]
the  homogeneous dimension of G.Ahomogeneous 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 radiusraboutx 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 |δt(E)|=tΔ|E|. All integrals on G are with respect to (the normalization of) Haar measure. Convolution is defined by
(3)f*g(x)=∫‍f(y)g(y-1x)dy=∫‍f(xy-1)g(y)dy.We consider𝔤 as the Lie algebra of all left-invariant vector fields on G, and fix a basis X1,…,Xn of 𝔤, obtained as a union of bases of the Vj. In particular, X1,…,Xν, with ν=dim(V1), is a basis of V1. We denote by Y1,…,Yn the corresponding basis for right-invariant vector fields, that is,
(4)Yjf(x)=ddtf(exp(tXj)x)∣t=0.
If I=(i1,…,in)∈ℕn is a multi-index we set XI=X1i1⋯Xnin and YI=Y1i1⋯Ynin. Moreover, we set
(5)|I|=∑k=1n‍ik,d(I)=∑k=1n‍dkik,
where the integers d1≤⋯≤dn are given according to that Xk∈Vdk. Then XI (resp., YI) is a left-invariant (resp., right-invariant) differential operator, homogeneous of degree d(I), with respect to the dilations δt, t>0.A complex-valued functionP on G is called a polynomial on G if P∘exp is a polynomial on 𝔤. Let ξ1,…,ξn be the basis for the linear forms on 𝔤 dual to the basis X1,…,Xn for 𝔤, and set ηj=ξj∘exp-1. From our definition of polynomials on G, η1,…,ηn are generators of the algebra of polynomials on G. Thus, every polynomial on G can be written uniquely as
(6)P=∑I‍aIηI,aI∈ℂ,
where all but finitely many of the coefficients vanish, and ηI=ηi1⋯ηin. A polynomial of the type (6) is called of homogeneous degreeL, where L∈ℕ, if d(I)≤L holds for all multi-indices I with aI≠0. We let 𝒫 denote the space of all polynomials on G, and let 𝒫L denote the space of polynomials on G of homogeneous degree L. Note that 𝒫L 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
(7)∀P∈𝒫L:∫G‍f(x)P(x)dx=0,
with the absolute convergence of the integral.The Schwartz class onG is defined by
(8)𝒮(G)={f∈C∞(G):P∂|I|f∂ηi1⋯∂ηin∈L∞(G),∀I∈ℕn,∀P∈𝒫∂|I|f∂ηi1…∂ηin};
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
(9)∥ϕ∥(L,N)=supd(I)≤L,x∈G(1+|x|)N(|XIϕ(x)|+|XIϕ~(x)|).
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 basisX1,…,Xn, and ν=dim(V1), the sub-Laplacian is defined by ℒ=-∑j=1ν‍Xj2. 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
(10)ℒ=∫0∞‍λdE(λ),
where dE(λ) is the projection measure. If m is a bounded Borel measurable function on [0,∞), the operator
(11)m(ℒ)=∫0∞‍m(λ)dE(λ)
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
(12)m(ℒ)f=f*M,∀f∈𝒮(G).
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,∞).Let𝒮(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.
Ifm∈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 in𝒮(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 functionh on G and t>0, we define the L1-normalized dilation of h by
(13)Dth(x)=tΔh(tx).
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.
LetL,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,
(14)|(D2jϕ)*(D2ℓψ)(x)|≤C∥ϕ∥(L+1,N)∥ψ∥(L+1,N)2-|j-ℓ|L2(j∧ℓ)Δ(1+2j∧ℓ|x|)N,
where j∧ℓ:=min{j,ℓ}.Proof.
Using dilations and the facts(D2jϕ)*(D2ℓψ)(x)=(D2ℓψ~)*(D2jϕ~)(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↦Px,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 ψ(15)|ϕ*(D2ℓψ)(x)|≤∫‍|ϕ(xy-1)-Px,ϕ(y-1)||(D2ℓψ)(y)|dy≡∫D1‍+∫D2‍+∫D3‍.
Fory∈D1, the stratified Taylor formula (cf. [3, Corollary 1.44]) yields that, with b a suitable positive constant,
(16)|ϕ(xy-1)-Px,ϕ(y-1)|≲|y|L+1sup|z|≤bL+1|y|d(I)=L+1|(XIϕ)(xz)|≲∥ϕ∥(L+1,N)sup|z|≤bL+1|y|L+1(1+|xz|)N≲∥ϕ∥(L+1,N)|y|L+1(1+|x|)N,
since 2bL+1(1+|xz|)≥2bL+1+|xz|≥2bL+1+|x|/γ-|z|≥bL+1+|x|/γ≳(1+|x|)/γ if |z|≤bL+1. Hence we have
(17)∫D1≲‍∥ϕ∥(L+1,N)∥ψ∥(0,N)2ℓΔ(1+|x|)N∫D1‍|y|L+1(1+2ℓ|y|)Ndy=∥ϕ∥(L+1,N)∥ψ∥(0,N)2-ℓ(L+1)(1+|x|)N∫G‍|2ℓy|L+1(1+2ℓ|y|)Nd(2ℓy)≲∥ϕ∥(L+1,N)∥ψ∥(0,N)2-ℓ(L+1)(1+|x|)N,
where for the last inequality we used [3, Corollary 1.17] and that N-L-1≥Δ+1.
Fory∈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
(18)1+2ℓ|y|≥2ℓ|y|≥2ℓ(1+|y|)2≥2ℓ(1+|x|)4γ,|y|L≤(γ+12)L|x|L.
