A convolution type characterization for L p-multipliers for the Heisenberg group

It is well known that if m is an Lp - multiplier for the Fourier transform on ℝn(1<p<∞) , then there exists a pseudomeasure σ such that Tmf  =σ*f. A similar result is proved for the group Fourier transform on the Heisenberg group Hn. Though this result is already known in generality for amenable groups, a simple proof is provided in this paper.


Introduction
One of the fundamental problems in Multiplier theory is the characterization of L p multipliers (1 < p < ∞) for the Fourier transform.Different types of such characterizations have been considered under various situations in the literature.One of the important characterizations is to obtain sufficient conditions on m ∈ L ∞ (R n ) such that T m f = (m f ) ∨ gives rise to a bounded operator T m on L p (R n ), which is popularly known as Hormander's multiplier theorem.According to another characterization, if m is an L p -multiplier for the Fourier transform, then T m can be expressed as a convolution operator using a pseudo measure σ .In particular if p = 1, σ turns out to be a finite Borel measure.Yet another type of characterization is to identify the class of multipliers (same as translation invariant operators on a locally compact abelian group) for L p as the dual space of a certain function space.These characterizations have been proved not only for R n but also for a general locally compact abelian group.For details, we refer to Hormander [12], Stein [18], Figà-Talamanca [8] and Larsen [13].The identification of the class of (right) translation invariant operators as the dual space for an amenable group has been studied by several authors.(See [3], [7], [10]).This type of result is shown for some non amenable groups by Cowling in [2] and for a general locally compact group in [5].Hormander's multiplier theorem has been proved for Weyl transform by Mauceri [14] in 1980 and for the Heisenberg group Fourier transform by Michele and Mauceri [15] in 1979.In 1998, it was proved by Radha and Thangavelu [17] that Weyl multiplier for L p (C n ) corresponds to a twisted convolution operator using a 'pseudomeasure'.
In this paper, we prove a similar result for an L p -multiplier for the group Fourier transform on the Heisenberg group H n .However this result is not new.In fact, the more general abstract result has already been established for amenable groups in [7] and [11].We also refer to Cowling [4] for further details and Pier [16] where the result of Herz is presented.We provide a simple proof in this paper.
It is important to mention here that unlike the Fourier transform on R n , the multiplier associated with the Heisenberg group Fourier transform is vector valued.

Notations and Background
Let H n denote the Heisenberg group.It is a unimodular nilpotent Lie group whose under lying manifold is C n × R and the group operation is defined by The Haar measure is given by dzdt.
By Stone-Von Neumann theorem (Folland [9]), the only infinite dimensional unitary irreducible representations (upto unitary equivalence) are given by π λ , λ ∈ R * , where π λ is defined by As in the case of R n , the group Fourier transform f satisfies the basic properties, namely if denotes the convolution, then (f * g) ∧ (λ) = f (λ)ĝ(λ).Under this convolution operation L 1 (H n ) becomes a non-commutative algebra.For further results of the group Fourier transform we refer to Thangavelu [19].

The Main Result
This definition is meaningful because This helps us to define a multiplier for L p (H n ).

Then it is easy to verify that A(H n ) is a Banach space under || • || A . Let P(H n ) denote the class of continuous linear functionals on A(H n ). viz. P(H
where * denotes the dual.Since the construction of P(H n ) is similar to that of the class of pseudomeasures on R n in the abstract sense, we call the members of P(H n ) as pseudomeasures on H n .
We shall show that there is an isometric isomorphism of It is important to notice that since the L 1 space under consideration consists of B 1 -valued functions (vector valued), the dual space is not known.In other words the well known relation (L p ) * = L p , 1 p + 1 p = 1, 1 ≤ p < ∞ for the complex-valued functions is not true in the case of vector valued functions.However, there is an isometric isomorphism of L p (R n , A * ) into a subspace of [L p (R n , A)] * defined as follows: For then L g is a bounded linear functional on L p (R n , A) and the mapping (Here A denotes a Banach space).For a more detailed study of vector valued functions and vector measures, we refer to Diestel and Uhl [6].
We now proceed towards establishing identification between P(H n ) and which shows that G ϕ is a continuous linear functional on A(H n ).Thus for σ ∈ P(H n ), there is a unique ϕ ∈ [L 1 (R * , B 1 , dμ)] * , which we denote by σ .Further from inequalities (1) and (2) it can be easily shown that ||σ|| = ||σ||.Thus σ −→ σ is a isometric isomorphism from P(H n ) onto [L 1 (R * , B 1 , dμ)] * .Now we are in a position to prove our main result of this section.Before stating the theorem, let us look at the following definitions.
where dX denotes the Lebesgue measure.This definition is meaningful.For, first notice that as g(λ The definition of ν g is meaningful.Further, by straight forward computations, it is easy to verify that ν , then ν g coincides with ordinary multiplication of ν and g .In fact,

The equation (3) follows from the following fact: If
Then B → F B is an isomorphism of B onto B * 1 (see Conway [1]).Theorem 2.5.Let m be an L p -multiplier for the group Fourier transform on the Heisenberg group H n .Then there exists a pseudomeasure σ such that Proof.
and L 2 (R * , B, dμ) denote the collection of square integrable B -valued functions on R * under the measure dμ, then the group Fourier transform is an isometric isomorphism between L 2 (H n ) and L 2 (R * , B, dμ).The inversion formula is given by