On some mean oscillation inequalities for martingales
Abstract
Kikuchi, Masato
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Publ. Mat. 50 (2006), 167–189 ON SOME MEAN OSCILLATION INEQUALITIES FOR MARTINGALES Masato Kikuchi Abstract Let (X, k · kX) be a Banach function space over a nonatomic probability space (Ω,Σ,P). If f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then we define θFf= sup 0≤n≤m<∞ Eˆ|fm−fn−1|˛ ˛Fn˜, where f−1≡0. In this paper, we give a necessary and sufficient condition for the existence of constants cand Csuch that for any martingale f= (fn)n∈Z+, clim n→∞ kfnkX≤ kθFfkX≤Clim n→∞ kfnkX. 1. Introduction Let (Ω,Σ,P) be a nonatomic probability space and let f= (fn)n∈Z+ be a martingale with respect to a filtration F= (Fn)n∈Z+(where by afiltration, we mean a nondecreasing sequence of sub-σ-algebras of Σ). We set f−1≡0 and define θFf= sup 0≤n≤m<∞ E|fm−fn−1|Fn. Let (X, k · kX) be a Banach function space over Ω (see Definition 1 below). In this paper, we consider the inequalities of the form (1) clim n→∞ kfnkX≤ kθFfkX≤Clim n→∞ kfnkX. If X=Lpfor some 1 < p < ∞, then (1) holds for any martingale f= (fn). Indeed, by using Theorem 7 of [7] and the estimate 2000 Mathematics Subject Classification. 60G42, 60G46, 46E30. Key words. Martingale, Banach function space, rearrangement-invariant function space, Boyd index. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research, No. 14540164, 2004.
168 M. Kikuchi θFf≤2 sup0≤n≤m<∞E|fm|Fn, we have that cp sup n |fn| p≤ kθFfkp≤2 sup 0≤n≤m<∞ E|fm|Fn p, which, together with the Doob inequality, implies that cplim n→∞ kfnkp=cpsup n kfnkp≤ kθFfkp ≤2p p−1sup n kfnkp=2p p−1lim n→∞ kfnkp. We can derive a similar result for Orlicz function spaces: if Φ is an N-function satisfying the ∆2and ∇2-conditions and if X=LΦ, then (1) holds for any martingale f= (fn). The purpose of this paper is to show that (1) holds for any martingale f= (fn) if and only if Xcan be renormed so that it is rearrangement-invariant and 0 < αX≤βX<1, where αXand βXdenote the lower and upper Boyd indices of X, respectively. An analogous problem has been studied in [4]. Combining our Main Theorem with the result of [4] shows that (1) holds for any martingale f= (fn) if and only if there exist constants kand Ksuch that for any uniformly integrable martingale f= (fn), kkf∞kX≤ kSf kX≤Kkf∞kX, where Sf denotes the square function of f, and f∞denotes the almost sure limit of f. 2. Preliminaries Let (Ω,Σ,P) be a fixed probability space. In this paper, we will deal with martingales on Ω with respect to various filtrations on (Ω,Σ,P) (and with respect to P). Assumption. We assume that the probability space (Ω,Σ,P) is nonatomic, that is, that (Ω,Σ,P) contains no atom. This assumption is essential and will be used implicitly throughout the paper. In addition to Ω, we have to deal with the canonical probability space (I, M, µ), where Idenotes the interval (0,1], Mdenotes the σ-algebra of Lebesgue measurable subsets of I, and µdenotes Lebesgue measure. We distinguish these two probability spaces. Although the reader may assume that (Ω,Σ,P) is the canonical probability space, our argument will not become so simple by doing so. Let Xand Ybe normed linear spaces of random variables. We write X ֒→Yto mean that Xis continuously embedded in Y, that is, that
