A note on real interpolation of L-P-spaces of vector measures on delta-rings

J. Ma h. Anal. Appl. 419 (2014) 995–1003 Con en s lis s a ailable a ScienceDi ec Jou nal o Ma hema ical Analysis and Applica ions www.else ie .com/loca e/jmaa A no e on eal in e pola ion o Lp-spaces o ec o measu es on δ- ings ✩ Rica do del Campo a, An onio Fe nández b,∗, Fe nando Mayo al b, F ancisco Na anjo b aDp o. Ma emá ica Aplicada I, Uni e sidad de Se illa, EUITA, C a. de U e a Km. 1, 41013 Se illa, Spain bDp o. Ma emá ica Aplicada II, Escuela Técnica Supe io de Ingenie os, Camino de los Descub imien os, s/n, 41092 Se illa, Spain a icle in o abs ac A icle his o y: Recei ed 5 No embe 2013 A ailable online 16 May 2014 Submi ed by B. Bongio no Keywo ds: Real in e pola ion me hods Lions–Pee e K- unc ional In eg able unc ion Vec o measu e δ-Ring Locally s ongly addi i e measu e We desc ibe he eal in e pola ion spaces ob ained when we apply he eal K-me hod o Lions–Pee e o Banach la ices o p-in eg able and weakly p-in eg able unc ions wi h espec o a Banach-space- alued measu e defined on a δ- ing. In gene al, he ob ained esul s a e qui e diffe en om hose in he case o ec o measu es on σ-algeb as desc ibed in [9]. Howe e , we find a wide class o ec o measu es on δ- ings o which he esul s on σ-algeb as hold ue. © 2014 Else ie Inc. All igh s ese ed. 1. In oduc ion A basic p oblem in in e pola ion heo y is o desc ibe he spaces ob ained by applying an in e pola- ion me hod o conc e e compa ible couples o spaces. Fo a Banach-space- alued measu e mdefined on aσ-algeb a, we ob ained in [11] he Calde ón in e pola ion spaces [X0,X 1][θ]and [X0,X 1][θ],andin[9] he eal in e pola ion spaces (X0,X 1)θ,q o he couples (X0,X 1), whe e X0and X1a e he Banach la ices Lp(m)o Lp w(m) o equi alence classes o scala p-in eg able o , espec i ely, weakly p-in eg able unc ions wi h espec o he measu e m. La e we in es iga ed in [5] he Calde ón in e pola ion me hods o he same spaces, bu o measu es defined on δ- ings. We showed in [5] ha he in e pola ion esul s o ec o measu es on δ- ings can be e y diffe en om hose on he con ex o σ-algeb as. Howe e , we iden ified ✩This esea ch has been pa ially suppo ed by La Jun a de Andalucía. The au ho s acknowledge he suppo o he Minis e io de Economía y Compe i i idad o Spain and FEDER, unde he p ojec MTM2012-36740-C02-01. *Co esponding au ho . E-mail add esses: camp[email p o ec ed] (R. del Campo), [email p o ec ed] (A. Fe nández), may[email p o ec ed] (F. Mayo al), [email p o ec ed] (F. Na anjo). h p://dx.doi.o g/10.1016/j.jmaa.2014.05.039 0022-247X/© 2014 Else ie Inc. All igh s ese ed. 996 