Pointwise Multipliers of Triebel-Lizorkin Spaces on Carnot-Carathéodory Spaces

Let X, d, μ be a Carnot-Carathéodory space, namely, X is a smooth manifold, d is a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. μ is a nonnegative Borel regular measure on X satisfying that there exists constant C0 ∈ 1,∞ such that for all x ∈ X and 0 < r <diamX, μ B x, 2r : μ {y ∈ X : d x, y < 2r} ≤ C0μ B x, r < ∞ doubling property . Using the discrete Calderón reproducing formula and the PlancherelPôlya characterization of the inhomogeneous Triebel-Lizorkin spaces developed in Han et al., in press and Han et al., 2008, pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces are obtained.


Introduction
The multiplier theory of function spaces has been studied for a long time, and a lot of results have been obtained.As we know, the multiplier theory is one of the important parts in the studies of the Gleason problem, function space properties, and general operator theory.The pointwise multipliers on R d are studied as a part of the researches of function spaces in several monographs, 1-8 .Pointwise multipliers have been found many important applications in partial differential equations.
However, it was not clear how to generalize the pointwise multipliers on R n to spaces of homogeneous type introduced by Coifman and Weiss see 9 because the Fourier transform is no longer available.The main purpose of this paper is to establish pointwise multipliers on inhomogeneous Triebel-Lizorkin spaces in the setting of Carnot-Carathéodory spaces.To be more precisely, we first recall some necessary definitions.In this paper, we always assume that X, d is a metric space with a regular Borel measure μ such that all balls defined by d have finite and positive measures.In what follows, set diam X ≡ sup{d x, y : x, y ∈ X}, and for any x ∈ X and r > 0, set B x, r ≡ {y ∈ X : d x, y < r}.Definition 1.1 see 10 .Let X, d be a metric space with a Borel regular measure μ such that all balls defined by d have finite and positive measures.The triple X, d, μ is called a space of homogeneous type if there exists a constant C 0 ∈ 1, ∞ such that for all x ∈ X and r > 0, μ B x, 2r ≤ C 0 μ B x, r doubling property . 1.1 Remark 1.2.We point out that the doubling condition 1.1 implies that there exist positive constants C and n such that for all x ∈ X and λ ≥ 1, where C is independent of x and r.Denote by n the homogeneous "dimension" of X as in 10 .
A space of homogeneous type is called an RD space, if there exist constants a 0 , C 0 ∈ 1, ∞ such that for all x ∈ X and 0 < r < diam X /a 0 , C 0 μ B x, r ≤ μ B x, a 0 r 1.3 that is, some "reverse" doubling condition holds.
Clearly, any Ahlfors n-regular metric measure space X, d, μ which means that there exists some n > 0 such that μ B x, r ∼ r n for x ∈ X and 0 < r < diam X /2 is a n, nspace see 10 , also is an RD space and space of homogeneous type in the sense of Coifman in 10 .In other words, μ satisfies the doubling condition which is weaker than Ahlfors nregular metric measure space and RD-spaces.
Another typical such a space is Carnot-Carathéodory space.One example with unbounded total measure studied in 11 is that X arises as the boundary of an unbounded model polynomial domain in C 2 .Let Ω { z, w ∈ C 2 : Im w > P z }, where P is a real, subharmonic, non-harmonic polynomial of degree m.Then X ∂Ω can be identified with C × R { z, t : z ∈ C, t ∈ R}.The basic 0, 1 Levi vector field is then Z ∂/∂z − i ∂P/∂z ∂/∂t , and we write Z X 1 iX 2 .The real vector fields {X 1 , X 2 } and their commutators of order ≤ m span the tangent space to X at each point.See 10, 12 for more details and references therein.
We will also suppose that μ X ∞, μ {x} 0 for all x ∈ X.For any x, y ∈ X and δ > 0, set V δ x ≡ μ B x, δ and V x, y ≡ μ B x, d x, y .It follows from 1.1 that V x, y ∼ V y, x .The following notion of approximations of the identity on RD spaces was first introduced in 10 .Let Z N ∪ {0}.We begin with recalling the definition of an approximation to the identity, which plays the same role as the heat kernel H s, x, y does in Nagel-Stein's theory 11 .
Definition 1.3 see 10, 12 .A sequence {S k } k∈Z of operators is said to be an approximation to the identity in short, ATI if there exists constant C 1 > 0 such that for all k ∈ Z and all x, x , y, and y ∈ X, S k x, y , the kernel of S k satisfies the following conditions: The space of test functions plays a key role in this paper; see 10 .
Definition 1.4.Fix two exponents 0 < β ≤ 1 and γ > 0. A function f defined on X is said to be a test function of type β, γ centered at x 0 ∈ X with width r > 0 if there exists a nonnegative constant C such that f satisfies the following conditions: If f is a test function of type β, γ centered at x 0 with width r > 0, we write f ∈ M x 0 , r, β, γ , and the norm of f in M x 0 , r, β, γ is defined by inf{C ≥ 0 : 1.8 and 1.9 hold}.