On Some Mean Oscillation Inequalities 169 X⊂Yand the inclusion map is continuous. If X ֒→Y ֒→X, then we write X≈Y. Thus X≈Yif and only if X=Y(as a set) and the norms of these spaces are equivalent. Definition 1. ABanach function space is a real Banach space of (equivalence classes of) random variables satisfying the following conditions: (B1) L∞֒→X ֒→L1; (B2) if |x| ≤ |y|a.s. and y∈X, then x∈Xand kxkX≤ kykX; (B3) if 0 ≤xn↑xa.s., xn∈Xfor all n, and supnkxnkX<∞, then x∈Xand kxkX= supnkxnkX. For convenience, we adopt the convention that if x6∈ X, then kxkX=∞. Thus kxkX<∞if and only if x∈X. Given random variables xand y, we write x≃dyto mean that x and yhave the same distribution. Definition 2. A Banach function space (X, k · kX) is said to be rearrangement-invariant (r.i.) provided that (RI) if x≃dyand y∈X, then x∈Xand kxkX=kykX. In this paper, a rearrangement-invariant Banach function space is called arearrangement-invariant space or an r.i. space. For example, Lebesgue, Orlicz, and Lorentz spaces are r.i. spaces. On the other hand, the weighted Lebesgue space Lp,w, with a suitable weight w, is a Banach function space that is not r.i. in general (see the remark following the Main Theorem). Definition 3. Let (X, k · kX) be a Banach function space over Ω. The associate space of Xis the Banach function space (X′,k · kX′) consisting of those random variables yfor which kykX′:= supE[xy]x∈X, kxkX≤1<∞. The associate space of a Banach function space over I= (0,1] is defined in the same way. For example, the associate space of L1is L∞, and the associate space of L∞is L1. For any Banach function space X, we have (X′)′=X; however, the associate space X′is not the dual space of Xin general. See [2, Chapter 1] for more details. By definition, we have that for any x∈Xand y∈X′, E|xy|≤ kxkXkykX′, which we call H¨older’s inequality.
170 M. Kikuchi It is known that a Banach function space (X, k · kX) is r.i. if and only if so is the associate space of X(see [2, p. 60]). Suppose that (X, k · kX) is an r.i. space over Ω. If A∈Σ and 1A denotes the indicator function of A, then the norm of 1Ain Xdepends only on the probability of A. Thus we may define a function ϕXon I by setting ϕX(t) = k1AkX,where A∈Σ and P(A) = t. We call ϕXthe fundamental function of X. It is clear that if 1 ≤p≤ ∞, then ϕLp(t) = t1/p (t∈I). Hence if we denote by p′the conjugate exponent of p, then ϕLp(t)ϕLp′(t) = tfor all t∈I. The same is true for any r.i. space Xand its associate space X′(see [2, p. 66]); that is, (2) ϕX(t)ϕX′(t) = tfor all t∈I. Now let xbe a random variable on Ω. The nonincreasing rearrangement of x, denoted by x∗, is a (unique) nonincreasing right-continuous function on Isuch that P(|x|> λ) = µ(x∗> λ) (λ > 0). Note that x∗is represented as x∗(t) = infλ > 0P(|x|> λ)≤t(t∈I), with the convention that inf ∅=∞. If φis a measurable function on I, then the nonincreasing rearrangement φ∗is defined by regarding φas a random variable on the canonical probability space. If φand ψare integrable functions on I, we write φ≺ψto mean that Zt 0 φ∗(s)ds ≤Zt 0 ψ∗(s)ds for all t∈I. Furthermore if xand yare integrable random variables on Ω and if x∗≺y∗, then we write x≺y. It is obvious that x≃dyif and only if x≺y≺x. In the literature