R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 a ce ain ype o ec o measu es on δ- ings (called locally s ongly addi i e measu es) which keep com- ple ely he same beha io ( o all he diffe en combina ions o couples) as measu es defined on σ-algeb as. In he p esen no e we comple e he pic u e wi h he s udy o eal in e pola ion me hods o Banach la ices o p-in eg able and weakly p-in eg able unc ions wi h espec o a Banach-space- alued measu e defined on a δ- ing. As in he case o complex me hods we can say ha ce ain in e pola ion equali ies o ec o measu es on σ-algeb as desc ibed in [9] emain ue o ec o measu es on δ- ings, bu some o he s cease o be ue o ec o measu es on δ- ings. Cu iously, o he same ype o measu es (locally s ongly addi- i e measu es) eal in e pola ion equali ies in he se ing o measu es defined on σ-algeb a emain ue o measu es on δ- ings. Howe e , he easons why his happens a e e y diffe en om hose on he con ex o complex in e pola ion me hods. 2. P elimina ies In his sec ion we es ablish he p elimina ies necessa ies abou in eg a ion o scala unc ions wi h espec o ec o measu es on δ- ings, in o de o make he pape mo e sel -con ained and eadable. The basic e e ences abou in eg a ion o us will be [7,12–14]. Th oughou his pape we will conside a ec o measu e ν:R→Xdefined on a δ- ing Ro subse s o some nonemp y se Ωwi h alues in a eal Banach space X, wi h dual X.Wedeno ebyRloc he σ-algeb a o subse s A⊆Ωsuch ha A∩B∈R o each B∈R. Measu abili y o unc ions :Ω→Rwill be conside ed wi h espec o he measu able space (Ω,Rloc). The semi a ia ion o νis he se unc ion ν:Rloc →[0,∞] defined by ν(A):=sup{|ν, x|(A):xX≤1}, whe e |ν, x| is he a ia ion o he scala measu e ν, x:A∈R→ν, x(A):=ν(A),x ∈R. Ase N∈Rloc is called ν-null i ν(N) = 0. A p ope y holds ν-almos e e ywhe e (ν-a.e.) i i holds excep on a ν-null se . A measu able unc ion :Ω→Ris called weakly in eg able (wi h espec o ν)i ∈L1(ν, x) o all x∈X. A weakly in eg able unc ion is said o be in eg able (wi h espec o ν) i , o each A∈Rloc he e exis s an elemen (necessa ily unique) A dν ∈X, sa is ying  A dν,x= A dν, x,x ∈X. I 1 ≤p<∞, a measu able unc ion :Ω→Ris called weakly p-in eg able (wi h espec o ν)i | |pis weakly in eg able and p-in eg able (wi h espec o ν)i | |pis in eg able. The space Lp w(ν)o all(ν-a.e. equi alence classes o ) weakly p-in eg able unc ions becomes a Banach la ice wi h he Fa ou p ope y when endowed wi h he usual ν-a.e. poin wise o de and he no m  Lp w(ν):= sup Ω | |pdν, x 1 p : x X≤1. Mo eo e , he space Lp(ν)o all(ν-a.e. equi alence classes o ) p-in eg able unc ions is a closed o de con inuous ideal o Lp w(ν). In ac , i is he closu e o S(R), he space o simple