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

Authors: Jarno Talponen
Journal: Journal of Function Spaces
 (2015)

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## Abstract

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


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## Body

## 1. Introduction

In this note we study the local theory of some very recently introduced varying exponentLp spaces.It is well-known that the classicalLp spaces are finitely representable in the respective lp 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 Lp(Lq) spaces in the corresponding double lp 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 spacesLp(·). Here we show that itdoes for a pair of recent classes of varying exponent Lp(·) and lp(·) 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 Lp spaces in the literature are related to Musielak-Orlicz spaces (cf. [3–7]) and in particular Nakano spaces whose norm is given by(1)f=inf⁡λ>0:∫Ωft/λptptdmt≤1,where p:Ω→[1,∞) is measurable. Such spaces have been studied indifferent connections (e.g., [8, 9]). However, the spaces considered here are defined bydifferent means.The varying exponentlp(·) space investigated here, with exponents(2)p1=p1,p2=p2,…∈1,∞,can be described naively as follows:(3)lp·=⋯R⊕p1R⊕p2R⊕p3R⊕p4⋯.Here R denotes a 1-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 Lp spaces whose norm f is governed by an ordinary d differential equation as follows:(4)φf0=0,φf′t=ftptptφft1-ptfora.e.t∈0,1.Here p:[0,1]→[1,∞) and f:[0,1]→R are measurable functions and φf is Carathéodory’s weak solution which exists and is unique for an initial value φf(0)=0+ (see the paper for details). In the case with esssupp<∞ the set(5)f∈L0:φfsolution  exists,φf1<∞becomes a Banach lattice with the usual pointwise operations defined a.e. and the norm(6)f≔φf1.In the constant p(·)=p∈[1,∞) case this construction reproduces the classical Lp 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 lp(·) and Lp(·) spaces, respectively.Ifp:[0,1]→[1,∞) is any measurable function there is a natural Banach function space L0p(·) such that p(·) is, intuitively speaking, almost bounded on this space. The space can be defined as the completion(7)⋃α<∞1pt<αf:f∈Lp·¯⊂Lp·.(The space on the right can be regarded as a metric space but it may be nonlinear for some cases of p(·).)The double varying exponentlp(·) spaces, that is, lp(·)(ls(·)), can be defined as follows. For infinite matrices (xn,m)n,m∈N⊂RN×N we define the values of the corresponding norms in 2 phases. First, we let ak:=xk,·ls(·) for all k∈N. Then we set(8)xn,mlp·ls·≔aklp·.In both phases we exclude the matrices (xn,m) producing infinite values. It is easy to see that this results in a Banach space and it is denoted by lp(·)(ls(·)).Fora,b≥0 and 1≤p<∞ we denote(9)a⊞pb=ap+bp1/p.IfF is a filter on N and an, a∈R, n∈N, we denote by limn,Fan=a the fact that(10)∀ɛ>0:n∈N:an-a<ɛ∈F.Recall that a Banach spaceX is finitely representable in a Banach space Y if for each finite-dimensional subspace E⊂X and ɛ>0 there is a finite-dimensional subspace F⊂Y and a linear isomorphism T:E→F with TT-1<1+ɛ.Given a Banach spaceX we denote(11)CX=l∞X/c0X,adopting the notation used for Calkin algebras.


## 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 lp(·) and Lp(·) spaces, respectively.Ifp:[0,1]→[1,∞) is any measurable function there is a natural Banach function space L0p(·) such that p(·) is, intuitively speaking, almost bounded on this space. The space can be defined as the completion(7)⋃α<∞1pt<αf:f∈Lp·¯⊂Lp·.(The space on the right can be regarded as a metric space but it may be nonlinear for some cases of p(·).)The double varying exponentlp(·) spaces, that is, lp(·)(ls(·)), can be defined as follows. For infinite matrices (xn,m)n,m∈N⊂RN×N we define the values of the corresponding norms in 2 phases. First, we let ak:=xk,·ls(·) for all k∈N. Then we set(8)xn,mlp·ls·≔aklp·.In both phases we exclude the matrices (xn,m) producing infinite values. It is easy to see that this results in a Banach space and it is denoted by lp(·)(ls(·)).Fora,b≥0 and 1≤p<∞ we denote(9)a⊞pb=ap+bp1/p.IfF is a filter on N and an, a∈R, n∈N, we denote by limn,Fan=a the fact that(10)∀ɛ>0:n∈N:an-a<ɛ∈F.Recall that a Banach spaceX is finitely representable in a Banach space Y if for each finite-dimensional subspace E⊂X and ɛ>0 there is a finite-dimensional subspace F⊂Y and a linear isomorphism T:E→F with TT-1<1+ɛ.Given a Banach spaceX we denote(11)CX=l∞X/c0X,adopting the notation used for Calkin algebras.