Also, we note that by [12, Proposition 20.3.14] the left Taylor polynomial Px,ϕ is of the form
(19)Px,ϕ(y)=ϕ(x)+∑h=1L‍∑k=1h‍∑1≤i1,…,ik≤ndi1+⋯+dik=h‍ηi1(y)…ηik(y)k!Xi1⋯Xikϕ(x),
where the integers d1≤⋯≤dn are given according to that Xj∈Vdj. From these remarks, it follows that
(20)∫D2‍≲∥ϕ∥(L,N)∥ψ∥(0,N)×∫D2‍[1(1+|xy-1|)N+∑0≤h≤L‍|y|h(1+|x|)N]×2ℓΔ(1+|2ℓy|)Ndy≲∥ϕ∥(L,N)∥ψ∥(0,N)∫D2‍[1(1+|xy-1|)N+|x|L(1+|x|)N]×2ℓΔ[2ℓ(1+|x|)]Ndy=∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓ(N-Δ)(1+|x|)N×[∫G‍1(1+|xy-1|)Ndy+|x|L(1+|x|)N∫|xy-1|≤|x|/2γ‍dy]≲∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓ(N-Δ)(1+|x|)N×[∫G‍1(1+|y|)Ndy+|x|L(1+|x|)N∫|y|≤γ(|x|+|x|/2γ)‍dy]≲∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓ(N-Δ)(1+|x|)N[1+|x|L+Δ(1+|x|)N]≲∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓL(1+|x|)N,
where we used that |{y:|y|≤γ(|x|+|x|/2γ)}|~|x|Δ and N-Δ>L.
Fory∈D3 we have |xy-1|>|x|/2γ, and, hence
(21)∫D3‍≲∥ϕ∥(L,N)∥ψ∥(0,N)×∫D3‍[1(1+|xy-1|)N+∑0≤h≤L‍|y|h(1+|x|)N]×2ℓΔ(1+|2ℓy|)Ndy≲∥ϕ∥(L,N)∥ψ∥(0,N)2ℓΔ(1+|x|)N∫|y|≥1‍|y|L(1+2ℓ|y|)Ndy≤∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓL(1+|x|)N∫G‍|2ℓy|L(1+2ℓ|y|)Nd(2ℓy)≲∥ϕ∥(L,N)∥ψ∥(0,N)2-ℓL(1+|x|)N,
where for the last inequality we used [3, Corollary 1.17] and N-L>Δ+1.
Combining the above estimates, we arrive at(22)|ϕ*(D2ℓψ)(x)|≤∥ϕ∥(L+1,N)∥ψ∥(L+1,N)2-ℓL1(1+|x|)N.
This is exactly what we need.Let𝒵(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.
Supposeℒ is a sub-Laplacian on G, and ϕ^∈𝒮(ℝ+) is a function with compact support, vanishing identically near the origin, and satisfying
(23)∑j∈ℤ‍ϕ^(2-2jλ)=1,∀λ∈ℝ+.
Then for all g∈𝒵(G), it holds that
(24)g=limk→∞∑|j|≤k‍g*(D2jϕ),
with convergence in 𝒵(G). Duality entails that, for all u∈𝒮'(G)/𝒫,
(25)u=limk→∞∑|j|≤k‍u*(D2jϕ),
and the convergence is in 𝒮'(G)/𝒫.Proof.
First note that the2-homogeneity of ℒ implies that the distribution kernel of ϕ^(2-2jℒ) coincides with D2jϕ. Let I∈ℕn 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
(26)(1+|x|)N|XI[g*(D2jϕ)](x)|=2d(I)j(1+|x|)N|g*[D2j(XIϕ)](x)|≤C2d(I)j2-|j|L2(j∧0)Δ(1+2(j∧0)|x|)-(L+Δ+2)(1+|x|)N≤C2d(I)j2-|j|L2(j∧0)Δ(1+2(j∧0)|x|)-N(1+|x|)N≤C2-|j|(L-N-d(I)),
where the constant C is a suitable multiple of ∥g∥(L+1,L+Δ+2)∥XIϕ∥(L+1,L+Δ+2). This implies that ∑j∈ℤ‍(1+|x|)N|XI[g*(D2jϕ)](x)| converges uniformly in x, for every I∈ℕn and every N∈ℕ. Consequently there exists h∈𝒮(G) such that ∑|j|≤k‍g*(D2jϕ) 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]),
(27)g=∑j∈ℤ‍ϕ^(2-2jℒ)g=∑j∈ℤ‍g*(D2jϕ)
holds in L2-norm. Therefore, h=g, which completes the proof.Let𝒜 denote the class of all functions ϕ^ in 𝒮(ℝ+) satisfying
(28)suppϕ^⊂[2-2,22],|ϕ^(λ)|≥C>0forλ∈[(35)2,(53)2].Definition 4.
Letα∈ℝ, 0<p<∞ and 0<q≤∞. Let ℒ be a sub-Laplacian on G and ϕ^∈𝒜. We define F˙p,qα(ℒ,ϕ^) as the space of all f∈𝒮'(G)/𝒫 such that
(29)∥f∥F˙p,qα(ℒ,ϕ^)≡∥(∑j∈ℤ‍(2jα|f*(D2jϕ)|)q)1/q∥Lp<∞,
with the usual modification for q=∞.We then introduce the Peetre type maximal functions: Givenf∈𝒮'(G), ϕ^∈𝒮(ℝ+), ℒ a sub-Laplacian, and a,t>0, we define
(30)Ma,t*(f,ℒ,ϕ^)(x)=supy∈G|f*(Dt-1ϕ)(x)|(1+t-1|y-1x|)aMa,t**(f,ℒ,ϕ^)(x)=supy∈G1≤k≤ν|Xk[f*(Dt-1ϕ)](x)|(1+t-1|y-1x|)a.Lemma 5.
