On some methods and applications of ordered Banach spaces

Basile, Achille (Ed.) Book — Published Version On some methods and applications of ordered Banach spaces Fuori Collana, No. 3 Provided in Cooperation with: FedOA Press (Federico II University Press) Suggested Citation: Basile, Achille (Ed.) (2014) : On some methods and applications of ordered Banach spaces, Fuori Collana, No. 3, ISBN 978-88-6887-003-4, FedOAPress, Napoli, https://doi.org/10.6093/978-88-6887-003-4 This Version is available at: https://hdl.handle.net/10419/200744 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Università degli Studi di Napoli Federico II Dipartimento di Scienze Economiche e Statistiche        On some methods and applications of ordered Banach spaces Edited by Achille Basile Università degli Studi di Napoli Federico II Fuori Collana 3 On some methods and applications of ordered Banach spaces Edited by Achille Basile Federico II Open Access University Press  On some methods and applications of ordered Banach spaces / edited by Achille Basile. – Napoli : FedOAPress, 2014. – (Fuori Collana ; 3). Accesso alla versione elettronica: http://www.fedoabooks.unina.it ISBN: 978-88-6887-003-4 DOI: 10.6093/978-88-6887-003-4 © 2014 FedOAPress - Federico II Open Access University Press Università degli Studi di Napoli Federico II Centro di Ateneo per le Biblioteche “Roberto Pettorino” Piazza Bellini 59-60 80138 Napoli, Italy http://www.fedoapress.unina.it/ Published in Italy Gli E-Book di FedOAPress sono pubblicati con licenza Creative Commons Attribution 4.0 International Contents Achille Basile Edito r ’ s Pre f ace 7 Ciro Tarantino Coalitional f airness with man y a g ents and commodities 9 Anna Canale - Ciro Tarantino Embedding and compactness results for multiplication operators in Sobolev s p aces 25   7 Editor’s Preface Achille Basile This book collects some recent contributions of Ciro Tarantino to mathematical research in infinite dimensional spaces. Spaces of infinite dimension have played an increasing role in the last four decades concerning their applications in models both for economics and for finance. The intertemporal allocation of resources, commodity differentiation, uncertainty, dynamics of the fundamental variables of financial markets, are some of the issues that can be properly captured by means of the mathematical techniques that are typical in such spaces. The contribution devoted to fairness properties of some kind of allocations that emerge in very general models of markets, the so-called mixed markets, mainly focuses on the absence of envy among coalitions of agents. Naturally the object of envy is something that can be subject to trade and not some moral entity (like beauty, say, or some other natural talent). Due to the exchange activities economic agents move from an initial endowment of goods to a new feasible one and may be possible that a group of agents envies the net trade of another coalition. When an allocation represents an efficient redistribution of resources and do not exhibit envy among coalitions, then we say that it is coalitionally fair, definitely a socially desirable property. Are competitive equilibria coalitionally fair? Is it the case that coalition proof allocations, namely the Core allocation, are also robust with respect to this further stability property? These are the themes addressed by Tarantino in Coalitional fairness with many agents and commodities. The answers that are provided improve the previous results that are known in the literature. Here markets consider the possibility of the interaction of oligopolies and price takers, under different level of information about the future states of the world and with reference to infinite dimensional random consumption.   Department of Economics and Statistics, University of Naples "Federico II", Via Cinthia, 80126 Napoli, Italy. Email: [email protected]. On some methods and applications of ordered Banach spaces 14 }.),,(;);,,();,{( Ttttt euBIT      TFE To a differential information economy E , we associate a family of complete information economies    )}({E by fixing for each state  the initial endowment ),(   e and the utility functions Ttt u  )),((  . 