of this subject, a Banach function space (X, k · kX) is said to be universally rearrangement-invariant (u.r.i.) provided that (URI) if x≺yand if y∈X, then x∈Xand kxkX≤ kykX. In our setting, however, there is no need to distinguish between r.i. spaces and u.r.i. spaces; a Banach function space is r.i. if and only if it is u.r.i., provided that the underlying probability space is nonatomic (see [2, Exercise 16, p. 90]). Now let Xbe an r.i. space over Ω. Then there exists a unique r.i. space b Xover I, equipped with the norm k · k b X, such that:
On Some Mean Oscillation Inequalities 171 •x∈Xif and only if x∗∈b X; • kxkX=kx∗kb Xfor all x∈X. In fact b Xconsists of those functions ϕfor which kϕkb X:= sup Z1 0 ϕ∗(s)y∗(s)ds y∈X′,kykX′≤1<∞. We call ( b X, k · k b X) the Luxemburg representation of (X, k · kX). For instance, the Luxemburg representation of Lp(Ω) is Lp(I). See [2, pp. 62–64] for details. In order to describe our results, we have to recall the notion of Boyd indices. Given any positive number sand any measurable function φ on I, we define (Dsφ)(t) = (φ(st) if st ∈I, 0 if st /∈I, (t∈I). If Zis an r.i. space over I, then each Ds(restricted to Z) is a bounded linear operator from Zinto itself and kDskB(Z)≤(1/s)∨1, where kDskB(Z)stands for the operator norm of Ds:Z→Z. The lower and upper Boyd indices of Zare defined by αZ= sup 0<s<1 log kDs−1kB(Z) log s= lim s↓0 log kDs−1kB(Z) log s and βZ= inf 1<s<∞ log kDs−1kB(Z) log s= lim s↑ ∞ log kDs−1kB(Z) log s, respectively. If Xis an r.i. space over Ω, then the Boyd indices of X are defined by αX=αb Xand βX=βb X, where b Xis the Luxemburg representation of X. For instance, αLp=βLp=1/p (1≤p≤∞). Note that 0≤αX≤βX≤1 for any r.i. space X. See [2, p. 149] for details. Now, let Z1and Z2be r.i. spaces over I, and let Tbe a linear operator on L1(I). We write T∈B(Z1, Z2) to mean that the restriction of Tto Z1 is a bounded operator from Z1into Z2. If Z1=Z2=Z, then we also write T∈B(Z) for T∈B(Z, Z). As shown in [4] and [5], there are deep connections between some martingale inequalities and the boundedness of some linear operators on L1(I). We will establish another connection between the inequalities
172 M. Kikuchi of the form (1) and the boundedness of the operators Pand Qdefined for φ∈L1(I) by (Pφ)(t) = 1 tZt 0 φ(s)ds (t∈I); (Qφ)(t) = Z1 t φ(s) sds (t∈I). Note that Qis the (formal) adjoint of P. It is well known that P ∈ B(Z) (resp. Q ∈ B(Z)) if and only if βZ<1 (resp. αZ>0). For a proof, (see [2, p. 150]) (cf. [8]). Now let (X, k · kX) be an r.i. space over Ω. For each random variable x, we let kxkH(X)=kPx∗kb Xand kxkK(X)=kQx∗kb X. Define H(X) (resp. K(X)) to be the set of all random variables xfor which kxkH(X)(resp. kxkK(X)) is finite. Then H(X) is an r.i. space equipped with the norm k · kH(X). Moreover, K(X) is an r.i. spaces if the function t7→ − log tis in b X; otherwise K(X) consists of the zero function only. Therefore we will assume that the function t7→ − log tis in b Xwhenever we consider the space K(X). See [5] for details. 3. Results Let f= (fn)n∈Z+be a martingale with respect to a filtration F= (Fn)n∈Z+. If fis uniformly integrable, then fn=E[f∞| Fn] a.s. for each n∈Z+, and moreover θFf= sup n∈Z+ E|f∞−fn−1|Fna.s. Here and in what follows f∞denotes the almost sure limit of f. Main Theorem. Let (X, k · kX)be a Banach function space over Ω. Then the following are equivalent: (i) there are positive constants cand Csuch that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then clim n→∞ kfnkX≤ kθFfkX≤Clim n→∞ kfnkX; (ii) there are positive constants cand Csuch that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then (3) clim n→∞ kfnkX≤ kθFfkX≤Clim n→∞ kfnkX;