Supposeℒ 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,
(31)Ma,2-j**(f,ℒ,ϕ^)(x)≤C2jMa,2-j*(f,ℒ,ϕ^)(x).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 ζ^(λ)=∑j=-11‍ϕ^(2-2jλ)ψ^(2-2jλ), λ∈ℝ+. Then ϕ^(λ)=ϕ^(λ)ζ^(λ) for all λ∈ℝ+. Consequently, for j∈ℤ and 1≤k≤ν,
(32)|Xk[f*(D2jϕ)](y)|=|Xk[f*(D2jϕ)*(D2jζ)](y)|≤∫‍|f*(D2jϕ)(z)||[Xk(D2jζ)](z-1y)|dz=2j(Δ+1)∫‍|f*(D2jϕ)(z)||(Xkζ)(2j(z-1y))|dz≲2jMa,2-j*(f,ℒ,ϕ^)(x)×∫‍2jΔ(1+2j|z-1x|)a(1+2j|z-1y|)-a-(Δ+1)dz≲2jMa,2-j*(f,ℒ,ϕ^)(x)(1+2j|y-1x|)a,
where for the last inequality we used [3, Corollary 1.17] and that (1+2j|z-1x|)a(1+2j|z-1y|)-a≲(1+2j|y-1x|)a. Dividing both sides of the above estimate by (1+2j|y-1x|)a, and then taking the supremum over y∈G and 1≤k≤ν, we obtain the desired estimate.Lemma 6.
Supposeℒ 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,
(33)Ma,2-j*(f,ℒ,ϕ^)(x)≤C[M(|f*(D2jϕ)|r)(x)]1/r,
where a=Δ/r, and M is the Hardy-Littlewood maximal operator on G.Proof.
Letg∈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,
(34)|g(y)-g(y0)|≤C|y0-1y|sup|z|≤b|y0-1y|1≤k≤ν|(Xkg)(y0z)|≤Cδsup|z|≤bδ1≤k≤ν|(Xkg)(y0z)|,
where b is a suitable positive constant. Hence we have
(35)|g(y0)|≤Cδsup|y-1y0|≤bδ1≤k≤ν|Xkg(y)|+δ-a(∫|y-1y0|≤δ‍|g(y)|rdy)1/r.
Putting g=f*(D2jϕ), dividing both sides by (1+2j|y0-1x|)a, and using Lemma 5, we have
(36)|f*(D2jϕ)(y0)|(1+2j|y0-1x|)a≤Cδsup|y-1y0|≤bδ1≤k≤ν|Xk[f*(D2jϕ)](y)|(1+2j|y-1x|)a(1+2j|y-1x|)a(1+2j|y0-1x|)a+δ-a(1+2j|y0-1x|)a(∫|y-1y0|≤δ‍|f*(D2jϕ)(y)|rdy)1/r≤Cδsup|y-1y0|≤bδ(1+2j|y-1y0|)aMa,2-j**(f,ℒ,ϕ^)(x)+δ-a(1+2j|y0-1x|)a×(∫|y-1x|≤γ(δ+|y0-1x|)‍|f*(D2jϕ)(y)|rdy)1/r≤Cδ(1+2jbδ)a2jMa,2-j*(f,ℒ,ϕ^)(x)+δ-aγa(δ+|y0-1x|)a(1+2j|y0-1x|)a[M(|f*(D2jϕ)|r)(x)]1/r≤Cε(1+bε)aMa,2-j*(f,ℒ,ϕ^)(x)+γa(1+ε-1)a[M(|f*(D2jϕ)|r)(x)]1/r,
where we have set δ=2-jε and used that (1+2jε-1|y0-1x|)/(1+2j|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.
Supposeℒ(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
(37)∥f∥F˙p,qα(ℒ(1),ϕ^(1))~∥f∥F˙p,qα(ℒ(2),ϕ^(2)),f∈𝒮′(G)/𝒫.Proof.
First we note that ifa>Δ/min{p,q} then we have
(38)∥(∑j∈ℤ‍(2jα|Ma,2-j*(f,ℒ(i),ϕ^(i))|)q)1/q∥Lp~∥(∑j∈ℤ‍(2jα|f*(D2jϕ(i))|)q)1/q∥Lp
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
(39)∥(∑j∈ℤ‍(2jα|Ma,2-j*(f,ℒ(1),ϕ^(1))|)q)1/q∥Lp~∥(∑j∈ℤ‍(2jα|Ma,2-j*(f,ℒ(2),ϕ^(2))|)q)1/q∥Lp.
To this end, letψ^(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
(40)f=∑j∈ℤ‍f*(D2jϕ(1))*(D2jψ(1))
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
(41)f*(D2ℓϕ(2))(y)=∑j∈ℤ‍f*(D2jϕ(1))*(D2jψ(1))*(D2ℓϕ(2))(y),∀y∈G.
It follows that
(42)|f*(D2ℓϕ(2))(y)|≤∑j∈ℤ‍∫‍|f*(D2jϕ(1))(z)|×|(D2jψ(1))*(D2ℓϕ(2))(z-1y)|dz≤∑j∈ℤ‍Ma,2-j*(f,ℒ(1),ϕ^(1))(y)×∫‍(1+2j|z|)a|(D2jψ(1))*(D2ℓϕ(2))(z)|dz≡∑j∈ℤ‍Ma,2-j*(f,ℒ(1),ϕ^(1))(y)Ij,ℓ.
