On spaces of integrable functions associated to vector measures and limiting real interpolation

J. Math. Anal. Appl. 540 (2024) 128573 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa Regular Articles On spaces of integrable functions associated to vector measures and limiting real interpolation Antonio Fernández a, Antonio Manzano b,∗,1 aDpto. Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092-Sevilla, Spain bDepartamento de Matemáticas y Computación, Escuela Politécnica Superior, Universidad de Burgos, Calle Villadiego s/n, 09001 Burgos, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 29 March 2024 Available online 3 June 2024 Submitted by H. Frankowska Keywords: Extreme interpolation spaces Vector measures Lorentz-Zygmund spaces Optimal domain p-th power factorable operators Bidual (p, q)-power-concave operators We investigate which spaces are obtained when considering the limiting class of real interpolation spaces (0, q; J)for ordered Banach couples of spaces of (scalar) integrable functions with respect to a vector measure m, defined on a σ-algebra, with values in a Banach space. If mis in particular a finite positive scalar measure, previous known results are derived from ours. Furthermore, we study the interpolation of p-th power factorable operators by the extreme real interpolation method (1, q; K). We also deduce interpolation results for the (1, q; K)-method that apply to other related classes of operators to p-th power factorable operators, such as bidual (p, q)-power-concave operators and q-concave operators. © 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons .org /licenses /by -nc /4 .0/). 1. Introduction Among all interpolation methods, the real method (A0, A1)θ,q, 0 <θ<1and 1 ≤q≤∞, has been the most studied. A reason for this is, probably, its flexible construction. Thus, there exist different equivalent definitions of this method, being specially prominent those ones using the K-and J-functionals of Peetre (see [3], [4], [5]and [45]). Moreover, the definition of the real method (A0, A1)θ,q has been generalized in several directions. For instance, an extension of (A0, A1)θ,q consists in replacing in its construction the function tθby a more general function f(t)(see [38], [39]and [44]). To illustrate an advantage of the real method with a function parameter (A0, A1)f,q we will mention that working with a couple of Lebesgue spaces one obtains a larger class of spaces than Lebesgue and Lorentz spaces provided by the real method (A0, A1)θ,q. In fact, if (Ω, Σ, μ)is a positive *Corresponding author. E-mail addresses: [email protected] (A. Fernández), [email protected] (A. Manzano). 1The second named author was supported in part by UCM Grant PR3/23-30811. https://doi.org/10.1016/j.jmaa.2024.128573 0022-247X/© 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons .org /licenses /by -nc /4 .0/). 2A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 σ-finite measure space, applying the real method with a function parameter to the couple (L∞, L1)it holds that L∞,L 1f,q =Lp,q(logL)α, when f(t) =t1 p(1 +| log t|)−α, 1 <p <∞, 1 ≤q≤∞and α∈R. Some authors have investigated (see [38], [31]and [32]) the logarithmic interpolation spaces (A0, A1)θ,q,A that correspond to the special case where f(t) =tθA(t), being A(t)a broken logarithmic function, that is, A =(α0,α ∞)∈R2, (t) =1 +| log t|and A(t):=α0(t),if 0 <t≤1, α∞(t),if 1 <t<∞. In the construction of (A0, A1)θ,q,Ait is possible to consider 1 ≤q≤∞and not only 0 <θ<1, but θeven taking the values 0and 1. For these limiting values θ=0and θ=1, the extra function A(t)is essential to get a meaningful definition of (A0, A1)θ,q,A. However, when (A0, A1)is an ordered Banach couple, that is when A0⊆A1(where “⊆” means continuous inclusion), then just making a natural modification in the definition of the real method (A0, A1)θ,q, it is possible to define two limiting classes of real interpolation spaces, (A0, A1)1,q;Kand (A0, A1)0,q;J, without involving any auxiliary function (see Section 2for precise definitions). The spaces (A0, A1)1,q;Kwere investigated for the first time by Gomez and Milman [37]. More recently, the extreme interpolation methods (A0, A1)1,q;Kand (A0, A1)0,q;Jhave been studied by Cobos, Fernández- Cabrera, Kühn and Ullrich [11], Cobos, Fernández-Cabrera and Mastyło [12], Cobos and Kühn [6]and Cobos, Fernández-Cabrera and Martínez [13](see also [10], [14], [15], [7], [8]and [9]). It is important to point out that (A0, A1)0,1;J=A0and for the rest of the values 1 <q≤∞, the space (A0, A1)0,q;Jis very close to A0. On the other hand, (A0, A1)1,∞;K=A1and for other values 1 ≤q<∞the space (A0, A1)1,q;K is very near A1. These facts constitute an important difference in the theory. As an intrinsic problem related to interpolation theory it has been constantly investigated the description of the spaces obtained by applying an interpolation method to concrete compatible couples of spaces. In the recent years, this question has awakened a lot of attention when the couple is formed by spaces of (scalar) integrable functions associated to vector measures (see, for instance, [34], [22]and [24]for the complex interpolation method; [35]and [25]for the real method; [27], [28]and [42]for the real method with a function parameter; and [21], [26], [29]and [30], among others, for related questions). This interest can be better understood by some applications that such spaces have. Thus, it is worthy to mention that the space L1(m), of scalar integrable functions with respect to a vector measure m,defines a broad class of Banach function spaces. More precisely, every order continuous Banach lattice Xwith a weak order unit can be represented (order isometrically) as an L1(m)space of a vector measure mdefined on a σ-algebra, as proved by Curbera [16, Theorem 8]. If the order continuity fails but Xhas a Fatou type property and a weak order unit belonging to its order continuous part, then X can be identified (order isometrically) with a space L1 w(m)of weakly integrable functions with respect to a vector measure mon a σ-algebra (see [18, Theorem 2.5]). For a p-convex Banach lattice X, analogous representations as an Lp(m)or an Lp w(m)space hold, depending on similar assumptions considered above (see [33]and [19]). Another significant application of integrable function spaces with respect to a vector measure refers to the so-called optimal domain of an operator Tacting from a Banach function space Xinto a Banach space E. Namely, if Xis an order continuous Banach function space over a finite measure space (Ω, Σ, μ)and T satisfies a natural condition which makes mTbe a vector measure (mTis associated to Tby mT(A) := T(χA), where χAis the characteristic function of A ∈Σ), the space L1(mT)is the optimal domain for T within the class of order continuous Banach function spaces, that is, L1(mT)is the largest space (in that class of spaces) to which Tcan be extended as a continuous operator, still with values in E(see [17]or [43, Theorem 4.14]). A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 3 The space Lp(mT)has also an important optimality property when Tis a p-th power factorable operator from Xinto E(see [43, Chapter 5]), in the sense that Lp(mT)is maximal among all order continuous Banach function spaces Ythat continuously contain Xand such that Thas an E-valued extension from Y into Ewhich is itself p-th power factorable (see [43, Theorem 5.11]). The class of p-th power factorable operators coincides with the class of operators that can be extended to the p-th power space X[p]of the Banach function space Xwhere the operator acts. These operators have turned out to be useful for analyzing some factorization properties of operators between Banach function spaces (see [43, Chapter 6] and [36]). In addition, relevant operators coming from Fourier analysis as convolution operators and the Fourier transform become p-th power factorable in certain cases (see [43, Section 7.5]). As far as we know, none description has been given in the literature regarding the spaces obtained when applying limiting interpolation methods to Banach couples formed by function spaces associated to a vector measure. Nothing is known either about interpolation of p-th power factorable operators by limiting methods. In this paper we start the research on these questions. Namely, after this introduction and some necessary preliminaries (Section 2), we establish results for the (0, q; J)-method in the case of the first question (Section 3), and theorems that apply to the (1, q; K)-method in relation to the second issue (Section 4). 