## 2. Results

### 2.1. Preparations: Banach Lattices of Ultraproducts

LetXn be a sequence of Banach lattices, each satisfying the property(12)x-y≤x-y.Observe that this condition immediately guarantees that the absolute value mapping x↦|x| is nonexpansive. We let(13)⨁nXnl∞sensebe the l∞ direct sum of the spaces. Write X=⨁nXn. Suppose that F is a filter on N, for example, a Fréchet filter. Then we let(14)NF=xn∈X:limn,F⁡xnXn=0⊂X.It is easy to check that this is a closed subspace. For example, ifXn=R for all n, a 1-dimensional Banach space, and F is the filter generated by cofinite subsets, then NF=c0.We may generate a vector lattice order⪯ on the space X/NF from the condition(15)n:xn≤Xnyn∈F⟹xn/~⪯yn/~.(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 X/NF by(16)xn/~⟼xnXn/~.Indeed, this is well defined since the absolute values·Xn satisfy (12). Then the condition |x|=x 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·Xn satisfy (12), respectively. Then X endowed with the partial order ⪯ is a Banach lattice whose absolute value coincides with the mapping ·.We denote byU a free ultrafilter on the natural numbers. Recall that the ultrapower of a Banach space Y is defined as(17)YU=l∞Y/NU.