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,
(43)Ij,ℓ≤C∫‍(1+2j|z|)a2-|j-ℓ|L2(j∧ℓ)Δ×(1+2j∧ℓ|z|)-(L+Δ+2)dz≤C∫‍(1+2j|z|)a2-|j-ℓ|L2(j∧ℓ)Δ×(1+2j∧ℓ|z|)-(a+Δ+1)dz≤C2-|j-ℓ|(L-a)∫‍2(j∧ℓ)Δ(1+2j∧ℓ|z|)-(Δ+1)dz≤C2-|j-ℓ|(L-a).
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
(44)Ma,2-j*(f,ℒ(1),ϕ^(1))(y)≤Ma,2-j*(f,ℒ(1),ϕ^(1))(x)(1+2j|y-1x|)a≲Ma,2-j*(f,ℒ(1),ϕ^(1))(x)×(1+2ℓ|y-1x|)amax(1,2(j-ℓ)a).
Putting these estimates into (42), multiplying both sides by 2ℓα, dividing both sides by (1+2ℓ|y-1x|)a and then taking the supremum over y∈G, we obtain
(45)2ℓαMa,2-ℓ*(f,ℒ(2),ϕ^(2))(x)≤∑j∈ℤ‍2-|j-ℓ|(L-2a-|α|)2jαMa,2-j*(f,ℒ(1),ϕ^(1))(x).
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 Theorem7 we see that the space F˙p,qα(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 F˙p,qα(G) instead of F˙p,qα(ℒ,ϕ^). Henceforth we shall fix any sub-Laplacian ℒ. Moreover, for the sake of briefness, we will write Ma,t*(f,ϕ^)(x) instead of Ma,t*(f,ℒ,ϕ^)(x).Proposition 9.
Forα∈ℝ, 0<p<∞ and 0<q≤∞, one has the continuous inclusion maps 𝒵(G)↪F˙p,qα(G)↪𝒮'(G)/𝒫.Proof.
Letg∈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
(46)|g*(D2jϕ)(x)|≤C∥g∥(L+1,L+Δ+2)2-|j|L2(j∧0)Δ(1+2j∧0|x|)-(L+Δ+2)≤C∥g∥(L+1,L+Δ+2)2-|j|L2(j∧0)Δ(1+2j∧0|x|)-N≤C∥g∥(L+1,L+Δ+2)2-|j|(L-N)(1+|x|)-N
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
(47)∥g∥F˙p,qα(G)≲∥g∥(L+1,L+Δ+2)(∑j∈ℤ‍2-|j|(L-N-|α|)q)1/q≲∥g∥(L+1,L+Δ+2),
which implies that 𝒵(G)↪F˙p,qs(G) continuously.
Now we show the other embedding. Letf∈F˙p,qα(G) and take any ψ∈𝒵(G). Let ϕ^∈𝒜, and then let ζ^∈𝒮(ℝ+) be a function with support in [2-2,22] such that ∑j∈ℤ‍ϕ^(2-2jλ)ζ^(2-2jλ)=1 for λ∈ℝ+. Then by Lemma 3 we have
(48)|〈f,ψ〉|=|〈∑j∈ℤ‍f*(D2jϕ)*(D2jζ),ψ〉|≤∑j∈ℤ‍|〈f*(D2jϕ),ψ*(D2jζ~)〉|≤∑j∈ℤ‍∥f*(D2jϕ)∥L∞∥ψ*(D2jζ~)∥L1.
To proceed we claim that
(49)∥f*(D2jϕ)∥L∞≤C2jΔ/p∥f*(D2jϕ)∥Lp
for all 0<p<∞. Assuming the claim for a moment, it follows from (48) that
(50)|〈f,ψ〉|≲(supj∈ℤ2jα∥f*(D2jϕ)∥Lp)×∑j∈ℤ‍2j[(Δ/p)-α]∥ψ*(D2jζ~)∥L1.
It is easy to see that
(51)supj∈ℤ2jα∥f*(D2jϕ)∥Lp≲∥f∥F˙p,qα(G).
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
(52)|ψ*(D2jζ~)(x)|≤C∥ψ∥(L+1,L+Δ+2)2-|j|(L-N)(1+|x|)-N,
where the constant C is a suitable multiple of ∥ζ~∥(L+1,L+Δ+2). From this it follows that
(53)∑j∈ℤ‍2j(Δ/p-α)∥ψ*(D2jζ~)∥L1≲∑j∈ℤ‍2-|j|[L-N-(Δ/p)-|α|]∥ψ∥(L+1,L+Δ+2)≲∥ψ∥(L+1,L+Δ+2).
Therefore,
(54)|〈f,ψ〉|≲∥f∥F˙p,qα(G)∥ψ∥(L+1,L+Δ+2).
This implies that F˙p,qα(G)↪𝒮'(G)/𝒫 continuously.
We are left with showing the claim. Indeed, ifr>0 is fixed then by Lemma 6 we have, for all x∈G,
(55)|f*(D2jϕ)(x)|~infy∈B(x,2-j)|f*(D2jϕ)(x)|(1+2j|x-1y|)Δ/r≤infy∈B(x,2-j)MΔ/r,2-j*(f,ϕ^)(y)≲infy∈B(x,2-j)[M(|f*(D2jϕ)|r)(y)]1/r.
