Journal of Function Spaces
Volume 2015 (2015), Article ID 186105, 6 pages
http://dx.doi.org/10.1155/2015/186105

Research Article

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

Jarno Talponen

Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 23 October 2014; Revised 3 April 2015; Accepted 5 April 2015

Academic Editor: Hugo Leiva

Copyright © 2015 Jarno Talponen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

P. Gomez Palacio, J. A. Lopez Molina, and M. J. Rivera, “Some applications of the lattice finite representability in spaces of measurable functions,” Turkish Journal of Mathematics, vol. 25, no. 4, pp. 475–490, 2001.

M. Levy and Y. Raynaud, “Ultrapuissances des espaces ,” Comptes Rendus de l'Académie des Sciences—Series I: Mathematics, vol. 299, pp. 81–84, 1984.

G. Loaiza, J. A. L. Molina, and M. J. Rivera, “Characterization of the maximal ideal of operators associated to the tensor norm defined by an Orlicz function,” Zeitschrift für Analysis und ihre Anwendungen, vol. 20, no. 2, pp. 281–293, 2001.

Z. Birnbaum, “Über die Verallgemeinerung des Begriffes der zueinander Konjugierten Potenzen,” Studia Mathematica, vol. 3, pp. 1–67, 1931.

L. Diening and M. Ruzicka, “Calderón-Zygmund operators on generalized Lebesgue spaces and problems related to fluid dynamics,” Journal für die Reine und Angewandte Mathematik, vol. 563, pp. 197–220, 2003.

A. Kamińska, “Strict convexity of sequence ORLicz-Musielak spaces with ORLicz norm,” Journal of Functional Analysis, vol. 50, no. 3, pp. 285–305, 1983.

E. Nakai and Y. Sawano, “Hardy spaces with variable exponents and generalized Campanato spaces,” Journal of Functional Analysis, vol. 262, no. 9, pp. 3665–3748, 2012.

Å. Hirvonen and T. Hyttinen, “Metric abstract elementary classes with perturbations,” Fundamenta Mathematicae, vol. 217, no. 2, pp. 123–170, 2012.

J. E. Jamison, A. Kamińska, and P.-K. Lin, “Isometries of Musielak-Orlicz spaces II,” Studia Mathematica, vol. 104, no. 1, pp. 75–89, 1993.

J. Talponen, “A natural class of sequential Banach spaces,” Bulletin of the Polish Academy of Sciences. Mathematics, vol. 59, no. 2, pp. 185–196, 2011.

G. Androulakis, C. D. Cazacu, and N. J. Kalton, “Twisted sums, Fenchel-Orlicz spaces and property (M),” Houston Journal of Mathematics, vol. 24, no. 1, pp. 105–126, 1998.

J. Talponen, “ODE representation for varying exponent norm,” http://arxiv.org/abs/1402.0528

S. Heinrich, “Ultraproducts in Banach space theory,” Journal für die Reine und Angewandte Mathematik, vol. 313, pp. 72–104, 1980.

R. Durrett, Probability: Theory and Examples, Cambridge University Press, Cambridge, UK, 4th edition, 2010.