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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 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.
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.  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  introduced fractional Sobolev spaces and Lipschitz spaces on stratified Lie groups. Later, Folland and Stein  established the theory of Hardy spaces on general homogeneous groups. Besov spaces on stratified Lie groups were first introduced by Saka , by means of the heat semigroup associated to the sub-Laplacian. Recently, Führ and Mayeli  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 . 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 , 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 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  (for ) and Korányi and Vági  (for ). 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 on stratified Lie groups, and give some basic properties of them. In Section 4 we show the -boundedness of a class of convolution singular integral operators. Throughout this paper the letter will denote a positive constant which is independent of the main variables involved but whose value may differ from line to line. The notation or for some variable quantities and means that for some constant ; stands for . We agree that the set of natural numbers contains .
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 . A Lie group is called a stratified Lie group if it is connected and simply connected, and its Lie algebra may be decomposed as a direct sum , with for and . Such a group is clearly nilpotent, and thus it may be identified with (as a manifold) via the exponential map . Examples of stratified Lie groups include Euclidean spaces and the Heisenberg group .
The algebra is equipped with a family of dilations which are the algebra automorphisms defined by Under our identification of with , 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 smooth away from (the group identity), vanishing only at , and satisfying and for all and . 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 such that for all . If and we define the ball of radius about by . The Lebesgue measure on induces a bi-invariant Haar measure on . As done in , we fix the normalization of Haar measure by requiring that the measure of be . We shall denote the measure of any measurable by . Clearly we have . All integrals on are with respect to (the normalization of) Haar measure. Convolution is defined by
We consider as the Lie algebra of all left-invariant vector fields on , and fix a basis of , obtained as a union of bases of the . In particular, , with , is a basis of . We denote by the corresponding basis for right-invariant vector fields, that is, If is a multi-index we set and . Moreover, we set where the integers are given according to that . Then (resp., ) is a left-invariant (resp., right-invariant) differential operator, homogeneous of degree , with respect to the dilations , .
A complex-valued function on is called a polynomial on if is a polynomial on . Let be the basis for the linear forms on dual to the basis for , and set . From our definition of polynomials on , 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 , if holds for all multi-indices with . We let denote the space of all polynomials on , and let denote the space of polynomials on of homogeneous degree . Note that is invariant under left and right translations (see [3, Proposition 1.25]). A function is said to have vanishing moments of order , if with the absolute convergence of the integral.
The Schwartz class on is defined by that is, if and only if . As is indicated in [3, p. 35], is a Fréchet space and several different choices of families of norms induce the same topology on . In this paper, for our purpose we use the family of norms given by Here and in what follows, we use the notation convention for any function . The dual space of is the space of tempered distributions on . If and we shall denote the evaluation of on by .
Using the above conventions for the choice of the basis , and , the sub-Laplacian is defined by . When restricted to smooth functions with compact support, is essentially self-adjoint. Its closure has domain , where is taken in the sense of distributions. We denote this extension still by the symbol . By the spectral theorem, admits a spectral resolution where is the projection measure. If is a bounded Borel measurable function on , the operator is bounded on , and commutes with left translations. Thus, by the Schwartz kernel theorem, there exists a tempered distribution on such that Note that the point may be neglected in the spectral resolution, since the projection measure of is zero (see [8, p. 76]). Consequently we should regard as functions on rather than on .
Let denote the space of restrictions to of functions in . An important fact proved by Hulanicki  is as in the following lemma.
Lemma 1. If then the distribution kernel of is in .
Moreover, from the proof of [10, Corollary 1] we see that if is a function in 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 , and use Greek alphabets without hats to denote the associated distribution kernels; for example, for we shall denote by the distribution kernel of the operator , where is a sub-Laplacian fixed in the context.
3. Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups
For any function on and , we define the -normalized dilation of by 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 with . Suppose both have vanishing moments of order . Then there exists a constant such that for all and all , where .
Proof. Using dilations and the facts and (see (9)), we may assume . To proceed we follow the idea in the proof of [11, Lemma B.1]. Let , and . Let be the left Taylor polynomial of at of homogeneous degree (see [3, pp. 26-27]). Then using vanishing moments of
For , the stratified Taylor formula (cf. [3, Corollary 1.44]) yields that, with a suitable positive constant, since if . Hence we have where for the last inequality we used [3, Corollary 1.17] and that .
For , we have . On the other hand, . Thus, we have Also, we note that by [12, Proposition 20.3.14] the left Taylor polynomial is of the form where the integers are given according to that . From these remarks, it follows that where we used that and .
For we have , and, hence where for the last inequality we used [3, Corollary 1.17] and .
Combining the above estimates, we arrive at This is exactly what we need.
Let denote the space of Schwartz functions with all moments vanishing. We then consider as a subspace of , including the topology. It is shown in [5, Lemma 3.3] that is a closed subspace of , and the topology dual of can be canonically identified with the factor space .
We now have the following Calderón type reproducing formula.