3. Coalitional fairness Let us take an economy E , an allocation x and a coalition S of agents. Definition 2. We say that the allocation x is blocked ex--post by the coalition S in the state o  if there exists an assignment y such that ;),(),( 00  dtedty SS   . a.a.for )),(,()),(,( 0000 Sttxutyu tt       Definition 3. An allocation x is an ex--post core allocation if there exists no coalition that blocks it in any state of nature. The set of all ex--post core allocations, the ex--post core of E , is denoted by )(EC (cf. Einy et al., Rational expectations). We move now our attention to a notion of coalitional fairness inspired to J. J. Gabszewicz, Coalitional Fairness. and already adapted by C. Donnini et al., Coalitional fairness. in differential information economies with reference to the interim case. Let us take an allocation x . Suppose that, corresponding to a nonnegligible coalition 1 S , we find a disjoint coalition 2 S , an assignment y and a state 0  such that 2) )),(,()),(,( 0000 txutyu tt      for a.a. 1 St  ; 3)  dtetxdtety SS ][][ ),(),(),(),( 0000 21     . It seems natural to say, under the above circumstances, that agents of 1 S in the state 0  discover ex-post that they envy the net trade2 of coalition 2 S . Indeed, they could redistribute it among themselves becoming better off.  2If x is an allocation and Sa coalition, the vector   dex S],[   is the net trade of coalition S at  . Ciro Tarantino, Coalitional fairness with many agents and commodities 15 We may well say, then, that under x there is an enviuos coalition. Consequently, the allocation x will be said ex-post c-fair (in the sense of Gabszewicz) if there are no envious coalitions. The formal definitions follow below. Definition 4. Given two disjoint coalitions 1 S and 2 S , an allocation x is ex--post coalitionally fair (c-fair) relative to 1 S and 2 S (or it does not show ex--post envy relative to 1 S and 2 S ), if there exist no state of nature  0  and no assignment y , such that for some i , 2,1  i : 1) 0)(  i S  ; 2) )),(,()),(,( 0000 txutyu tt      for a.a. i St  ; 3) ijdexdey ji SS         ][][ 0000 . Definition 5. Given a class S of coalitions, an allocation x is an ex--post c-fair allocation with respect to S if it doesn't show ex--post envy relative to 1 S and 2 S , whenever 1 S and 2 S belong to S . The set of all ex-post c-fair allocations with respect to S is denoted by S - )(ECfair . The set of all ex-post c-fair allocations with respect to T is shortly denoted by )(ECfair and we usually omit to say with respect to T . If no ex--post envy emerges relative to any possible pair of disjoint coalitions 21,SS with 11 S  S and 22 S  S , then we speak of c--fairness with respect to the classes 1 S and 2 S of coalitions. It follows directly from the definitions that Proposition 6. Any ex--post coalitionally fair allocation in T is in the ex-- post core: )(ECfair )(EC Proof: We suppose that x is in )(ECfair , but it does not belong to the expost core. Then, there exist a coalition A and an assignment y such that in a fixed state of nature A blocks x . Setting AS  1 and 2 S empty we get a contradiction. The inclusion above cannot be reversed in general. In the case of complete information economies, this is shown to be true in J. J. Gabszewicz, Coalitional Fairness Proposition 2. For atomless economies, however, ex-post On some methods and applications of ordered Banach spaces 16 core and ex-post c-fair allocations do coincide since one can properly extend both J. J. Gabszewicz, Coalitional Fairness, Theorem 1 and E. Einy et al, Rational expectations, Theorem 3.1. In general mixed market with infinitely many commodities, generalizing J. J. Gabszewicz, Coalitional Fairness, Theorem 2, one can show that Theorem 7. Let x be a strictly positive ex--post core allocation. Then x is ex--post coalitionally fair with respect to }|{ 00 TSS    TT and }|{ 11 TSS   TT . The result can be interpreted by saying that, in a market economy with many commodities and atoms, under a core allocation coalitions of negligible traders do not envy ex--post coalitions containing all large traders and conversely. First, observe that analogously to E. Einy et al, Rational expectations, Theorem 3.1, we can obtain the following characterization: Theorem 8. Let E be an economy. For any class of coalitions S , we get }.))