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

J. Math. Anal. Appl. 419 (2014) 995–1003 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa A note on real interpolation of Lp-spaces of vector measures on δ-rings ✩ Ricardo del Campo a, Antonio Fernández b,∗, Fernando Mayoral b, Francisco Naranjo b aDpto. Matemática Aplicada I, Universidad de Sevilla, EUITA, Ctra. de Utrera Km. 1, 41013 Sevilla, Spain bDpto. Matemática Aplicada II, Escuela Técnica Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain article info abstract Article history: Received 5 November 2013 Available online 16 May 2014 Submitted by B. Bongiorno Keywords: Real interpolation methods Lions–Peetre K-functional Integrable function Vector measure δ-Ring Locally strongly additive measure We describe the real interpolation spaces obtained when we apply the real K-method of Lions–Peetre to Banach lattices of p-integrable and weakly p-integrable functions with respect to a Banach-space-valued measure defined on a δ-ring. In general, the obtained results are quite different from those in the case of vector measures on σ-algebras described in [9]. However, we find a wide class of vector measures on δ-rings for which the results on σ-algebras hold true. © 2014 Elsevier Inc. All rights reserved. 1. Introduction A basic problem in interpolation theory is to describe the spaces obtained by applying an interpolation method to concrete compatible couples of spaces. For a Banach-space-valued measure mdefined on aσ-algebra, we obtained in [11] the Calderón interpolation spaces [X0,X 1][θ]and [X0,X 1][θ],andin[9] the real interpolation spaces (X0,X 1)θ,q of the couples (X0,X 1), where X0and X1are the Banach lattices Lp(m)orLp w(m) of equivalence classes of scalar p-integrable or, respectively, weakly p-integrable functions with respect to the measure m. Later we investigated in [5] the Calderón interpolation methods of the same spaces, but for measures defined on δ-rings. We showed in [5] that the interpolation results for vector measures on δ-rings can be very different from those on the context of σ-algebras. However, we identified ✩This research has been partially supported by La Junta de Andalucía. The authors acknowledge the support of the Ministerio de Economía y Competitividad of Spain and FEDER, under the project MTM2012-36740-C02-01. *Corresponding author. E-mail addresses: rcamp[email protected] (R. del Campo), [email protected] (A. Fernández), may[email protected] (F. Mayoral), [email protected] (F. Naranjo). http://dx.doi.org/10.1016/j.jmaa.2014.05.039 0022-247X/© 2014 Elsevier Inc. All rights reserved. 996 R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 a certain type of vector measures on δ-rings (called locally strongly additive measures) which keep completely the same behavior (for all the different combinations of couples) as measures defined on σ-algebras. In the present note we complete the picture with the study of real interpolation methods of Banach lattices of p-integrable and weakly p-integrable functions with respect to a Banach-space-valued measure defined on a δ-ring. As in the case of complex methods