Construction of Frames on the Heisenberg Groups

The theory of wavelet analysis has played an important role in many di erent branches of science and technology; see, for instance, [1, 2, 7, 8, 11] and the references therein. Wavelet analysis provides a simpler and more e cient way to analyze functions and distributions that have been studied through Fourier series and integrals. R. CoifmanandG.Weiss invented the atoms andmolecules (cf. [6, 21])which formed the basic building blocks of various function spaces. The atom decomposition can be obtained by using a discrete version of a well-known identity, due to A. Calderón ([3]), in whichwavelets were implicitly involved. The wavelet series decompositions are nowadays e ective expansion by unconditional bases in various function spaces arising from the theory of harmonic analysis. Let us now recall the frame constructed by Frazier and Jawerth in [12] using the Fourier transform. Let ψ ∈ S(Rn) with


Introduction
The theory of wavelet analysis has played an important role in many di erent branches of science and technology; see, for instance, [1,2,7,8,11] and the references therein. Wavelet analysis provides a simpler and more e cient way to analyze functions and distributions that have been studied through Fourier series and integrals. R. Coifman and G. Weiss invented the atoms and molecules (cf. [6,21]) which formed the basic building blocks of various function spaces. The atom decomposition can be obtained by using a discrete version of a well-known identity, due to A. Calderón ([3]), in which wavelets were implicitly involved. The wavelet series decompositions are nowadays e ective expansion by unconditional bases in various function spaces arising from the theory of harmonic analysis.
If f is a test function as above, we denote f ∈ M(β, γ, r, z , t ) and the norm of f ∈ M(β, γ, r, z , t ) is de ned to be the smallest constant C given in (i) and (ii) above.
The following is our main result, whose proof is based on the Calderón-Zygmund operator theory.

Theorem 1.2. Suppose that ϕ(z, t) is a function satisfying the above conditions (i) and (ii) with
Then there exist families of functions ψ j (z, t; w, s) and ψ * j (z, t; w, s) such that for all f ∈ L (H n ), where Q represents a dyadic cube on H n of side length (Q) = −j−N with N being a large xed positive integer, and ψ j (z, t, z Q , t Q ) and ψ * j (z Q , t Q , z, t) satisfy the similar smoothness and cancellation conditions as ψ j ((z, t) · (z Q , t Q ) − ).
For clari cation purposes, we point out that the number N must be su ciently large, depending on ϕ and the dimension. As mentioned before, the key tool used in the proof for Frazier-Jawerth's result is the Fourier transform. In contrast, Theorem 1.2 is proven through the Calderón-Zygmund operator theory.
Throughout the article, let X Y mean that there is a constant c > such that X ≤ cY. This article is organized as follows. In Section 2, we present preliminaries and some lemmas. The main result, Theorem 1.2 is proven in Section 3.

Preliminaries and some lemmas
The n-dimensional Heisenberg group H n consists of the set C n × R = {(z, t) : z ∈ C n , t ∈ R} with the multiplication law (z, t) · (w, s) = (z + w, t + s + Im(zw)).
We de ne balls B((z, t), δ) in H n by These balls are left-invariant under the action of H n . We refer the reader to Stein [20] for more background about the Heisenberg groups. Notice that (H n , ρ, dzdt) is a space of homogeneous type in the sense of Coifman and Weiss [9]. Christ [5] provided a dyadic grid in a space of homogeneous type.
We may think of I k α as being a dyadic cube with side-length (I k α ) = −k centered at z k α . The following is the de nition of generalized approximations to the identity of the Heisenberg group, whose kernels have only Lipschitz smoothness.

De nition 2.2.
Let < ϵ ≤ . A sequence {S k } k∈Z of operators is said to be an ϵ-approximation to the identity (or simply an approximation to the identity) if there exists < C < ∞ such that for all (z, t), (w, s) ∈ H n , S k (z, t; w, s), the kernel of S k , are functions from H n × H n into C satisfying ((z, t), (w, s))) n+ +ϵ ; (2.1) Note that if ϕ(z, t) is a function satisfying the above conditions (i) and (ii) with H n ϕ(z, t)dzdt = , then ϕ k ((z, t) · (w, s) − ) is an ϵ-approximation to the identity. Let D k = S k − S k− and D k (x, y) be the kernel for k ∈ Z. Then D k (·, y) ∈ M(ϵ, ϵ, −k ) for any xed y = (w, s) and k, and similarly, D k (x, ·) ∈ M(ϵ, ϵ, −k ) for any xed x = (z, t) and k.
The following result follows from a more general one in [11, Lemmas 3.7 and 3.11].
Then for < ϵ < ϵ, there exists a constant C which depends on ϵ and ϵ, but not on k, l such that for l ≤ k and d((w, s), (w , s )) ≤ a ( −l + d((z, t), (w, s)))

De nition 2.4. An approximation to the identity {S k } k∈Z is said to satisfy the double Lipschitz condition if
Lemma 2.5. [11,Lemma 3.12] Suppose that {S k } k∈Z is an approximation to the identity and S k ((z, t), (w, s)), the kernels of S k , satisfy the condition (2.6). Set D k = S k − S k− for all k ∈ Z. Then for any < ϵ < ϵ, there exists a constant C which depends on ϵ and ϵ, but not on k or l, such that De nition 2.6. Let D(R n ) be the space of smooth functions on H n with compact support, and let D (H n ) denote its dual space. An operator T is said to be a Calderón-Zygmund singular integral operator with the kernel K if s)} and satis es the following estimates: For some < ϵ ≤ and some c > , The main tool we will use to show Theorem 1.2 is the following result.
Theorem 2.7. [15] Suppose that T is a Calderón-Zygmund singular integral operator and extends to be a bounded operator on L (H n ). Furthermore, if T( ) = T * ( ) = and the kernel K(z, t; w, s) satis es the following double di erence condition: where T = T L →L + T CZ , the T CZ being the smallest constants in the de nition of Calderón-Zygmund kernels and (2.7).
We would also like to mention a result of Meyer. To study the property of the Besov spaceḂ , (R n ), Meyer introduced the following de nition of smooth atoms.
De nition 2.8. [19] A function f (x) is said to be a smooth atom if there exist < β ≤ , γ, r > , and a constant C such that M(β,γ,r) .

If f is a smooth atom as above, then the norm of f is de ned by the smallest constant C given in (i) and (ii) above and is denoted by f
We would like to point out that if f ∈ M(β, γ, r), then f ∈ M(β, γ, r). Meyer in [19] proved the following theorem. The extension of this result to spaces of homogeneous type can be found in [11]. Theorem 2.9. [19] If T is a Calderón-Zygmund singular integral operator and extends to be a bounded operator on L (R n ) and T( ) = T * ( ) = , then there exists a constant C such that M(β,γ,r) for < β < β < ϵ, < γ < γ < ϵ, and all r > .

Proof of Theorem 1.2
Let ϕ j (z, t) = j( n+ ) ϕ( j z, j t), and S j (f )(x) = ϕ j * f (x). Then {S j } is an ϵ-approximation to the identity satisfying the double Lipschitz condition (2.6), and the following properties  (z, t). Then we will apply Coifman's decomposition of the identity operator as follows.
For any large positive integer N, we write To prove Theorem 1.2, we write f as follows: where R N (f )(z, t) = j,k:|j−k|>N ψ k * ψ j * f (z, t). For any j ∈ Z, denote ψ N j = k:|k|≤N ψ j+k . We write where Q denotes dyadic cubes in the sense of Lemma 2.1, and (z Q , t Q ) stands for any xed point in Q.