Note on order-isomorphic isometric embeddings of some recent function spaces

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


Introduction
In this note we study the local theory of some very recently introduced varyingexponent L p spaces.
It is well-known that the classical L p spaces are finitely representable in the respective ℓ p 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 L p (L q ) spaces in the corresponding double ℓ p spaces has been recently studied, see [3] (cf. [5], [7]). 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, i.e. for spaces L p(·) . Here we show that it does for a pair of recent classes of varying-exponent L p(·) and ℓ p(·) spaces. We also prove some results involving ultraproducts but it turns out that there are versions of these results which are actually not specific to the ultrapower methods.
The varying-exponent ℓ p(·) space investigated here, with exponents can be described naively as follows: Here R denotes a 1-dimensional Banach space and the construction of this space in [11] is rigorous. There is a natural 'continuous version' of the above space. The author introduced in [12] a class a varying-exponent L p spaces whose norm f is governed by an ordinary differential equation as follows: Here p : [0, 1] → [1, ∞) and f : [0, 1] → R are measurable functions and ϕ f is Carathéodory's weak solution which exists and is unique for an initial value ϕ f (0) = 0 + (see the paper for details). In the case with ess sup p < ∞ the set becomes a Banach lattice with the usual point-wise operations defined a.e. and the norm In the constant p(·) = p ∈ [1, ∞) case this construction reproduces the classical L p 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 [4] for a suitable background information. The reader should consult both [11] and [12] on the details on ℓ p(·) and L p(·) spaces, respectively.
(The space on the right can be regarded as a metric space but it may be non-linear for some cases of p(·).) The double varying-exponent ℓ p(·) spaces, i.e. ℓ p(·) (ℓ s(·) ) can be defined as follows. For infinite matrices (x n,m ) n,m∈N ⊂ R N×N we define the values of the corresponding norms in 2 phases. First, we let a k := x k,· ℓ s(·) for all k ∈ N. Then we set (x n,m ) ℓ p(·) (ℓ s(·) ) := (a k ) ℓ p(·) .
In both phases we exclude the matrices (x n,m ) producing infinite values. It is easy to see that this results in a Banach space and it is denoted by ℓ p(·) (ℓ s(·) ).
If F is a filter on N and a n , a ∈ R, n ∈ N, we denote by lim n,F a n = a the fact that Recall that a Banach space X is finitely representable in a Banach space Y if for each finite-dimensional subspace E ⊂ X and ε > 0 there is a finite-dimensional subspace F ⊂ Y and a linear isomorphism T : Given a Banach space we denote adopting the notation used for Calkin algebras.

2.1.
Banach lattices of ultraproducts. Let X n be a sequence of Banach lattices, each satisfying the property Observe that this condition immediately guarantees that the absolute value mapping x → |x| is non-expansive. We let n X n (ℓ ∞ sense) be the ℓ ∞ direct sum of the spaces. Write X = n X n . Suppose that F is a filter on N, e.g. a Fréchet filter. Then we let Recall that the ultrapower of a Banach space Y is It is easy to check that this is a closed subspace. For example, if X n = R for all n, a 1-dimensional Banach space, and F is the filter generated by cofinite subsets, then N F = c 0 .
We may define a partial order on the space X/N F as follows An alternative approach is that we may define an absolute value | · | on X/N F by Indeed, this is well defined since the absolute values | · | Xn are non-expansive. Then the condition |x| = x characterizes a positive cone which can be used in recovering the order . Proposition 2.1. Let us retain the above notations and assume that the absolute values | · | Xn satisfy (2.1), respectively. Then X endowed with the partial order is a Banach lattice whose absolute value coincides with the mapping | · |.