Taking r<p in the above estimate and using Hardy-Littlewood maximal inequality, we have
(56)|f*(D2jϕ)(x)|p≲1|B(x,2-j)|∫B(x,2-j)‍[M(|f*(D2jϕ)|r)(y)]p/rdy≤2jΔ∫‍|f*(D2jϕ)(y)|pdy.
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.
Forα∈ℝ, 0<p<∞ and 0<q≤∞, F˙p,qα(G) is a quasi-Banach space.Let us introduce a class of functions. We say thatf∈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.
Letα∈ℝ and 0<p,q<∞. Then ℛ(G) is dense in F˙p,qα(G). In particular, 𝒵(G) is dense in F˙p,qα(G).Proof.
Take anyu∈F˙p,qα(G) and any ε>0. In the appendix we show that F˙p,qα(G) admits smooth atomic decomposition. By the smooth atomic decomposition, we see that C0∞(G)∩F˙p,qα(G) is dense in F˙p,qα(G), for α∈ℝ and 0<p,q<∞. Thus, we can find g∈C0∞(G)∩F˙p,qα(G) such that ∥g-u∥F˙p,qα(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-∑j=-NN‍g*(D2jϕ)∥F˙p,qα(G)<ε/2. Now we put
(57)h=∑j=-NN‍ϕ^(2-2jℒ)g=∑j=-NN‍g*(D2jϕ).
Then h∈ℛ(G), and we have
(58)∥h-u∥F˙p,qα(G)≲∥h-g∥F˙p,qα(G)+∥g-u∥F˙p,qα(G)<ε.
This proves the claimed statement.We next consider lifting property ofF˙p,qα(G). For σ∈ℝ, the power ℒσ is naturally given by
(59)ℒσf=∫0∞‍λσdE(λ)f,f∈Dom(ℒσ)⊂L2(G).Remark 12.
By [17, Theorem 13.24], we have ℛ(G)⊂Dom(ℒσ) for all σ∈ℝ. As a consequence, Dom(ℒσ)∩F˙p,qα(G) is dense in F˙p,qα(G), for all α,σ∈ℝ and 0<p,q<∞.
We now have the lifting property ofF˙p,qα(G).Theorem 13.
Letα,σ∈ℝ and 0<p, q<∞.(i)
The operatorℒσ, initially defined on Dom(ℒσ)∩F˙p,qα(G), extends to a continuous operator from F˙p,qα(G) to F˙p,qα-2σ(G).(ii)
LetTσ:F˙p,qα(G)→F˙p,qα-2σ(G) denote the continuous extension of ℒσ. Then Tσ is an isomorphism, and ∥Tσf∥F˙p,qα-2σ(G) is an equivalent quasi-norm of F˙p,qα(G).Proof.
(i) Letϕ^∈𝒜. Set ζ^(λ)=λσ and ψ^(λ)=ϕ^(λ)ζ^(λ)=ϕ^(λ)λσ. Clearly ψ^ is also in 𝒜, and 22jσψ^(2-2jλ)=ϕ^(2-2jλ)ζ^(λ). By [17, Theorem 13.24], we have ϕ^(2-2jℒ)∘ζ^(ℒ)⊂(ϕ^(2-2j·)ζ^(·))(ℒ), and moreover
(60)Dom(ϕ^(2-2jℒ)∘ζ^(ℒ))=Dom(ζ^(ℒ))∩Dom((ϕ^(2-2j·)ζ^(·))(ℒ))=Dom(ℒσ)∩L2(G)=Dom(ℒσ).
Hence, for every f∈Dom(ℒσ), we have 22jσψ^(2-2jℒ)f=ϕ^(2-2jℒ)ℒσf.
Now letf∈Dom(ℒσ)∩F˙p,qs(G). By the above remarks, we have 22jσ[f*(D2jψ)]=(ℒσf)*(D2jϕ). It follows that
(61)∥ℒσf∥F˙p,qα-2σ(G)~∥(∑j∈ℤ‍(2j(α-2σ)|(ℒσf)*(D2jϕ)|)q)1/q∥Lp=∥(∑j∈ℤ‍(2jα|f*(D2jψ)|)q)1/q∥Lp~∥f∥F˙p,qα(G).
Since Dom(ℒσ)∩F˙p,qα(G) is dense in F˙p,qα(G) (see Remark 12), the mapping ℒσ:Dom(ℒσ)∩F˙p,qα(G)↦F˙p,qα(G) extends to a continuous operator from F˙p,qα(G) to F˙p,qα-2σ(G). We denote this extension by Tσ:F˙p,qα(G)↦F˙p,qα-2σ(G).
(ii) Let us first show that the mappingTσ:F˙p,qα(G)→F˙p,qα-2σ(G) is injective. Indeed, assume f∈F˙p,qα(G) such that Tσf is the zero element of F˙p,qα-2σ(G). By Remark 12 we can find a sequence fℓ in Dom(ℒσ)∩F˙p,qα(G) which converges in F˙p,qα(G) to f. Then applying (i) to ℒσ yields that ℒσfℓ converges in F˙p,qs-2σ(G) to the zero element. Since ℒσfℓ∈Dom(ℒ-σ)∩F˙p,qα-2σ(G) and fℓ=ℒ-σ(ℒσfℓ), applying (i) to the operator ℒ-σ we see that fℓ converges in F˙p,qα(G) to the zero element. Therefore, f is the zero element in F˙p,qα(G). This proves that Tσ is injective.
