Decompositions of Nakano norms by ODE techniques
aa r X i v : . [ m a t h . C A ] F e b DECOMPOSITIONS OF NAKANO NORMSBY ODE TECHNIQUES
JARNO TALPONEN
Abstract.
We study decompositions of Nakano type varying-exponent Lebesguenorms and spaces. These function spaces are represented here in a natural wayas tractable varying-exponent ℓ p sums of projection bands. The main resultsinvolve embedding the variable Lebesgue spaces to such sums, as well as thecorresponding isomorphism constants.The main tool applied here is an equivalent variable Lebesgue norm which isdefined by a suitable ordinary differential equation introduced recently by theauthor. We also analyze the effect of transformations changing the ordering ofthe unit interval on the values of the ODE-determined norm. Introduction
In this paper we analyze Nakano type varying-exponent L p norms and theirdecompositions. Alternatively, we study embedding results of the above Nakanospaces to some more tractable Banach spaces which arise as varying ℓ p summandsof classical L p spaces. We approach this topic via an alternative equivalent normon the Nakano space defined by the means of weak solutions to suitable ordinarydifferential equations (ODE). These can be easily analyzed by studying the prop-erties of the corresponding ODEs and thus by virtue of equivalence of the normswe obtain strong estimates for Nakano type variable Lebesgue norms.Motivated by Nakano’s work ([Nak50]), we call the following special case ofLuxemburg ([Lux55]) norm on a Musielak-Orlicz space ([Mus83], cf. [BO31]) aNakano norm: ||| f ||| p p¨q : “ inf t λ ą ρ p f { λ q ď u where ρ p g q : “ ż p p t q | g p t q| p p t q dt. See [Mal11] for a historical account.The Luxemburg norm clearly arises as an application of the Minkowski func-tional. It is not that easy to analyze the Luxemburg norms, even the numericalvalue of the norm of a constant function cannot be calculated immediately basedon the definition of the norm. Here we consider an alternative approach to variableLebesgue norms which is rather recursive than global.The above weight w p t q “ p p t q , appearing in Nakano’s work, catches the eye(see also discussion in [DHHR11, Sect. 3.1]). In a sense it involves the continuityproperties of the mapping λ ÞÑ ρ p f { λ q . In Section 5.2 we provide further motivation Date : February 9, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Variable Lebesgue space, Musielak-Orlicz space, Nakano norm, ordi-nary differential equation, norm inequality, embedding theorem. and discussion. This weight also has a special role in our ODEs studied. Theseprovide a novel approach to the varying-exponent Lebesgue spaces which do not involve the Minkowski functional at all.There is a vast literature involving inequalitites and embeddings on the variableLebesgue spaces. For an important paper initiating the investigation of varying-exponent Sobolev spaces, see [KR91]. Different kinds of inequalities are in thecore of the study of Lebesgue and Sobolev spaces. These are essential for instancein classical integral operators, harmonic analysis and spaces of analytic functions.These spaces involve for instance the Poincar´e-like inequalities carrying the namesof Morrey, Nash and Gagliardo-Nirenberg-Sobolev. The norms of a Riesz operatorand a potential are related by the Hardy-Littlewood-Sobolev lemma. We refere to[CF07, CFMP06, CWD17, MSS16] for samples of papers concentrating on theseand other important inequalities. See also [CFN03, DR03, DHH09, Ho15, RS09]for works which rely on techniques with central inequalities. All the mentionedinequalities deal with L p structures which is a very particular case in the realm ofBanach spaces. The exponent p may additionally vary locally but nevertheless thesestructures are detectable. Apart from the typical applications in mathematicalanalysis, the variable Lebesgue spaces have been recently an object of study inmathematical logic, see e.g. [Yaa09].The main purpose of this paper is to provide natural tools for treating the L p structure of variable Lebesgue spaces and to demonstrate the usefulness of ODE-techniques in this connection.Because of the way Nakano norms, and, more generally, Luxemburg norms aredefined, it is not so easy to identify a priori the ’contribution’ of different parts ofthe function to the norm; that is, how the restriction of the support of the functionaffects the norm. Namely, if f P L p p¨q r , s (with any weight) and ∆ Ă r , s is measurable, possibly disjoint from the support of f , it is not easy in general toevaluate efficiently the relationship between ||| f ||| p p¨q and ||| f ` ∆ ||| p p¨q quantitatively.It seems reasonable to ask how the Nakano norm can be decomposed in a naturalway. For instance, if p p¨q has constant values, say, p , . . . , p n , on some sets of positivemeasure, one might ask what is the quantitative relationship between the followingNakano norms: ||| f ||| p p¨q and ||| p p¨q‰ p ,...,p n f ||| p p¨q , ||| p p¨q“ p f ||| p p¨q , . . . , ||| p p¨q“ p n f ||| p p¨q ?If the correspondence is natural enough, it should not depend on n , since themeasure space is atomless. The relationship should be somehow formalized math-ematically and the simplest way is stating ||| f ||| p p¨q „ ||| p p¨q‰ p ,...,p n f ||| p p¨q ` ||| p p¨q“ p f ||| p p¨q ` . . . ` ||| p p¨q“ p n f ||| p p¨q , which, unfortunately, turns out to be the wrong approach after a short reflection,unless p p¨q “