unc ions suppo ed on R. The Banach la ices Lp(ν)andLp w(ν) o equi alence classes o scala p-in eg able and weakly p-in eg able unc ions we e ini ially s udied in [10] o ec o measu es νon a σ-algeb a and i s basic p ope ies can be ex ended and emain ue o ec o measu es on δ- ings (see [4]). Also we can find in [15, Chap e 3] a e y good ma e ial abou spaces o in eg able unc ions wi h espec o a ec o measu e on a σ-algeb a. Finally, le us conside wo mo e spaces s ongly ela ed wi h he spaces o p-in eg able unc ions wi h espec R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 997 o a ec o measu e. Deno e by L∞(ν) he space o classes o essen ially bounded measu able unc ions :Ω→Rwi h he essen ial sup emum no m. Conside also he ec o space L0(ν) o all classes o measu able unc ions :Ω→R. I he ec o measu e νis defined on a σ-algeb a i is well-known (see [10, Co olla y 3.2]) ha he ollowing inclusions hold o all p>1 L∞(ν)⊆Lp(ν)⊆Lp w(ν)⊆L1(ν)⊆L1 w(ν)⊆L0(ν),(1) and all o hem a e con inuous inclusions, whe e he opology o con e gence in measu e is conside ed on L0(ν). When he ec o measu e νis defined on a δ- ing ins ead o a σ-algeb a, he inclusions (1) a e in gene al alse, bu we can sa e some hing (see, o example, P oposi ion 2.2 and Rema k 3.3). In wha ollows we will always conside ec o measu es ν:R→Xwhich a e σ-fini e, ha is, he e exis a pai wise disjoin sequence (Ωk)kin R,andaν-null se N∈Rloc, such ha Ω=( k≥1Ωk)∪N.The simples example o a σ-fini e ec o measu e on a δ- ing is gi en by he Lebesgue measu e λdefined on he δ- ing R:= {A∈M:λ(A)<∞}, whe e Mis he σ-algeb a o all Lebesgue measu able subse s o he eal line R. I we conside he ec o measu e ν:A∈R→ν(A)=λ(A)∈R, hen Lp w(ν)=Lp(ν)=Lp(R) o all p≥1. In he con ex o in e pola ion i is well-known ha we need a opological ec o space as an en i onmen space in o de o conside couples o Banach spaces. In ou case i is he linea space L0(ν), endowed wi h he opology o con e gence in measu e on each subse Ωk. This opology is gene a ed by he F-no m · L0(ν) ha we shall now desc ibe. Fo each k=1,2,... conside he σ-algeb a Σk:= {A∈R:A⊆Ωk}o subse s o Ωkand he ec o measu e νk:A∈Σk→νk(A)=ν(A)∈X, ha is, he es ic ion o ν o Σk.Now define  L0(ν):= ∞  k=1 1 2k(1 + ν(Ωk))    | | 1+| |χΩk   L1 w(νk) , ∈L0(ν). Fo de ails see [5, Lemmas 3.2, 3.3 and 3.4]. In pa icula , le us men ion ha each pai o spaces Lp w(ν) o Lp(ν) o msacompa ible couple o Banach spaces, ha is, hey a e imbedded con inuously in he same opological ec o space L0(ν). Gi en ∈L0(ν), we shall conside i s dis ibu ion unc ion (wi h espec o he ec o measu e ν) defined by ν :s∈[0,∞)→ν (s):=νw∈Ω: (w)>s ∈[0,∞], whe e νis he semi a ia