ABSTRACT

Let be the -dimensional Heisenberg group. In this paper, we shall give among other things, the properties of some homogeneous norms relative to dilations on the and prove the equivalence of these norms.

1. Introduction

The Heisenberg group (of order), is a noncommutative nilpotent Lie group whose underlying manifold is with coordinates and group law given by

Setting, then

forms a real coordinate system for. In this coordinate system, we define the following vector fields:

The set forms basis for the left invariant vector fields on [1]. These vector fields span the Lie algebra of and the following commutation relations hold:

Similarly, we obtain the complex vector fields by setting

In the complex coordinate, we also have the commutation relations

If we identify with then each element of is given by and the group law becomes

where denotes the scalar product of The neutral element of is of the form and the inverse element

The centre of is given by

and therefore isomorphic to the additive locally compact topological group The Haar measure on is the Lebesgue measure on [1].

On the group, we introduce the group

of dilations defined for each element of

by on the complex coordinates and by on the real coordinates. The family of dilations forms a one-parameter group of automorphisms of Indeed, we have the following properties of this family of dilations.

(i)

(ii) Moreover(iii) Properties (i) and (iii) can be easily seen [2,3]. To see (ii), we notice that: For and we have

With these dilations as automorphisms of becomes a stratified Lie group whose generators are the defined vector fields [4]. Similarly, and its Lie structure equipped with this family of dilations is a homogeneous group of dimension [5].

2. Homogeneous Norms on

Definition 2.1: A norm on the Heisenberg group, is a function

(2.1)

satisfying the following properties:

(i),

(ii),

(iii)(iv) for all and where

The value is called the Heisenberg distance of from the origin and

is the Heisenberg unit ball [6]. We say the norm in is homogeneous of degree with respect to the dilations if for any we have. The value given by

is the popular Koranyi norm on which is always positive definite [7].

Property (i) is the homogeneity of the Heisenberg norm while property (iv) indicates the subadditivity of the Heisenberg norm. The proof of properties (i)-(iii) is trivial and that of (iv) can be found in [8].

Following [9], we shall further define the following norms on. For define

(2.2)

We notice that gives a choice which is not smooth away from the origin. The norm

and the properties above do not uniquely determine the norm. For if is positive, smooth away from 0, and homogeneous of degree 0 in the Heisenberg group dilation structure, then gives another norm [10].

Since it can be equipped with the Euclidean norm in denoted by and defined by

We have the following:

Proposition 2.3 [10]: For we have

We notice however, that this norm is not homogeneous. In what follows, we show that homogeneous norms on the Heisenberg group are equivalent following [10].

Lemma 2.4: Let be a homogeneous norm on Then, there is a constant such that

where is as defined in (2.2).

Proof: Now observe that is homogeneous of degree and by hypothesis, is homogeneous. Let

and set

Now, if we identify as then sup is actually a maximum and inf is a minimum. Thus exists and the inequality in the theorem holds. This is possible since and follows from the fact that is a compact subset of not containing the origin and is a continuous function which is strictly positive in

Corollary 2.5: For every fixed homogeneous norm on there exists a constant such that

Proof: We notice that the norm function is continuous and therefore, Now consider the the group of dilations on Then

is an automorphism of Therefore, by Lemma 2.4, the result follows.

Theory 2.6: Any two homogeneous norms on are equivalent.

Proof: We apply the previous method as follows: Let

and define by

Then

is obviously continuous by the homogeneity property with respect to Since is bounded with respect to attains it bounds and therefore, exists. Thus, such that If

then there exists such that so that

The theorem then follows by Lemma 2.4.

REFERENCES