1. A reasonable approach is replacing above the addition operationsby suitable operations corresponding to ℓ p -summands applying the respective valuesof p i . It may be instructive to observe that } f } p “ ` } ∆ f } pp ` } r , sz ∆ f } pp ˘ p for any f P L p r , s , 1 ď p ă 8 , and a measurable subset ∆ Ă r , s . Thus thecorrect operation in this simple case appears to be a ⊞ p b : “ p a p ` b p q p , a, b ě , ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 3 instead of the ` operation. This leads to the relationship ||| f ||| p p¨q „ p . . . pp||| p p¨q‰ p ,...,p n f ||| p p¨q ⊞ p ||| p p¨q“ p f ||| p p¨q q ⊞ p . . .. . . ⊞ p n ´ ||| p p¨q“ p n ´ f ||| p p¨q q ⊞ p n ||| p p¨q“ p n f ||| p p¨q . We will prove in this paper several inequalities which substantiate the above equiv-alence as a simple case of a more general natural principle (see Theorem 4.1). Theobtained isomorphism constants do not depend on n . We also analyze the effect ofchanging the order in taking the operations above.The main tool is an equivalent ODE-determined Lebesgue norm introduced bythe author in [Tal17], extending the class of classical L p norms. This constructioncan be seen as a continuous version of a recursively constructed varying-exponent ℓ p norm. It appeared in [Sob41] as a remark and was later studied in [Tal11], cf.[ACK98, Kal07]. Unlike the Luxemburg style variable Lebesgue norm, the ODE-determined norm satisfies properly the H¨older inequality and the dualities are neat.The norm determining ODE is rather simple: ϕ f p q “ , ϕ f p t q “ | f p t q| p p t q p p t q ϕ f p t q ´ p p t q for almost every t P r , s . It turns out that in some cases arguing by means of the above differential equationmakes the analysis of the norms tractable.Representing the variable Lebesgue norms by using projection bands is useful inthe analysis of operators on these spaces. Recall that a projection band is a subspaceof functions supported on a given measurable subset, cf. [LT13]. For instance, the p -convexifications or ℓ p -type sums of functions are instrumental in [DHH09] and[MSS16]. We obtain natural decompositions of Lebesgue spaces which yield upperand lower norm estimates for variable Lebesgue norms. This likely simplifies theanalysis of the various operators acting on these spaces and otherwise reduces normestimates to a combination of well-understood ‘blocks’ with classical L p norms.2. On variable Lebesgue spaces
We rely heavily on the properties and the theoretical framework of ODE-determinednorms appearing in [Tal17]. See [CL55, DHHR11, FHHMPZ01, Mal11, Mus83,RR91] for other suitable background information.In what follows f, p P L r , s , i.e. f, p : r , s Ñ R are measurable functions,and p p¨q ě
1. We denote by p “ ess sup p p¨q if p p¨q is essentially bounded. Denote a ⊞ b “ max p a, b q for a, b ě ||| f ||| p p¨q : “ inf " λ ą ż p p t q | g p t q| p p t q λ p p t q dt ď * . In the literature the variable Lebesgue spaces are considered over n -dimensionalspace, say, expressed in a rather high degree of generality, spaces of the type L p p¨q p R n , Bor p R n q , m n , ω q where p R n , Bor p R n q , m n q is a standard measure spaceinvolving the Lebesgue measure and the Borel σ -algebra on R n , ω : R n Ñ r , isa measurable ’weight’ function (possibly restricting the essential domain) and themodular is given by ρ p g q : “ ż R n ω p t q| g p t q| p p t q dm n p t q . JARNO TALPONEN
Alternatively, one could apply in the integral a measure µ : Bor p R n q Ñ R definedby µ p A q “ ż A ω p t q dm n p t q , A P Bor p R n q . Note that this defines a σ -finite measure. We refer to [Fre00] for advanced measuretheory. Our methodology in this paper, being an application of ODEs, requirestotally ordered measure spaces, essentially the case n “
1. This feature appears tonarrow down the applicability of these ODE-defined spaces.Unexpectedly, it turns out that this is not the case, at least in what comes tothe order-isometric structure of these spaces, which is the topic here.
Corollary 2.1.
The dimension n in L p p¨q p R n , Bor p R n q , m n , ω q is isometricallyorder-isomorphically redundant due to suitable rearrangements of the underlyingmeasure spaces. More precisely, there exists an isometric isomorphism T : L p p¨q p R n , Bor p R n q , m n , ω q Ñ L r p¨q pr , s , Bor pr , sq , m , ω q having the form p T f qr h p x qs “ a p x q f r x s for µ -a.e. x P R n where r p h p x qq “ p p x q and ω p h p x qq “ ω p x q for µ -a.e. x P R n ,a : supp p µ q Ñ p , is a measurable function, and h : supp p µ q Ñ r , s a bijection. This follows form the results given in the Final Remarks (section 5.1) where wealso provide a comprehensive description of the functions above.2.1.
ODE-determined norms.
The main tool in analyzing the Nakano normshere is passing to a tractable norm, defined by means of an ODE, such that thenew norm } ¨ } L p p¨q ODE is equivalent to the Nakano norm: ||| f ||| p p¨q ď } ¨ } L p p¨q ODE ď ||| f ||| p p¨q , see [Tal17, Prop. 3.3]. The ODE-determined varying-exponent Lebesgue class L p p¨q ODE r , s can in principle be defined for any measurable p : r , s Ñ r , . How-ever, if p p¨q is essentially unbounded it may happen that the class fails to be a linearspace. Therefore, for the purposes in this paper, we may restrict to the case where p p¨q is essentially bounded, since the corresponding Nakano norms can in any casebe approximated pointwise by suitably truncating supports.The strategy in [Tal17] is to design the ODE in such a way that its solution ϕ f,p p¨q , corresponding to the function and the exponent, models the accumulationof the norm as follows: ϕ f,p p¨q p t q “ } r ,t s f } p p¨q , so that in particular ϕ f,p p¨q p q “ ϕ f,p p¨q p q becomes the definition for the newnorm “ } f } p p¨q . The ODE which defines the non-decreasing absolutely continoussolution is(2.1) ϕ f p q “ ` , ϕ f p t q “ | f p t q| p p t q p p t q ϕ f p t q ´ p p t q for almost every t P r , s . These are weak solutions in the sense of Carath´eodory with a minor modification;the asymptotic initial condition is to provide the uniqueness of the solution and itis useful in other ways as well. These solutions are further discussed in [Tal17].
ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 5
The ODE-determined variable Lebesgue class is defined as L p p¨q ODE r , s “ f P L r , s : ϕ f exists ( where the solution to (2.1) exists in the required sense.Recall from [Tal17] that the ϕ f p t q ´ p p t q part in the ODE makes it very stable,so that for positive initial values x ą ` initialvalue solution in (2.1) means that we take first all positive-inital-value-solutions,provided that they exist, and then take their limit pointwise as x Œ
0. This leadsto a rather natural unique solution which solves (2.1) for 0 initial value. For apositive inital value x ą p p¨q we have ϕ x ,f p t q ď | f p t q| p p t q p p t q p x q ´ p from which we can deduce the existence of the solution if ż | f p t q| p p t q p p t q dm p t q ă 8 . The solutions can be constructed and approximated by using simple semi-norms([Tal17]) which are motivated by variable ℓ p spaces (see [Tal11], cf. [Tal15]). Thesimple semi-norms here have a role analogous to the simple functions in the con-struction of the Lebesgue integral.The absolute continuity of the solutions implies that ϕ f p q ă 8 . In some cases,for instance if p p¨q is essentially bounded, this class becomes automatically a linearspace. Then the solutions define a complete norm (see [Tal17]): } f } p p¨q : “ } f } L p p¨q ODE : “ ϕ f p q , f P L p p¨q ODE r , s . For a constant exponent case, p p¨q “ p P r , , the above ODE (2.1) be-comes a separable one and solving it yields the classical L p r , s norm: p ϕ f p qq p “ ş | f p t q| p dt . It is worth noting that unlike the usual Luxemburg type variableLebesgue norms, the ODE-determined norms satisfy properly the H¨older inequal-ity. The dualities work nicely as well, see [Tal18].2.2. Some useful estimates.
Let a « .
76 be the solution to a a “ e . This numbersatisfies that b x x is increasing on x ě b ě a . Namely, the constant a satisfiesthat dda a x | x “ “
1. Let us recall the following useful fact from [Tal17]:
Proposition 2.2.
Let f, p P L where ď p p¨q and r P p , . The followinginequalities hold whenever defined: (1) ` a } p p¨qě r f } r ď } p p¨qě r f } p p¨q , (2) ` ae } p p¨qď p p¨q f } p p¨q ď } p p¨qď p p¨q f } p p¨q , (3) } f } p p¨q ă e } f } . Although we are here mainly insterested in the relationship between Nakano andODE-determined type variable Lebesgue norms, the results have also some bearingon the most typical Luxemburg type variable Lebesgue norms given by ||| f ||| MO ,p p¨q : “ inf " λ ą ż | g p t q| p p t q λ p p t q dt ď * , denoted here after Musielak and Orlicz. Indeed, despite the weight in the Nakanonorm, it is equivalent to the MO norm. This is known (see e.g. [DHHR11, (3.2.2)])and below we provide a better isomorphism constant. JARNO TALPONEN
Proposition 2.3.
Given a measurable p : r , s Ñ r , then f P L p p¨q in the senseof Nakano norm if and only if the same holds in the sense of the above MO norm.Moreover, in these equivalent cases with f P L p p¨q we have a ||| f ||| MO ,p p¨q ď ||| f ||| p p¨q ď ||| f ||| MO ,p p¨q . Proof.
We apply the following inequalities ż | f p t q| p p t q p aλ q p p t q dt ď ż p p t q | f p t q| p p t q λ p p t q dt ď ż | f p t q| p p t q λ p p t q dt where the left inequality follows from the above property of the constant a . (cid:3) Below we consider a constant b p depending on p defined as follows: b “ b “ b p for 1 ă p ă 8 is the unique solution to b ` b ´ p “ , ă b ă . Below we will apply the norm notation without distinguishing whether the L p p¨q ODE class is a linear space or not. The following result improves some estimates in[Tal17].
Proposition 2.4.
Let f P L p p¨q ODE . Then ||| f ||| p p¨q ď } f } p p¨q ď b p ||| f ||| p p¨q . Proof.
Let f P L p p¨q ODE . These inequalities were proved in [Tal17], except that aweaker version of the right-hand side was shown with 2 in place of b p .To check the latter inequality, we may assume without loss of generality that ||| f ||| p p¨q “
1. By using the Monotone Convergence Theorem we see that ż p p t q | f p t q| p p t q dt “ . Let b ď } f } p p¨q : “ ϕ f p q ą t P p , q is such that ϕ f p t q “
1. We will apply the well-knownfact (see [DHHR11]) that ||| f ||| p p¨q ď ˆż p p t q | f p t q| p p t q dt ˙ p for every f P L p p¨q (Nakano space), so that ||| r ,t s f ||| p p¨q ď ˆż t p p t q | f p t q| p p t q dt ˙ p . Since } r ,t s f } p p¨q “ b that b ´ ď ||| r ,t s f ||| p p¨q thus b ´ p ď ż t p p t q | f p t q| p p t q dt. Hence ż t p p t q | f p t q| p p t q dt ď ´ b ´ p . ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 7
Note that ϕ ´ p p t q f p t q ď t P r t , s . Thus ϕ f p q “ ż ϕ f dt ď ` ż t p p t q | f p t q| p p t q dt ď ´ b ´ p . Since ||| f ||| p p¨q “ b is the best constant, this implies b ď ´ b ´ p . Thus b isdominated by the solution to b p ` b ´ pp “ , ă b p ă . (cid:3) Proposition 2.5.