ion o he measu e ν. This dis ibu ion unc ion has simila p ope ies as in he scala case (see [9]). Fo ins ance, ν is non-inc easing and igh -con inuous. The dec easing ea angemen o (wi h espec o he measu e ν)isgi enby ∗: ∈(0,∞)→ ∗( ):=in s>0:ν (s)≤ ∈[0,∞]. Some p ope ies o ∗can be ound in [9] when he measu e is defined on a σ-algeb a. Ne e heless, i is no difficul o see (e en o measu es on δ- ings) ha ∗is a non-inc easing, igh -con inuous unc ion. Mo eo e he ollowing wo equali ies hold ∞  0 ν ( )d = ∞  0 ∗( )d and sup >0 ν ( )=sup >0 ∗( ). Fo 1 ≤p, q ≤∞ he Lo en z space Lp,q(ν) wi h espec o he ec o measu e νconsis s o all unc ions ∈L0(ν) o which he quan i y 998 R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003  Lp,q (ν):= (∞ 0(s1 p ∗(s))qds s)1 q(1 ≤q<∞) sups>0s1 p ∗(s)(q=∞) is fini e. The unc ional → Lp,q (ν)is no always a no m, e en when p, q ≥1, because he iangle inequali y ails. Ne e heless i is no difficul o p o e ha  +gLp,q (ν)≤C( Lp,q (ν)+gLp,q (ν)), whe e C≥1 is a cons an depending on pand q. The e o e · Lp,q (ν)is only a quasi-no m. We also no e ha Lp,q(ν) is a quasi-Banach la ice wi h he Fa ou p ope y. Fo he special case p=q, we deno e he space Lp,p(ν)simplybyLp(ν). As i has been poin ed ou in [9], in gene al, he spaces Lp(ν) and Lp(ν) do no coincide. Fo p>1and1≤q≤∞ he Lo en z spaces Lp,q(ν) a e in e media e spaces o he couple (L1(ν),L ∞(ν)), ha is, L1(ν)∩L∞(ν)⊆Lp,q(ν)⊆L1(ν)+L∞(ν). Mo eo e , i he measu e νis defined on a σ-algeb a, i holds he ollowing inclusions, o all 1 ≤p<∞(see [9, P oposi ion 7]) L∞(ν)⊆Lp,1ν⊆Lpν⊆Lp(ν)⊆Lp w(ν)⊆Lp,∞ν,(2) and all hese inclusions a e con inuous. Howe e , i he ec o measu e νis defined on a δ- ing ins ead o a σ-algeb a, he inclusion Lp(ν)⊆Lp(ν) is in gene al alse as Example 2.1 below poin s ou . The inclusion L∞(ν)⊆Lp,1(ν) is alse e en o a non-fini e posi i e scala measu e. On he con a y he o he s inclusions o he chain (2) emain ue as we shall see wi h he ollowing P oposi ion 2.2 (see also Rema k 3.3). Example 2.1. Le Rbe he δ- ing o fini e subse s o na u al numbe s N, and conside he σ-fini e ec o measu e ν:A∈R→ν(A):=χA∈c0, whe e c0is he space o null sequences. Fo e e y 1 ≤p<∞, i is easy o check ha Lp w(ν)=∞, he space o bounded sequences, and Lp(ν)=c0. In wha ollows i will be in e es ing o no e ha ν(A) = 1, o e e y nonemp y A⊆N,andν(∅) = 0. This means, in pa icula , ha ν =χ[0,∞)i is an unbounded sequence, bu ν =χ[0, ∞)i ∈∞.Consequen ly, we ha e o an unbounded sequence ha ∗( )=∞i ∈(0,1) and ∗( )=0i ≥1. On he o he hand, ∗= ∞χ(0,1) i ∈∞.ThusL1(ν)=∞=L1 w(ν), and L1(ν)L1(ν). P oposi ion 2.2. The ollowing con inuous inclusions hold L1ν⊆L1 w(ν)⊆L1,∞ν⊆L0(ν).