((-),( and assignment an is |{)(-         ECSECS fairfair xxx Then we proceed with the proof of Theorem 7. Proof of Theorem 7: Since both, ex--post core and c--fair allocations are selections (E. Einy et al, Rational expectations., Theorem 3.1 and Theorem 8), it is enough to prove the statement for an economy with complete information or, in other words, to extend J. J. Gabszewicz, Coalitional Fairness, Theorem 2 to the case of a general commodity space B I for which int(  BI ) is non-empty. Let, then, 01 T  S , 12 T  S , y be such that 0)( 1S  , ))(())(( txutyu tt  , a.e. in 1 S and   dexdey SS )()( 21      . Consider the assignment )( 2 1yxz   . We have that it is strictly positive and ))(())(( txutzu tt  , a.e. in 1 S . Moreover .)( 2 1 )( 2 1 )( 2 1 )( 21111  dexdeydexdez SSSSS    Since 0 1   dz S and by continuity of preferences, there exist 0  and a subset C of 1 S such that ))(())(( txutzu tt   , a.e. in C . Define the func- Ciro Tarantino, Coalitional fairness with many agents and commodities 17 tion z to be the same as z on CS  1 and equal to z  on C . Then ))(())(( txutzu tt  , a.e. in 1 S and , 111 \  dzdzdzdz SCSCS   so .)( 2 1 )()( 2111  dexdezdez SSSS    Since 0)()(: 121 2 1            dezdexv SSS we can take 0  in such a way that the disk centered in v and radius  is fully contained into int )(  BI . Observe now that since x is a core allocation, the monotonicity gives that the feasibility actually guarantees   edxd TT    . Now, if the set 21 SS  exhausts T , then the integral  dex SS )( 21   is zero and 0)( 1   dez S gives the feasibility of z , namely we violate that x is in the core. We can therefore assume that the set )( 21 SST  , which does not contain atoms, is of positive  -measure. By the relative convexity of the range of the integral function over 0 T , we find a set .)( 2 1 )(with)( )(\ 21 21    dexdexSSTS SSTS Define s to be the same as z on the coalition 1 S and the same as x on S , then ))(())(( txutsu tt  , a.e. in SS  1 and ))(())(( txutsu tt  , a.e. in 1 S . Moreover    ))( 2 1 )(()( 2111  dexdezdes SSSSS .))( 2 1 )( 2 1 )( )(\ 21    dexdexdex TSSTS It follows that 0)( 1      des SS so we can take a vector 0w such that   dedsw SSSS      11 . Finally, modifying s by taking )(S w x   instead of x on S , we violate that x is in the core since the coalition SS  1 blocks x via the new assignment s . Indeed ))(())(( txutsu tt , a.e. in SS  1 and . 111     dewdxdzds SSSSSS         For the case where 11 T  S , 02 T  S a similar proof applies. On some methods and applications of ordered Banach spaces 18 4. Fairness of Rational Expectations equilibria We analyze now coalitional fairness of rational expectations equilibria of the economy E . In this context agents restrict their consumption choices to budget sets defined as follows: )},,()()(:{),( tepapBIapBt           for the agent t , in the state  and with respect to a prevailing system of prices p belonging to }F:)({ measurable  ispBIp 3. Moreover, since agents evaluate choices with a state by state comparison of the conditional expectation of their utility, taking into account both private information and the information revealed by prices, we assume that every agent t is given with a strictly positive probability over F . Consequently we denote, with reference to such probability, by )|( GfEt the conditional expectation of an F -measurable function f given the subalgebra G of F . We denote by )( p  the smallest  -algebra of F that makes the function p measurable ( )( p  represents the information contained in p ). F being generated by a partition  , the  --algebra )( p  is generated by the partition of  , coarser than  , in its turn generated by the function p . In this case, moreover, )|( GfEt is a G --measurable random variable having the same mean of f on the elements of the partition that generates G . Definition 9. A rational expectations equilibrium (RE equilibrium) is a pair ),( xp where p is a price system, and x is an allocation such that: (i) ),( tx  is ))(( t pF  -measurable for all Tt  ; (ii) ),(),( pBtx t    for each    and for each Tt  ; (iii) almost everywhere in T , if y is ))(( t pF  --measurable and satisfies ),()( pBy t    for each    , then ))(|))(,(())(|)),(,(( tttttt pyuEptxuE FF         pointwise on  .  3Here we use standard notation for the topological dual of B I , its positive cone and the duality mapping. Ciro Tarantino, Coalitional fairness with many agents and commodities 19 In the above definition, the private information t F of agent t is refined by means of the information generated by p . In case the price reveals all the information, i.e. F  )( p  , the RE equilibrium is said fully revealing. The set of allocations that are RE equilibria for a suitable price is denoted by )(ERE 4. Given the inclusions above, we see that: .)()( ECE fair RE  therefore, a RE equilibrium needs not be coalitionally fair. This can only happen when agents do not know their utility functions. In fact, we shall show next that assuming the functions ),( tu  to be t F --measurable, then any RE equilibrium allocation is ex--post coalitionally fair. Theorem 10. Assume that in the economy E the functions ),( xut  are t F - -measurable. Then .)()( ECE fair RE  Proof: Let ))((  EW be the Walrasian allocations of )(  E , the using A. De Simone et al, Some, Theorem 11 and Theorem 8, we have to prove that       }))((),( and assignment an is |{    EWxxx }))((),( and assignment an is |{         ECfair xxx So, it is sufficient to show that in a complete information economy any Walrasian allocation is also coalitionally fair, i.e. .fixedany for ))(())((       ECE fair W Let us consider a fixed state of nature 0  and the corresponding deterministic information economy )( 0  E. Let x be a competitive allocation in ))(( 0  EW, and assume by way of contradiction that ))(( 0  ECfair x  . A contradiction easily follows like in J. J. Gabszewicz, Coalitional Fairness. In the case of atomless economies, under suitable hypotheses, RE equilibrium allocations are exactly ex-post c-fair allocations.  4In E. Einy et al, Rational expectations, Example 4.2, an economy is constructed with a RE equilibrium, which can be blocked by an ex--post core allocation. On some methods and applications of ordered Banach spaces 20 Theorem 11. Let us assume that the set 1 T of large traders is empty. Moreover, for any agent and any state, suppose that ),( xut  are t F --measurable. Then .)()( ECE fair RE  Proof: Only the inclusion )()( ECE fair RE  has to be justified. But if x is an ex--post c-fair allocation, and therefore in the ex--post core, then E. Einy et al, Rational expectations., Theorem 3.1, applies. So fix a state  . Hence, ))((),(   ECx   and using the core equivalence Theorem in A. Rustichini et al, Edgeworth’s conjecture, we have ))((),(   EWx   and by A. De Simone et al, Some new characterization, Theorem 11, )(EREx  . In the case of mixed markets one has to observe that ex--post c-fair allocation need not be RE equilibrium allocations in general. Such a result is too strong for oligopolistic models ( 1 T non-empty), even in the case of complete information economies (see J. J. Gabszewicz, Coalitional Fairness, Proposition 2). If 1 T consists of a single atom, it is possible to prove that the RE equilibrium allocations are exactly ex-post c-fair allocations with respect to the class of coalitions which do not contain the atom. Theorem 12. Let E be a mixed market with only one atom, 1||.. 1Tei . Moreover, for any agent and any state, suppose that ),( xut  are t F --measurable. Then )(-)( 0ECTE fair RE  . Proof: By Theorem 8 we can write .