1.10
We denote by M β, γ the class of all f ∈ M x 0 , 1, β, γ .It is easy to see that M x 1 , r, β, γ M β, γ with the equivalent norms for all x 1 ∈ X and r > 0. Furthermore, it is also easy to check that M β, γ is a Banach space with respect to the norm in M β, γ .
In what follows, for given ∈ 0, 1 , we let M β, γ be the completion of the space M , in M β, γ when 0 < β, γ ≤ .Obviously M , M , .Moreover, f ∈ M β, γ if and only if f ∈ M β, γ when 0 < β, γ ≤ and there exists {f j } j∈N ⊂ M , such that M β, γ is a Banach space, and we also have f M β,γ lim j → ∞ f j M β,γ for the above chosen {f j } j∈N .
We denote by M β, γ the dual space of M β, γ consisting of all linear functionals L from M β, γ to C with the property that there exists a constant C such that for all f ∈ M β, γ , 1.11 We denote by h, f the natural pairing of elements h ∈ M β, γ and f ∈ M β, γ .Since M x 1 , r, β, γ M β, γ with the equivalent norms for all x 1 ∈ X and r > 0. Thus, for all h ∈ M β, γ , h, f is well defined for all f ∈ M x 0 , r, β, γ with x 0 ∈ X and r > 0.
The following constructions, which provide an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type, were given by Christ in 13 .Lemma 1.5.Let X be a space of homogeneous type.Then there exist a collection open subsets, where I k is some (possible finite) index set, and constants δ ∈ 0, 1 and ii for any α, β, k, l with l ≥ k, either iii for each k, α and each l < k there is a unique In fact, we can think of Q k α as being a dyadic cube with a diameter roughly δ k and centered at z k α .In what follows, we always suppose δ 1/2.See 14 for how to remove this restriction.Also, in the following, for k ∈ Z , τ ∈ I k , we will denote by Q k,ν τ , ν 1, . . ., N k, τ, M , the set of all cubes where M is a fixed large positive integer.Now, we can introduce the inhomogeneous Triebel-Lizorkin spaces F α,q p X via the approximation in Definition 1.3.Note that the Triebel-Lizorkin spaces have been already investigated for decades in the study of partial differential equations, interpolation theory, and approximation theory.Definition 1.6.Suppose that {S k } k∈Z is an ATI and let D 0 S 0 , and Let M be a fixed large positive integer, Q 0,ν τ be as above.Suppose that −1 < s < 1.The inhomogeneous Triebel-Lizorkin space F α,q p where The restrictions 1.12 guarantee that the definitions of the inhomogeneous Triebel-Lizorkin space F s,q p X for max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞ are independent of the choices of β and γ satisfying these conditions and F The classical scale of inhomogeneous Triebel-Lizorkin spaces contains many wellknown function spaces.For example, if α > 0, p q ∞, one recovers the H ölder-Zygmund spaces The space C α X is defined as the collection of f such that p X h p X are the inhomogeneous Hardy spaces, which are closely related to the Hardy spaces H p X in 15, 16 more precisely: the homogeneous spaces Ḟ0,2 p X coincide with the usual Hardy spaces H p X .We will use here the notation F Then these spaces will be denoted as the inhomogeneous Hardy-Sobolev spaces, which include the above Lebesgue-Sobolev spaces for 1 < p < ∞.
The inhomogeneous Triebel-Lizorkin spaces have the following Plancherel-P ôlya characterizations in 10 , which will be one of the the basic tools to prove the main results of this paper.