(3) P oo . Fi s we check he inclusion L1(ν)⊆L1 w(ν). Take ∈L1(ν), and choose any x∈X,wi h x≤1. Fo he posi i e σ-addi i e measu e |ν, x| we ha e (see [2, P oposi ion II.1.8] o he fi s equali y)  Ω | |dν, x= ∞  0ν, x ( )d ≤ ∞  0 ν ( )d = L1(ν).(4) Taking sup emum in (4) when x≤1, we ob ain  L1 w(ν)≤ L1(ν). Now we p o e second inclusion L1 w(ν)⊆L1,∞(ν). Take ∈L1 w(ν)andle >0. Then χ{w∈Ω:| (w)|> }≤| |,andso χ{w∈Ω:| (w)|> }∈L1 w(ν). Mo eo e ν ( )= χ{w∈Ω:| (w)|> }L1 w(ν)≤ L1 w(ν).(5) Taking sup emum in (5) we ob ain  L1,∞(ν):= sup >0 ν ( )≤ L1 w(ν). The con inui y o he las inclusion L1,∞(ν)⊆L0(ν) (and also all o he inclusions in he pape ) ollows om [1, Theo em 16.6].2 R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 999 3. Real in e pola ion esul s o measu es on δ- ings Le us ecall b iefly he cons uc ion o he eal in e pola ion me hod o Lions–Pee e.Le (A0,A 1)bea quasi-Banach couple, ha is, wo quasi-Banach spaces A0,A1which a e con inuously embedded in some Hausdo ff opological ec o space. The Pee e K- unc ional is defined, o >0and ∈A0+A1,by K( , ;A0,A 1):=in  0A0+  1A1: = 0+ 1, 0∈A0, 1∈A1. Fo 0 <θ<1and1≤q≤∞, hespace(A0,A 1)θ,q is o med by all hose elemen s ∈A0+A1such ha he quasi-no m  (A0,A1)θ,q := (∞ 0( −θK( , ;A0,A 1))qd )1 q,i 1 ≤q<∞, sup >0 −θK( , ;A0,A 1),i q=∞, is fini e. One o he main esul s in [9] is Co olla y 17 which assu es, o 0 <θ<1≤q≤∞,1≤p0= p1≤∞, and a ec o measu e νdefined on a σ-algeb a ha Lp0(ν),L p1(ν)θ,q =Lp0 w(ν),L p1 w(ν)θ,q =Lp,qν,(6) whe e 1 p=1−θ p0+θ p1.AsExample 2.1 shows, he abo e equali ies a e no longe ue i he measu e ν is defined on a δ- ing. Tha is, o such a measu e (Lp0(ν),L p1(ν))θ,q =c0bu (Lp0 w(ν),L p1 w(ν))θ,q =∞. Ne e heless, he e a e cases whe e he si ua ion is simila o he case o σ-algeb as, desc ibed in (6),e en o measu es genuinely defined on δ- ings. The e is a b oad class o ec o measu es o which his occu s: locally s ongly addi i e ec o measu es. Recall ha a ec o measu e ν:R→Xis called locally s ongly addi i e i limn→∞ ν(An)X= 0 o all disjoin sequences (An)nin Rsuch ha ν(n≥1An)<∞.No e ha he ec o measu e we ha e conside ed in Example 2.1 is no locally s ongly addi i e. In wha ollows we con inue wi h a σ-fini e ec o measu e ν:R→X. Locally s ongly addi i e ec o measu es we e cha ac e ized in [5] in he ollowing o m. Lemma 3.1. (See Lemma 4.1 in [5].) The ollowing condi ions a e equi alen : A) The measu e νis locally s ongly addi i e. B) I B∈Rloc and χB∈L1 w(ν), hen χB∈L1(ν). Now we add some mo e equi alen condi ions o ha cha ac e iza ion. P oposi ion 3.2. The ollowing condi ions a e equi alen : A) The measu e νis locally s ongly addi i e. C) I B∈Rloc and