Hardy’s inequalities for the twisted convolution with Laguerre functions

In this article, two types of Hardy’s inequalities for the twisted convolution with Laguerre functions are studied. The proofs are mainly based on an estimate for the Heisenberg left-invariant vectors of the special Hermite functions deduced by the Heisenberg group approach.


Introduction
The classical Hardy's inequalities on C state that if f (z) = ∞ k= a k z k belongs to the ordinary Hardy space H p (C),  < p ≤ , then one has the following results for the coefficients: In [], Radha and Thangavelu gave the following theorem, but the proof is left to the interested reader.
Here H p a (C n ) is the atomic Hardy space defined in terms of an atomic decomposition. When p = , this inequality has been established by Thangavelu []. The key tool they provided to prove this result was the estimate for the ordinary differential of Laguerre functions In this article we shall use an estimate related to the Heisenberg left-invariant vectors of the special Hermite functions, together with Taylor formula of Heisenberg group, to gain a new proof of this theorem. Moreover, we build another type of Hardy's inequality.
Comparing Theorems . and . with the classical Hardy's inequalities, we find that the varieties of indexes for k + n and k +  have the same similarity. We shall show this fact in the last section.
The outline of this paper is as follows. In Section  we briefly summarize the harmonic analysis on the Heisenberg group and the atomic theory needed in the sequel. Section  is devoted to the proof of our main result. In order to do this, some lemmas are stated in this section. We will adopt the convention that c denotes constants which may be different from one statement to another.

Preliminaries
The (n + )-dimensional Heisenberg group H n is a Lie group structure on C n × R with the multiplication law where zz = n j= z jz j . The Lie algebra G of H n , which admits a stratification by G = V  ⊕V  , is generated by the left-invariant vector fields and T = ∂ ∂t . The horizontal layer is just the first layer V  generated by Z j ,Z j ,  ≤ j ≤ n. Now let σ = (σ  , . . . , σ n , σ n+ ) = (, . . . , , ), I = (i  , . . . , i k ) with i  , . . . , i k ∈ {, , . . . , n + }, and use the notation σ (I) = σ i  + · · · + σ i k to denote the homogeneous length of I. We also write X j = Z j , X n+j =Z j ,  ≤ j ≤ n, X n+ = T and use the notation X I = X i  · · · X i k to denote the left-invariant differential operator.
A function P on H n is called a polynomial if P • exp is a polynomial on G. Every polynomial on H n can be written uniquely as a finite sum where η j = ζ j • log, ζ  , . . . , ζ n+ is the basis for G * dual to the basis X  , . . . , X n+ for G. The monomial η J is homogeneous of degree d(J) = n i= j i +j n+ and the homogeneous degree of P is given by d(P) = max{d(J) : a J = }. For s ∈ N = {, , , . . .}, we denote by P s the space of polynomials whose homogeneous degree ≤ s.
, then by Bonfiglioli [] one has the following horizontal Taylor formula with integral remainder for the Heisenberg group: where the notation log(z, ) = j≤n ξ j X j . Now we pick up the orthonormal basis in L  (R n ) to be the Hermite functions given by where H k (s) = (-) k e s  ( d ds ) k (e -s  ) is the Hermite polynomial. For λ ∈ R\{}, the Schrödinger representation λ of H n acts on L  (R n ) by where z = x + iy. Moreover, λ (z, t) is a strongly continuous unitary representation satisfying and λ (z, t)ϕ converges to ϕ in L  (R n ) as (z, t) →  in H n . If we set α,β (z, t) = (  (z, t) α , β ) and (z) =  (z, ), then α,β (z) = α,β (z, ) = ( (z) α , β ) is called the special Hermite function and { α,β (z)} forms an orthonormal basis for L  (C n ). Note that the operator is an irreducible projective representation of C n into L  (R n ) such that For various results related to these expansions, readers can refer to [, ]. Now we are going to introduce the atomic Hardy spaces H p a (C n ). For  < p ≤  ≤ q ≤ ∞, p = q, s ∈ N and s ≥ [n(/p -)], a function a is called the (p, q, s)-atom with the center z  if it satisfies (i) supp(a) ⊂ B r (z  ), (ii) a q ≤ |B r (z  )| /q-/p , and (iii) the cancelation conditions C n a(z)P(zz  )e i  Im zz  dz =  for any polynomial P whose degree ≤ s. For  < p ≤ , a tempered distribution f is said to be an element of the atomic Hardy space H p a (C n ) if it can be characterized by the decomposition where a j 's are (p, q, s)-atoms and ∞ j= |λ j | p < ∞. Moreover, the space H p a (C n ) can be made into a metric space by means of the quasi-norm defined by Note that the cancelation condition of (p, q, s)-atom is defined in consideration of the Weyl transform, and thus the atomic Hardy space H p a (C n ) defined above is different from the ordinary Hardy space H p (C n ) for  < p <  (see []).

Proofs of the main results
We are now in a position to give the proofs of the main results. First we state some crucial lemmas. For any (z, t) ∈ H n and α ∈ N n , we have
Then, for (z, ) = exp( j≤n ξ j X j ), by the horizontal Taylor formula and Lemma ., we where we have used the notation (z s , ) = exp( j≤n sξ j X j ) and the fact |ξ i l | ≤ c|(z, )| = c|z| (see Lemma  of []) in the second inequality.
Lemma . Suppose that f is a (p, , s)-atom supported in a ball B r (z  ). Then we have Proof From p. of [] we see that It follows that Now let P s ( α,γ , )(z, ) be the Taylor polynomial of α,γ (z, ) at the origin of homogeneous degree s. Then, by the cancelation condition of p-atom and Lemma ., we have where in the last inequality we have used (ii) of p-atom.
Proof of Theorem . For f ∈ H p a (C n ), it follows that f = ∞ j= λ j a j , where a j 's are (p, , s)atoms supported in B r (z  ). Then we get