1.16
We now introduce the following definition of the pointwise multiplier.
Definition 1.8.Suppose that g is a given function on X.Then g is called a pointwise multiplier for F s,q p X if f → gf admits a bounded linear mapping from F s,q p X into itself.
The main result in this paper is the following.
Theorem 1.9.Let −1 < s < 1, max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞ and α > max s, n/ min{p, q, 1} − n − s , then g ∈ C α X with 0 < α < 1 is a multiplier for F s,q p X .In other words, f → gf yields a bounded linear mapping from F s,q p X into itself and there is a positive constant C such that 1.17 holds for all g ∈ C α X and f ∈ F s,q p X .
We would like to point out that the study of pointwise multipliers is one of important problems in the theory of function spaces.It has attracted a lot of attention in the decades since starting with 7 .Pointwise multipliers in general spaces F s,q p R n , where 0 < p < ∞, 0 < q ≤ ∞, s ∈ R have been studied in great detail in 4, 5, 8 .Theorem 1.9 was proved in 8 for pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces on R n based on the Fourier transform.In the present setting, however, we do not have the Fourier transform at our disposal.Since the Fourier transform on Carnot-Carathéodory spaces is not available and hence the idea used in 8 does not work for this more general setting.A new idea to prove Theorem 1.9 is to use the discrete Calder ón reproducing formula, which was developed in 10 .Therefore, this scheme easily extends to geometrical settings where the Fourier transform does not exist.The Fourier transform is missing but a version of pointwise multiplier is still present.
Throughout, we also denote by C a positive constant independent of main parameters involved, which may vary at different occurrences.Constants with subscripts do not change through the whole paper.We use f g and f g to denote f ≤ Cg and f ≥ Cg, respectively.If f g f, we then write

Proof of Theorem 1.9
In this section, we will prove Theorem 1.9.Since there is no the Fourier transforms on spaces of homogeneous type, the proof of Theorem 1.9 is quite different from the proof of Theorem where the series converges in the norm of Journal of Function Spaces and Applications when k ∈ N; X D k x, y dμ y X D k x, y dμ x 1 when k 0. To prove Theorem 1.9, we first show the following lemma.Lemma 2.2.Let {S k x, y } k∈Z and {G k x, y } k∈Z be two approximations to the identity as in Lemma 2.1 above and where k, k ∈ Z .
Proof.We first show the inequality 2.5 of the case k k 0 above.In fact, in this case, it follows that

2.6
We next consider the case k ≥ k for k > 0 and k ≥ 0 or k ≥ 0 and k > 0. We write

2.7
For E, we can obtain where k, k ∈ Z .

Journal of Function Spaces and Applications 9
We may rewrite this integral as 2.9 where Observe further, a simple argument yields the same estimate for k, k ∈ Z as follows:

2.11
Combining the estimate for E, F, we conclude that if k ≥ k, then 2.5 holds.The case k ≥ k is similar to above.This finishes the proof of Lemma 2.2.Now we show the following technical version of Theorem 1.9.Proposition 2.3.For any g ∈ C α X , f ∈ M β, γ with β and γ satisfying 1.12 , then where −1 < s < 1, max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞ and α > max s, n/min{p, q, 1} − n − s .

Journal of Function Spaces and Applications
Proof.Using the Calder ón reproducing formula, for any g ∈ C α X , f ∈ M β, γ , We write

Applying the H ölder inequality for p > 1 and
for all a k ∈ C and p ≤ 1, for L 1 , it follows that where we used the fact that for any y 0,ν For L 2 , in fact for all k ∈ N, τ ∈ I k and ν 1, 2, . . ., N k , τ , M , and all x ∈ X, z ∈ From the inequality 2.5 , the Hölder inequality for q > 1 and 2.14 for q ≤ 1, the Fefferman-Stein vector-valued maximal function inequality in 17 and Lemma 1.7, it follows that where we choose r satisfying max n/ n , n/ n s α < r < min{1, p, q}.By the inequality 2.5 , similarly,

Journal of Function Spaces and Applications 13
We now consider the estimate of L 4 , Similar to the estimate of L 2 , using the equality 2.5 , the Hölder inequality if q > 1 and 2.14 if q ≤ 1, the Fefferman-Stein vector-valued maximal function inequality in 17 , we obtain

2.20
where the first and the second inequalities follow from the estimates with max n/ n 1 , n/ n s α < r < min{1, p, q}, s can be any number in − , α , which verifies Proposition 2.3.
To introduce our definition of pointwise multiplication, the interesting estimate is needed.
Lemma 2.4.Let {S k x, y } k∈Z be a approximation to the identity as in Lemma 2.1 above and D k S k − S k−1 for k ∈ N and D 0 S 0 .For any g ∈ C α X with 0 < α < 1, h ∈ M β, γ with β and γ satisfying 1.12 .Then where k ∈ Z .
Proof.We first prove inequality 2.22 with k 0. In fact, since D 0 z, y 0 if d z, y ≥ 2C 1 , then

2.25
That is, 2.22 with k 0 holds.To prove 2.22 with k ∈ N, we write

y g y h x − h y dμ x
: I II.