Next we show thatTσ:F˙p,qα(G)→F˙p,qα-2σ(G) is surjective. Indeed, given f∈F˙p,qα-2σ(G), we let fℓ be a sequence in Dom(ℒ-σ)∩F˙p,qα-2σ(G) which converges in F˙p,qα-2σ(G) to f. Then from (i) we see that ℒ-σfℓ converges in F˙p,qα(G). Denote this limit by g. We claim that Tσg=f. Indeed, since fℓ=ℒσ(ℒ-σfℓ), it follows from (i) that fℓ converges to Tσg in F˙p,qα(G). Hence Tσg=f in F˙p,qα-2σ(G). This proves that Tσ is surjective.
The above arguments also show that bothT-σ∘Tσ and Tσ∘T-σ are identity operators on F˙p,qα(G). Furthermore, by an easy density argument we see that (61) holds for all f∈F˙p,qα(G), provided that ℒσ in (61) is replaced by Tσ. Thus, Tσ:F˙p,qα(G)→F˙p,qα-2σ(G) is an isomorphism, and ∥Tσf∥F˙p,qα-2σ(G) is an equivalent quasi-norm of F˙p,qα(G).Our next goal is to show the Lusin and Littlewood-Paley function characterizations ofF˙p,qα(G). If α∈ℝ, 0<q<∞, λ>0, b∈ℝ, and u(x,k) is a function on G×ℤ, we define
(62)gqα(u)(x)=(∑k=-∞∞‍(2kα|u(x,k)|)q)1/q,Gλ,qα(u)(x)=[∑k=-∞∞‍∫G‍(2kα|u(y,k)|)q×(1+2k|y-1x|)-λq2kΔdy∑k=-∞∞‍∫G‍(2kα|u(y,k)|)q]1/q,Sb,qα(u)(x)=(∑k=-∞∞‍∫|y-1x|≤2b-k‍(2kα|u(y,k)|)q2(k-b)Δdy)1/q.
The following proposition shows that the spaces F˙p,qα(G) are characterized by Lusin and Littlewood-Paley functions.Proposition 14.
Letα∈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*(D2kϕ)(x), for x∈G and k∈ℤ. Then one has
(63)∥gqα(u)∥Lp~∥Sb,qα(u)∥Lp~∥Gλ,qα(u)∥Lp.Proof.
Step  1. Show that if λ>max{Δ/p,Δ/q} then
(64)∥gqα(u)∥Lp≲∥Gλ,qα(u)∥Lp≲∥Sb,qα(u)∥Lp.
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=ℝn×(0,∞), the arguments there can also be adapted to functions u which are defined on G×ℤ.
Step  2. We prove that ∥Sb,qα(u)∥Lp≲∥gqα(u)∥Lp. First note that, by an argument similar to the proofs of [19, Theorems 1 and 2], it follows that ∥Sb,qα(u)∥Lp~∥S0,qα(u)∥ for every fixed b∈ℝ. Hence, in view of (38), it is enough to show that
(65)∥S0,qα(u)∥Lp≲∥(∑j∈ℤ‍(2jα|Mλ,2-j*(f,ϕ^)|)q)1/q∥Lp.
But this is a consequence of the following elementary estimate:
(66)∫|y-1x|<2-j‍(2jα|f*(D2jϕ)(y)|)q2jΔdy≲supy∈B(x,2-j)(2jα|f*(D2jϕ)(y)|)q(1+2j|y-1x|)λq≤supy∈G(2jα|f*(D2jϕ)(y)|)q(1+2j|y-1x|)λq=[2jαMλ,2-j*(f,ϕ^)(x)]q.
The proof of Proposition 14 is thus complete.Corollary 15.
LetG be a stratified Lie group. Then F˙p,20(G)=Hp(G) with equivalent (quasi-)norms, for 0<p<∞. Here Hp(G) are Hardy spaces on G.Proof.
In [3, Chapter 7], Folland and Stein proved the characterization of Hardy spaces Hp(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 F˙p,20(G) with Hp(G) for 0<p<∞.


## 4. Convolution Singular Integral Operators onF˙p,qα(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.
Letr be a positive integer. A kernel of order r is a distribution K∈𝒮'(G) with the following properties:
(i)K coincides with a Cr function K(x) away from the group identity 0 and enjoys the regularity condition:
(67)|XIK(x)|≤CI|x|-Q-d(I),for|I|≤r,x≠0.
(ii)K satisfies the cancellation condition: For all normalized bump function ϕ and all R>0, we have
(68)|〈K,ϕR〉|≤C,
where ϕR(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
(69)|XIϕ(x)|≤1,∀|I|≤N,∀x∈G,
for some fixed positive integer N.
The convolution operatorT 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:
(70)|YIK(x)|≤CI|x|-Q-d(I),for|I|≤r,x≠0.Examples of such kernels include the class of distributions which arehomogeneous 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|<L‍K(x)dx=0 for all 0<ε<L<∞. Hence, for every normalized bump function ϕ, by the homogeneity of K we have
(71)|〈K,ϕR〉|=|〈K,ϕ〉|=|limε→0∫ε<|x|<2‍K(x)[ϕ(x)-ϕ(0)]dx|≤∫|x|<2‍|K(x)||ϕ(x)-ϕ(0)|dx.
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.
Letα∈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 F˙p,qα(G).IfK∈𝒮'(G) and t>0, we define DtK as the tempered distribution given by 〈DtK,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.
Letr be a positive integer. Suppose K is a kernel of order r, and ϕ is a smooth function supported in B(0,1/4γ2br) and having vanishing moments of order r-1. Then, there exists a constant C>0 such that for all j∈ℤ, all I∈ℕn with |I|≤r, and all x∈G(72)max{|XI[ϕ*(D2jK)](x)|,|YI[ϕ*(D2jK)](x)|}≤C(1+|x|)-Δ-r.