χB∈L1(ν), hen χB∈L1(ν). D) L∞(ν)∩L1(ν)⊆L∞(ν)∩L1(ν). E) L1(ν)⊆L1(ν). P oo . A) ⇔C). Fo a se B∈Rloc no e ha χB∈L1(ν) i and only i χB∈L1 w(ν), because χBL1(ν)=χBL1 w(ν)=ν(B). Then, he equi alence A) ⇔C) ollows om Lemma 3.1. C) ⇒D). Take ∈L∞(ν)∩L1(ν). Fo e e y k≥1 conside he subse s Bk:= {w∈Ω:1 k≤| (w)|} ∈ Rloc.No e ha 1 kχBk≤| |.ThusχBk∈L1(ν), and by he hypo hesis χBk∈L1(ν). Conside o all k≥1 he unc ions gk:= χBk, and no e ha gk≤ L∞(ν)χBk.Thusgk∈L1(ν) o all k≥1 and mo eo e ( aking in o accoun P oposi ion 2.2)wege 1000 R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003  −gkL1 w(ν)≤ −gkL1(ν)= ∞  0 ν −gk( )d ≤ 1 k  0 ν ( )d →0, as k→∞because ∈L1(ν). Then ∈L1(ν)sinceL1(ν)⊆L1 w(ν)isclosed. D) ⇒E). Take 0 ≤ ∈L1(ν). Fo e e y n≥1 conside he unc ions n:= min{ ,n}∈L∞(ν). No e ha n∈L∞(ν)∩L1(ν), and so n∈L1(ν) o all n≥1. Again aking in o accoun P oposi ion 2.2 we ge  − nL1 w(ν)≤ − nL1(ν)= ∞  0 ν − n( )d = n  0 ν − n( )d + ∞  n ν − n( )d ≤ n  0 ν (n)d + ∞  n ν ( )d =nν (n)+ ∞  n ν ( )d →0, as n→∞because ∈L1(ν). Then ∈L1(ν)sinceL1(ν)⊆L1 w(ν)isclosed. The implica ion E) ⇒C) is ob ious. 2 Rema k 3.3. Le ν:R→Xbe a locally s ongly addi i e σ-fini e ec o measu e. Then, o all 1 ≤p<∞, we ha e he ollowing con inuous inclusions Lp,1ν⊆Lpν⊆Lp(ν)⊆Lp w(ν)⊆Lp,∞ν.(7) The chain o inclusions Lp,1(ν)⊆Lp(ν)⊆Lp,∞(ν) is simila o he case o a posi i e scala measu e (see [2, P oposi ion IV.4.2]). The es o he inclusions in (7) ollow om he equi alence E) o P oposi ion 3.2 and also he con inuous inclusions (3) in P oposi ion 2.2,byno ing ha Lp(ν)={ ∈L0(ν):| |p∈ L1(ν)}. Rema k 3.4. Lewis p o ed in [12, Theo em 5.1] he equi alence o he ollowing asse ions: i)The Banach space Xhas no subspace isomo phic o c0. ii)L1(ν)=L1 w(ν) o e e y X- alued ec o measu e νdefined on a δ- ing. Thus Lemma 3.1 ells us ha e e y σ-fini e measu e ν:R→Xis locally s ongly addi i e i he Banach space Xhas no subspace isomo phic o c0. This esul is a so o Dies el–Fai es heo em o measu es on δ- ings (see [8, Theo em I.4.2]). In wha ollows we need some es ima es o he K- unc ional ha will be use ul o es ablish ou in e - pola ion esul s. These es ima es can be ob ained ollowing he same echniques used in [9] wi h mino modifica ions (see [9, Lemma 3 and P oposi ions 8 and 10] o de ails). Le us also men ion ha simila es ima es we e ob ained independen ly by Ce dà, Ma ín and Sil es e in [6] o capaci ies. As usual, in wha ollows abmeans ha a≤cb o some posi i e cons an cindependen o he quan i ies aand b. R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 1001 P oposi ion 3.5. Fo a σ-fini e ec o measu e ν:R→X he ollowing es ima es o he K- unc ional hold: i)I ∈L1(ν)+L∞(ν), hen