Hilbert-substructure of Real Measurable Spaces on Reductive Groups, I;Basic Theory

This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, Fn(G), of real Lp(G)−spaces (when G is a reductive group in the Harish-Chandra class and p=2n), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, L2(G). This success opens the door for harmonic analysis of unitary representations, G→End(Fn(G)), of G on the Hilbert-substructure Fn(G), which has hitherto been considered impossible.


Introduction
Let G be a reductive group in the Harish-Chandra class and denote L p (G, ) as the Lebesque spaces of −valued functions on G, 0<p<∝. The harmonic analysis of L 2 (G, ) is complete with the successful decomposition of unitary representations of G in [1]. However not much, as known for L 2 (G, ), could be said about other members of the Lebesque spaces on G. This is largely due to the absence of a manageable 'Hilbert space theory' (which made the discussion of unitary representations of G on L p>2 (G, ) a forbidden concept) and has led to the employment of indirect techniques to extract important results out of them. The most celebrated of these successful indirect techniques at harmonic analysis of L p (G, ) is the Trombi-Varadarajan theory [2] which entails refining the decay estimates of L 2 (G, ) for L p (G, ) − Schwartz-like functions (for only 0<p ≤ 2) in order to match the asymptotic estimates of the corresponding Fourier transform on G.
These refinements could however hardly hold nor match for other values of p, especially p>2. Had it been that (any of) the L p>2 − spaces are Hilbert spaces, with respect to which unitary representations could be discussed, a direct approach to such analysis would have been possible, general and more satisfying than that of only L 2 (G, ) and would have subsumed the Trombi-Varadarajan theory as well.
The modalities for conducting harmonic analysis on L p (G, ) have been roughly and immaturely spelt out in [3]. where (as it will be shown in the course of this paper) an inner product which was proved to be consistent with the norm-convergence in L 2n ([a,b], ), n∈, a,b∈, and which led to a general Cauchy-Schwartz inequality and construction of higher orthogonal polynomials, was made available. In this paper we shall employ the techniques of [3] to initiate discussion on harmonic analysis of L 2n (G, ) by showing explicitly that each of these real measurable spaces on G contains a Hilbert space substructure (thus correcting the outlook in [3] where the substructure was wrongly placed on all of L 2n ([a,b], ), thereby making the techniques of this earlier paper of the author available to a wider audience) and that this substructure is rich enough to allow L p −harmonic analysis on G. The results of this paper lay a foundation for the successful treatments of unitary representations of G on F n (G).