}))((-),( and assignment an is |{)(- 00        ECTECT fairfair xxx Let us fix a state  . Using the weak form of Lyapunov's convexity theorem, we can reproduce the proof of B. Shitovitz, Coalitional fair, Corollary A*, and obtain that .}))((),( and assignment an is |{)(- 0        EECT Wxxx fair The statement follows by E. Einy et al, Rational expectations, Theorem 4.3. Ciro Tarantino, Coalitional fairness with many agents and commodities 21 So, in the case of just one large trader, under a RE equilibrium allocation coalitions made by small traders do not envy each other ex--post. Let us discuss now the general case of mixed markets. Moving from the observation that core allocations are not necessarily c-fair, we have proved in Theorem 7 that a strictly positive allocation x which is in the ex--post core is ex-post cfair with respect to 0 T and 1 T . Here we wish to point out that if the (ex-- post) core allocation x is also a restricted rational expectation equilibrium, then it comes out to be c-fair with respect to the class 10 TT  of coalitions. In the following, we limit consideration to the case of finitely many commodities. By a restricted equilibrium we mean what follows. Definition 13. A restricted rational expectations equilibrium (RRE equilibrium) is a pair ),( xp where p is a price system, and x is an allocation such that: (i) ),( tx  is ))(( t pF  --measurable for all Tt  ; (ii) ),()(),()( teptxp        for each    and for each 0 Tt ; (iii) almost everywhere in T , if y is ))(( t pF  --measurable and satisfies )},,()()(:{:),()( txpapBIapBy t     for each    , then ))(|))(,(())(|)),(,(( tttttt pyuEptxuE FF         pointwise on  . The usual interpretation goes as follows: under a RRE equilibrium allocation x , there exists a price system p such that: 1) Each trader maximizes (ex--post) over his/her efficiency set; 2) small traders are in a RE equilibrium with respect to p . As for restricted competitive equilibria of complete information economies, in general, it is not true that the restriction of the allocation x on 0 T is a RE equilibrium, since the feasibility condition in the atomless sub-economy may be violated. However, RRE equilibria which are also in the ex-post core satisfy additional fairness properties. On some methods and applications of ordered Banach spaces 22 Proposition 14. Let B I be finite dimensional. Every strictly positive, ex-- post core allocation x which is also a RRE equilibrium is ex--post coalitionally fair with respect to 10 TT  . Proof: Let p be the equilibrium price. Since )(ECx  , then, for any   ˆ , we have that )) ˆ (( ˆ   ECx  and therefore  ˆ x is a restricted competitive equilibrium in the economy ) ˆ (  E with respect to the price ) ˆ (  p . To check latter statement it is enough to go like in the proof of E. Einy et al, Rational expectations…, Theorem 4.3, by taking ), ˆ (pBa t    and replacing ), ˆ (),( tete    by ), ˆ (),( txtx    . Of course the equalities ), ˆ () ˆ (), ˆ () ˆ (teptxp        over 0 T are due to the definition of RRE equilibrium. Now,  ˆ x is a restricted competitive equilibrium in ) ˆ (  E and also belongs to the core. Since we have proved the c--fairness of x with respect to 0 T and 1 T in Theorem 7, then the concluding argument goes like in the proof of J. J. Gabszewicz, Coalitional Fairness, Theorem 3. The previous result implies that under conditions ensuring that ex-post core allocations are RRE equilibria, the ex--post core consists of allocations coalitionally fair with respect to the whole class 10 TT  . Such conditions are well known in the case of finitely many commodities (see for example J. J. Gabszewicz, Coalitional Fairness, B. Shitovitz, Coalitional fair, M. G. Graziano et al, A note on the private...). We leave the identification of analogous conditions in the presence of infinitely many commodities as subject of future investigation. A further line of open research is represented by the case in which the commodity space B I does not exhibit interior points in its positive cone. Such case