K( , ;L1(ν),L ∞(ν))  0 ∗(s)ds. ii)I ∈L1,∞(ν)+L∞(ν), hen ∗( )K( , ;L1,∞(ν),L ∞(ν)). Theo em 3.6. Le ν:R→Xbe a σ-fini e ec o measu e, and 0<θ<1≤q≤∞.Then (L1(ν),L ∞(ν))θ,q =(L1,∞(ν),L ∞(ν))θ,q =L1 1−θ,q(ν). P oo . The inclusion (L1(ν),L ∞(ν))θ,q ⊆(L1,∞(ν),L ∞(ν))θ,q ollows om he inclusion L1(ν)⊆ L1,∞(ν), and since his las inclusion is con inuous we ha e he inequali y  (L1,∞(ν),L∞(ν))θ,q  (L1(ν),L∞(ν))θ,q , ∈L1ν,L ∞(ν)θ,q.(8) We ha e also he inclusion (L1,∞(ν),L ∞(ν))θ,q ⊆L1 1−θ,q(ν) as a consequence o he inequali y ii)in P oposi ion 3.5. In pa icula , we ob ain  L 1 1−θ,q (ν) (L1,∞(ν),L∞(ν))θ,q , ∈L1,∞ν,L ∞(ν)θ,q.(9) In o de o check ha he inclusion L1 1−θ,q(ν)⊆(L1(ν),L ∞(ν))θ,q holds, we assume fi s ha q<∞. P oposi ion 3.5.i) and he Ha dy inequali y (see [2, Lemma III.3.9])gi e, o any ∈L1 1−θ,q(ν),  (L1(ν),L∞(ν))θ,q =∞  0 −θK , ;L1ν,L ∞(ν)qd  1 q ∞  0 −θ  0 ∗(u)duqd  1 q =∞  0 1−θ1  0 ∗(u)duqd  1 q ∞  0 1−θ ∗( )qd  1 q = L 1 1−θ,q (ν).(10) This implies ha L1 1−θ,q(ν)⊆(L1(ν),L ∞(ν))θ,q. Fo he case q=∞, he inclusion L1 1−θ,∞(ν)⊆ (L1(ν),L ∞(ν))θ,∞can be ob ained by using he es ima e i)inP oposi ion 3.5 and no ing ha −θK , ;L1ν,L ∞(ν) −θ  0 ∗(s)ds = −θ  0 s1−θ ∗(s)sθ−1ds ≤1 θ L 1 1−θ,∞(ν). Taking sup emum, we ob ain L1 1−θ,∞(ν)⊆(L1(ν),L ∞(ν))θ,∞,and  (L1(ν),L∞(ν))θ,∞ L 1 1−θ,∞(ν), ∈L1 1−θ,∞ν.(11) Finally no e ha we ge he equali y be ween he h ee spaces (e en o q=∞) as me ic spaces. The equi alence o hei quasi-no ms is gi en by (8),(9) and (10),o (11) o q=∞.2 1002 R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 Co olla y 3.7. Le ν:R→Xbe a σ-fini e locally s ongly addi i e ec o measu e, and 0<θ<1≤q≤∞. Then L1(ν),L ∞(ν)θ,q =L1 w(ν),L ∞(ν)θ,q =L1 1−θ,qν. P oo . Fo a σ-fini e locally s ongly addi i e ec o measu e ν:R→X ake in o accoun he equi alence E) o P oposi ion 3.2 and also he con inuous inclusions (3) in P oposi ion 2.2, ha is, L1ν⊆L1(ν)⊆L1 w(ν)⊆L1,∞ν⊆L0(ν), and apply he abo e Theo em 3.6.2 Rema k 3.8. 1) No e ha he second equali y in Co olla y 3.7 holds e en o a non-locally s ongly addi i e σ-fini e ec o measu e νdue o he inclusions (3) in P oposi ion 2.2.Theni νis a ec o measu e o which L1(ν)=L1 w(ν) ( ecall Rema k 3.4) bo h equali ies in Co olla y 3.7 hold. 2) On he o he hand, i νis as ongly addi i e, ha is, limn→∞ ν(An)X= 0 o all disjoin sequences (An)nin R, henL1(ν), L1 w(ν)andalsoLp,q(ν), 1 ≤p, q ≤∞coincide wi h he co esponding spaces o ce ain ec o measu e defined on a σ-algeb a, see [7, Co olla y 3.2.a)].In ha case,Co olla y 3.7 ollows om [9, Co olla y 13]. Ne e heless, he e a e locally s ongly addi i e σ-fini e ec o measu es νwhich a e no s ongly addi i e and such ha L1(ν)L1 w(ν) as he