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A note about associate and dual spaces to the Grand Lebesgue ones

We describe in this short article the associate and dual (conjugate) spaces to the Grand Lebesgue Spaces by means of its embedding to the suitable exponential Orlicz ones.


1
Definitions. Notations. Previous results. Statement of problem.
The separability is understood relative the usually distance function Note in particular that despite these measures are atomless, but the random values (measurable functions) defined on these spaces may have a discrete distribution.
Furthermore, let now η = η(z), z ∈ S be arbitrary family of random variables defined on any set z ∈ S such that The function p → ψ S (p) is named as a natural function for the family of random variables S. Obviously, sup z∈S ||η(z)||GΨ S = 1.
The family S may consists on the unique r.v., say ∆ : if of course the last function is finite for some values p ∈ (p 1 , p 2 ); p 0 > 1, These spaces appearers at first in [15]; see also [11]- [13], [4], [6]- [7], [10], [20] etc. They are applied in statistics, theory of random processes and fields, theory of Partial Differential Equations, theory of operators and so one.
One unexpected new application of these spaces may be found in a recent article [1].
The space B(φ), φ ∈ Φ with respect to the norm ||ξ||B(φ) and ordinary algebraic operations is a rearrangement invariant Banach space in the classical sense [3], chapters 1,2; which is in turn isomorphic to the subspace of the space Gψ φ , where consisting on all the centered r.v. having a finite norm ||ξ||Gψ φ < ∞.
Our aim in this short report is description of the associate and dual spaces for the Grand Lebesgue Spaces.

2
Main result: structure of associate spaces.
Recall first of all that the associate space Y ′ for arbitrary rearrangement one Y builded over source measurable space (X, B, µ) consists by definition on all the linear functionals of the forms for (certain) measurable function g : X → R. It will be presumed that this functional l g (·) is bounded: The detail description of these spaces is explained in the classical monograph of C.Bennet and R.Sharpley [3], chapters 1,2.
We will identify in this case as usually for convenience the linear functional l g with its generating function g : Let now Y = Gψ for some ψ = ψ(·) ∈ Ψ(a, b), where as above a = const ≥ 1, b = const ∈ (a, ∞]. Let also 0 = f ∈ Gψ; one can suppose without loss of generality ||f ||Gψ = 1. Therefore ∈ (a, b).

Denote as usually
Let us estimate the functional l g (f ) from the relation (2.1). We apply the classical Hölder's inequality for the values correspondingly p ∈ (a, b) and q = q(p) ∈ (b ′ , a ′ ) .
Define a following functional
In this case Thus, in this example the associate space is quite equal to the its conjugate. The case m = 2 correspondent to the subgaussian case. We have Notice that in the considered case the the estimate (2.4a) gives a weak result. Namely, as long as for any r.v. η

3
Main result: structure of dual spaces.
We need to introduce the following addition notations, conditions and definitions. Recall that the well -known Young-Fenchel, or Legendre transform h * for the real valued function h = h(z), not necessary to be convex, is defined as follows Further, denote by Γ = { γ } the collection of all finite-additive numerical valued functions defined on the sigma -field B.
Of course, the finite additivity does not exclude the "ordinary" countably sigma -additivity.
Define following [18], see also [19], norm on the set Γ : where the integral in (3.2) may be understood in the sense of R.G.Bartle [2]; see also [18]. On the other hands, one can choose only the function f (·) to be simple, or on the other words, step function, i.e. taking only finite possible values on the sets of finite "measure" γ : where as ordinary χ D (x) denotes an indicator function of the measurable set D : Theorem 3.1 Assume in addition to the foregoing conditions imposed on the function ψ(·) that the function V (x) = V [ψ](x) satisfies the following restriction: which is in turn a slight analog of the famous ∆ 2 condition.
We assert that arbitrary continuous linear functional l(f ), f ∈ Gψ has an unique representation of the form Proof. I. Suppose γ ∈ S and following |||γ||| ψ < ∞. The equality (3.5) follows immediately from the direct definition of the spaces S(ψ); obviously, it is sufficient to verify (3.5) for all the step functions.
It is worth to note that this conclusion is true still without the condition (3.3)
It is proved in particular in [12] that the Grand Lebesgue Space Gψ in this case quite coincides with the so-called exponential Orlicz space L(N[ψ]) builded over our measurable triplet (X, B, µ) equipped with the correspondent Young-Orlicz function The norm in this space may be used the ordinary Luxemburg norm.
The conjugate spaces for these ones are calculated in [18]; see also [19]. They have the form (3.5).
This completes the proof of theorem 3.1.
In turn, the last condition is satisfied if the function L(x) is in addition strictly increasing.

4
Again about associate space.
Let us return to the problem of finding of associate space to the Grand Lebesgue ones. We intend to apply the described above approach throughout embedding into Orlicz spaces [13], [14], [17]. In detail, let µ(X) = 1 and let the function ψ(·) satisfy the condition (3.3). Suppose also that the function f : X → R be from the space Gψ; then it belongs also to the correspondent Orlicz's space L(N[ψ]) and herewith and inverse inequality is also true.

Concluding remarks.
A. It is no hard by our opinion to generalize obtained here results into a multidimensional case of Gψ spaces.
B. As long as the B(φ) spaces are particular cases of Grand Lebesgue ones, we obtained on the way also the associate and dual for these spaces, as well.
C. It is interest by our opinion to extend the integral representation for dual and associate linear continuous functionals for many wide classes of rearrangement invariant spaces.