we can say that certain interpolation equalities for vector measures on σ-algebras described in [9] remain true for vector measures on δ-rings, but some others cease to be true for vector measures on δ-rings. Curiously, for the same type of measures (locally strongly additive measures) real interpolation equalities in the setting of measures defined on σ-algebra remain true for measures on δ-rings. However, the reasons why this happens are very different from those on the context of complex interpolation methods. 2. Preliminaries In this section we establish the preliminaries necessaries about integration of scalar functions with respect to vector measures on δ-rings, in order to make the paper more self-contained and readable. The basic references about integration for us will be [7,12–14]. Throughout this paper we will consider a vector measure ν:R→Xdefined on a δ-ring Rof subsets of some nonempty set Ωwith values in a real Banach space X, with dual X.WedenotebyRloc the σ-algebra of subsets A⊆Ωsuch that A∩B∈Rfor each B∈R. Measurability of functions f:Ω→Rwill be considered with respect to the measurable space (Ω,Rloc). The semivariation of νis the set function ν:Rloc →[0,∞] defined by ν(A):=sup{|ν, x|(A):xX≤1}, where |ν, x| is the variation of the scalar measure ν, x:A∈R→ν, x(A):=ν(A),x ∈R. AsetN∈Rloc is called ν-null if ν(N) = 0. A property holds ν-almost everywhere (ν-a.e.) if it holds except on a ν-null set. A measurable function f:Ω→Ris called weakly integrable (with respect to ν)iff∈L1(ν, x) for all x∈X. A weakly integrable function fis said to be integrable (with respect to ν) if, for each A∈Rloc there exists an element (necessarily unique) Afdν ∈X, satisfying  A fdν,x= A fdν, x,x ∈X. If 1 ≤p<∞, a measurable function f:Ω→Ris called weakly p-integrable (with respect to ν)if|f|pis weakly integrable and p-integrable (with respect to ν)if|f|pis integrable. The space Lp w(ν)ofall(ν-a.e. equivalence classes of) weakly p-integrable functions becomes a Banach lattice with the Fatou property when endowed with the usual ν-a.e. pointwise order and the norm fLp w(ν):= sup Ω |f|pdν, x 1 p : x X≤1. Moreover, the space Lp(ν)ofall(ν-a.e. equivalence classes of) p-integrable functions is a closed order continuous ideal of Lp w(ν). In fact, it is the closure of S(R), the space of simple functions supported on R. The Banach lattices Lp(ν)andLp w(ν) of equivalence classes of scalar p-integrable and weakly p-integrable functions were initially studied in [10] for vector measures νon a σ-algebra and its basic properties can be extended and remain true for vector measures on δ-rings (see [4]). Also we can find in [15, Chapter 3] avery good material about spaces of integrable functions with respect to a vector measure on a σ-algebra. Finally, let us consider two more spaces strongly related with the spaces of p-integrable functions with respect R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 997 to a vector measure. Denote by L∞(ν) the space of classes of essentially bounded measurable functions f:Ω→Rwith the essential supremum norm. Consider also the vector space L0(ν) of all classes of measurable