On Some Mean Oscillation Inequalities 173 (iii) there are positive constants cand Csuch that if f= (fn)n∈Z+is a uniformly integrable martingale with respect to a filtration F= (Fn)n∈Z+, then (4) ckf∞kX≤ kθFfkX≤Ckf∞kX; (iv) there exists a norm ||| · |||Xon Xwhich is equivalent to k · kXand with respect to which Xis a rearrangement-invariant space such that 0< αX≤βX<1. Remarks. (a) Recall from [5, Remark 4.3] that if f= (fn) is a martingale, then fn≺fn+1 for all n∈Z+. Hence if (iv) of the Main Theorem holds, then |||fn|||X≤ |||fn+1|||X(n∈Z+). Thus if the equivalent conditions of the Main Theorem hold, then csup n∈Z+ |||fn|||X≤ |||θFf|||X≤Csup n∈Z+ |||fn|||X for any martingale f= (fn). (b) Let 1 < p < ∞and let wbe a strictly positive random variable. The weighted Lebesgue space Lp,w consists of those random variables x for which xpwis integrable with respect to P. If w−1/(p−1) ∈L1, then L∞֒→Lp,w ֒→L1and Lp,w is a Banach function space (with respect to P). In the case where X=Lp,w, the equivalent conditions of the Main Theorem hold if and only if there are strictly positive constants a and bsuch that a≤w≤ba.s. There is a similar result for weighted Orlicz spaces (see [4, Section 4]). As shown in the final section, the Main Theorem is a consequence of Propositions 1, 2, and 3 below. Proposition 1. Let (X, k · kX)be a Banach function space over Ω. Suppose that there is a positive constant Csuch that for any x∈Xand for any sub-σ-algebra Gof Σ, (5) E[xG] X≤CkxkX. Then there exists a norm ||| · |||Xon Xwhich is equivalent to k · kXand with respect to which Xis a rearrangement-invariant space. If the second inequality of (3) holds for any martingale f= (fn)n∈Z+, then (5) holds for any x∈Xand any sub-σ-algebra Gof Σ. Given a martingale f= (fn)n∈Z+, we denote by Mf the maximal function of f;Mf = supn∈Z+|fn|.
174 M. Kikuchi Proposition 2. Let (X, k · kX)and (Y, k · kY)be rearrangement-invariant spaces over Ω. Then the following are equivalent: (i) P ∈ B(b Y , b X); (ii) there is a positive constant Csuch that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then (6) kθFfkX≤Csup n∈Z+ kfnkY; (iii) there is a constant Csuch that if f= (fn)n∈Z+is a martingale, then (7) kMf kX≤Csup n∈Z+ kfnkY. Moreover, if these equivalent conditions hold, then Y ֒→H(X). Corollary 1. Let (X, k · kX)be a rearrangement-invariant space over Ω. Then, for any martingale f= (fn)n∈Z+with respect to a filtration F= (Fn)n∈Z+, kθFfkX≤2 sup n∈Z+ kfnkH(X)and kMf kX≤sup n∈Z+ kfnkH(X). From Proposition 2 and Corollary 1, it follows that H(X) is maximal among all r.i. spaces Ywhich satisfies the inequality of the form (6). Recall that the Zygmund space L(log L) is an r.i. space equipped with the norm defined by kxkL(log L):= Z1 0 (1 −log s)x∗(s)ds, and recall also that x∈L(log L) if and only if |x|log1 + |x|∈L1. If X=L1, then H(X) coincides with L(log L) (see [5, Section 5]) and hence kθFfk1≤2 sup n∈Z+ kfnkL(log L) for any martingale f= (fn) with respect to F= (Fn). From Proposition 2, we can derive an extension of the results of Antipa [1]. Corollary 2 (cf. [1]).Let (X, k · kX)be a Banach function space over Ω. The following are equivalent: (i) there is a positive constant Csuch that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then (8) kMf kX≤Clim n→∞ kfnkX;
