Note on Order-Isomorphic Isometric Embeddings of Some Recent Function Spaces

Abstract

We investigate certain recently introduced ODE-determined varying exponent spaces. It turns out that these spaces are finitely representable in a concrete universal varying exponent space. Moreover, this can be accomplished in a natural unified fashion. This leads to order-isomorphic isometric embeddings of all of the above spaces to an ultrapower of the above varying exponent space.

1. Introduction

In this note we study the local theory of some very recently introduced varying exponent spaces.

It is well-known that the classical spaces are finitely representable in the respective space. Moreover, the relevant finite-dimensional isomorphisms witnessing the finite representability can be chosen in such a way that they preserve bands, very roughly speaking. For example, the finite representability of Bochner spaces in the corresponding double spaces has been recently studied; see [1] (cf. [2, 3]). Recall that there is a known connection between finite representability and ultraproducts, and, in fact, in the above mentioned paper the local theory of these spaces is investigated by means of ultraproducts.

It is reasonable to ask if an analogous finite representability result holds in the varying exponent case, that is, for spaces . Here we show that it does for a pair of recent classes of varying exponent and spaces. We also prove some results involving ultrapowers but it turns out that there are versions of these results which are actually not specific to the ultrapower methods. The varying exponent spaces in the literature are related to Musielak-Orlicz spaces (cf. [3–7]) and in particular Nakano spaces whose norm is given by where is measurable. Such spaces have been studied in different connections (e.g., [8, 9]). However, the spaces considered here are defined by different means.

The varying exponent space investigated here, with exponents can be described naively as follows: Here denotes a -dimensional Banach space and the construction of the above space in [10] is rigorous. See [11] for similar constructions.

There is a natural “continuous version” of the above space. The author introduced in [12] a class of varying exponent spaces whose norm is governed by an ordinary differential equation as follows: Here and are measurable functions and is Carathéodory’s weak solution which exists and is unique for an initial value (see the paper for details). In the case with the set becomes a Banach lattice with the usual pointwise operations defined a.e. and the norm In the constant case this construction reproduces the classical spaces.

This paper also illustrates the inner workings of the above recent classes of spaces.

1.1. Preliminaries

We refer to the monographs in the references and the survey [13] for a suitable background information. Throughout we are assuming the familiarity with papers [10, 12] regarding the construction, notations, and basic facts involving and spaces, respectively.

If is any measurable function there is a natural Banach function space such that is, intuitively speaking, almost bounded on this space. The space can be defined as the completion (The space on the right can be regarded as a metric space but it may be nonlinear for some cases of .)

The double varying exponent spaces, that is, , can be defined as follows. For infinite matrices we define the values of the corresponding norms in phases. First, we let for all . Then we set In both phases we exclude the matrices producing infinite values. It is easy to see that this results in a Banach space and it is denoted by .

For and we denote

If is a filter on and , , , we denote by the fact that

Recall that a Banach space is finitely representable in a Banach space if for each finite-dimensional subspace and there is a finite-dimensional subspace and a linear isomorphism with .

Given a Banach space we denote adopting the notation used for Calkin algebras.

2. Results

2.1. Preparations: Banach Lattices of Ultraproducts

Let be a sequence of Banach lattices, each satisfying the property Observe that this condition immediately guarantees that the absolute value mapping is nonexpansive. We let be the direct sum of the spaces. Write . Suppose that is a filter on , for example, a Fréchet filter. Then we let

It is easy to check that this is a closed subspace. For example, if for all , a -dimensional Banach space, and is the filter generated by cofinite subsets, then .

We may generate a vector lattice order on the space from the condition (We are not claiming reverse implication above. This is so, for instance, because the equivalence classes do not determine the corresponding sequences uniquely.)

An alternative approach is that we may define an absolute value on by

Indeed, this is well defined since the absolute values satisfy (12). Then the condition characterizes a positive cone which can be used in recovering the order . It is not hard to verify that these separate constructions result in the vector lattice order.

Proposition 1. Let one retain the above notations and assume that the absolute values satisfy (12), respectively. Then endowed with the partial order is a Banach lattice whose absolute value coincides with the mapping .

We denote by a free ultrafilter on the natural numbers. Recall that the ultrapower of a Banach space is defined as

2.2. Order-Isomorphic Isometric Embeddings

Let be a bijection, that is, an enumeration of the rationals . Denote . It is known that (resp., ) contains almost isometrically all the spaces of the type (resp., ), in particular the spaces (resp., for ); see [10].

Theorem 2. Let be as above. The space is universal for spaces of the type . More precisely, the latter spaces considered with their a.e. pointwise order, can be mapped by a linear order-preserving isometry into , endowed with the Banach lattice order , as described above. Moreover, the same conclusion holds if one considers the ultrapower in place of for any free ultrafilter on .