### 2.2. Order-Isomorphic Isometric Embeddings

Letr:N→Q∩[1,∞) be a bijection, that is, an enumeration of the rationals q≥1. Denote l0r(·)=[(en)]⊂lr(·). It is known that lr(·) (resp., l0r(·)) contains almost isometrically all the spaces of the type lq(·) (resp., l0q(·)), in particular the spaces lp (resp., lp for p<∞); see [10].Theorem 2.
Letr(·) be as above. The space C(l0r(·)) is universal for spaces of the type L0p(·)[0,1]. More precisely, the latter spaces considered with their a.e. pointwise order, can be mapped by a linear order-preserving isometry into C(l0r(·)), endowed with the Banach lattice order ⪯, as described above. Moreover, the same conclusion holds if one considers the ultrapower (l0r(·))U in place of C(l0r(·)) for any free ultrafilter U on N.Proof.
We prove the latter statement involving the ultrapower which is more abstract (if not more complicated). Letp:[0,1]→[1,∞) be a measurable function and U a free ultrafilter on N.
Note thatY⊕pLp(μ) can be written isometrically as (Y⊕pLp(μ1))⊕pLp(μ2) where μ=μ1+μ2 is a decomposition such that max⁡supp⁡μ1≤min⁡supp⁡μ2.
According to Lusin’s theorem there is a sequence of compact setsCn⊂[0,1] such that p|Cn are uniformly continuous for each n and m(Cn)→1. Since the norm-defining solutions φf are assumed to be absolutely continuous and taking into account the basic properties of the solutions (see [12]), we may identify(18)fLp·=supn⁡1CnfLp·,f∈L0p·.Indeed, let us recall the justification for this. It was proved in [12] that ·Lp(·) is a lattice norm and moreover that φf≤φg pointwise if |f|≤|g| pointwise a.e. Since ess⁡sup⁡p(·)<∞, it is known that since f∈Lp(·) then also 1Cnf∈Lp(·), and φ1Cnf≤φf pointwise; see [10]. Then, inspecting the governing differential equation (4), we get immediately that(19)φ1Cnf′≥φf′a.e. on Cn and of course φ1Cnf′=0 a.e. in the complement of Cn. On the other hand, the solution φf, by its definition, is absolutely continuous which implies(20)∫0,1∖Cnφf′tdt⟶0as n→∞. Thus, using (19) we get(21)supn⁡1CnfLp·=supn⁡∫0,1φ1Cnf′tdt≥∫0,1φf′tdt=fLp·.Since 1CnfLp(·)≤fLp(·) for all n∈N, we observe that the above inequality becomes equality.
Step 1 (approximation of the norm by simple seminorms). First we assume that f∈L∞. This makes sense because it was shown in [12] that L∞ is dense in Lp(·) in the case where ess⁡sup⁡p<∞.
Consider simple seminorms (as in [12]):(22)fN≔fLp1μ1⊕q2Lp2μ2⊕q3Lp3μ3)⊕q4⋯)⊕qnLpnμn.
Here the measuresμi:Σi→[0,1] are obtained as restrictions of the Lebesgue measure to compact subsets Ii⊂[0,1] where max⁡Ii≤min⁡Ii+1. Thus Σi={A∩Ii:A∈Σ} where Σ is the σ-algebra of the completed Lebesgue measure on the unit interval and μi(B)=m(B) for all B∈Σi.
Let(Nn) be a sequence of such seminorms with pi≤p(·)≤qi on supp(μi). Then by the construction of the ·Lp(·) norm we have that |f|Nn≤fLp(·) for each n and f∈L0p(·). Indeed, this is due to the fact that fLp(·) is essentially defined as a supremum of such seminorms.
By a diagonal argument we may chooseNn in such a way that(23)limn→∞⁡Nn1Cmf=1CmfLp·for each f∈L∞ and m∈N. Indeed, since p is bounded and uniformly continuous on Cm we may find for each ɛ>0 numbers 0=a1<a2<⋯<aj=1 such that (1)
the corresponding supports for measures satisfysupp(μi)⊂[ai,ai+1];(2)
qi=supCm∩[ai,ai+1]p(·), qi=supCm∩[ai,ai+1]p(·);(3)