Lemma 3. Suppose is a sub-Laplacian on , and is a function with compact support, vanishing identically near the origin, and satisfying Then for all , it holds that with convergence in . Duality entails that, for all , and the convergence is in .
Proof. First note that the -homogeneity of implies that the distribution kernel of coincides with . Let and be arbitrarily chosen. Then take such that . Since both and are Schwartz functions with all moments vanishing, it follows by Lemma 2 that where the constant is a suitable multiple of . This implies that converges uniformly in , for every and every . Consequently there exists such that converges in the topology of to , as . On the other hand, by (23) and the spectral theorem (cf. [13, Theorem VII.2]), holds in -norm. Therefore, , which completes the proof.
Let denote the class of all functions in satisfying
Definition 4. Let , and . Let be a sub-Laplacian on and . We define as the space of all such that with the usual modification for .
We then introduce the Peetre type maximal functions: Given , , a sub-Laplacian, and , we define
Lemma 5. Suppose is a sub-Laplacian and . Then for every there is a constant such that for all , all , and all ,
Proof. Because of (28) it is possible to find a function supported in such that for . Set , . Then for all . Consequently, for and , where for the last inequality we used [3, Corollary 1.17] and that . Dividing both sides of the above estimate by , and then taking the supremum over and , we obtain the desired estimate.
Lemma 6. Suppose is a sub-Laplacian, , and . Then there exists a constant such that for all , all , and all , where , and is the Hardy-Littlewood maximal operator on .
Proof. Let and . The stratified mean value theorem (cf. [3, Theorem 1.41]) gives that for every and every with , where is a suitable positive constant. Hence we have Putting , dividing both sides by , and using Lemma 5, we have where we have set and used that . Finally, taking sufficiently small (such that ), and taking the supremum over , we get the desired estimate.
Theorem 7. Suppose are any two sub-Laplacians on , and are any two functions in . Then, for , and , we have the (quasi-)norm equivalence
Proof. First we note that if then we have
for , where denotes the distribution kernel of . 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., ). Thus, to prove (37), it suffices to show that
To this end, let be a function in with support in such that for . For , by Lemma 3 we have with convergence in . Here is the distribution kernel of . Hence, since , we have the pointwise representation It follows that Since both and are Schwartz functions with all moments vanishing, we can use Lemma 2 to estimate that, with sufficiently large, Here, the constant is a suitable multiple of . On the other hand we observe that Putting these estimates into (42), multiplying both sides by , dividing both sides by and then taking the supremum over , we obtain In view of [15, Lemma 2], taking 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 is actually independent of the choice of and . Thus, in what follows we don't specify the choice of and and write instead of . Henceforth we shall fix any sub-Laplacian . Moreover, for the sake of briefness, we will write instead of .
Proposition 9. For , and , one has the continuous inclusion maps .
Proof. Let and . Choose and . Since both and are Schwartz functions with all moments vanishing, it follows by Lemma 2 that
for all and all , where the constant is a suitable multiple of . This together with [3, Corollary 1.17] give that
which implies that continuously.
Now we show the other embedding. Let and take any . Let , and then let be a function with support in such that for . Then by Lemma 3 we have To proceed we claim that for all . Assuming the claim for a moment, it follows from (48) that It is easy to see that To estimate the sum in (50), we note that if we choose and then similarly to (46) we have where the constant is a suitable multiple of . From this it follows that Therefore, This implies that continuously.
We are left with showing the claim. Indeed, if is fixed then by Lemma 6 we have, for all , Taking in the above estimate and using Hardy-Littlewood maximal inequality, we have 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].
Proposition 10. For , and , is a quasi-Banach space.
Let us introduce a class of functions. We say that , if there exists whose support is compact and which vanishes identically near the origin, and , such that . Clearly .
Lemma 11. Let and . Then is dense in . In particular, is dense in .
Proof. Take any and any . In the appendix we show that admits smooth atomic decomposition. By the smooth atomic decomposition, we see that is dense in , for and . Thus, we can find such that . 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 such that . Now we put Then , and we have This proves the claimed statement.
We next consider lifting property of . For , the power is naturally given by
Remark 12. By [17, Theorem 13.24], we have for all . As a consequence, is dense in , for all and .
We now have the lifting property of .
Theorem 13. Let and , .(i)The operator , initially defined on , extends to a continuous operator from to .(ii)Let denote the continuous extension of . Then is an isomorphism, and is an equivalent quasi-norm of .
Proof. (i) Let . Set and . Clearly is also in , and . By [17, Theorem 13.24], we have , and moreover
Hence, for every , we have .
Now let . By the above remarks, we have . It follows that Since is dense in (see Remark 12), the mapping extends to a continuous operator from to . We denote this extension by .
(ii) Let us first show that the mapping is injective. Indeed, assume such that is the zero element of . By Remark 12 we can find a sequence in which converges in to . Then applying (i) to yields that converges in to the zero element. Since and , applying (i) to the operator we see that converges in to the zero element. Therefore, is the zero element in . This proves that is injective.
Next we show that is surjective. Indeed, given , we let be a sequence in which converges in to . Then from (i) we see that converges in