Proof. We prove the latter statement involving the ultrapower which is more abstract (if not more complicated). Let be a measurable function and a free ultrafilter on .
Note that can be written isometrically as where is a decomposition such that .
According to Lusin’s theorem there is a sequence of compact sets such that are uniformly continuous for each and . Since the norm-defining solutions are assumed to be absolutely continuous and taking into account the basic properties of the solutions (see [12]), we may identify Indeed, let us recall the justification for this. It was proved in [12] that is a lattice norm and moreover that pointwise if pointwise a.e. Since , it is known that since then also , and pointwise; see [10]. Then, inspecting the governing differential equation (4), we get immediately that a.e. on and of course a.e. in the complement of . On the other hand, the solution , by its definition, is absolutely continuous which implies as . Thus, using (19) we get Since for all , we observe that the above inequality becomes equality.
Step 1 (approximation of the norm by simple seminorms). First we assume that . This makes sense because it was shown in [12] that is dense in in the case where .
Consider simple seminorms (as in [12]): Here the measures are obtained as restrictions of the Lebesgue measure to compact subsets where . Thus where is the -algebra of the completed Lebesgue measure on the unit interval and for all .
Let be a sequence of such seminorms with on . Then by the construction of the norm we have that for each and . Indeed, this is due to the fact that is essentially defined as a supremum of such seminorms.
By a diagonal argument we may choose in such a way that for each and . Indeed, since is bounded and uniformly continuous on we may find for each numbers such that (1)the corresponding supports for measures satisfy ;(2), ;(3)intuitively, the differences are negligibly small;(4)  a.e. on for such that .
This is due to the fact that uniformly for , as , . The diagonal argument is then applied to choose the sequence of seminorms as to eventually cover all cases , and for all .
For each let be the finite -algebra generated by all the supports of corresponding to for . Without loss of generality we may assume by adding suitable finitely many sets (e.g., dyadic decompositions of the unit interval) to each that and that -generates the Borel -algebra on the unit interval. By an atom of an algebra of sets we mean such that if , , then or .
Next we study the conditional expectation operators . Here where is considered as the average integral (operator) over where are atoms with respect to the finite algebra of sets . We use the convention that whenever .
Restricting to the support of any of the , it follows from the martingale convergence principle that almost everywhere as and also in the -sense; see, for example, [14, Ch. 5.4.]. Consequently, putting the pieces together, we obtain that Define versions of the seminorms by replacing with . By the uniform continuity of on the sets , (25), (26), and (28) we may choose subsequences with , as such that where need not be strictly increasing. In fact, the above equality clearly holds for any .
Step 2 (approximation of the required operator by tame nonlinear operators). Consider a sequence and nonlinear operators given by for a.e. . Thus the following condition holds: Indeed, is dense whenever and consequently it follows from the definition of the space that is dense in it as well. We may additionally choose the above sequences of , conditional expectation operators, and the seminorms in such a way that This can be established by using the uniform continuity of on the compact sets , using the fact that in such a case the simple seminorms converge uniformly, and invoking the martingale -convergence fact above. Note that are order-preserving although they are nonlinear.
Next we analyze the simple seminorms chosen and in particular the exponents . We obtain that for each for the exponents corresponding to there are with rational exponents very close to the corresponding . Indeed, by repeating the almost isometric embedding construction in [10] we may pick the in such a way that Here that is, with the norm We can find for each a finite-dimensional varying exponent space and a natural linear order-preserving linear bijection where is the finite dimension of the space of simple functions of the form , such that where and . Indeed, this applies the fact that contains all the supports of corresponding to and we may write where the subsets are successive () and are -atomic subsets of the supports of corresponding to . Recall that in a canonical way. The mapping is given by and it is easy to see that it is well-defined, linear, and bijective.
Let be a natural linear order-preserving mapping corresponding to the arrangement in (32). Next, we define a mapping as follows: , where Note that by the absolute continuity of the solutions , as observed above.
The mapping required in the statement is the induced mapping , mapping to the quotient space (in this case the ultrapower). This mapping is clearly order-preserving.
It is also norm-preserving, since . Indeed, this follows by using (31), (32), (36), (41), and the fact that is dense in .
To verify the linearity of , observe that for any and , there are , such that . Thus, selecting in such a way that , we obtain that Here we are using the fact that is a Banach lattice in its usual order (see [12]). This means that as . Recalling (31) and the construction of , it follows that This shows that for all , . The homogeneity of is seen similarly. This completes the proof.

Theorem 3. Let be as above. The space is universal for spaces of the type . More precisely, the latter spaces considered with their matrix entrywise order can be mapped by a linear order-preserving isometry into . Moreover, the same conclusion holds if we consider the ultrapower in place of . In particular, each space is finitely representable in .

Proof (sketch.). Consider each element of as a sequence with for . We may consider these elements as infinite matrices .
Let . By repeating inductively the observation involving (32) we can find and such that the mapping and setting other coordinates to satisfies Here the domain of is and we denote by the canonical projection to this domain.
The required linear isometry is induced by the operator

We note that the previous result holds also as a left-handed version, where we consider all the varying exponent -spaces formally as

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work has been financially supported by Väisälä Foundation’s and the Finnish Cultural Foundation’s research grants and Academy of Finland Project no. 268009.