Suppose that p p¨q is essentially bounded or non-decreasing. Thenthe corresponding L p p¨q classes, in the ODE, Nakano and MO sense coincide andthe norms are mutually equivalent.Proof. In the case with essentially bounded exponent f P L p p¨q ODE r , s if and only if ż p p t q | f p t q| p p t q dt ă 8 see [Tal17, Prop. 3.8.]. Combining this fact with Proposition 2.3 yields that the L p p¨q classes coincide. The equivalence of the norms then follows from Proposition2.3 and Proposition 2.4.The case with non-decreasing exponent becomes an easy adaptation of the aboveargument, since in each proper initial segment of the unit interval the exponent isbounded. (cid:3) For p a i q P ℓ p p N q , 1 ď p ă 8 , with a i ě i , we define p ð i P N a i “ }p a i q} ℓ p . By inductively applying Proposition 2.1 in [Tal18] we see the following fact.
Proposition 2.6. If ď r ď s ă 8 and t x N u N P N n ` is a family of non-negativenumbers, then p n ð i n ` P N . . . p k ` ð i k ` P N s ð i k ` P N r ð i k ` P N p k ð i k P N . . . p ð i P N x N ď p n ð i n ` P N . . . p k ` ð i k ` P N r ð i k ` P N s ð i k ` P N p k ð i k P N . . . p ð i P N x N where we consider N “ p i , i , . . . , i n ` q . (cid:3) Rearrangements
In the Nakano space a simultaneous measure-preserving permutation of the ex-ponent p p¨q and the function f does not affect the value of the norm. This is not thecase in an ODE-determined variable L p space, since the order of the arrangementaffects the accumulation of the solutions ϕ f .It is easy to see that a ⊞ p b ⊞ c q ď p a ⊞ b q ⊞ c JARNO TALPONEN for all a, b, c ě
0, and, more generally, a ⊞ r p b ⊞ p c q ď p a ⊞ r b q ⊞ p c holds for all 1 ď p ď r ă 8 and a, b, c ě
0. This inequality generalizes considerably,as we saw in Proposition 2.6.It appears natural to ask whether a similar conclusion holds for ODE-determinedvariable Lebesgue spaces. Namely, if we have a simultaneous rearrangement of theexponent and the function, does the increasing (resp. decreasing) arrangementyield the minimal (resp. the maximal) value of the norm? If so, is the ratio of themaximal value and the minimal value bounded and by what constant?3.1.
Heuristic motivation via an auxiliary transformation.
We will beginwith a useful transformation. Suppose that p : r , s Ñ r , is a measurablefunction and f P L p p¨q ODE . It follows from the equivalence of the ODE-determinednorm } ¨ } p p¨q and the corresponding Nakano norm ||| ¨ ||| p p¨q that ż | f p t q| p p t q p p t q dt ă 8 . Let us assume that f ‰ ϕ f the norm-determining weaksolution with initial value 0 ` .Let us define an absolutely continuous increasing bijective transform T : r , s Ñr , α s as follows T p t q “ ż t | f p s q| p p s q p p s q ds, ď t ď α “ T p q .Define ˆ ϕ : r , α s Ñ r , and ˆ p : r , α s Ñ r , by ˆ ϕ p t q : “ ϕ f p T ´ p t qq andˆ p p t q : “ p p T ´ p t qq . Observe thatˆ ϕ p t q “ ddt ϕ f p T ´ p t qq “ ˆ ddt T ´ p t q ˙ ϕ f p T ´ p t qq“ ˜ | f p T ´ p t qq| p p T ´ p t qq p p T ´ p t qq ¸ ´ | f p T ´ p t qq| p p T ´ p t qq p p T ´ p t qq ϕ ´ p p T ´ p t qq f p T ´ p t qq “ ˆ ϕ ´ ˆ p p t q p t q . This explains heuristically why decreasing (resp. increasing) arrangement of theexponent yields the greatest (resp. the least) norm. Namely, due to the exponent1 ´ ˆ p p t q ď ϕ when: ‚ ˆ p is large for small values of ˆ ϕ , ‚ ˆ p is small for large values of ˆ ϕ .3.2. Quantitative estimates for rearranged norms.
By a measure-preservingtransform on the unit interval we mean a bijection π : r , s Ñ r , s such thatfor each Lebesgue measurable subset E Ă r , s the preimage π ´ p E q is Lebesguemeasurable and m p π ´ p E qq “ m p E q . See [Fre00, Vol III] for discussion on repre-sentations of measure algebra isomorphims. Theorem 3.1.
Let p : r , s Ñ r , be a measurable function. Let π : r , s Ñr , s be a measure-preserving transform. Suppose that f P L p p¨q ODE and f ˝ π ´ P L p ˝ π ´ ODE . Then b p } f } p p¨q ď } f ˝ π ´ } p ˝ π ´ ď b p } f } p p¨q . ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 9
Moreover, if π and π , as above, are chosen such that p ˝ π ´ (resp. p ˝ π ´ ) isan increasing (resp. decreasing) rearrangement of p , then } f ˝ π ´ } p ˝ π ´ ď } f } p p¨q ď } f ˝ π ´ } p ˝ π ´ . We note that the constant b “ p p¨q case. This is seen by studying the norms of the constant function considered in the natural way in L p p , p q ‘ L p p , q , and, alternatively, in L p , p q ‘ p L p p p , q . Then } } L p p , p q‘ L p p , q Ñ } } L p , p q‘ p L p p p , q Ñ p Õ 8 . The latter equation line in Theorem 3.1 tells us that the value of the norm of isbetween 1 and 2 for any rearrangement of the above simple functions p p¨q . Proposition 3.2. If p p¨q is increasing then } } p p¨q ď and if p p¨q is decreasing then } } p p¨q ě . Moreover, if p p¨q is additionally (essentially) a non-constant functionthen the inequalities hold strictly. In particular, if p p¨q is monotone and } } p p¨q “ then p p¨q is a constant function.Proof. Suppose that 1 ď p ď p ď . . . ď p n and 0 ă a , a , . . . , a n ă a ` . . . ` a n ď
1. Then p a p ⊞ p a p q ⊞ p a p “ pp a p p ` a q p p ` a q p ď p a ` a ` a q p . Similarly p . . . p a p ⊞ p a p q ⊞ p . . . q ⊞ p n a pn n ď p a ` . . . ` a n q pn ď . An approximation argument with semi-norms (recall in [Tal17]) then yields that if p p¨q is increasing then } } p p¨q ď
1. Similarly we see the other direction.To check the strict inequality, let ϕ and ψ be the solutions corresponding todecreasing and increasing rearrangements of p . Suppose that ϕ p q ď ψ p q andlet t ă ϕ p t q ď ψ p t q for t ď t ď