functions f:Ω→R. If the vector measure νis defined on a σ-algebra it is well-known (see [10, Corollary 3.2]) that the following inclusions hold for all p>1 L∞(ν)⊆Lp(ν)⊆Lp w(ν)⊆L1(ν)⊆L1 w(ν)⊆L0(ν),(1) and all of them are continuous inclusions, where the topology of convergence in measure is considered on L0(ν). When the vector measure νis defined on a δ-ring instead of a σ-algebra, the inclusions (1) are in general false, but we can save something (see, for example, Proposition 2.2 and Remark 3.3). In what follows we will always consider vector measures ν:R→Xwhich are σ-finite, that is, there exist a pairwise disjoint sequence (Ωk)kin R,andaν-null set N∈Rloc, such that Ω=( k≥1Ωk)∪N.The simplest example of a σ-finite vector measure on a δ-ring is given by the Lebesgue measure λdefined on the δ-ring R:= {A∈M:λ(A)<∞}, where Mis the σ-algebra of all Lebesgue measurable subsets of the real line R. If we consider the vector measure ν:A∈R→ν(A)=λ(A)∈R, then Lp w(ν)=Lp(ν)=Lp(R) for all p≥1. In the context of interpolation it is well-known that we need a topological vector space as an environment space in order to consider couples of Banach spaces. In our case it is the linear space L0(ν), endowed with the topology of convergence in measure on each subset Ωk. This topology is generated by the F-norm · L0(ν) that we shall now describe. For each k=1,2,... consider the σ-algebra Σk:= {A∈R:A⊆Ωk}of subsets of Ωkand the vector measure νk:A∈Σk→νk(A)=ν(A)∈X, that is, the restriction of νto Σk.Now define fL0(ν):= ∞  k=1 1 2k(1 + ν(Ωk))    |f| 1+|f|χΩk   L1 w(νk) ,f∈L0(ν). For details see [5, Lemmas 3.2, 3.3 and 3.4]. In particular, let us mention that each pair of spaces Lp w(ν) or Lp(ν)formsacompatible couple of Banach spaces, that is, they are imbedded continuously in the same topological vector space L0(ν). Given f∈L0(ν), we shall consider its distribution function (with respect to the vector measure ν) defined by νf:s∈[0,∞)→νf(s):=νw∈Ω:f(w)>s ∈[0,∞], where νis the semivariation of the measure ν. This distribution function has similar properties as in the scalar case (see [9]). For instance, νfis non-increasing and right-continuous. The decreasing rearrangement of f(with respect to the measure ν)isgivenby f∗:t∈(0,∞)→f∗(t):=inf s>0:νf(s)≤t∈[0,∞]. Some properties of f∗can be found in [9] when the measure is defined on a σ-algebra. Nevertheless, it is not difficult to see (even for measures on δ-rings) that f∗is a non-increasing, right-continuous function. Moreover the following two equalities hold ∞  0 νf(t)dt = ∞  0 f∗(t)dt and sup t>0 tνf(t)=sup t>0 tf∗(t). For 1 ≤p, q ≤∞the Lorentz space Lp,q(ν) with respect to the vector measure νconsists of all functions f∈L0(ν) for which the quantity 998 R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 fLp,q (ν):= (∞ 0(s1 pf∗(s))qds s)1 q(1 ≤q<∞) sups>0s1 pf∗(s)(q=∞) is finite. The functional f→fLp,q (ν)is not always a norm, even when p, q ≥1, because the triangle inequality fails. Nevertheless it is not difficult to prove that f+gLp,q (ν)≤C(fLp,q (ν)+gLp,q (ν)), where C≥1 is a constant depending on pand q. Therefore · Lp,q (ν)is only a quasi-norm. We also note that Lp,q(ν) is