On Some Mean Oscillation Inequalities 175 (ii) there exists a norm ||| · |||Xon Xwhich is equivalent to k · kXand with respect to which Xis a rearrangement-invariant space such that βX<1. Suppose that these equivalent conditions hold. Then for any martingale f= (fn)n∈Z+, (9) kMf kX≤Csup n∈Z+ kfnkX. Proposition 3. Let (X, k · kX)and (Y, k · kY)be as in Proposition 2. (i) Assume that Q ∈ B(b Y , b X). Then there is a positive constant C such that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then (10) sup n∈Z+ kfnkX≤ kMf kX≤CkθFfkY. (ii) Assume that the following two conditions are satisfied: (a) there is a positive constant Csuch that if f= (fn)n∈Z+is a martingale with respect to a filtration F= (Fn)n∈Z+, then (11) sup n∈Z+ kfnkX≤CkθFfkY; (b) βY<1, or equivalently P ∈ B(b Y). Then Q ∈ B(b Y , b X)and Y ֒→K(X). Thus (11) holds for any martingale f= (fn)n∈Z+if and only if Q ∈ B(b Y , b X), provided that βY<1. Corollary 3. Let (X, k · kX)be a rearrangement-invariant space over Ω. Then, for any martingale f= (fn)n∈Z+with respect to a filtration F= (Fn)n∈Z+, (12) kMf kX≤16 kθFfkK(X). Given a∈(0,∞), we denote by Lexp:athe r.i. space consisting of those random variables xfor which kxkexp:a:= sup t∈I 1 t(1 −log t)1/a Zt 0 x∗(s)ds < ∞. Then x∈Lexp:aif and only if expλ|x|a∈L1for some λ > 0. It is not difficult to verify that K(Lexp:1)≈L∞and that K(Lexp:a)≈Lexp: a 1−a for a∈(0,1) (see [5, Section 5]). Hence it follows from (12) that kMf kexp:a≤CakθFfkexp: a 1−a(0 < a < 1); kMf kexp:1 ≤C1kθFfk∞.
182 M. Kikuchi Now let us estimate the norm of fnin Y. If tn= 0, then fn=xa.s. and hence kfnkY=kxkY=kφkb Y. So let us assume that tn>0. Using H¨older’s inequality and (2), we see that (Pφ)(tn) = 1 tnZA(tn) x dP≤kxkYϕY′(tn) tn =kxkY ϕY(tn). Therefore it follows from (18) that kfnkY≤kxkY ϕY(tn)· 1A(tn) Y+ x1Ω\A(tn) Y≤2kxkY. Thus sup n∈Z+ kfnkY≤2kxkY= 2 kφkb Y. Combining this with (20), we have the estimate kPφkb X≤2(2C+d)kφkb Y, and thus P ∈ B(b Y , b X). We now show that (i) implies (iii). Let f= (fn)n∈Z+be a martingale. Then, for each k∈Z+ (Mkf)∗(t)≤Pf∗ k(t) (t∈I), where Mkf= sup0≤n≤k|fn|(see the proof of Proposition 3 of [4] or the proof of Theorem 4.1 of [5]). Since P ∈ B(b Y , b X) by assumption, kMkfkX=k(Mkf)∗kb X≤ kPf∗ kkb X ≤Ckf∗ kkb Y=CkfkkY≤Csup n∈Z+ kfnkY, where C=kPkB( b Y , b X). By letting k→ ∞ we obtain (7), as desired. Finally, we need to prove the last statement of Proposition 2. Let P ∈ B(b Y , b X) and C=kPkB( b Y , b X). Then, for any x∈Y, kxkH(X)=kPx∗kb X≤Ckx∗kb Y=CkxkY, which shows that Y ֒→H(X). Proof of Corollary 1: From the proof of Proposition 2, we already know that if P ∈ B(b Y , b X), then (7) holds with C=kPkB( b Y , b X). We also know that if (7) holds with a constant C, then (6) holds with Creplaced by 2C. Therefore, to prove the corollary, it suffices to show that P ∈ BHb(X),b Xand kPkB(H b (X), b X)≤1. Here Hb(X) denotes the Luxemburg representation of H(X). Suppose φ∈Hb(X). Then