2.26
For I, since D k x, y 0 if d x, y ≥ 2C 1 2 −k , we obtain

2.27
For I 1 , by d x, y ≤ 1/2 1 d y, x 0 implies that d y, x 0 ≤ 2 d x, x 0 1 and

Journal of Function Spaces and Applications
To obtain the estimate of the II, we write where 1 {x ∈ X : d x, y ≤ 1/2 1 d y, x 0 } and 2 {x ∈ X : d x, y > 1/2 1 d y, x 0 }.
For II 1 , we have

2.33
For II 2 , similar to I 2 , we have

2.36
Combining the estimate of I and II, we obtain that for k ∈ N

2.37
Thus, 2.22 also holds.This finishes the proof of Lemma 2.4.
s < q ≤ ∞, it is not clear in general what is meant by gf pointwise multiplication .Our approach is the following.Lemma 2.5.For any f ∈ F s,q p X with max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞, −1 < s < 1, and g ∈ C α X with 1 > α > max s, n/min{p, q, 1} −n−s .There exist a constant C and a sequence {f j } j∈N such that f j ∈ M , , f j F s,q p X ≤ C f F s,q p X and lim j → ∞ gf j , h converges for any h ∈ M β, γ with β and γ satisfying 1.12 .
Proof.For any f ∈ F s,q p X with max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞, −1 < s < 1, and g ∈ C α X with α > max s, n/min{p, q, 1} − n − s , denote It is easy to see that f j ∈ M , .Applying a similar proof as in Proposition 2.3 with g 1 and f f j gives f j F s,q p X ≤ C f F s,q p X .We write

2.39
The fact 2.22 implies

2.40
Thus this finishes the proof of R → 0 as j, m → ∞.
We now consider the estimate of T .By Hölder inequality if p, q > 1, and 2.14 if p, q < 1 and 2.22 , it follows that

2.42
where let ε be a positive number with s − ε n 1 − 1/p α ∧ β > 0 when p ≤ 1 and s α ∧ β > 0 if p > 1, and the second inequality can be obtained by using the fact that X if s ∈ −1, 1 , p, q > 1.Thus by the Calder ón reproducing that the series converges in the norm of F −s,q p X for gh ∈ F −s,q p X with s ∈ −1, 1 , p, q > 1 in Lemma 2.1, which proves T → 0 as j, m → ∞ when max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞, −1 < s < 1, and g ∈ C α X with α > max s, n/min{p, q, 1} − d − s .
For Y , applying the H ölder inequality for p, q > 1 and 2.14 for p, q ≤ 1 and 2.22 , we also have

2.44
where let ε be a positive number with s−ε n 1−1/p α∧β > 0 when p ≤ 1 and s α∧β > 0 if p > 1, and we also used the fact that F s,q p X ⊂ B s,max p,q p X ⊂ B s−ε,∞ p X , when − < s−ε.This finishes the proof of Y → 0 as j, m → ∞, and hence the proof of Lemma 2.5 is concluded.
The above estimate shows lim j → ∞ gf j , h exists and the limit is independent of the choice of f j .Therefore, for g ∈ C α X , f ∈ F s,q p X with max n/ n 1 , n/ n 1 s < p < ∞ and max n/ n 1 , n/ n 1 s < q ≤ ∞ s ∈ −1, 1 , α > max s, n/min{p, q, 1} − n − s .We define gf, h lim j → ∞ gf j , h 2.45 for any h ∈ M β, γ with β and γ satisfying 1.12 , f j is fundamental sequence defined in Lemma 2.5.We now prove Theorem 1.9.
Proof of Theorem 1.9.By Proposition 2.3 and Lemma 2.5, for any g ∈ C α X , f ∈ F s,q p X , Fatou's lemma implies that g C α X f F s,q p X .

2.46
We complete the proof of Theorem 1.9.
and M β , γ for f ∈ M β, γ with β < β and γ < γ, and M β , γ for f ∈ M β, γ with > β > β and > γ > γ.Moreover, D k x, y and D k x, y , the kernels of D k and D k , satisfy the similar estimates but with x and y interchanged in 2.3 : for 0 < < 1,
Lemma 2.1.Suppose that {S k } k∈Z is an approximation to the identity as in Definition 1.3.Set D k S k − S k−1 for k ∈ N and D 0 S 0 .Then for any fixed M ∈ N large enough, there exists a family of functions { D k x, y } k∈Z and { D k x, y } k∈Z such that for any fixed y k,ν τ