comprises interesting models for applications like investigation of infinite-horizon economies, asset pricing models, differentiated commodity models. To cover this general case, the set of assumptions imposed in section 2 will require a modification inspired by classical properness assumptions on preferences. In particular, we expect that the commodity space enjoys a Riesz space structure. This requirement together with the lattice structure of the price space seems to be indispensable to carry over lattice theoretical arguments connected with properness conditions. Ciro Tarantino, Coalitional fairness with many agents and commodities 23 References C. D. Aliprantis - D. J. Brown, Equilibria in markets with a Riesz space of commodities, in «Journal of Mathematical Economics», 11, 1988, pp. 189207. C. D. Aliprantis - D. J. Brown - O. Burkinshaw, Existence and optimality of competitive equilibria, Springer-Verlag, New York, 1990. R. J. Aumann, Existence of competitive equilibria in markets with a continuum of traders, in «Econometrica», 34, 1966, pp. 39-50. A. Basile - A. De Simone - M. G. Graziano, On the Aubin--like characterization of competitive equilibria in infinite dimensional economies, in «Rivista di Matematica per le Scienze Economiche e Sociali», 19, 1996, pp. 187213. A. De Simone - C. Tarantino, Some new characterization of rational expectation equilibria in economies with asymmetric information, in «Decisions in Economics and Finance», 33, 2010, pp. 7-21. G. Debreu, Valuation Equilibrium and Pareto Optimum, in Mathematical Economics: Twenty Papers of Gerard Debreu, Cambridge University Press, New York, 1984, pp. 98-104. C. Donnini - M.G. Graziano - M. Pesce, Coalitional fairness in interim differential information economies, in «Journal of Economics», 111, 2014, pp. 55-68. E. Einy - D. Moreno - B. Shitovitz, Rational expectations equilibria and the ex-post core of an economy with asymmetric information, in «Journal of Mathematical Economics», 34, 2000, pp. 527-535. D. Foley, Resource allocation and the public sector, in «Yale Econ. Essays», 7, 1967, pp. 45-98. J. J. Gabszewicz, Coalitional Fairness of Allocations in Pure Exchange Economies, in «Econometrica», 43, 1975, pp. 661-668. T. Gajdos - J.M. Tallon, Fairness under Uncertainty, in «Economics Bulletin», 4, 2002, pp. 1-7. M. G. Graziano - M. Pesce, A note on the private core and coalitional fairness under asymmetric information, in «Mediterranean Journal of Mathematics», 7, 2010, pp. 573-601. A. Rustichini - N.C. Yannelis, Edgeworth's conjecture in economies with a continuum of agents and commodities, in «J. Math. Econ.», 20, 1991, pp. 307-326. On some methods and applications of ordered Banach spaces 30 3. Inclusion properties The following Theorem states inclusion properties of the spaces )( p M . Theorem 3.1 The following inclusions hold: a) )()(  p LM [,1[     p . b)  qp pq 1),()( MM . c) ),( ~ )(  pq MM      qp1 . Proof. a) If )(  Lg we get    )( /1/ ],0] )( ))(( / ],0] )( |||||)(|sup|||| ||||sup||||              L ppn d x L xL pn d x gcxg gg pp       M (3.1) where, if n  denotes the volume of the unit ball )0( 1 B , we get p n c1   . b) By Hölder inequality it follows ,||||sup )|)(|||||(sup|||| ))(( ],0] 1 ))(( ],0] )( 11 xL d x xL d x q q n qp q p n p gc xgg                    M (3.2) with qp n c11 1    . c) We observe that, if )( q gM , from )b we have )( p gM . Furthermore Canale - Tarantino, Embedding and compactness results for multiplication operators in Sobolev spaces 31  ,|)(|sup|||| |||||)(|sup ||||sup|||| 11 11 ],0] )( ))(( / ],0] ))(( / ],0] )( qp q q qp pp Exg gEx gg n d x M ExL p pn d x ExL pn d x E                            M (3.3) and we deduce that the function p g  , defined by (sigma), is continuous in zero. From Lemma 2.2 it follows that )( p gM . 4. Embedding result Let us consider the function          ).(if0 )(if1 ),(: xy xy yx    (4.1) and, for any   x , we set )}.(:{)( yxyxE       (4.2) Lemma 4.1 For any   x , )(xE is a measurable set and there exist  Rcc 21, such that .