1. Then ϕ p t q ě ψ p t q for t ď t ď r s, s if p is non-constant. (cid:3) Sketch of proof of Theorem 3.1.
To justify the first inequality line, it suffices toprove the right hand inequality, as the other estimate then follows. We applythe equivalence of ODE-determined norms and Nakano norms together with therearrangement invariance of the Nakano norms. That is, one may rearrange thefunction and the exponent in a similar fashion without affecting the norm. Thuswe obtain } f ˝ π ´ } p ˝ π ´ ď b p ||| f ˝ π ´ ||| p ˝ π ´ “ b p ||| f ||| p ď b p } f } p . Towards the second claim, it follows as a special case from Proposition 2.6 that p a ⊞ p b q ⊞ r c ď p a ⊞ r c q ⊞ p b, for all a, b, c ą , ď p ď r. We may apply the above fact recursively for simple semi-norms, changing the placesof 2 successive summands, 1 pair at a time, in checking the following claim: |p , f j , f j , . . . , f j n q| p L p p µ q‘ pj L pj p µ j qq‘ pj ... ‘ pjn L pjn p µ jn q ď |p , f , . . . , f n q| p L p p µ q‘ p L p p µ qq‘ p ... ‘ pn L pn p µ n q ď |p , f k , f k , . . . , f k n q| p L p p µ q‘ pk L pk p µ k qq‘ pk ... ‘ pkn L pkn p µ kn q where we assume µ pr , sq “ t p j i u ni “ (resp. t p k i u ni “ )is increasing (resp. decreasing) permutation of t p i u ni “ .To extend the above observation to the general setting, with decreasing exponentand maximal norm, we approximate p p¨q by simple seminorms N n such that ˜ p N n Õ p p¨q in measure. It is known that then also N n p f q Ñ } f } p p¨q for any f P L p p¨q . Itsuffices to consider essentially bounded p and f .Let π n be simple measurable measure-preserving transformations such that ˜ p N n ˝ π ´ n become decreasing. Let ψ n be the corresponding solutions. Note that by theabove observations regarding the arrangements of the simple semi-norms, we have N n p f q ď ψ n p q .Suppose, as in the assumptions, that π is a measurable measure-preserving trans-form such that p ˝ π ´ is decreasing. Let ϕ be the corresponding solution and recallthat the solution ϕ is (absolutely) continuous.Then for any p ă p and ε ą f and p p¨q that for sufficiently large n we have(3.1) ż t t : p ă p ˝ π ´ n p t qă p u ψ n dt ď ż t t : p ă p ˝ π ´ p t qă p u ϕ dt ` ε (3.2) if ψ n p t q ě ϕ p t q for t “ ess inf t t : p ă p ˝ π ´ p t q ă p u . Indeed, by considering the distribution function s ÞÑ m pt t : p p t q ě s uq we obtainthat ess inf t t : p ă p ˝ π ´ n p t q ă p u Ñ ess inf t t : p ă p ˝ π ´ p t q ă p u , ess sup t t : p ă p ˝ π ´ n p t q ă p u Ñ ess sup t t : p ă p ˝ π ´ p t q ă p u as n Ñ 8 . It follows that ż t t : p ă p ˝ π ´ p t qă p u ˇˇˇˇˇ | f ˝ π ´ n p t q| p ˝ π ´ n p t q p ˝ π ´ n p t q ´ | f ˝ π ´ p t q| p ˝ π ´ p t q p ˝ π ´ p t q ˇˇˇˇˇ dt Ñ n Ñ 8 by Lusin’s theorem. Then (3.1) follows by including the respective terms ψ ´ p ˝ π ´ n n and ϕ ´ p ˝ π ´ according to (3.2) (roughly ψ ´ p n ď ϕ ´ p in the subsetunder consideration). Thenlim sup n Ñ8 ψ n p q “ lim sup n Ñ8 ż | f ˝ π ´ n p t q| p ˝ π ´ n p t q p ˝ π ´ n p t q ψ ´ p ˝ π ´ n n dt ď ż | f ˝ π ´ p t q| p ˝ π ´ p t q p ˝ π ´ p t q ϕ ´ p ˝ π ´ dt “ ϕ p q . We conclude that } f } p p¨q “ lim n Ñ8 N n p f q ď ϕ p q . (cid:3) ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 11 Decompositions
Next we arrive to the main topic of this paper which involves representing Nakanospaces in a natural tractable way as summands of their projection bands. We willapply the observations from the previous section.The decompositions of classical L p spaces provide some insight on what to expect,albeit, in some ways, too simplistic a view in variable Lebesgue space setting. Forinstance, in the classical setting we have L p r , s “ L p r , { s ‘ p L p r { , s and } f } p ď } f } r , p ď r, therefore } r , s f } p ⊞ r } r , s f } r ď } f } r . However, only a quasi-version of the above inequality holds in the variable Lebesguespace setting, and, in fact, the norm is not monotone with respect to the exponents p p¨q . Theorem 4.1.