a quasi-Banach lattice with the Fatou property. For the special case p=q, we denote the space Lp,p(ν)simplybyLp(ν). As it has been pointed out in [9], in general, the spaces Lp(ν) and Lp(ν) do not coincide. For p>1and1≤q≤∞the Lorentz spaces Lp,q(ν) are intermediate spaces of the couple (L1(ν),L ∞(ν)), that is, L1(ν)∩L∞(ν)⊆Lp,q(ν)⊆L1(ν)+L∞(ν). Moreover, if the measure νis defined on a σ-algebra, it holds the following inclusions, for all 1 ≤p<∞(see [9, Proposition 7]) L∞(ν)⊆Lp,1ν⊆Lpν⊆Lp(ν)⊆Lp w(ν)⊆Lp,∞ν,(2) and all these inclusions are continuous. However, if the vector measure νis defined on a δ-ring instead of a σ-algebra, the inclusion Lp(ν)⊆Lp(ν) is in general false as Example 2.1 below points out. The inclusion L∞(ν)⊆Lp,1(ν) is false even for a non-finite positive scalar measure. On the contrary the others inclusions of the chain (2) remain true as we shall see with the following Proposition 2.2 (see also Remark 3.3). Example 2.1. Let Rbe the δ-ring of finite subsets of natural numbers N, and consider the σ-finite vector measure ν:A∈R→ν(A):=χA∈c0, where c0is the space of null sequences. For every 1 ≤p<∞, it is easy to check that Lp w(ν)=∞, the space of bounded sequences, and Lp(ν)=c0. In what follows it will be interesting to note that ν(A) = 1, for every nonempty A⊆N,andν(∅) = 0. This means, in particular, that νf=χ[0,∞)if fis an unbounded sequence, but νf=χ[0,f∞)if f∈∞.Consequently, we have for an unbounded sequence fthat f∗(t)=∞if t∈(0,1) and f∗(t)=0ift≥1. On the other hand, f∗=f∞χ(0,1) if f∈∞.ThusL1(ν)=∞=L1 w(ν), and L1(ν)L1(ν). Proposition 2.2. The following continuous inclusions hold L1ν⊆L1 w(ν)⊆L1,∞ν⊆L0(ν).(3) Proof. First we check the inclusion L1(ν)⊆L1 w(ν). Take f∈L1(ν), and choose any x∈X,with x≤1. For the positive σ-additive measure |ν, x| we have (see [2, Proposition II.1.8] for the first equality)  Ω |f|dν, x= ∞  0ν, xf(t)dt ≤ ∞  0 νf(t)dt =fL1(ν).(4) Taking supremum in (4) when x≤1, we obtain fL1 w(ν)≤fL1(ν). Now we prove second inclusion L1 w(ν)⊆L1,∞(ν). Take f∈L1 w(ν)andlett>0. Then tχ{w∈Ω:|f(w)|>t}≤|f|,andsotχ{w∈Ω:|f(w)|>t}∈L1 w(ν). Moreover tνf(t)=tχ{w∈Ω:|f(w)|>t}L1 w(ν)≤fL1 w(ν).(5) Taking supremum in (5) we obtain fL1,∞(ν):= supt>0tνf(t)≤fL1 w(ν). The continuity of the last inclusion L1,∞(ν)⊆L0(ν) (and also all other inclusions in the paper) follows from [1, Theorem 16.6].2 R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 999 3. Real interpolation results for measures on δ-rings Let us recall briefly the construction of the real interpolation method of Lions–Peetre.Let(A0,A 1)bea quasi-Banach couple, that is, two quasi-Banach spaces A0,A1which are continuously embedded in some Hausdorff topological vector space. The Peetre K-functional is defined, for t>0andf∈A0+A1,by K(t, f;A0,A 1):=inf f0A0+tf1A1:f=f0+f1,f 0∈A0,f 1∈A1. For 0 <θ<1and1≤q≤∞,thespace(A0,A 1)θ,q is formed by all those elements f∈A0+A1such that the quasi-norm f(A0,A1)θ,q := (∞ 0(t−θK(t, f;A0,A 1))qdt t)1 q,if 1 ≤q<∞, supt>0t−θK(t, f;A0,A 1),if q=∞, is finite. One of the main results in [9] is Corollary 17 which assures, for 0 <θ<1≤q≤∞,1≤p0= p1≤∞, and a vector measure νdefined on a σ-algebra that