|)(| 21  xcxEc nn  (4.3) Proof. Clearly the function  is a measurable function. Then, for any fixed y , the function ),(: yxx y   is measurable. Since y  is the characteristic function of )(yE , we have that )(yE is measurable. Now we prove that (4.3) holds. The inequality on the right is easily proved. We will prove the inequality on the left. On some methods and applications of ordered Banach spaces 32 Let us consider ),(   xC such that ),(   xC . We get ).(),( xExC    In fact let ),(   xCy . Then there exists a cone ),,( hC     such that Cyx , . So ).(),( xEyyBx       Thus the inequality (4.3) is stated. Now we state a Lemma which we will use in the proof of the embedding result. Lemma 4.2 If ) 1 h holds, then we have )( 1  Lv if and only if the map ))(( 1 |||| xL nvx      belongs to )( 1L . Therefore there exist  Rcc 21, such that ).(|||||||||||| 1 )( 2 ))(()( 1111     Lvvcdxvvc LxL n L   (4.4) Proof. The result is a consequence of the relation dxdyyv dxdyyxyvdxv n yE n xL n                )( ))(( |)(| ),(|)(||||| 1 (4.5) and of the Lemma 4.1. Let q p r ,, be real number with the condition .1if,,1,) 2 p r n r n q r n qqpNrh Canale - Tarantino, Embedding and compactness results for multiplication operators in Sobolev spaces 33 Let )( , pr Wu . For any   x we set .))(()(,: *xx xy xy p xx      .))(()(:)( xzxuxzuu x    We note that )).(( ,xWu pr    We also note that, in consequence of ) 1 h )(x    has the cone property, with the characteristic cone having height and opening independent of x . On the other hand, if pq/   , from ) 2 h we get . 11 ,1if1,1 n r pp p r n      From well-known embedding theorems of Sobolev spaces (see, e.g., R.A. Adams, Sobolev ……), we deduce that ))(( 1xLu p      and the following bound holds ,|||||||| ))(( 0 ))(( , 1xW xL prp ucu           (4.6) where ),,,( 00 nrqpcc  is a constant independent of x and  u . From (4.6) easily it follows that .|||||||| ))(( || || 0 ))(( 1 )1( xL r xL n p p n p pucu                   (4.7) Theorem 4.3 If ) 1 h and ) 2 h hold, then for any )( q gM and for any )( , pr Wu we get )( p Lgu and ,|||||||||||| )()()( ,  prqp WL ugcgu M (4.8) On some methods and applications of ordered Banach spaces 34 where the constant ),,,( nrqpcc  is independent of g and u . Proof. Let )( , pr Wu and )( q gM . By (4.4) and by Hölder inequality it follows that  .|||||||| |||||||||||| ))(( )( 1 ))(( ))(( 1 )( 1 1 1 1 dxugc dxugcdxdygucdxgu p xL n p p xL p xL np x np pq pp                              M (4.9) On the other hand from (4.7) and Lemma (4.2) we obtain .|||| |||||||| )( || || 1 ))(( || || 0 ))(( 1 )1( p L p r p xL np r p xL n p pp uc dxucdxu                             (4.10) From (4.9) and (4.10) the inequality (4.8) follows. The following theorem is a consequence of the embedding result stated in the Theorem 4.3 (see result of C. Fefferman, The uncertainty, and also F. Chiarenza - M. Frasca, A remark on, for a simplified proof). Theorem 4.4 If ) 1 h and ) 2 h hold then for any )( q gM , we get ).(|||||||||||| ,1 0 )()()(   p LL Wuugcgu pqp M where the constant ),,( nqpcc  is independent of g and u . Proof. Taking in mind the Poincaré inequality, the proof is a direct consequence of the Theorem 4.3 when 1  r . The p L estimates below are obtained when the function g belongs to the closure of )(  L in )( q M or to the closure of )( 0  C in )( q M . In the Canale - Tarantino, Embedding and compactness results for multiplication operators in Sobolev spaces 35 applications to the elliptic differential equations we deal with multiplication operators belonging to these spaces. Lemma 4.5 If ) 1 h , ) 2 h hold and )( q gM , then for any  R  there exists  Rc )(  such that ).(||||)(|||||||| , )()()( ,  pr LWL Wuucugu pprp  Proof. Let )(  L   such that ,/|||| )( cg q      M (4.11) where the constant c is the constant in the bound (4.8). Then by Theorem 4.3 )()( )()()()( )()()( ||||)(|||| ||||||||||||||)(|| ||||||)(|||||| , ,           ppr pprq ppp LW LLW LLL ucu uuugc uuggu       M (4.12) with )( ||||)(   L c    . Lemma 4.6 If ) 1 h , ) 2 h hold and )( 0 q gM , then for any  R  there exist  Rc )(  and an open set     with cone property such that ).