intuitively, the differencesqi-pi are negligibly small;(4)
(24)ddt1Cm∩0,tfN≥φf′t-ɛa.e. onCm for f such that supt|f(t)|=1.
This is due to the fact that(25)ddtAqi+txpiqi/pi1/qit=0⟶xppA1-puniformly for 0≤A, |x|≤1 as pi↗p, qi↘p. The diagonal argument is then applied to choose the sequence of seminorms Nn as to eventually cover all cases ɛ=1/m, Cm and |f|≤m for all m∈N.
For eachk∈N let Fk be the finite σ-algebra generated by all the supports of μi corresponding to Nn for 1≤n≤k. Without loss of generality we may assume by adding suitable finitely many sets (e.g., dyadic decompositions of the unit interval) to each Fk that(26)limk→∞⁡sup⁡diam⁡Δ:Δ∈Fkatom=0and that ⋃kFkσ-generates the Borel σ-algebra on the unit interval. By an atom of an algebra of sets F we mean Δ∈F such that if Δ′∈F, Δ′⊂Δ, then Δ′=Δ or Δ′=∅.
Next we study the conditional expectation operatorsE(f∣C,Fk). Here(27)Ef∣C,F=∑ΔAΔ∩C∗f1Δ∩C,where AΔ∩C∗∈L∞ is considered as the average integral (operator) over Δ∩C where Δ∈F are atoms with respect to the finite algebra of sets F. We use the convention that AΔ∩C∗(f)=0 whenever m(Δ∩C)=0.
Restrictingf to the support of any of the μi:s, it follows from the martingale convergence principle that E(f∣C,Fk)→1Cf almost everywhere as k→∞ and also in the Lpi(μi)-sense; see, for example, [14, Ch. 5.4.]. Consequently, putting the pieces together, we obtain that(28)Ef∣Cm,FkNn⟶1CmfNn,ask⟶∞,∀n,m∈N,f∈L∞.
Define versionsNn′ of the seminorms Nn by replacing qi with pi. By the uniform continuity of p on the sets Cm, (25), (26), and (28) we may choose subsequences (mn),(kn)⊂N with mn, kn→∞ as n→∞ such that(29)limn→∞⁡Ef∣Cmn,FknNn′=fLp·,f∈SL∞,where n↦mn need not be strictly increasing. In fact, the above equality clearly holds for any f∈L∞.
Step 2 (approximation of the required operator by tame nonlinear operators). Consider a sequence 0≤αn↗∞ and nonlinear operators Tn:L0p(·)[0,1]→L0p(·)[0,1] given by Tn(f)[t]=min(αn,max(-αn,f(t))) for a.e. t. Thus the following condition holds: (30)f-TnfLp·⟶0asn⟶∞for  eachf∈L0p·.Indeed, L∞⊂Lp(·) is dense whenever esssupp<∞ and consequently it follows from the definition of the space L0p(·) that L∞ is dense in it as well. We may additionally choose the above sequences of α:s, conditional expectation operators, and the seminorms in such a way that (31)limn→∞⁡supf∈L0p·⁡ETnf∣Cmn,FknNn′-1CmnfLp·=0.This can be established by using the uniform continuity of p on the compact sets Cm, using the fact that in such a case the simple seminorms converge uniformly, and invoking the martingale Lp-convergence fact above. Note that Tn:s are order-preserving although they are nonlinear.
Next we analyze the simple seminorms chosen and in particular the exponentspi:s. We obtain that for each n for the exponents p2(n),p3(n),…,pj(n) corresponding to Nn′ there are i2<i3<⋯<ij with rational exponents ri2(n),ri3(n),…,rij(n) very close to the corresponding p:s. Indeed, by repeating the almost isometric embedding construction in [10] we may pick the rik:s in such a way that(32)·lri2n,ri3n,…,rijn≤·lp2n,p3n,…,pjn≤n+1n·lri2n,ri3n,…,rijn.Here(33)lq2,q3,…,qj=⋯R⊕q2R⊕q3R⊕q4⋯⊕qj-1R⊕qjR,that is, Rj+1 with the norm(34)xnn=1j+1lq2,q3,…,qj=⋯x1⊞p2x2⊞p3x3⊞p4⋯⊞pjxj+1.