Lp0(ν),L p1(ν)θ,q =Lp0 w(ν),L p1 w(ν)θ,q =Lp,qν,(6) where 1 p=1−θ p0+θ p1.AsExample 2.1 shows, the above equalities are not longer true if the measure ν is defined on a δ-ring. That is, for such a measure (Lp0(ν),L p1(ν))θ,q =c0but (Lp0 w(ν),L p1 w(ν))θ,q =∞. Nevertheless, there are cases where the situation is similar to the case of σ-algebras, described in (6),even for measures genuinely defined on δ-rings. There is a broad class of vector measures for which this occurs: locally strongly additive vector measures. Recall that a vector measure ν:R→Xis called locally strongly additive if limn→∞ ν(An)X= 0 for all disjoint sequences (An)nin Rsuch that ν(n≥1An)<∞.Note that the vector measure we have considered in Example 2.1 is not locally strongly additive. In what follows we continue with a σ-finite vector measure ν:R→X. Locally strongly additive vector measures were characterized in [5] in the following form. Lemma 3.1. (See Lemma 4.1 in [5].) The following conditions are equivalent: A) The measure νis locally strongly additive. B) If B∈Rloc and χB∈L1 w(ν), then χB∈L1(ν). Now we add some more equivalent conditions to that characterization. Proposition 3.2. The following conditions are equivalent: A) The measure νis locally strongly additive. C) If B∈Rloc and χB∈L1(ν), then χB∈L1(ν). D) L∞(ν)∩L1(ν)⊆L∞(ν)∩L1(ν). E) L1(ν)⊆L1(ν). Proof. A) ⇔C). For a set B∈Rloc note that χB∈L1(ν) if and only if χB∈L1 w(ν), because χBL1(ν)=χBL1 w(ν)=ν(B). Then, the equivalence A) ⇔C) follows from Lemma 3.1. C) ⇒D). Take f∈L∞(ν)∩L1(ν). For every k≥1 consider the subsets Bk:= {w∈Ω:1 k≤|f(w)|} ∈ Rloc.Notethat1 kχBk≤|f|.ThusχBk∈L1(ν), and by the hypothesis χBk∈L1(ν). Consider for all k≥1 the functions gk:= fχBk, and note that gk≤fL∞(ν)χBk.Thusgk∈L1(ν) for all k≥1 and moreover (taking into account Proposition 2.2)weget 1000 R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 f−gkL1 w(ν)≤f−gkL1(ν)= ∞  0 νf−gk(t)dt ≤ 1 k  0 νf(t)dt →0, as k→∞because f∈L1(ν). Then f∈L1(ν)sinceL1(ν)⊆L1 w(ν)isclosed. D) ⇒E). Take 0 ≤f∈L1(ν). For every n≥1 consider the functions fn:= min{f,n}∈L∞(ν). Note that fn∈L∞(ν)∩L1(ν), and so fn∈L1(ν) for all n≥1. Again taking into account Proposition 2.2 we get f−fnL1 w(ν)≤f−fnL1(ν)= ∞  0 νf−fn(t)dt = n  0 νf−fn(t)dt + ∞  n νf−fn(t)dt ≤ n  0 νf(n)dt + ∞  n νf(t)dt =nνf(n)+ ∞  n νf(t)dt →0, as n→∞because f∈L1(ν). Then f∈L1(ν)sinceL1(ν)⊆L1 w(ν)isclosed. The implication E) ⇒C) is obvious. 2 Remark 3.3. Let ν:R→Xbe a locally strongly additive σ-finite vector measure. Then, for all 1 ≤p<∞, we have the following continuous inclusions Lp,1ν⊆Lpν⊆Lp(ν)⊆Lp w(ν)⊆Lp,∞ν.(7) The chain of inclusions Lp,1(ν)⊆Lp(ν)⊆Lp,∞(ν) is similar to the case of a positive scalar measure (see [2, Proposition IV.4.2]). The rest of the inclusions in (7) follow from the equivalence E) of Proposition 3.2 and also the continuous inclusions (3) in Proposition 2.2,bynotingthatLp(ν)={f∈L0(ν):|f|p∈ L1(ν)}. Remark 3.4. Lewis proved in [12, Theorem 5.1] the equivalence of the following assertions: i)The Banach space Xhas no subspace isomorphic to c0. ii)L1(ν)=L1 w(ν) for every X-valued vector measure νdefined on a δ-ring. Thus Lemma 3.1 tells us