(||||)(|||||||| , )()()( ,  pr LWL Wuucugu pprp   (4.13) Proof. Let )( 0  C   be such that (4.11) holds. Reasoning as in the proof of the Lemma 4.5 we get ,||||)(|||||||| sup, )()( ,     pp WL ucugu prp   (4.14) with )(  c as in the Lemma 4.5. On some methods and applications of ordered Banach spaces 36 Then let us fix [2/,0]    and [2/)supp,(dist,0]      h . If we denote by   the open set of n R union of the cones ),,(   hC    such that  gC supp is not empty, then (4.13) follows from (4.14). 5. Compactness result Now we state the compactness result. Theorem 5.1 If ) 1 h , ) 2 h hold and )( 0 q gM , then the operator )()( , ppr LguWu is compact. Proof. We remark that for any open set     the operator )(|)( ,,   prpr WuWu is linear and bounded. On the other hand, if   verifies the cone property too, by RellichKondrachov's theorem the operator )()'( ,  ppr LuWu is compact. Then also the operator )(|)( ,  ppr LuWu is compact. Therefore if )(}{ , pr nWu is a bounded sequence there exists a subsequence }{ k n u converging to u in )( p L . By Lemma 4.6 we get )(||||)(||||||)(|| , )()()( ,   pr L n W n L nWuuucuuuug p k pr k p k  from which we can deduce the result. Canale - Tarantino, Embedding and compactness results for multiplication operators in Sobolev spaces 37 References R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. H. Brezis, Analisi funzionale, Liguori Ed., Napoli, 1986. A. Canale, A priori bounds in weighted spaces, in «J. Math. Anal. Appl.», Vol. 287, (4), 2003, pp. 101-117. A. Canale, On some results in weighted spaces under Chicco type conditions, in «Int. J. Pure Appl. Math. », Vol. 31, (2), 2006, pp. 185-202. A. Canale On some results in weighted spaces under Cordes type conditions, in «J. Interdiscip. Math. », Vol. 10, (2), 2007, pp. 245-261. A. Canale, Bounds in spaces of Morrey under Cordes type conditions, in «J. Appl. Funct. Anal.», Vol. 3, (1), 2008, pp. 11-32. A. Canale, Bounds in spaces of Morrey under Chicco type conditions, in «Math. Ineq. and Appl. », Vol. 12, (2), 2009, pp. 265-278. F. Chiarenza - M. Franciosi, A generalization of a theorem by C.Miranda, in «Ann. Mat. Pura Appl. », Vol. 161, 1992, pp. 285-297. F. Chiarenza - M. Frasca, A remark on a paper by C.Fefferman, in «Proc. Amer. Math. Soc.», Vol. 108, (2), 1990, pp. 407-409. C. Fefferman, The uncertainty principle, in «Bull. Am. Math. Soc. », (9), 1983, ,pp. 129-206. A. Kufner, Weighted Sobolev spaces, in «Teubner Texte zur Math. », Band 31, 1980. J. Necas, Les methodes directes en theorie des equations elliptiques, Masson et C.ie Editeurs, Paris, 1967.  Università degli Studi di Napoli Federico II Dipartimento di Scienze Economiche e Statistiche      Spaces of infinite dimension have played an increasing role in the last four decades concerning their applications in models both for economics and for finance. The intertemporal allocation of resources, commodity differentiation, uncertainty, dynamics of the fundamental variables of financial markets, are some of the issues that can be properly captured by means of the mathematical techniques that are typical in such spaces. This collection contains some recent contributions in this area.       Achille Basile is Professor of Mathematical Methods for Economics at Department of Economics and Statistics of University of Naples Federico II ([email protected]). He served as Vice President of the Division of Human and Social Sciences and has been Dean of the Faculty of Economic s. Currently he coordinates the Presidio della qualità of the University Federico II and is a member of the Nucleo di valutazione at the University of Bologna. He is also Chairman of the Italian Association for Mathematics Applied to Economics and Social Sciences (AMASES). He is the author of scientific publications on themes of general economic equilibrium, game theory, functional analysis and measure theory.            ISBN: 978-88-6887-003-4 DOI: 10.6093/ 978-88-6887-003-4