Let p p¨q be essentially bounded and f P L p p¨q ODE . Let “ r ă r ă . . . ă r n “ p . Then p ` ae q p||| p ´ p r n ´ ,r n s f ||| r n ´ ⊞ r n ´ ||| p ´ p r n ´ ,r n ´ s f ||| r n ´ q ⊞ r n ´ . . . ⊞ r ||| p ´ r r ,r s f ||| r ď ||| f ||| p p¨q ď p ` ae qp||| p ´ r r ,r q f ||| r ⊞ r ||| p ´ r r ,r q f ||| r q ⊞ r . . . ⊞ r n ||| p ´ r r n ´ ,r n s f ||| r n . More generally, if ∆ i , i “ , . . . , n , form a measurable decomposition of r , s , and r i “ ess inf ∆ i p , s i “ ess sup ∆ i p , then p ` ae q p||| ∆ f ||| r ⊞ r ||| ∆ f ||| r q ⊞ r . . . ⊞ r n ||| ∆ n f ||| r n ď ||| f ||| p p¨q ď p ` ae qp||| ∆ f ||| s ⊞ s ||| ∆ f ||| s q ⊞ s . . . ⊞ s n ||| ∆ n f ||| s n . Moreover, b p p} ∆ f } ⊞ s } ∆ f }q ⊞ s . . . ⊞ s n } ∆ n f } ď } f } p p¨q ď b p p} ∆ f } ⊞ r } ∆ f }q ⊞ r . . . ⊞ r n } ∆ n f } and b p p||| ∆ f ||| ⊞ s ||| ∆ f |||q ⊞ s . . . ⊞ s n ||| ∆ n f ||| ď ||| f ||| p p¨q ď b p p||| ∆ f ||| ⊞ r ||| ∆ f |||q ⊞ r . . . ⊞ r n ||| ∆ n f ||| . Admittedly, the formulas appear complicated in the general case with n fixedconstant exponents. The point here is that the multiplicative constants above donot depend on n . Note that in view of Theorem 3.1 the ordering of r i in the first partof the statement is chosen to be ’the worst possible’, so that the given multiplicativeconstants apply to all possible orderings. The illustrate the first claim, we have112 ` ||| p p¨qě r f ||| r ` ||| p p¨qă r f ||| ˘ ď ||| f ||| p p¨q ď ´ ||| p p¨qă r f ||| r ⊞ p ||| p p¨qě r f ||| p ¯ . Proof of Theorem 4.1.
To prove the first inequality, we may apply a measure-preserving rearrangement of the unit interval such that p becomes decreasing underthe new ordering. Indeed, by an approximation argument employing Lusin’s theo-rem we may reduce to the case where p p¨q is continuous and apply methods similarto the proof of Theorem 3.1.The Nakano norm is invariant under simultaneous measure-preserving rearrange-ments of the function and the exponent, that is, ||| f ˝ h ||| p ˝ h “ ||| f ||| p based on the change of variable ż p p t q | f p t q| p p t q λ p p t q dm p t q “ ż p p h p t qq | f p h p t qq| p p h p t qq λ p p h p t qq dm p t q . Here h : r , s Ñ r , s is a mapping such that h p A q is measurable if and only if A is measurable and m p h p A qq “ m p A q for each measurable A . The Radon-Nikodymderivative satisfies d p m ˝ h q dm ” p is decreasing.Let us choose a varying-exponent ˜ p corresponding to the space p L r n ´ p p ´ p r n ´ , r n sq ‘ r n ´ L r n ´ p p ´ p r n ´ , r n ´ sqq ‘ r n ´ . . . ‘ r L r p p ´ r r , r sq . Recall Propositions 2.2 and 2.4. Then p||| p ´ p r n ´ ,r n s f ||| r n ´ ⊞ r n ´ ||| p ´ p r n ´ ,r n ´ s f ||| r n ´ q ⊞ r n ´ . . . ⊞ r ||| p ´ r r ,r s f ||| r ď p} p ´ p r n ´ ,r n s f } r n ´ ⊞ r n ´ } p ´ p r n ´ ,r n ´ s f } r n ´ q ⊞ r n ´ . . . ⊞ r } p ´ r r ,r s f } r “ } f } ˜ p p¨q ď p ` ae q} f } p p¨q ď p ` ae q||| f ||| p p¨q . The right-hand inequality is seen in a similar manner.
The second part of the statement is also seen in a similar way by first rearrangingthe unit interval such that the images of ∆ i become successive. The third claim follows inductively by considering first the case with 2 summandsonly: } ∆ f } ⊞ r } ∆ f } ě } ∆ Y ∆ f } for the right-hand side. The argument uses approximating simple semi-norms andthe following observations, where r ď min k p k : p . . . p a ⊞ p a q ⊞ p . . . ⊞ p k a k q ⊞ p k ` a k ` q ⊞ p k ` a k ` . . . ď p . . . p a ⊞ p a q ⊞ p . . . ⊞ p k a k q ⊞ r a k ` q ⊞ p k ` a k ` . . . ď p . . . p a ⊞ p a q ⊞ p . . . ⊞ p k a k q ⊞ r p a k ` ⊞ p k ` a k ` . . . q Indeed, the last inequality follows from Proposition 2.6 by putting x , , , , ,... “ a , x , , , , ,... “ a , x , , , , ,... “ a , x , , , , ,... “ a ,x , , , , ,... “ a , . . . , x , , ,..., , , , ,... “ a k ,x , , ,..., , , , , ,... “ a k ` , x , , ,..., , , , , , ,... “ a k ` , x , , ,..., , , , , , ,... “ a k ` , . . . and 0 for other entries. The Ð r operation can be inductively moved to be the firstoperation on the left, thus producing the required inequality. ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 13
If the intervals are successive, then we are done, otherwise we apply Theorem3.1 which involves the constant b p . The left-hand inequality is seen similarly. The last claim is verified by using the previous claim with successive intervals andthe invariance property of the Nakano norm together with the inequality ||| f ||| p p¨q ď } f } p p¨q ď b p ||| f ||| p p¨q from Proposition 2.4. (cid:3) Final Remarks
Reduction to the n “ case. Next we will give two results which yieldCorollary 2.1 in the beginning of the paper. We use the same notations.