that every σ-finite measure ν:R→Xis locally strongly additive if the Banach space Xhas no subspace isomorphic to c0. This result is a sort of Diestel–Faires theorem for measures on δ-rings (see [8, Theorem I.4.2]). In what follows we need some estimates for the K-functional that will be useful to establish our interpolation results. These estimates can be obtained following the same techniques used in [9] with minor modifications (see [9, Lemma 3 and Propositions 8 and 10] for details). Let us also mention that similar estimates were obtained independently by Cerdà, Martín and Silvestre in [6] for capacities. As usual, in what follows abmeans that a≤cb for some positive constant cindependent of the quantities aand b. R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 1001 Proposition 3.5. For a σ-finite vector measure ν:R→Xthe following estimates for the K-functional hold: i)If f∈L1(ν)+L∞(ν), then K(t, f;L1(ν),L ∞(ν)) t 0f∗(s)ds. ii)If f∈L1,∞(ν)+L∞(ν), then tf∗(t)K(t, f;L1,∞(ν),L ∞(ν)). Theorem 3.6. Let ν:R→Xbe a σ-finite vector measure, and 0<θ<1≤q≤∞.Then (L1(ν),L ∞(ν))θ,q =(L1,∞(ν),L ∞(ν))θ,q =L1 1−θ,q(ν). Proof. The inclusion (L1(ν),L ∞(ν))θ,q ⊆(L1,∞(ν),L ∞(ν))θ,q follows from the inclusion L1(ν)⊆ L1,∞(ν), and since this last inclusion is continuous we have the inequality f(L1,∞(ν),L∞(ν))θ,q f(L1(ν),L∞(ν))θ,q ,f∈L1ν,L ∞(ν)θ,q.(8) We have also the inclusion (L1,∞(ν),L ∞(ν))θ,q ⊆L1 1−θ,q(ν) as a consequence of the inequality ii)in Proposition 3.5. In particular, we obtain fL 1 1−θ,q (ν)f(L1,∞(ν),L∞(ν))θ,q ,f∈L1,∞ν,L ∞(ν)θ,q.(9) In order to check that the inclusion L1 1−θ,q(ν)⊆(L1(ν),L ∞(ν))θ,q holds, we assume first that q<∞. Proposition 3.5.i) and the Hardy inequality (see [2, Lemma III.3.9])give,foranyf∈L1 1−θ,q(ν), f(L1(ν),L∞(ν))θ,q =∞  0t−θKt, f;L1ν,L ∞(ν)qdt t 1 q ∞  0t−θ t  0 f∗(u)duqdt t 1 q =∞  0t1−θ1 t t  0 f∗(u)duqdt t 1 q ∞  0t1−θf∗(t)qdt t 1 q =fL 1 1−θ,q (ν).(10) This implies that L1 1−θ,q(ν)⊆(L1(ν),L ∞(ν))θ,q. For the case q=∞, the inclusion L1 1−θ,∞(ν)⊆ (L1(ν),L ∞(ν))θ,∞can be obtained by using the estimate i)inProposition 3.5 and noting that t−θKt, f;L1ν,L ∞(ν)t−θ t  0 f∗(s)ds =t−θ t  0 s1−θf∗(s)sθ−1ds ≤1 θfL 1 1−θ,∞(ν). Taking supremum, we obtain L1 1−θ,∞(ν)⊆(L1(ν),L ∞(ν))θ,∞,and f(L1(ν),L∞(ν))θ,∞fL 1 1−θ,∞(ν),f∈L1 1−θ,∞ν.(11) Finally note that we get the equality between the three spaces (even for q=∞) as metric spaces. The equivalence of their quasi-norms is given by (8),(9) and (10),or(11) for q=∞.2 1002 R. del Campo et al. / J. Math. Anal. Appl. 419 (2014) 995–1003 Corollary 3.7. Let ν:R→Xbe a σ-finite locally strongly additive vector measure, and 0<θ<1≤q≤∞. Then L1(ν),L ∞(ν)θ,q =L1 w(ν),L ∞(ν)θ,q =L1 1−θ,qν. Proof. For a σ-finite locally strongly additive vector measure ν:R→Xtake into account the equivalence E) of Proposition 3.2 and also the continuous inclusions (3) in Proposition 2.2,thatis, L1ν⊆L1(ν)⊆L1 w(ν)⊆L1,∞ν⊆L0(ν), and apply the above Theorem 3.6.2 Remark 3.8. 