Hilbert-Substructure of L p (G, )
Let G denote a reductive group in the Harish-Chandra class and let ( , ) c C G ∝  represent the space of smooth real-valued functions on G, [4]. Let L p (G, ) denote the Lebesque space of real-valued functions on G, where 0 < p < ∝. We shall write ‫›܁,܁‹‬ 2 for the inner product on the Hilbert space L 2 (G, ) It is well-known that each member of L p (G, ) is a completion of The defining requirement, and that (when endowed with the sup-norm) * ( ) n G F is an incomplete normed linear space over . We shall denote the completion of * ( ) n G F under the L p − norm, ‫܁‬ p=2n , simply by F n (G). A first property of F n (G) giving its relationship with L p=2n (G, ) is proved as follows.
, for all n∈, with equality when, and only when, n=1.
Proof: Let f ∈F n (G), then which, when combined with the fact that The details of the proof above show that each F n (G) is an ‫܁‬ 2n − normed linear subspace of L 2n (G, ) and that F 1 (G) is a real Hilbert space. That each of L 2n (G, ), for n>1 is not an inner product space does not preclude this possibility for each of F n (G) In fact we may convert each * ( ) n G F into an inner product space in the following defined manner. we set the pairing ‹f, g› 2n as ‹f, g› 2n :=‹f, g 2n−1 › 2 .
We shall refer to the pair (F n (G), ‫)‪›2n‬܁,܁‹‬ as a Hilbert-substructure of L 2n (G, ). It is our modest aim in this paper to use the present general outlook (afforded by F n (G)) on Hilbert (function) spaces to prove some results about F n (G) in order to convince the mathematical public of the necessity of doing analysis on (F n (G), ‫,)‪›2n‬܁,܁‹‬ as against the consideration of only (F 1 (G), ‫‪›2)=(L‬܁,܁‹‬ 2 (G,),‹‫.)2›܁,܁‬ We shall establish the foundation on which each of F n (G), n∈, would be seen to possess a generalization of the fine structure of F 1 (G)=L 2 (G,). A first among the fine structure well-known for L 2 (G,) is the contribution of its inner product, ‫›܁,܁‹‬ 2 , in the proof of the triangle-inequality axiom of an ‫܁‬ 2n − norm. We are here referring to the direct contribution of Cauchy-Schwartz inequality in the proof of the triangle inequality  f+g 2 ≤ f  2 +g 2 , for all f,g∈L 2 (G,). Even though we already know that the Minkowski inequality  f+g 2n ≤ f  2n +g 2n , holds for all f,g∈L 2 (G,), via the truth of Holder's inequality and that hence  f+g 2n ≤ f  2n +g 2n holds for all f,g∈F n (G) (by obvious restriction), it would be necessary (as usually performed for the ‫܁‬ 2n − norm via the Cauchy-Schwartz inequality) to have what we may call an inner product proof of the Minkowski inequality for all the Hilbertsubstructures, F n (G), n ≥1 (and not just for F 1 (G)=L 2 (G,) only). Indeed, if accomplished, this will give credence to the independence of each of F n (G) from (the normed linear space) L 2n (G,), a reminiscence of the importance of Cauchy-Schwartz inequality and the independence of L 2 (G,) from all L p (G,) − spaces.
In order to achieve the feat outlined above we need a (general) Cauchy-Schwartz inequality for members of F n (G), n ≥1. It happens that the much we need in order to achieve our aim is contained in the following.

Proof:
The classical Cauchy-Schwartz inequality implies that |‹f p−k ,g k › 2 |≤f p−k  2 g k  2 , for all f,g∈F 1 (G). We only need to show that To this end we see that Since k∈ we may set p=2k∈, so that 2(p−k)=2(2k−k)=2k=p. Hence   as required. On setting p=2 and k=1 in Lemma 2.5 we see that n=1 and we arrive at the classical Cauchy-Schwartz inequality for F 1 (G).
It may have been expected that a generalization of the classical Cauchy-Schwartz inequality for F 1 (G) to all of F n (G), n≥1, would be that of finding a bound for |‹f,g› 2n |, for f,g∈F n (G). We are however not motivated by blind generalization but by seeking an inequality that would serve the L 2n −norm on F n (G) (in exactly the same way the classical Cauchy-Schwartz inequality serves the L 2 −norm on F 1 (G)) in the proof of Minkowski inequality. We shall advise this cautionary measure in generalizing other inequalities (like the Bessel's inequality) of inner product spaces to all of n (G). This use for Lemma 2.5 is, in this wise, contained in the following. Theorem 2.6: (An inner-product proof of Minkowski inequality on n (G)).
Given that f,g∈F n (G), then for all p=2n.

Proof:
The reader may check that the above computations go through even when n=1(i.e.,p=2). This shows the universality of Lemma 2.5 and Theorem 2.6.
One of the cornerstones of inner product spaces, in particular of the space L 2 (G,)=F 1 (G), is the parallelogram equality; so named because of its geometric contents. It is customary, in the F F general theory of inner product spaces, to verify this equality for any given norm in order to ascertain if the corresponding normed linear space could also be an inner product space. It was on this basis that each of the L p>2 − spaces was rightly knocked out of the race for the possession of an inner product. However Theorem 2.4 has now shown that it is unfair to force all the L p>2 − spaces to be induced by the inner product, ‫›܁,܁‹‬ 2 , of the L 2 − space.
In the light of Theorem 2.4 it would therefore be necessary to reconsider the properties of the polynomial map,