Theorem 5.1.
Consider the setting of Corollary 2.1 with µ p R n q ą . Then thereis a bijection h : supp p µ q Ñ I , where I “ r , µ p R n qq , such that p T f qr h p x qs “ f r x s for µ -a.e. x P R n defines an isometric isomorphism T : L p p¨q p R n , Bor p R n q , m n , ω q Ñ L r p¨q p I, Bor p I q , m , ω q where r p h p x qq “ p p x q and ω p h p x qq “ ω p x q for µ -a.e. x P R n . Proof.
Let us consider completed versions of both the measure spaces, p R n , Σ , µ q and p I, Σ , m q .The heart of the argument is Maharam’s celebrated classification of measurealgebras and their representation theory. The topic of measure algebras is ratherinvolved and it cannot be covered here in a self-contained fashion, thus we refer to[Fre00, Vol III] for a detailed discussion. Measure algebras p A , ν q can be realized(via Stone spaces) as quotients of suitable measure spaces p Ω , F , ν q , A “ F { Null p ν q , where Null p ν q is the sub- σ -ring of F consisting of sets A such that ν p A q “ A becomes a σ -complete Boolean algebra in a natural way and a σ -additivemapping ν : A Ñ r , is canonically defined by ν pr A sq “ µ p A q . Note that Ωabove has hardly any role in the measure algebras. However, when it does play asignificant role, this tends to be subtle.Consider the probability spaces pr , s n , Bor pr , s n q , m n q and let A and B bethe corresponding measure algebras with n “ n “
1, respectively. Thesemeasure algebras are so-called Maharam homogenous and Maharam’s classificationstates that they are Boolean isomorphic to t , u ω normal form measure algebra where ω is the least infinite cardinal. This particularmeasure algebra is the classification of the above measure algebras and is generallyknown as the measure algebra . More generally, the number n P N does not affect theMaharam classification and this is crucial here. The measure algebra correspondsto the standard probability space; the notation is instructive in this regard in thesense that it can be modeled by an i.i.d. sequence of tosses of fair coins. Consideringdydadic decompositions of, say, the unit square, and using it to model a sequenceof independent coins gives some insight, why, modulo null sets, n is irrelevant. Taking a quotient with respect to the null sets loses lots of information andthis is the very reason why a general and elegent classification of measure algebrasis possible, that is, with infinite cardinals κ in place of ω in the homogenous case.Here we require a full isomorphism between measure spaces, rather than an inducedBoolean isomorphism between the corresponding measure algebras.Measure spaces p X, F , ν q and p Y, F , ν q are said to be isomorphic if thereis a bijection h : X Ñ Y such that f p A q P F if and only if A P F and then ν p h p A qq “ ν p A q holds.We will apply the following isomorphism result (see [Fre00, Vol. III, Thm. 344I]):Let p X, F , ν q and p Y, F , ν q be atomless, perfect, complete, strictly localizable,countably separated measure spaces of the same non-zero magnitude. Then theyare isomorphic.This of course implies that their measure algebras are isomorphic by a Booleanisomorphism induced by the measure space isomorphism. Let us comment on thesenotions:(1) A measure space p X, F , ν q is atomless if for each A P F , ν p A q ą
0, thereare disjoint
B, C P F with B Y C “ A and ν p B q ą ν p C q ą X , a Borel measure space p X, F , ν q is perfect iffor each A P F there are Borel sets A , A Ă X such that A Ă A Ă A and µ p A z A q “ p X, F , ν q is σ -finite if there are A n P F , n P N , with µ p A n q ă 8 and X “ Ť n A n . This property implies strict localizability.(4) A measure space p X, F , ν q is complete if A Ă B P F with ν p B q “ A P F .(5) A measure space p X, F , ν q is countably separated if there is a countableset A Ă F separating the points of X in the sense that for any distinct x, y P X there is an E P A containing one but not the other. (Of coursethis is a property of the structure p X, F q rather than of p X, F , ν q .)(6) Two measure spaces have the same magnitude (e.g.) if they have the sametotal measure or if they both have infinite total measure and are σ -finite.It is well-known that a completed Lebesgue measure space has these properties(regardless of the dimension n ). Then it is easy to see from the construction ofthe spaces p R n , Σ , µ q and p I, Σ , m q with µ p R n q “ m p I q that they satisfy theseconditions as well. Thus, according to the above measure space isomorphism result, p R n , Σ , µ q » ` I, Σ , m ˘ holds in the above sense and let h : R n Ñ I be the bijection involved in the isomor-phism. Now, if p p¨q , ω p¨q and f P L p p¨q p R n , Bor p R n q , m n , ω q are given, then r p t q “ p p h ´ p t qq , ω p t q “ ω p h ´ p t qq , p T f qr t s “ f r h ´ p t qs become measurable functions since h ´ maps the sets in Σ to sets in Σ. By usingthe fact that h is µ - m -measure-preserving, we obtain by a change of variable forevery λ ą ż R n ω p x q ˇˇˇˇ f p x q λ ˇˇˇˇ p p x q dµ p x q “ ż I ω p h ´ p t qq ˇˇˇˇ f p h ´ p t qq λ ˇˇˇˇ p p h ´ p t qq dm p t q“ ż I ω p t q ˇˇˇˇ p T f qr t s λ ˇˇˇˇ r p t q dm p t q ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 15 (possibly infinite) holds, where the Radon-Nikodym derivative d p µ ˝ h ´ q dm ” T is a linear isometry L p p¨q p R n , Bor p R n q , m n , ω q Ñ L r p¨q p I, Bor p I q , ω q . In fact this is an isomorphism,surjectivity is easiest to check by noting that T has an inverse given by the formula T ´ p g qr x s “ g r h p x qs . (cid:3) Moreover, with a similar reasoning as above, any Musielak-Orlicz space on p R n , Bor p R n q , m n q allows a dimension reduction to the real line. Next we willreduce the measure space of our function space to the unit interval. We will con-sider the more complicated case with µ p R n q “ 8 , if µ is finite then a similar factholds. Proposition 5.2.
