Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of LittlewoodPaley-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.


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  and the summability parameter  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  or the summability parameter  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   -boundedness of convolution operators with homogeneous distribution kernels on Lie groups endowed with suitable homogeneous structure was proved by Knapp and Stein [6] (for  = 2) and Korányi and Vági [7] (for 1 <  < ∞).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  , () on stratified Lie groups, and give some basic properties of them.In Section 4 we show the Ḟ  , ()-boundedness of a class of convolution singular integral operators.Throughout this

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  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 =  1 ⊕ ⋅ ⋅ ⋅ ⊕   , with [ 1 ,   ] =  +1 for 1 ≤  ≤  − 1 and [ 1 ,   ] = 0.Such a group  is clearly nilpotent, and thus it may be identified with g (as a manifold) via the exponential map exp : g → .
Examples of stratified Lie groups include Euclidean spaces R  and the Heisenberg group H  .
The algebra g is equipped with a family of dilations {  :  > 0} which are the algebra automorphisms defined by Under our identification of  with g,   may also be viewed as a map  → .We generally write  instead of   (), for  ∈ .We shall denote by the homogeneous dimension of .
A homogeneous norm on G is a continuous function   → || from  to [0, ∞) smooth away from 0 (the group identity), vanishing only at 0, and satisfying | −1 | = || and || = || for all  ∈  and  > 0. Homogeneous norms on  always exist and any two of them are equivalent.We assume  is provided with a fixed homogeneous norm.It satisfies a triangle inequality: there exists a constant  ≥ 1 such that || ≤ (||+||) for all ,  ∈ .If  ∈  and  > 0 we define the ball of radius  about  by (, ) = { ∈  : | −1 | < }.The Lebesgue measure on g induces a bi-invariant Haar measure on .As done in [3], we fix the normalization of Haar measure by requiring that the measure of (0, 1) be 1.We shall denote the measure of any measurable  ⊂  by ||.(3) We consider g as the Lie algebra of all left-invariant vector fields on , and fix a basis  1 , . . .,   of g, obtained as a union of bases of the   .In particular,  1 , . . .,  ] , with ] = dim( 1 ), is a basis of  1 .We denote by  1 , . . .,   the corresponding basis for right-invariant vector fields, that is, where the integers  1 ≤ ⋅ ⋅ ⋅ ≤   are given according to that   ∈    .Then   (resp.,   ) is a left-invariant (resp., rightinvariant) differential operator, homogeneous of degree (), with respect to the dilations   ,  > 0.
A complex-valued function  on  is called a polynomial on  if  ∘ exp is a polynomial on g.Let  1 , . . .,   be the basis for the linear forms on g dual to the basis  1 , . . .,   for g, and set   =   ∘ exp −1 .From our definition of polynomials on ,  1 , . . .,   are generators of the algebra of polynomials on .Thus, every polynomial on  can be written uniquely as where all but finitely many of the coefficients vanish, and A polynomial of the type ( 6) is called of homogeneous degree , where  ∈ N, if () ≤  holds for all multi-indices  with   ̸ = 0. We let P denote the space of all polynomials on , and let P  denote the space of polynomials on  of homogeneous degree .Note that P  is invariant under left and right translations (see [3,Proposition 1.25]).A function  :  → C is said to have vanishing moments of order , if with the absolute convergence of the integral.The Schwartz class on  is defined by Using the above conventions for the choice of the basis  1 , . . .,   , and ] = dim( 1 ), the sub-Laplacian is defined by =1  2  .When restricted to smooth functions with compact support, L is essentially self-adjoint.Its closure has domain D = { ∈  2 () : L ∈  2 ()}, where L 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 where () is the projection measure.If  is a bounded Borel measurable function on [0, ∞), the operator is bounded on  2 (), and commutes with left translations.Thus, by the Schwartz kernel theorem, there exists a tempered distribution  on  such that Note that the point  = 0 may be neglected in the spectral resolution, since the projection measure of {0} is zero (see [8, p. 76]).Consequently we should regard  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.
Moreover, from the proof of [10, Corollary 1] we see that if  is a function in S(R + ) which vanishes identically near the origin, then  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  the distribution kernel of the operator φ(L), where L is a sub-Laplacian fixed in the context.
For  ∈ Also, we note that by [12,Proposition 20.3.14] the left Taylor polynomial  , is of the form where the integers  1 ≤ ⋅ ⋅ ⋅ ≤   are given according to that   ∈    .From these remarks, it follows that where we used that |{ : For  ∈  3 we have | −1 | > ||/2, and, hence where for the last inequality we used [3, Corollary 1.17] and  −  > Δ + 1.
Combining the above estimates, we arrive at This is exactly what we need.
Let Z() denote the space of Schwartz functions with all moments vanishing.We then consider Z() as a subspace of S(), including the topology.It is shown in [5,Lemma 3.3] that Z() is a closed subspace of S(), and the topology dual Z  () of Z() can be canonically identified with the factor space S  ()/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
Then for all  ∈ Z(), it holds that with convergence in Z().Duality entails that, for all  ∈ S  ()/P, and the convergence is in S  ()/P.
Let A denote the class of all functions φ in S(R ] . (28) with the usual modification for  = ∞.
We then introduce the Peetre type maximal functions: Given  ∈ S  (), φ ∈ S(R + ), L a sub-Laplacian, and ,  > 0, we define Lemma 5. Suppose L is a sub-Laplacian and φ ∈ A. Then for every  > 0 there is a constant  > 0 such that for all  ∈ S  ()/P, all  ∈ Z, and all  ∈ , Proof.Because of (28) it is possible to find a func- where for the last inequality we used [3, Corollary 1.17] and that (1+2 where we have set  = 2 −  and used that Finally, taking  sufficiently small (such that (1 + )  < 1/2), and taking the supremum over  0 ∈ , we get the desired estimate.
Remark 8. From Theorem 7 we see that the space Ḟ  , (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  , () instead of Ḟ  , (L, φ).Henceforth we shall fix any sub-Laplacian L.Moreover, for the sake of briefness, we will write  * , (, φ)() instead of  * , (, L, φ)().