We can find for eachn a finite-dimensional varying exponent lp space lp2(n),…,pj(n) and a natural linear order-preserving linear bijection(35)Bn:ImageE·∣Cmn,Fkn⟶lp2n,…,pjn,where j is the finite dimension of the space of simple functions of the form E(f∣Cmn,Fkn), such that(36)Ef∣Cmn,FknNn′=BnEf∣Cmn,Fknlp2n,…,pjn,where n∈N and f∈L0p(·). Indeed, this applies the fact that Fkn contains all the supports of μi:s corresponding to Nn′ and we may write(37)Ef∣Cmn,Fkn=∑i=1jAΔi∗f1Δi,Ef∣Cmn,FknNn′=AΔ1∗f1Δ1Lp1nΔ1⊞p2nAΔ2∗f1Δ2Lp1nΔ2⊞p3n⋯⊞pjnAΔj∗f1ΔjLpjnΔj,where the subsets Δi⊂[0,1] are successive (supΔi≤infΔi+1) and are Fkn-atomic subsets of the supports of μi:s corresponding to Nn′. Recall that(38)LpΔi⊕pLpΔi+1=LpΔi∪Δi+1in a canonical way. The mapping Bn is given by(39)Ef∣Cmn,Fkn⟼mΔ11/p1·AΔ1∗f,mΔ21/p2AΔ2∗f,…,mΔj1/pj·AΔj∗fand it is easy to see that it is well-defined, linear, and bijective.
Letιn:lp2(n),p3(n),…,pj(n)→lr(·) be a natural linear order-preserving mapping corresponding to the arrangement in (32). Next, we define a mapping S:L0p(·)→l∞(lr(·)) as follows: S(f)=(xn), where(40)xn=ιnBnETnf∣Cmn,Fkn.
Note that(41)1CmnfLp·⟶fLp·,n⟶∞by the absolute continuity of the solutions φf, as observed above.
The mapping required in the statement is the induced mappingS^:f↦S(f)+NF, mapping to the quotient space (in this case the ultrapower). This mapping is clearly order-preserving.
It is also norm-preserving, sincelimn→∞xnlr(·)=f. Indeed, this follows by using (31), (32), (36), (41), and the fact that L∞ is dense in L0p(·).
To verify the linearity ofS^, observe that for any ɛ>0 and f, g∈L0p(·) there are f0, g0∈L∞ such that max(f-f0Lp·,g-g0Lp(·))<ɛ. Thus, selecting n∈N in such a way that αn≥max(fL∞,gL∞), we obtain that(42)max⁡f-TnfLp·,g-TngLp·≤max⁡f-f0Lp·,g-g0Lp·<ɛ.Here we are using the fact that L0p(·) is a Banach lattice in its usual order (see [12]). This means that (43)Tnf+g-Tnf+TngLp·≤Tnf+g-f+gLp·+Tnf-fLp·+Tng+gLp·⟶0as n→∞. Recalling (31) and the construction of S, it follows that(44)Snf+g-Snf+Snglr·⟶0,n⟶∞.This shows that S^(f+g)=S^(f)+S^(g) for all f, g∈L0p(·). The homogeneity of S^ is seen similarly. This completes the proof.Theorem 3.
Letr(·) be as above. The space C(lr(·)(lr(·))) is universal for spaces of the type lq(·)(ls(·)). More precisely, the latter spaces considered with their matrix entrywise order can be mapped by a linear order-preserving isometry into C(lr(·)(lr(·))). Moreover, the same conclusion holds if we consider the ultrapower (lr(·)(lr(·)))U in place of C(lr(·)(lr(·))). In particular, each space lq(·)(ls(·)) is finitely representable in lr(·)(lr(·)).Proof (sketch.).
Consider each element oflq(·)(ls(·)) as a sequence (xn) with xn∈ls(·) for n∈N. We may consider these elements as infinite matrices (xn,m)n,m∈N.
Letk∈N. By repeating inductively the observation involving (32) we can find n1,n2,…,nk and m1,m2,…,mk such that the mapping ιk:(xi,j)↦(yni,mj) and setting other coordinates yn,m to 0 satisfies(45)rkxi,jlq·ls·≤ιkrkxi,jlr·lr·≤k+1krkxi,jlq·ls·.Here the domain of ιk is {(1n,m≤kxn,m):xn,m∈R,n,m∈N} and we denote by rk:lq(·)(ls(·))→lq(·)(ls(·)) the canonical projection to this domain.
The required linear isometry is induced by the operator(46)S:lq·ls·⟶l∞lr·lr·,xn,m⟼ιkrkxn,mk∈N.We note that the previous result holds also as a left-handed version, where we consider all the varying exponentlp-spaces formally as(47)R⊕p1R⊕p2R⊕p3⋯⋯⋯.