On Some Mean Oscillation Inequalities 183 there is a random variable xsuch that x∗=φ∗(see [3, p. 44]). Since |Pφ| ≤ Pφ∗=Px∗on I, kPφkb X≤ kPx∗kb X=kxkH(X)=kφkH b (X). This completes the proof. Proof of Corollary 2: (i) ⇒(ii). Let f= (fn)n∈Z+be a martingale with respect to F= (Fn)n∈Z+. For fixed k∈Z+, we define a martingale g= (gn)n∈Z+as in the proof of Proposition 2. Suppose that (i) holds. Then, θFf(k) X≤2kMgkX≤2Clim n→∞ kgnkX= 2CkfkkX, where the first inequality follows from (16) and the second inequality follows from (8) applied to g= (gn). Letting k→ ∞, we see that (21) kθFfkX≤2Clim k→∞ kfkkX. Hence, by Proposition 1, there is a norm ||| · |||Xon Xwhich is equivalent to the original norm of Xand with respect to which Xis an r.i. space. Since fn≺fn+1 for all n∈Z+, inequality (21) can be rewritten as |||θFf|||X≤C′sup n |||fn|||X. It then follows from Proposition 2 that P ∈ B(b X), or equivalently that βX<1. Thus (i) implies (ii). (ii) ⇒(i). Assume that (ii) holds. Then P ∈ B(b X), since βX<1. Hence Proposition 2 implies that, for any martingale f= (fn) (22) |||Mf|||X≤C′sup n |||fn|||X=C′lim n→∞ |||fn|||X with a positive constant C′, independent of f. Since the norms k · kX and ||| · |||Xare equivalent, (22) can be rewritten as (8). Thus (ii) implies (i). Moreover, since (22) can also be rewritten as (9), the last statement follows. 6. Proof of Proposition 3 In order to prove Proposition 3, we will use the fact that if f= (fn) is a martingale with respect to F= (Fn), then (23) E[Mf]≤16 E[θFf]. A more general inequality was established by R. L. Long in [7]. For an elementary proof of (23) (for uniformly integrable martingales), (see [5, Appendix A]).
184 M. Kikuchi Proof of Proposition 3: (i) Let f= (fn)n∈Z+be a martingale with respect to F= (Fn)n∈Z+such that θFf∈Y. To prove the second inequality of (10), it suffices to show that (24) (Mf)∗≺16 Q(θFf)∗. Indeed, since Q ∈ B(b Y , b X) by assumption, estimate (24) implies that kMf kX=k(Mf)∗kb X≤16 kQ(θFf)∗kb X ≤16 kQkB( b Y , b X)k(θFf)∗kb Y = 16 kQkB( b Y , b X)kθFfkY. If we can show that (25) E[Mf −Mk−1f| Fk]≤16 E[θFf| Fk] a.s. (k∈Z+), then (24) will follow from Theorem 3.3 of [5] (or Lemma 4 of [4]). Notice that (25) is the conditional form of (23). To prove (25), we can use a standard way to derive the conditional form from a inequality for processes. Fix k∈Z+and let A∈ Fk. We define a filtration F′= (F′ n) and a process f′= (f′ n) by setting F′ n=Fk+nand f′ n= (fk+n−fk−1) 1A(n∈Z+). Then f′= (f′ n) is a martingale with respect to F′= (F′ n). Since Mf ≤(Mk−1f) + sup n∈Z+ |fk+n−fk−1|, we see that (Mf −Mk−1f) 1A≤sup n∈Z+ |fk+n−fk−1|1A=Mf′. On the other hand, θF′f′= sup k≤n≤m<∞ E|fm−fn−1|Fn1A≤(θFf) 1A. Applying (23) to the martingale f′(with respect to F′), we conclude that E(Mf −Mk−1f) 1A≤16 E(θFf) 1A(A∈ Fk), which implies (25). (ii) Suppose that (a) and (b) in (ii) hold. We want to show that kQψkb X≤Kkψkb Yfor all ψ∈b Y, with a positive constant Kindependent of ψ. According to Lemma 3, we may assume that ψ∈ D(b Y) and ψ6≡ 0. Given ε > 0, we can find a sequence {tn}n∈Z+in Isuch that (26) t0= 1 and (Qψ)(tn) = (Qψ)(tn−1) + ε(n= 1,2,...), since limt↓0(Qψ)(t) = ∞. It is clear that tn>0 for all nand tn↓0.
