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# Decomposition of Analytic Measures

Define measures and by their Fourier transforms: , and . Then we have the following decomposition theorem.

Theorem 5.1   Let be a locally compact abelian group with an ordered dual group . Suppose that is a sup path attaining representation of in . Then for any weakly analytic measure we have that the set of for which is countable, and that

 (32)

where the right side converges unconditionally in norm in . Furthermore, there is a positive constant , depending only upon , such that for any signs we have

 (33)

One should compare this theorem to the well-known results from Littlewood-Paley theory on , where (see Edwards and Gaudry [11]). For with , it is well-known that the subgroups form a Littlewood-Paley decomposition of the group , which means that the martingale difference series

converges unconditionally in to . Thus, Theorem 5.1 above may be considered as an extension of Littlewood-Paley Theory to spaces of analytic measures.

The next result, crucial to our proof of Theorem 5.1, is already known in the case that with the lexicographic order on the dual. This is due to Garling [15], and is a modification of the celebrated inequalities of Burkholder. Our result can be obtained directly from the result in [15] by combining the techniques of [3] with the homomorphism theorem 4.5. However, we shall take a different approach, in effect reproducing Garling's proof in this more general setting.

Theorem 5.2   Suppose that is a locally compact group with ordered dual . Then for , for any set of indices less than , and for any numbers ( ), there is an absolute constant such that

 (34)

Furthermore,

 (35)

where the right hand side converges unconditionally in the norm topology on .

Proof. The second part of Theorem 5.2 follows easily from the first part and Fourier inversion.

Now let us show that if we have the result for compact , then we have it for locally compact . Let denote the quotient homomorphism of onto the discrete group (recall that is open), and define a measurable order on to be . By Remarks 2.2 (c), the decomposition of the group that we get by applying Theorem 2.1 to that group, is precisely the image by of the decomposition of the group . Let denote the compact dual group of . Thus if Theorem 5.2 holds for , then applying Theorem 4.5, we see that Theorem 5.2 holds for .

Henceforth, let us suppose that is compact. We will suppose that the Haar measure on is normalized, so that with Haar measure is a probability space.

Since each one of the subgroups , and ( ) is open, it follows that their annihilators in , , and , are compact. Let and denote the normalized Haar measures on and , respectively. We have (for all ), and (for all ), so that .

For each , let denote the -algebra of subsets of of the form , where is a Borel subset of . We have , whenever . It is a simple matter to see that for , the conditional expectation of with respect to is equal to (see [11, Chapter 5, Section 2]).

We may suppose without loss of generality that . Thus the -algebras form a filtration, and the sequence is a martingale difference sequence with respect to this filtration.

In that case, we have the following result of Burkholder [7, Inequality (1.7)], and [8]. If , then there is a positive constant , depending only upon , such that

 (36)

Lemma 5.3   For any index , , and , we have almost everywhere on

 (37)

where is the normalized Haar measure on the compact subgroup .

Proof. The dual group of is and can be ordered by the set , where is the natural homomorphism of onto .

Next, by convolving with an approximate identity for consisting of trigonometric polynomials, we may assume that is a trigonometric polynomial. Then we see that for each that the function , , is in . To verify this, it is sufficient to consider the case when is a character in . Then

and by definition is in .

Now we have the following generalization of Jensen's Inequality, due to Helson and Lowdenslager [16, Theorem 2]. An independent proof based on the ideas of this section is given in [3]. For all

 (38)

Apply (38) to , to obtain

Extending the integrals to the whole of (since is supported on ), raising both sides to the th power, and then applying the usual Jensen's inequality for the logarithmic function on finite measure spaces, we obtain

Changing to , we obtain the desired inequality.

Let us continue with the proof of Theorem 5.2. We may suppose that is a mean zero trigonometric polynomial, and that the spectrum of is contained in , that is to say

By Lemma 5.3, we have that
 (39)

Also, we have that is a martingale with respect to the filtration . Hence, by Doob's Maximal Inequality [10, Theorem (3.1), p. 317] we have that
 (40)

The desired inequality follows now upon combining Burkholder's Inequality (36) with (39), and (40).

Proof of Theorem 5.1. Transferring inequality (34) by using Theorem 1.8, we obtain that for any set of indices less than , and for any numbers ( ), there is a positive constant , depending only upon the representation , such that

 (41)

Now suppose that is a countable collection of indices less than . Then by Bessaga and Peczynski [5], the series is unconditionally convergent. In particular, for any , for only finitely many do we have that . Since this is true for all such countable sets, we deduce that the set of for which is countable.

Hence we have that is unconditionally convergent to some measure, say . Clearly is weakly measurable. To prove that , it is enough by Proposition 1.4 to show that for every , we have for almost all .

We first note that since for every the series converges to in , it follows that, for every , the series converges to in the weak-* topology of . Now on the one hand, for and , we have , because of the (unconditional) convergence of the series to . On the other hand, by considering the function , we have that , weak *. Thus for almost all , and the proof is complete.

Next: Generalized F. and M. Up: Decomposition of analytic measures Previous: Homomorphism theorems
Stephen Montgomery-Smith 2002-10-30