## 2.1. Preparations: Banach Lattices of Ultraproducts

LetXn be a sequence of Banach lattices, each satisfying the property(12)x-y≤x-y.Observe that this condition immediately guarantees that the absolute value mapping x↦|x| is nonexpansive. We let(13)⨁nXnl∞sensebe the l∞ direct sum of the spaces. Write X=⨁nXn. Suppose that F is a filter on N, for example, a Fréchet filter. Then we let(14)NF=xn∈X:limn,F⁡xnXn=0⊂X.It is easy to check that this is a closed subspace. For example, ifXn=R for all n, a 1-dimensional Banach space, and F is the filter generated by cofinite subsets, then NF=c0.We may generate a vector lattice order⪯ on the space X/NF from the condition(15)n:xn≤Xnyn∈F⟹xn/~⪯yn/~.(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 X/NF by(16)xn/~⟼xnXn/~.Indeed, this is well defined since the absolute values·Xn satisfy (12). Then the condition |x|=x 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·Xn satisfy (12), respectively. Then X endowed with the partial order ⪯ is a Banach lattice whose absolute value coincides with the mapping ·.We denote byU a free ultrafilter on the natural numbers. Recall that the ultrapower of a Banach space Y is defined as(17)YU=l∞Y/NU.


## 2.2. Order-Isomorphic Isometric Embeddings

Letr:N→Q∩[1,∞) be a bijection, that is, an enumeration of the rationals q≥1. Denote l0r(·)=[(en)]⊂lr(·). It is known that lr(·) (resp., l0r(·)) contains almost isometrically all the spaces of the type lq(·) (resp., l0q(·)), in particular the spaces lp (resp., lp for p<∞); see [10].Theorem 2.
Letr(·) be as above. The space C(l0r(·)) is universal for spaces of the type L0p(·)[0,1]. More precisely, the latter spaces considered with their a.e. pointwise order, can be mapped by a linear order-preserving isometry into C(l0r(·)), endowed with the Banach lattice order ⪯, as described above. Moreover, the same conclusion holds if one considers the ultrapower (l0r(·))U in place of C(l0r(·)) for any free ultrafilter U on N.Proof.
We prove the latter statement involving the ultrapower which is more abstract (if not more complicated). Letp:[0,1]→[1,∞) be a measurable function and U a free ultrafilter on N.
Note thatY⊕pLp(μ) can be written isometrically as (Y⊕pLp(μ1))⊕pLp(μ2) where μ=μ1+μ2 is a decomposition such that max⁡supp⁡μ1≤min⁡supp⁡μ2.
According to Lusin’s theorem there is a sequence of compact setsCn⊂[0,1] such that p|Cn are uniformly continuous for each n and m(Cn)→1. Since the norm-defining solutions φf are assumed to be absolutely continuous and taking into account the basic properties of the solutions (see [12]), we may identify(18)fLp·=supn⁡1CnfLp·,f∈L0p·.Indeed, let us recall the justification for this. It was proved in [12] that ·Lp(·) is a lattice norm and moreover that φf≤φg pointwise if |f|≤|g| pointwise a.e. Since ess⁡sup⁡p(·)<∞, it is known that since f∈Lp(·) then also 1Cnf∈Lp(·), and φ1Cnf≤φf pointwise; see [10]. Then, inspecting the governing differential equation (4), we get immediately that(19)φ1Cnf′≥φf′a.e. on Cn and of course φ1Cnf′=0 a.e. in the complement of Cn. On the other hand, the solution φf, by its definition, is absolutely continuous which implies(20)∫0,1∖Cnφf′tdt⟶0as n→∞. Thus, using (19) we get(21)supn⁡1CnfLp·=supn⁡∫0,1φ1Cnf′tdt≥∫0,1φf′tdt=fLp·.Since 1CnfLp(·)≤fLp(·) for all n∈N, we observe that the above inequality becomes equality.
Step 1 (approximation of the norm by simple seminorms). First we assume that f∈L∞. This makes sense because it was shown in [12] that L∞ is dense in Lp(·) in the case where ess⁡sup⁡p<∞.
Consider simple seminorms (as in [12]):(22)fN≔fLp1μ1⊕q2Lp2μ2⊕q3Lp3μ3)⊕q4⋯)⊕qnLpnμn.