2. Notation and basic definitions Throughout the paper all functions will be R-valued and all Banach spaces will be over R. If Xand Y are Banach spaces, we will put X=Ywhenever Xand Yare equal in the algebraic and topological sense (their norms are equivalent). By X⊆Ywe will mean that Xis continuously embedded into Y. As usual, given two quantities ν, τdepending on certain parameters, we will write ντif there exists a constant c >0 independent of appropriate parameters such that ν≤cτ. When ντand also τν, we will put ν≃τ. We will consider ordered Banach couples, that is to say couples ¯ A=(A0, A1)such that A0, A1are Banach spaces with A0⊆A1. We recall that, for each t >0, Peetre’s K-and J-functionals are defined by K(t, a)=K(t, a;A0,A 1):=inf{a0A0+ta1A1:a=a0+a1,a j∈Aj},a∈A1, and J(t, a)=J(t, a;A0,A 1):=max{aA0,taA1},a∈A0. The space ¯ A0,q;J=(A0, A1)0,q;J, 1 ≤q≤∞, is formed (see [11, Definition 3.1]) by all the elements a ∈A1for which there exists a strongly measurable function u(t)with values in A0such that a =∞ 1u(t)dt t (convergence in A1) and ⎛ ⎝ ∞ 1 J(t, u(t))qdt t⎞ ⎠ 1 q <∞sup t>1 {J(t, u(t))}<∞,if q=∞. We set a¯ A0,q;Jas the infimum of the last quantity over all possible representations u(t)of asatisfying the above conditions. Note that (A0, A1)0,q;Jis an intermediate Banach space with respect to (A0, A1)for any 1 ≤q≤ ∞and (A0, A1)0,1,J =A0(see [11, Lemma 3.2]). Furthermore, (A0, A1)0,q;Jcoincides (with the usual modification if q=∞) with the collection of all those a ∈A1such that a =∞ n=1 un(convergence in A1), with (un) ⊂A0and (∞ n=1 J(2n,u n)q)1 q<∞, being the norm a¯ A0,q;Jequivalent to a0,q := 4A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 inf ∞ n=1 J(2n,u n)q1 q:a=∞ n=1 un. As a straightforward consequence, (0, q; J)-spaces are ordered in the sense that (A0, A1)0,p;J⊆(A0, A1)0,q;Jif p ≤q. We also recall that the space ¯ A1,q;K=(A0, A1)1,q;K, 1 ≤q≤∞, consists of those a ∈A1which have a finite norm a1,q;K:= ⎛ ⎝ ∞ 1K(t, a) tqdt t⎞ ⎠ 1 qa1,q;K:= sup t>1K(t, a) t,if q=∞. The space (A0, A1)1,q;Kis an intermediate Banach space with respect to (A0, A1)for every 1 ≤q≤∞ and (A0, A1)1,∞,K =A1(see [37]and [11, Section 7]). Furthermore, (A0, A1)1,q;Kcoincides (with the usual modification if q=∞) with the set of elements a ∈A1such that a1,q := ∞ n=1 [2−nK(2n,a)]q1 qis finite. In addition, the norm a1,q;Kis equivalent to a1,q. Hence, it follows that (1, q; K)-spaces are increasing with q, that is, (A0, A1)1,p;K⊆(A0, A1)1,q;Kif p ≤q. It is worth mentioning that (0, q; J)-spaces have an equivalent description by using the K-functional, as it is shown in [11, Theorem 4.2]. Namely, it holds by [11, Theorem 4.2] that (A0,A 1)0,q;J=(A0,A 1)log,q;K,1<q≤∞,(1) where (see [11, Definition 4.1]) the space ¯ Alog,q;K=(A0, A1)log,q;K, 1 <q≤∞, consists of all elements a ∈A1for which the following norm is finite alog,q;K:= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ ∞ 1K(t, a) 1+logtqdt t⎞ ⎠ 1 q ,1<q<∞, sup t>1K(t, a) 1+logt,q=∞. Analogously, the space (A0, A1)1,q;Kcan be described by means of the J-functional (see [11, Theorem 7.6]). In what follows, by a Banach function space Xon a given finite measure space (Ω, Σ, μ), or on μfor short, we mean that Xis a lattice ideal of the space of (equivalence classes of) measurable functions L0(μ), endowed with a complete norm  · Xthat is compatible with the μ-a.e. order and such that L∞(μ) ⊆X. A Banach function space Xis said to be order continuous if for every sequence (fn)in Xsuch that 0 ≤fn↓0 pointwise, it holds that fnX↓0. Next we will recall some basic definitions and results on integration with respect to vector measures. Let Ebe a Banach space and m :Σ →Ebe a countably additive vector measure, where Σis a σ-algebra of subsets of some nonempty set Ω. Let E∗denote the dual space of E. The semivariation of mis the set function m :Σ −→ [0, ∞) defined by m(A):=sup{|m, x∗|(A):x∗E∗≤1},A∈Σ, where |m, x∗| is the total variation measure of the scalar measure m, x∗, given by m, x∗(A) := m(A), x∗for A ∈Σ. It is well-known