ollowing example shows. Example 3.9. (See Example 3.11 in [5].) Le Rbe he δ- ing o fini e subse s o na u al numbe s N,andle α:= (αn)nbe a sequence wi hou any bounded subsequence. Conside he σ-fini e ec o measu e ν:A∈R→ν(A):=α·χA∈c0. I is easy o check ha L1 w(ν)=( n)n:( nαn)n∈∞, L1(ν)=( n)n:( nαn)n∈c0. On he o he hand i is no difficul o see ha ν(A)=sup n∈A|αn|, o e e y nonemp y A⊆N.This means, in pa icula , ha ν(A)<∞i and only i A∈Rsince (αn)nhas no bounded subsequences, and consequen ly νis locally s ongly addi i e. Finally no e ha νis clea ly no s ongly addi i e and L1(ν)L1 w(ν). Rema k 3.10. Le 1 <p<∞and ake θ=1−1 p. Pu ing q=1in heabo eCo olla y 3.7 we ob ain, in pa icula , (L1(ν),L ∞(ν))θ,1=Lp,1(ν). Simila ly, we ha e (L1(ν),L ∞(ν))θ,∞=Lp,∞(ν)i we ake q=∞in Co olla y 3.7.Now om(7) in Rema k 3.3 we conclude ha L1(ν),L ∞(ν)θ,1⊆Lp(ν)⊆Lp w(ν)⊆L1(ν),L ∞(ν)θ,∞.(12) In he e minology o [3, Theo em 3.5.2] he inclusions abo e say ha he spaces Lp(ν)andLp w(ν)belong bo h o he bo h classes CJ(θ, L1(ν),L ∞(ν)) and CK(θ, L1(ν),L ∞(ν)). See [3,p.49]jus a e Defini- ion 3.5.1. Also no e ha o a gene al ec o measu e νi ollows ha Lp w(ν) belongs o he classes CJ(θ,L1 w(ν),L ∞(ν)) and CK(θ, L1 w(ν),L ∞(ν)). Co olla y 3.11. Le ν:R→Xbe a σ-fini e locally s ongly addi i e ec o measu e, 0<η<1≤q≤∞, and 1<p 0=p1<∞.Then R. del Campo e al. / J. Ma h. Anal. Appl. 419 (2014) 995–1003 1003 Lp0(ν),L p1(ν)η,q =Lp0 w(ν),L p1 w(ν)η,q =Lp,qν, whe e 1 p=1−η p0+η p1. P oo . Ha ing in mind he inclusions (12) in he p e ious ema k, we can apply he ei e a ion heo em [3, Theo em 3.5.3] wi h pa ame e s θ0=1−1 p0and θ1=1−1 p1. The ei e a ion heo em ells us ha Lp0(ν),L p1(ν)η,q =Lp0 w(ν),L p1 w(ν)η,q =L1(ν),L ∞(ν)θ,q, whe e θ=(1−η)θ0+ηθ1,inwhichcase1−θ=1 p. Finally he abo e Co olla y 3.7 gi es (L1(ν),L ∞(ν))θ,q = L1 1−θ,q(ν)=Lp,q(ν), which is he las equali y. 2 Rema k 3.12. No e ha he second equali y in Co olla y 3.11 holds e en o non-locally s ongly addi i e ec o measu es. Acknowledgmen The au ho s a e g a e ul o he unknown e e ee o his aluable sugges ions imp o ing he eading o he pape and, especially, o epo ing Example 3.9. Re e ences [1] C.D. Alip an is, O. Bu kinshaw, Locally Solid Riesz Spaces, Pu e Appl. Ma h., ol. 76, Academic P ess [Ha cou B ace Jo ano ich Publishe s], New Yo k, 1978, MR0493242 (58 #12271). [2] C. Benne , R. Sha pley, In e pola ion o Ope a o s, Pu e Appl. Ma h., ol. 129, Academic P ess Inc., Bos on, MA, 1988, MR928802 (89e:46001). [3] J. Be gh, J. Lö s öm, In e pola ion Spaces: An In oduc ion, G undleh en Ma h. Wiss., ol. 223, Sp inge -Ve lag, Be lin, 1976, MR0482275. 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