Let L r p¨q pr , , Bor pr , , ω q be as above and h : r ,
8q Ñr , q , h p t q “ ´ ` t . Then p Sf qr h p t qs “ p ` t q r p t q f r t s for m - a.e. t P r , defines an isometric isomorphism T : L r p¨q pr , , Bor pr , , m, ω q Ñ L q p¨q pr , s , Bor pr , sq , m, w q where q p h p t qq “ r p t q and w p h p t qq “ ω p t q for m -a.e. t P r , . We mainly omit the proof, just note that after taking the q p h p t qq “ r p t q powerin the target space the weight becomes p ` t q , this is the inverse of the Radon-Nikodym derivative of the measure transformation (plainly | h p t q| ) which is requiredas a ’compensator’ in the change of variable.Next we comment on the weight appearing in the Nakano norm. On one hand,this weight corresponds to a special state in our ODE of interest, where the value ofthe solution ϕ p t q “ Continuity of the modulars.
Note that the map λ ÞÑ ż ˇˇˇˇ f p t q λ ˇˇˇˇ p dt, λ ą L p norm is in fact continuous with respectto λ . This appears more generally a reasonable requirement on the modulars, say,at least for bounded functions f . Proposition 5.3.
We investigate below measurable functions f : r , s Ñ R , p : r , s Ñr , , w : r , s Ñ p , and reals λ ą . Consider the following conditions: (1) There exists c ą and C ą such that w p t q ď Cc p p t q for a.e. t (2) For each f such that ş | f p t q| p p t q dt ď the mapping λ ÞÑ ż w p t q | f p t q| p p t q λ p p t q dt is continuous at λ “ . (3) There exists D ą such that w p t q ď Dp p t q for a.e. t. Then p q ùñ p q ùñ p q . Moreover, if p is essentially bounded and ş | f p t q| p p t q dt “ , then ddλ ż p p t q | f p t q| p p t q λ p p t q dt ˇˇˇˇ λ “ “ ´ . The latter observation above does not fully motivate by itself the use of theweight w , since we made the ad hoc assumption that ş | f p t q| p p t q dt ď
1. However,continuing our heuristic line of reasoning, the constant functions should be canonicalenough a test bed for assessing the behavior of weight functions. We note that thecontinuity of λ ÞÑ ż w p t q λ p p t q dt at λ “ ż p p t q w p t q dt ă 8 . Here the value λ “ ż w p t q λ p p t q dt ą ô λ ă , for instance, if C p p p t qq α ď w p t q ď C, α ą p tends to suitably slowly.5.3. Connections between the ODE-determined norm and the modular.
We may express the ODE-determined norm in a manner similar to the Luxemburgnorm. Admittedly, this involves choosing the weight in a very liberal way, somewhatin the same style as in Proposition 5.3. We may write } f } p p¨q “ ϕ f p q“ inf " λ ą λ ϕ f p q ď * “ inf " λ ą λ ż | f p t q| p p t q p p t q ϕ ´ p p t q f p t q dt ď * “ inf " λ ą ż p ϕ f p t q{ λ q ´ p p t q p p t q p| f p t q|{ λ q p p t q dt ď * “ inf " λ ą ż p p t qp ϕ f { λ p t qq p p t q´ p| f p t q|{ λ q p p t q dt ď * . Here ddλ ż p p t qp ϕ f { λ p t qq p p t q´ p| f p t q|{ λ q p p t q dt “ ´ λ ż | f p t q| p p t q p p t q ϕ ´ p p t q f p t q dt ECOMPOSITIONS OF NAKANO NORMS BY ODE TECHNIQUES 17 and for λ “ } f } p p¨q the above reads “ ´ } f } p p¨q . So, in this case the modular doesnot merely define the norm by means of a level set, but it actually behaves locallyaccording to the required norm.Approaching the connection between ODE-determined norms and Luxemburgnorms from another direction, suppose that for some weight function w p t q ą λ ą ż w p t q | f p t q| p p t q λ p p t q dt “ . This can be rewritten as ż w p t q | f p t q| p p t q λ ´ p p t q dt “ λ. Replacing f with 1 r ,t s f , 0 ď t ď
1, leads to separate respective solutions λ t with λ t “ ż t w p s q | f p s q| p p s q p λ t q ´ p p s q ds. Heuristically speaking, the scalars λ t may be considered some kind of averages ofmore localized constants or a function with a similar role. Let us further localizethese scalars by defining ’a varying lambda’ function λ p t q as the solution to λ p t q “ ż t w p s q | f p s q| p p s q p λ p s qq ´ p p s q ds, if such a weak solution exists, thus with a weak formulation(5.1) λ p t q “ w p t q | f p t q| p p t q p λ p t qq ´ p p t q a.e.Let us see what happens if p p t q “ p is a constant and we choose w p t q “ p . Then p p λ p t qq p ´ λ p t q “ | f p t q| p p t q a.e.This yields ż t p p λ p s qq p ´ λ p s q ds “ ż t | f p s q| p ds, p λ p t qq p “ ż t | f p s q| p dsλ p t q “ } r ,t s f } p . This is compatible with the philosophy of how the λ :s are defined and used, ›››› r ,t s fλ p t q ›››› “ . The conclusion is that our solutions ϕ f and the ’varying lambdas’ coincide whenthe weight in (5.1) is chosen to be w p t q “ p p t q . Acknowledgments.
This work has received financial support from the V¨ais¨al¨afoundation, the Finnish Cultural Foundation and the Academy of Finland Project
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