2.2.
Order isomorphic isometric embeddings. Let r : N → Q ∩ [1, ∞) be a bijection, i.e. an enumeration of the rationals q ≥ 1. Denote by ℓ It is known that ℓ r(·) (resp. ℓ r(·) 0 ) contains almost isometrically all the spaces of the type ℓ q(·) (resp. ℓ q(·) 0 ), in particular the spaces ℓ p (resp. ℓ p for p < ∞). Proof. We prove the latter statement involving the ultrapower which is more abstract (if not more complicated). Let p : [0, 1] → [1, ∞) be a measurable function and U a non-principal ultrafilter on N. According to Lusin's theorem there is a sequence of compact sets C n ⊂ [0, 1] such that p| Cn are uniformly continuous for each n and m(C n ) → 1. Since the norm-defining solutions ϕ f are assumed to be absolutely continuous (and taking into account their basic properties of the solutions), we may identify Consider simple semi-norms, as in [12], |f | N = |f | (L p 1 (µ1)⊕q 2 L p 2 (µ2))⊕q 3 L p 3 (µ3))⊕q 4 ...)⊕q n L pn (µn) .
Let (N n ) be a sequence of such semi-norms with p i ≤ q i . Then by the construction of the · L p(·) norm we have that |f | Nn ≤ f L p(·) for each n and f ∈ L p(·) 0 . By a diagonal argument we may choose N n in such a way that 0 . Above we have max suppµ i ≤ min suppµ i+1 . We may also assume that the supports σ-generate the Borel σ-algebra on the unit interval.
For each k ∈ N let F k be the finite σ-algebra generated by the supports of µ i for 1 ≤ i ≤ k. Without loss of generality we may assume that lim k→∞ sup{diam(∆) : ∆ ∈ F k atom} = 0.
Next we study the conditional expectation operators E(f |C, F k ). Here E(f |C, F ) = ∆ A * ∆∩C (f )1 ∆∩C where A * ∆∩C ∈ L ∞ is considered as the average integral (operator) over ∆ ∩ C where ∆ ∈ F are atoms with respect to this σ-algebra. We use the convention that A * ∆∩C (f ) = 0 whenever m(∆ ∩ C) = 0. Restricting f to the support of any of the µ i :s, it follows from the martingale convergence principle that E(f |C, F k ) → 1 C f almost everywhere as k → ∞ and also in the L pi (µ i )-sense. Consequently, putting the pieces together, we obtain that Define versions N ′ n of the semi-norms N n by replacing q i with p i . We may choose subsequences (m n ), (k n ) ⊂ N with m n , k n → ∞ as n → ∞ such that Consider a sequence 0 ≤ α n ր ∞ and non-linear operators T n : L p(·) 1] given by T n (f )[t] = min(α n , max(−α n , f (t))) for a.e. t. Thus the following condition holds: (a) f − T n f L p(·) → 0 as n → ∞ for each f ∈ L p(·) 0 . Indeed, L ∞ ⊂ L p(·) is dense whenever ess sup p < ∞ and consequently it follows from the definition of the space L p(·) 0 that L ∞ is dense in it as well. We may additionally choose the above sequences of α:s, conditional expectation operators and the semi-norms in such a way that This can be established by using the uniform continuity of p on the compact sets C m , using the fact that in such a case the simple semi-norms converge uniformly and invoking the martingale L p -convergence fact above. Note that T n :s are orderpreserving although they are non-linear. Next we analyze the simple semi-norms chosen and in particular the exponents p i :s. We obtain that for each n for the exponents p ij very close to the corresponding p:s. Indeed, by repeating the almost isometric embedding construction in [11] we may pick the q i k :s in such a way that Note that Y ⊕ p L p (µ) can be written isometrically as (Y ⊕ p L p (µ 1 )) ⊕ p L p (µ 2 ) where µ = µ 1 + µ 2 is a decomposition such that max supp µ 1 ≤ min supp µ 2 . By using this observation inductively, we can find for each n a finite-dimensional varying exponent ℓ p space ℓ → ℓ r(·) be a natural linear order-preserving mapping corresponding to the arrangement in (2.2). Next, we define a mapping S : L p(·) 0 → ℓ ∞ (ℓ r(·) ) as follows: S(f ) = (x n ) where x n = ι n B n E(T n f |C mn , F kn ).
Observe that 1 Cm n f L p(·) → f L p(·) as n → ∞ by the absolute continuity of the solutions ϕ f , similarly as in [12].
The mapping required in the statement is the induced mapping S : f → S(f ) + N F , mapping to the quotient space (in this case the ultrapower). Indeed, this mapping is clearly order-preserving. It is also norm-preserving, since lim n→∞ x n ℓ r(·) = f . To verify the linearity of S, observe that for any ε > 0 and f, g ∈ L p(·) 0 there are f 0 , g 0 ∈ L ∞ such that max( f − f 0 L p(·) , g − g 0 L p(·) ) < ε. Thus, selecting n ∈ N in such a way that α n ≥ max( f L ∞ , g L ∞ ), we obtain that Here we are using the fact that L p(·) 0 is a Banach lattice in its usual order. This means that + T n f − f L p(·) + T n g + g L p(·) → 0 as n → ∞. Recalling (b) and the construction of S, it follows that S n (f + g) − (S n (f ) + S n (g)) ℓ r(·) → 0, n → ∞.
Proof. (Sketch.) Consider each element of ℓ q(·) (ℓ s(·) ) as a sequence (x n ) with x n ∈ ℓ s(·) for n ∈ N. We may consider these elements as infinite matrices (x n,m ) n,m∈N .