1) Note that the second equality in Corollary 3.7 holds even for a non-locally strongly additive σ-finite vector measure νdue to the inclusions (3) in Proposition 2.2.Thenifνis a vector measure for which L1(ν)=L1 w(ν) (recall Remark 3.4) both equalities in Corollary 3.7 hold. 2) On the other hand, if νis astrongly additive, that is, limn→∞ ν(An)X= 0 for all disjoint sequences (An)nin R,thenL1(ν), L1 w(ν)andalsoLp,q(ν), 1 ≤p, q ≤∞coincide with the corresponding spaces for certain vector measure defined on a σ-algebra, see [7, Corollary 3.2.a)].Inthatcase,Corollary 3.7 follows from [9, Corollary 13]. Nevertheless, there are locally strongly additive σ-finite vector measures νwhich are not strongly additive and such that L1(ν)L1 w(ν) as the following example shows. Example 3.9. (See Example 3.11 in [5].) Let Rbe the δ-ring of finite subsets of natural numbers N,andlet α:= (αn)nbe a sequence without any bounded subsequence. Consider the σ-finite vector measure ν:A∈R→ν(A):=α·χA∈c0. It is easy to check that L1 w(ν)=(fn)n:(fnαn)n∈∞, L1(ν)=(fn)n:(fnαn)n∈c0. On the other hand it is not difficult to see that ν(A)=sup n∈A|αn|, for every nonempty A⊆N.This means, in particular, that ν(A)<∞if and only if A∈Rsince (αn)nhas not bounded subsequences, and consequently νis locally strongly additive. Finally note that νis clearly not strongly additive and L1(ν)L1 w(ν). Remark 3.10. Let 1 <p<∞and take θ=1−1 p. Putting q=1intheaboveCorollary 3.7 we obtain, in particular, (L1(ν),L ∞(ν))θ,1=Lp,1(ν). Similarly, we have (L1(ν),L ∞(ν))θ,∞=Lp,∞(ν)ifwetake q=∞in Corollary 3.7.Nowfrom(7) in Remark 3.3 we conclude that L1(ν),L ∞(ν)θ,1⊆Lp(ν)⊆Lp w(ν)⊆L1(ν),L ∞(ν)θ,∞.(12) In the terminology of [3, Theorem 3.5.2] the inclusions above say that the spaces Lp(ν)andLp w(ν)belong both to the both classes CJ(θ, L1(ν),L ∞(ν)) and CK(θ, L1(ν),L ∞(ν)). See [3,p.49]just after Definition 3.5.1. Also note that for a general vector measure νit follows that Lp w(ν) belongs to the classes CJ(θ,L1 w(ν),L ∞(ν)) and CK(θ, L1 w(ν),L ∞(ν)). Corollary 3.11. Letν:R→Xbe a σ-finite locally strongly additive vector measure, 0<η<1≤q≤∞, and 1<p 0=p1<∞.Then R. del Campo et al. / J. Math. 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, where 1 p=1−η p0+η p1. Proof. Having in mind the inclusions (12) in the previous remark, we can apply the reiteration theorem [3, Theorem 3.5.3] with parameters θ0=1−1 p0and θ1=1−1 p1. The reiteration theorem tells us that Lp0(ν),L p1(ν)η,q =Lp0 w(ν),L p1 w(ν)η,q =L1(ν),L ∞(ν)θ,q, where θ=(1−η)θ0+ηθ1,inwhichcase1−θ=1 p. Finally the above Corollary 3.7 gives (L1(ν),L ∞(ν))θ,q = L1 1−θ,q(ν)=Lp,q(ν), which is the last equality. 2 Remark 3.12. Note that the second equality in Corollary 3.11 holds even for non-locally strongly additive vector measures. Acknowledgment The authors are grateful to the unknown referee for his valuable suggestions improving the reading of the paper and, especially, for reporting Example 3.9. References [1] C.D. Aliprantis, O. Burkinshaw, Locally Solid Riesz Spaces, Pure Appl. Math., vol. 76, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978, MR0493242 (58 #12271). [2] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press Inc., Boston, MA, 1988, MR928802 (89e:46001). [3] J. Bergh, J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. 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