Proof
Since  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].
Let us introduce a class of functions.We say that  ∈ R(), if there exists φ ∈ S(R + ) whose support is compact and which vanishes identically near the origin, and  ∈ S(), such that  = φ(L).Clearly R() ⊂ Z().
We next consider lifting property of Ḟ  , ().For  ∈ R, the power L  is naturally given by Hence, for every  ∈ Dom(L  ), we have , () and  ℓ = L − (L   ℓ ), applying (i) to the operator L − we see that  ℓ converges in Ḟ  , () to the zero element.Therefore,  is the zero element in Ḟ  , ().This proves that   is injective.
Our next goal is to show the Lusin and Littlewood-Paley function characterizations of Ḟ  , ().If  ∈ R, 0 <  < ∞,  > 0,  ∈ R, and (, ) is a function on  × Z, we define The following proposition shows that the spaces Ḟ  , () are characterized by Lusin and Littlewood-Paley functions.
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  considered in [19] is defined on the half space R +1 + = R  × (0, ∞), the arguments there can also be adapted to functions  which are defined on  × Z.
Step 2. We prove that ‖  , ()‖   ≲ ‖   ()‖   .First note that, by an argument similar to the proofs of [19, But this is a consequence of the following elementary estimate: The proof of Proposition 14 is thus complete.
Proof.In [3, Chapter 7], Folland and Stein proved the characterization of Hardy spaces   () by continuous version Lusin function, for 0 <  < ∞.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 Ḟ 0 ,2 () with   () for 0 <  < ∞.

Convolution Singular Integral Operators on Ḟ 𝛼 𝑝,𝑞 (𝐺)
In this section we study boundedness of convolution singular integral operators on homogeneous 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  ∞ functions away from 0. Indeed, assume  ∈ S  () is such a distribution, then it is easy to verify that  satisfies the regularity condition (i) in Definition 16; moreover, from [3,Proposition 6.13] we see that  is a principle value distribution such that ∫ <||< () = 0 for all 0 <  <  < ∞.If  ∈ S  () and  > 0, we define    as the tempered distribution given by ⟨  , ⟩ = ⟨, ( −1 ⋅)⟩ (∀ ∈ S()).For the proof of Theorem 18, we will need the following lemma, in which  is the positive constant as in [3,  Proof.Recall that the convolution of  ∈ S() with  ∈ S  () is defined by  * () := ⟨, (  ) ∼ ⟩, where   is the function given by  () = (), and as before f() := ( −1 ) for any function  :  → C. From [3, p. 38] we see that  * ( 2  ) are  ∞ functions,  ∈ Z.We claim that for every  with || ≤ 1/2, the function   → (  ) ∼ () is a normalized bump function multiplied with a constant independent of .Indeed, using the quasi-triangle inequality satisfied by the homogeneous norm it is easy to verify that the function   → (  ) ∼ () is supported in (0, 1); moreover, since || ≤ 1/2 and since where  , are polynomials of homogeneous degree ()−() (see [3,     It is straightforward to verify that  * ( 2  ) have vanishing moments of the same order as .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.Proof.From the appendix of [22] we see that there exists θ ∈ S(R + ) such that θ(0) = 1 and  has compact support.Now let us define ζ() = ( −2 )  θ( −2 ),  ∈ R + .Here  > 0 and  is a nonnegative integer.Then () =   (L  )() =  Δ (L  )().Hence, if we take ,  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  0 is sufficiently large.
We are now ready to prove Theorem 18.
Definition 16.Let  be a positive integer.A kernel of order  is a distribution  ∈ S  () with the following properties:(i)  coincides with a   function () away from the group identity 0 and enjoys the regularity condition:        ()      ≤   || −−() , for || ≤ ,  ̸ The convolution operator  with kernel of order  is called a singular integral operator of order .Remark 17.Using [3, Proposition 1.29], it is easy to verify that (67) is equivalent to the following condition:         ()      ≤   || −−() , for || ≤ ,  ̸ = 0.
Hence, for every normalized bump function , by the homogeneity of  we have      ⟨,   ⟩ , it is easy to verify that the last integral converges absolutely and is bounded by a constant independent of  and .Hence  satisfies the condition (ii) in Definition 16.Now we state the main result of this section.