On Some Mean Oscillation Inequalities 185 Let φ=Qψ−ψ, and define xand A(t)t∈Ias in Lemma 4. Define a filtration F= (Fn) and a martingale f= (fn) by setting Fn=σΛ\A(tn)Λ∈Σand fn=E[x| Fn] a.s. (n∈Z+). Because Pφ=P(Qψ)− Pψ=Qψ, fn= (Qψ)(tn)1A(tn)+x1Ω\A(tn)a.s. (n∈Z+). Hence for each n≥1, f∞−fn−1=x−(Qψ)(tn−1)1A(tn−1)a.s., and thus E|f∞−fn−1|Fn =1A(tn) tnZA(tn)x−(Qψ)(tn−1)dP+x−(Qψ)(tn−1)1A(tn−1)\A(tn) ≡E(n) 1+E(n) 2a.s. To estimate E(n) 1, note that (Qψ)(1 −ξ)≥(Qψ)(tn)≥(Qψ)(tn−1) on the set A(tn) = {1−ξ < tn}. Then we see that ZA(tn)x−(Qψ)(tn−1)dP =ZA(tn)(Qψ)(1 −ξ)−ψ(1 −ξ)−(Qψ)(tn−1)dP ≤Z{1−ξ<tn}n(Qψ)(1 −ξ)−(Qψ)(tn−1)odP +Z{1−ξ<tn} ψ(1 −ξ)dP =Ztn 0 (Qψ)(s)ds +Ztn 0 ψ(s)ds −tn(Qψ)(tn−1).
186 M. Kikuchi Hence by (26) E(n) 1≤1A(tn) tnZtn 0 (Qψ)(s)ds +Ztn 0 ψ(s)ds −tn(Qψ)(tn−1) =nP(Qψ)(tn) + (Pψ)(tn)−(Qψ)(tn−1)o1A(tn) =2(Pψ)(tn) + (Qψ)(tn)−(Qψ)(tn−1)1A(tn) =2(Pψ)(tn) + ε1A(tn). To estimate E(n) 2, observe that on the set A(tn−1)\A(tn), x−(Qψ)(tn−1) = (Qψ)(1 −ξ)−ψ(1 −ξ)−(Qψ)(tn−1)≥ −ψ(1 −ξ) and x−(Qψ)(tn−1)≤(Qψ)(tn)−(Qψ)(tn−1) = ε. Then we have that E(n) 2=x−(Qψ)(tn−1)1A(tn−1)\A(tn) ≤ψ(1 −ξ) + ε1A(tn−1)\A(tn). As a result, for each n≥1, E|f∞−fn−1|Fn ≤2(Pψ)(tn) + ε1A(tn)+ψ(1 −ξ) + ε1A(tn−1)\A(tn) ≤2(Pψ)(tn)1A(tn)+ψ(1 −ξ)1A(tn−1)\A(tn)+εa.s. Moreover, if n= 0, then E|f∞−fn−1|Fn =kxk1≤ kQψk1+kψk1= 2 kψk1= 2(Pψ)(t0) a.s. It then follows that θFf≤2 sup n∈Z+ (Pψ)(tn)1A(tn)+ψ(1 −ξ) + ε = 2 ∞ X k=1 (Pψ)(tk−1)1A(tk−1)\A(tk)+ψ(1 −ξ) + εa.s. Thus (27) kθFfkY≤2 ∞ X k=1 (Pψ)(tk−1)1A(tk−1)\A(tk) Y +kψ(1 −ξ)kY+εk1kb Y.