Here the measuresμi:Σi→[0,1] are obtained as restrictions of the Lebesgue measure to compact subsets Ii⊂[0,1] where max⁡Ii≤min⁡Ii+1. Thus Σi={A∩Ii:A∈Σ} where Σ is the σ-algebra of the completed Lebesgue measure on the unit interval and μi(B)=m(B) for all B∈Σi.
Let(Nn) be a sequence of such seminorms with pi≤p(·)≤qi on supp(μi). Then by the construction of the ·Lp(·) norm we have that |f|Nn≤fLp(·) for each n and f∈L0p(·). Indeed, this is due to the fact that fLp(·) is essentially defined as a supremum of such seminorms.
By a diagonal argument we may chooseNn in such a way that(23)limn→∞⁡Nn1Cmf=1CmfLp·for each f∈L∞ and m∈N. Indeed, since p is bounded and uniformly continuous on Cm we may find for each ɛ>0 numbers 0=a1<a2<⋯<aj=1 such that (1)
the corresponding supports for measures satisfysupp(μi)⊂[ai,ai+1];(2)
qi=supCm∩[ai,ai+1]p(·), qi=supCm∩[ai,ai+1]p(·);(3)
intuitively, the differencesqi-pi are negligibly small;(4)
(24)ddt1Cm∩0,tfN≥φf′t-ɛa.e. onCm for f such that supt|f(t)|=1.
This is due to the fact that(25)ddtAqi+txpiqi/pi1/qit=0⟶xppA1-puniformly for 0≤A, |x|≤1 as pi↗p, qi↘p. The diagonal argument is then applied to choose the sequence of seminorms Nn as to eventually cover all cases ɛ=1/m, Cm and |f|≤m for all m∈N.
For eachk∈N let Fk be the finite σ-algebra generated by all the supports of μi corresponding to Nn for 1≤n≤k. Without loss of generality we may assume by adding suitable finitely many sets (e.g., dyadic decompositions of the unit interval) to each Fk that(26)limk→∞⁡sup⁡diam⁡Δ:Δ∈Fkatom=0and that ⋃kFkσ-generates the Borel σ-algebra on the unit interval. By an atom of an algebra of sets F we mean Δ∈F such that if Δ′∈F, Δ′⊂Δ, then Δ′=Δ or Δ′=∅.
Next we study the conditional expectation operatorsE(f∣C,Fk). Here(27)Ef∣C,F=∑ΔAΔ∩C∗f1Δ∩C,where AΔ∩C∗∈L∞ is considered as the average integral (operator) over Δ∩C where Δ∈F are atoms with respect to the finite algebra of sets F. We use the convention that AΔ∩C∗(f)=0 whenever m(Δ∩C)=0.
Restrictingf to the support of any of the μi:s, it follows from the martingale convergence principle that E(f∣C,Fk)→1Cf almost everywhere as k→∞ and also in the Lpi(μi)-sense; see, for example, [14, Ch. 5.4.]. Consequently, putting the pieces together, we obtain that(28)Ef∣Cm,FkNn⟶1CmfNn,ask⟶∞,∀n,m∈N,f∈L∞.
Define versionsNn′ of the seminorms Nn by replacing qi with pi. By the uniform continuity of p on the sets Cm, (25), (26), and (28) we may choose subsequences (mn),(kn)⊂N with mn, kn→∞ as n→∞ such that(29)limn→∞⁡Ef∣Cmn,FknNn′=fLp·,f∈SL∞,where n↦mn need not be strictly increasing. In fact, the above equality clearly holds for any f∈L∞.
Step 2 (approximation of the required operator by tame nonlinear operators). Consider a sequence 0≤αn↗∞ and nonlinear operators Tn:L0p(·)[0,1]→L0p(·)[0,1] given by Tn(f)[t]=min(αn,max(-αn,f(t))) for a.e. t. Thus the following condition holds: (30)f-TnfLp·⟶0asn⟶∞for  eachf∈L0p·.Indeed, L∞⊂Lp(·) is dense whenever esssupp<∞ and consequently it follows from the definition of the space L0p(·) that L∞ is dense in it as well. We may additionally choose the above sequences of α:s, conditional expectation operators, and the seminorms in such a way that (31)limn→∞⁡supf∈L0p·⁡ETnf∣Cmn,FknNn′-1CmnfLp·=0.This can be established by using the uniform continuity of p on the compact sets Cm, using the fact that in such a case the simple seminorms converge uniformly, and invoking the martingale Lp-convergence fact above. Note that Tn:s are order-preserving although they are nonlinear.
Next we analyze the simple seminorms chosen and in particular the exponentspi:s. We obtain that for each n for the exponents p2(n),p3(n),…,pj(n) corresponding to Nn′ there are i2<i3<⋯<ij with rational exponents ri2(n),ri3(n),…,rij(n) very close to the corresponding p:s. Indeed, by repeating the almost isometric embedding construction in [10] we may pick the rik:s in such a way that(32)·lri2n,ri3n,…,rijn≤·lp2n,p3n,…,pjn≤n+1n·lri2n,ri3n,…,rijn.Here(33)lq2,q3,…,qj=⋯R⊕q2R⊕q3R⊕q4⋯⊕qj-1R⊕qjR,that is, Rj+1 with the norm(34)xnn=1j+1lq2,q3,…,qj=⋯x1⊞p2x2⊞p3x3⊞p4⋯⊞pjxj+1.