Moreover,ϕ*(D2jK) have vanishing moments of the same order as ϕ.Proof.
Recall that the convolution ofϕ∈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*(D2jK) 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
(73)YI=∑|J|≤|I|d(J)≥d(I)‍PI,JXJ,
where PI,J are polynomials of homogeneous degree d(J)-d(I) (see [3, Proposition 1.29]), we have
(74)|XI[(xϕ)~](z)|=|YI(xϕ)(z-1)|≲∑|J|≤|I|d(J)≥d(I)‍|PI,J(z-1)||XJ(xϕ)(z-1)|=∑|J|≤|I|d(J)≥d(I)‍|PI,J(z-1)||(XJϕ)(xz-1)|≤CI.
Here CI 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∈ℕn, [x(YIϕ)]~ is a normalized bump function multiplied with a constant CI′ 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∈ℕn with |I|≤r, and all x with |x|≤1/2γ(75)|YI[ϕ*(D2jK)](x)|=|(YIϕ)*(D2jK)](x)|=|〈D2jK,[x(YIϕ)]~〉|=|〈K,[x(YIϕ)]~(2-j·)〉|≤C.
From this and [3, Proposition 1.29], we also get that, for all j∈ℤ, all I∈ℕn with |I|≤r, and all x with |x|≤1/2γ(76)|XI[ϕ*(D2jK)](x)|≤C∑|I′|≤|I|d(I′)≥d(I)‍|x|d(I′)-d(I)|YI′[ϕ*(D2jK)](x)|≤C.
Let now|x|>1/2γ. Let y∈suppϕ. Let I∈ℕn with |I|≤r, and denote by P~x,YI(D2jK) the right Taylor polynomial of YI(D2jK) 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
(77)|XI(D2jK)(y-1x)-P~x,XI(D2jK)(y-1)|≤C|y|r-|I|sup|z|≤br-|I||y|d(J)=r-|I||YJXI(D2JK)(zx)|.
Observe that D2jK satisfies (67) with the bound C independent of j∈ℤ. Also note that for y∈suppϕ and |z|≤br-|I||y| we have zx∈G∖{0}. Thus, for all J with d(J)=r-|I| and all z with |z|≤br-|I||y|, by using (73) and (67) (with K replaced by D2jK) we have
(78)|YJXI(D2jK)(zx)|≤∑|J′|≤|J|d(J′)≥d(J)‍|PJ,J′(zx)||XJ′+I(D2jK)(zx)|≲∑|J′|≤|J|d(J′)≥d(J)‍|zx|d(J′)-d(J)|zx|-Δ-d(J′+I)≲|zx|-Δ-r+|I|-d(I).
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
(79)|XI(D2jK)(y-1x)-P~x,XI(D2jK)(y-1)|≤C|y|r-|I|sup|z|≤br-|I||y||zx|-Δ-r+|I|-d(I).
Notice that for |x|≥1/2γ, y∈suppϕ and |z|≤br-|I||y|, we have |zx|~|x|. Thus, by using the vanishing moments of ϕ and (79), we have
(80)|XI[ϕ*(D2jK)](x)|=|∫‍ϕ(y)XI(D2jK)(y-1x)dy|≤∫‍|ϕ(y)||XI(D2jK)(y-1x)-P~x,XI(D2jK)(y-1)|dy≤Csup|z|≤br-|I||y|∫‍|zx|-Δ-r+|I|-d(I)|y|r-|I||ϕ(y)|dy≤C|x|-Δ-r+|I|-d(I)∫‍|y|r-|I||ϕ(y)|dy≤C|x|-Δ-r+|I|-d(I).
Combining (76) and (80), we see that, for all j∈ℤ, all I∈ℕn with |I|<r, and all x∈G(81)|XI[ϕ*(D2jK)](x)|≤C(1+|x|)-Δ-r+|I|-d(I).
From this and (73), we also get that, for all j∈ℤ, all I∈ℕn with |I|<r, and all x∈G(82)|YI[ϕ*(D2jK)](x)|≤∑|I′|≤|I|d(I′)≥d(I)‍|PI,I′(x)|XI′[ϕ*(D2jK)](x)|≤C∑|I′|≤|I|d(I′)≥d(I)‍|x|d(I′)-d(I)(1+|x|)-Δ-r+|I′|-d(I′)≤C(1+|x|)-Δ-r+|I|-d(I).
Since |I|≤d(I), (81) along with (82) yield (72).
It is straightforward to verify thatϕ*(D2jK) have vanishing moments of the same order as ϕ. The proof of Lemma 19 is therefore complete.The proof of Theorem18 also relies on the existence of smooth functions with compact support and having arbitrarily high order vanishing moments.Lemma 20.
Given any nonnegative integerL and any positive number δ, there exists a function ζ^∈𝒮(ℝ+) with the following properties:(i)
|ζ^(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 ζ^(λ)=(t-2λ)kθ^(t-2λ), λ∈ℝ+. Here t>0 and k is a nonnegative integer. Then ζ(x)=Dt(ℒkθ)(x)=tΔ(ℒkθ)(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 Theorem18.Proof of Theorem18.