On Some Mean Oscillation Inequalities 187 Since ψ(1 −ξ)∗(t) = ψ(t) and since ∞ X k=1 (Pψ)(tk−1)1A(tk−1)\A(tk)∗ (t) = ∞ X k=1 (Pψ)(tk−1)1[tk,tk−1)(t)≤(Pψ)(t) (t∈I), we see from (27) that (28) kθFfkY≤2kPψkb Y+kψkb Y+εk1kb Y. On the other hand, (29) kQψkb X− kψkb X≤ (Qψ)−ψ b X=kφkb X=kxkX≤sup n kfnkX, where the last inequality follows from (B3) and the fact that xis the almost sure limit of f= (fn). Combining (11), (28) and (29), we have that kQψkb X≤C2kPψkb Y+kψkb Y+εk1kb Y+kψkb X. Since b Y ֒→b Xby Lemma 2, the norm kψkb Xon the right-hand side may be replaced by a constant multiple of kψkb Y. It follows that kQψkb X≤kkψkb Y+kPψkb Y+ε with a positive constant k, independent of ψ. Letting ε↓0, we have the estimate kQψkb X≤kkψkb Y+kPψkb Y. Since P ∈ B(b Y), we conclude that kQψkb X≤Kkψkb Y, as desired. To complete the proof, it only remains to show that Y ֒→K(X). Suppose x∈Y. Then kxkK(X)=kQx∗kb X≤ kQkB( b Y , b X)kx∗kb Y=kQkB( b Y , b X)kxkY. Thus Y ֒→K(X), as desired. Proof of Corollary 3: From (24) we see that kMf kX=k(Mf)∗kb X≤16 kQ(θFf)∗kb X= 16 kθFfkK(X), as desired.
188 M. Kikuchi 7. Proof of the Main Theorem This final section is devoted to the proof of our Main Theorem. Proof of the Main Theorem: (i) ⇒(ii). Obvious. (ii) ⇒(iv). Suppose that (ii) holds. Then, by Proposition 1, there exists a norm ||| · |||Xon Xwhich is equivalent to k · kXand with respect to which Xis r.i. To show that 0 < αX≤βX<1, it suffices to show that P,Q ∈ B(b X). Since |||fn|||X≤ |||fn+1|||Xfor each n∈Z+, (3) can be rewritten as (30) csup n∈Z+ |||fn|||X≤ |||θFf|||X≤Csup n∈Z+ |||fn|||X. From Proposition 2 and (30), it follows that P ∈B(b X), and hence βX<1, where b Xis the Luxemburg representation of the r.i. space (X, ||| · |||X). It also follows from Proposition 3 that Q ∈ B(b X). Thus 0 < αX≤βX<1, as desired. (iv) ⇒(iii). Suppose that (iv) holds. Then P,Q ∈ B(b X). Hence we obtain (30) by using Propositions 2 and 3. If f= (fn) is a uniformly integrable martingale, then supn|||fn|||X≤ |||f∞|||Xsince fn≺f∞for all n∈Z+. On the other hand, by (B3), |||f∞|||X≤limn|||fn|||X= supn|||fn|||X. Thus |||f∞|||X= supn|||fn|||Xand (30) can be rewritten as (4). (iii) ⇒(i). Suppose that (iii) holds, and let f= (fn)n∈Z+be a martingale. Applying (4) to the stopped martingale f(k)= (fn∧k)n∈Z+, we see that ckfkkX≤ θFf(k) X≤CkfkkX(k∈Z+). Since θFf(k)↑θFfas k↑ ∞, we conclude from (B3) that clim k→∞ kfkkX≤ kθFfkX≤Clim k→∞ kfkkX, as desired. References [1] A. Antipa, Doob’s inequality for rearrangement-invariant function spaces, Rev. Roumaine Math. Pures Appl. 35(2) (1990), 101–108. [2] C. Bennett and R. Sharpley,“Interpolation of operators”, Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988.
On Some Mean Oscillation Inequalities 189 [3] K. M. Chong and N. M. Rice,“Equimeasurable rearrangements of functions”, Queen’s Papers in Pure and Applied Mathematics 28, Queen’s University, Kingston, Ont., 1971. [4] M. Kikuchi, Characterization of Banach function spaces that preserve the Burkholder square-function inequality, Illinois J. Math. 47(3) (2003), 867–882. [5] M. Kikuchi, New martingale inequalities in rearrangement-invariant function spaces, Proc. Edinb. Math. Soc. (2) 47(3) (2004), 633–657. [6] M. Kikuchi,On the Davis inequality in Banach function spaces, submitted. [7] R. L. Long, Rearrangement techniques in martingale setting, Illinois J. Math. 35(3) (1991), 506–521. [8] T. Shimogaki, Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965), 365–373. Department of Mathematics Toyama University Toyama 930–8555 Japan E-mail address:[email protected] Rebut el 18 d’abril de 2005.