We can find for eachn a finite-dimensional varying exponent lp space lp2(n),…,pj(n) and a natural linear order-preserving linear bijection(35)Bn:ImageE·∣Cmn,Fkn⟶lp2n,…,pjn,where j is the finite dimension of the space of simple functions of the form E(f∣Cmn,Fkn), such that(36)Ef∣Cmn,FknNn′=BnEf∣Cmn,Fknlp2n,…,pjn,where n∈N and f∈L0p(·). Indeed, this applies the fact that Fkn contains all the supports of μi:s corresponding to Nn′ and we may write(37)Ef∣Cmn,Fkn=∑i=1jAΔi∗f1Δi,Ef∣Cmn,FknNn′=AΔ1∗f1Δ1Lp1nΔ1⊞p2nAΔ2∗f1Δ2Lp1nΔ2⊞p3n⋯⊞pjnAΔj∗f1ΔjLpjnΔj,where the subsets Δi⊂[0,1] are successive (supΔi≤infΔi+1) and are Fkn-atomic subsets of the supports of μi:s corresponding to Nn′. Recall that(38)LpΔi⊕pLpΔi+1=LpΔi∪Δi+1in a canonical way. The mapping Bn is given by(39)Ef∣Cmn,Fkn⟼mΔ11/p1·AΔ1∗f,mΔ21/p2AΔ2∗f,…,mΔj1/pj·AΔj∗fand it is easy to see that it is well-defined, linear, and bijective.
Letιn:lp2(n),p3(n),…,pj(n)→lr(·) be a natural linear order-preserving mapping corresponding to the arrangement in (32). Next, we define a mapping S:L0p(·)→l∞(lr(·)) as follows: S(f)=(xn), where(40)xn=ιnBnETnf∣Cmn,Fkn.
Note that(41)1CmnfLp·⟶fLp·,n⟶∞by the absolute continuity of the solutions φf, as observed above.
The mapping required in the statement is the induced mappingS^:f↦S(f)+NF, mapping to the quotient space (in this case the ultrapower). This mapping is clearly order-preserving.
It is also norm-preserving, sincelimn→∞xnlr(·)=f. Indeed, this follows by using (31), (32), (36), (41), and the fact that L∞ is dense in L0p(·).
To verify the linearity ofS^, observe that for any ɛ>0 and f, g∈L0p(·) there are f0, g0∈L∞ such that max(f-f0Lp·,g-g0Lp(·))<ɛ. Thus, selecting n∈N in such a way that αn≥max(fL∞,gL∞), we obtain that(42)max⁡f-TnfLp·,g-TngLp·≤max⁡f-f0Lp·,g-g0Lp·<ɛ.Here we are using the fact that L0p(·) is a Banach lattice in its usual order (see [12]). This means that (43)Tnf+g-Tnf+TngLp·≤Tnf+g-f+gLp·+Tnf-fLp·+Tng+gLp·⟶0as n→∞. Recalling (31) and the construction of S, it follows that(44)Snf+g-Snf+Snglr·⟶0,n⟶∞.This shows that S^(f+g)=S^(f)+S^(g) for all f, g∈L0p(·). The homogeneity of S^ is seen similarly. This completes the proof.Theorem 3.
Letr(·) be as above. The space C(lr(·)(lr(·))) is universal for spaces of the type lq(·)(ls(·)). More precisely, the latter spaces considered with their matrix entrywise order can be mapped by a linear order-preserving isometry into C(lr(·)(lr(·))). Moreover, the same conclusion holds if we consider the ultrapower (lr(·)(lr(·)))U in place of C(lr(·)(lr(·))). In particular, each space lq(·)(ls(·)) is finitely representable in lr(·)(lr(·)).Proof (sketch.).
Consider each element oflq(·)(ls(·)) as a sequence (xn) with xn∈ls(·) for n∈N. We may consider these elements as infinite matrices (xn,m)n,m∈N.
Letk∈N. By repeating inductively the observation involving (32) we can find n1,n2,…,nk and m1,m2,…,mk such that the mapping ιk:(xi,j)↦(yni,mj) and setting other coordinates yn,m to 0 satisfies(45)rkxi,jlq·ls·≤ιkrkxi,jlr·lr·≤k+1krkxi,jlq·ls·.Here the domain of ιk is {(1n,m≤kxn,m):xn,m∈R,n,m∈N} and we denote by rk:lq(·)(ls(·))→lq(·)(ls(·)) the canonical projection to this domain.
The required linear isometry is induced by the operator(46)S:lq·ls·⟶l∞lr·lr·,xn,m⟼ιkrkxn,mk∈N.We note that the previous result holds also as a left-handed version, where we consider all the varying exponentlp-spaces formally as(47)R⊕p1R⊕p2R⊕p3⋯⋯⋯.



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*Source: 186105-2015-06-08.xml*