Choose a functionζ^∈𝒮(ℝ+) which satisfies conditions (i)–(iii) in Lemma 20 with L=r-1 and δ=1/4γ2br. The condition (i) guarantees the existence of a function ψ^∈𝒮(ℝ+) with the following properties:
(83)suppψ^⊂[2-2(k0+1),2-2(k0-1)],|ψ^(λ)|>0forλ∈(2-2(k0+1),2-2(k0-1)),∑j∈ℤ‍ψ^(2-2jλ)ζ^(2-2jλ)=1,∀λ∈ℝ+.
Note that ψ^(2-2k0·)∈𝒜. For f∈𝒵(G), by Lemma 3 we have
(84)f=∑j∈ℤ‍f*(D2jψ)*(D2jζ)
with convergence in 𝒵(G). Let ϕ^∈𝒜, and let K be the convolution kernel of the operaotor T. Then we have the representation
(85)f*K*(D2ℓϕ)=∑j∈ℤ‍f*(D2jψ)*(D2jζ)*K*(D2ℓϕ)=∑j∈ℤ‍f*(D2jψ)*D2j[ζ*(D2-jK)]*(D2ℓϕ),
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
(86)|D2j[ζ*(D2-jK)]*(D2ℓϕ)(y)|≤C2-|j-ℓ|(r-2)2(j∧ℓ)Δ(1+2j∧ℓ|y|)r+Δ.
This together with (85) gives that
(87)|f*K*(D2ℓϕ)(x)|≤∑j∈ℤ‍∫‍|f*(D2jψ)(z)|×|D2j[ζ*(D2-jK)]*(D2ℓϕ)(z-1x)|dz≲∑j∈ℤ‍2-|j-ℓ|(r-2)∫‍2(j∧ℓ)Δ|f*(D2jψ)(z)|(1+2j∧ℓ|z-1x|)r+Δdz≲∑j∈ℤ‍2-|j-ℓ|(r-2-a)[supz∈G|f*(D2jψ)(z)|(1+2j|z-1x|)a]×∫‍2(j∧ℓ)Δ(1+2j∧ℓ|z-1x|)r+Δ-adz,
where for the last inequality we used that (1+2j∧ℓ|z-1x|)-a≤2|j-ℓ|a(1+2j|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
(88)2ℓα|f*K*(D2ℓϕ)(x)|≲∑j∈ℤ‍2-|j-ℓ|(r-2-a-|α|)2jαMa,2-j*(f,ψ^)(x).
Hence, it follows by [15, Lemma 2] that
(89)∥(∑ℓ∈ℤ‍(2ℓα|f*K*(D2ℓϕ)(x)|)q)1/q∥Lp≲∥(∑j∈ℤ‍(2jα|Ma,2-j*(f,ψ^)|)q)1/q∥Lp.
This together with (38) imply that ∥f*K∥F˙p,qα(G)≲∥f∥F˙p,qα(G) for all f∈𝒵(G). Since 𝒵(G) is dense in F˙p,qα(G), f↦f*K extends to an bounded operator on F˙p,qα(G). This completes the proof of Theorem 18.Corollary 21.
Letα∈ℝ, 0<p,q<∞, and let k be a nonnegative integer. Then
(90)∥f∥F˙p,qα(G)~∑d(I)=k‍∥XIf∥F˙p,qα-k(G).Proof.
Note that by the Poincaré-Birkhoff-Witt theorem (cf. [23, I.2.7]), the operators XI 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
(91)∥f∥F˙p,qα(G)~∑j=1ν‍∥Xjf∥F˙p,qα-1(G).
To this end, we first note that when restricted to Schwartz functions, Xjℒ-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, Xjℒ-1/2 extend to bounded operators on F˙p,qα(G). From this fact and the lifting property (Theorem 7), we deduce that
(92)∥Xjf∥F˙p,qα-1(G)=∥(Xjℒ-1/2)ℒ1/2f∥F˙p,qα-1(G)≲∥ℒ1/2f∥F˙p,qα-1(G)~∥f∥F˙p,qα(G).
Hence ∑j=1ν‍∥Xjf∥F˙p,qα-1(G)≲∥f∥F˙p,qα(G). To see the converse, we need to use [2, Lemma 4.12], which asserts that there exists tempered distributions K1,…,Kν homogeneous of degree -Δ+1 and coinciding with smooth functions in G∖{0} such that f=∑j=1ν‍(Xjf)*Kj for all f∈𝒮(G). By this result and Theorem 7, we have, at least for f∈ℛ(G) (⊂Dom(ℒ-1/2)),
(93)∥f∥F˙p,qα(G)=∥ℒ(ℒ-1/2f)∥F˙p,qα-1(G)=∥ℒ(f*R1)∥F˙p,qα-1(G)=∥ℒ(∑j=1ν‍(Xjf)*Kj*R1)∥F˙p,qα-1(G)=∥∑j=1ν‍(Xjf)*ℒ(Kj*R1)∥F˙p,qα-1(G),
where R1 is the convolution kernel of the operator ℒ-1/2. As is indicated in [2, p. 190], ℒ(Kj*R1) are distributions homogeneous of degree -Δ and coincide with smooth functions away from 0. Thus it follows by Theorem 18 that
(94)∥(Xjf)*ℒ(Kj*R1)∥F˙p,qα-1(G)≲∥Xjf∥F˙p,qα-1(G).
Inserting this into (93), we obtain ∥f∥F˙p,qα(G)≲∑j=1ν‍∥Xjf∥F˙p,qα-1(G) for all f∈ℛ(G). Since ℛ(G) is dense in F˙p,qα(G), the latter inequality also holds for all f∈F˙p,qα(G). This completes the proof.



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*Source: 475103-2013-03-31.xml*