that 1 2m(A)≤sup {m(B)E:B⊆A, B ∈Σ}≤m(A), for every set A ∈Σ. For the particular case when mis a finite positive scalar measure, the semivariation mand the measure mcoincide. A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 5 A set A ∈Σis called m-null if m(A) =0. A Rybakov (control) measure for mis a measure defined as |m, x∗| for some x∗∈E∗, satisfying that |m, x∗|(A) =0if and only if Ais a m-null set. Such a measure always exists (see for example [43, p. 108]). Let L0(m)denote the space of all measurable functions f:Ω −→ R. Two functions f, g∈L0(m) will be identified if are equal m-a.e., that is, if {w∈Ω :f(w) =g(w)}is an m-null set. We also recall that f∈L0(m)is said to be weakly integrable (with respect to m) if f∈L1(|m, x∗|)for all x∗∈E∗. The space L1 w(m)of all (m-a.e. equivalence classes of) weakly integrable functions is a Banach function space on every Rybakov (control) measure for m, when endowed with the norm fL1 w(m):= sup ⎧ ⎨ ⎩ Ω |f|d|m, x∗| :x∗E∗≤1⎫ ⎬ ⎭. A weakly integrable function fis said to be integrable (with respect to m) if for every A ∈Σ there exists an element of Edenoted by Afdm (called integral of fover A) such that Afdm,x∗=Afdm, x∗for all x∗∈E∗. The space L1(m)of all (m-a.e. equivalence classes of) integrable functions is an order continuous closed ideal of L1 w(m), and in general L1(m) L1 w(m). If 1 <p <∞, a function f∈L0(m)is said to be weakly p-integrable (with respect to m) if |f|p∈L1 w(m), and p-integrable (with respect to m) if |f|p∈L1(m). We will denote by Lp w(m)the space of (m-a.e. equivalence classes of) weakly p-integrable functions and by Lp(m)the space of (m-a.e. equivalence classes of) p-integrable functions. Obviously we have that Lp(m) ⊆Lp w(m). The natural norm for both spaces is given by fLp w(m):= sup ⎧ ⎪ ⎨ ⎪ ⎩⎛ ⎝ Ω |f|pd|m, x∗|⎞ ⎠ 1 p :x∗E∗≤1⎫ ⎪ ⎬ ⎪ ⎭,f∈Lp w(m). We remark that, when mis a finite positive scalar measure, both spaces Lp w(m)and Lp(m)coincide with the classical (scalar) Lebesgue space Lp. In addition, let L∞(m)be the Banach space of all (m-a.e. equivalence classes of) essentially bounded functions equipped with the supremum norm. It holds that L∞(m) ⊆Lp(m), for any 1 ≤p <∞. We refer to the paper [33]and the monograph [43]for much more additional information on spaces Lp(m)and Lp w(m). Finally, we recall the definition of the Lorentz-Zygmund space with respect to a vector measure m. For 1 ≤p, q≤∞and α∈R, the space Lp,q(log L)α(m)is defined as those functions f∈L0(m)for which the quantity fLp,q (log L)α(m):= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ ∞ 0t1 p(1 + |log t|)αf∗(t)qdt t⎞ ⎠ 1 q ,1≤q<∞, sup t>0t1 p(1 + |log t|)αf∗(t),q=∞, is finite. Here f∗stands for the decreasing rearrangement of fwith respect to the vector measure m,given by f∗(t) := inf{s >0 :mf(s) ≤t}, for t >0, where mf(s) := m({w∈Ω :|f(w)| >s}). Note that the function f∗is non-increasing, right-continuous, and also f∗(t) =0for any t ≥m(Ω). If α=0, the space Lp,q(log L)α(m)coincides with the space Lp,q(m), introduced in [35]. For p =q, the space Lp,p(m)is simply denoted by Lp(m). As it has been pointed out in [35], in general, the spaces Lp(m)and Lp(m)do not coincide. However, for every 1 ≤p <∞, all the following continuous inclusion hold (see [35, Proposition 7]) 6A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 L∞(m)⊆Lp,1(m)⊆Lp(m)⊆Lp(m)⊆Lp w(m)⊆Lp,∞(m). 3. Interpolation of integrable function spaces with respect to vector measures by limiting (0, q; J)-methods We begin by remembering two estimates for the K-functional that we will use in this section. In the next proposition inequality (2) (respectively, inequality (3)) is a consequence of [35, Proposition 8 and Remark 9] (respectively, [35, Proposition 10 and Lemma 3]) and [3, Chapter 5, Proposition 1.2]. Proposition 3.1. If f∈L1 w(m), then f∗(t)Kt−1,f;L∞(m),L 1 w(m).(2) On the other hand, if f∈L1(m), then Kt, f;L∞(m),L 1(m)t t−1 0 f∗(s)ds. (3) Our first theorem shows that when interpolating the couples L∞(m),L 1(m)and L∞(m),L 1 w(m)by the (0, q; J)-method, we obtain the Lorentz-Zygmund space L∞,q(log L)−1(m). The proof uses some ideas of [27, Theorem 3] and also equality (1)and Proposition 3.1. Theorem 3.2 provides a version of [11, Corollary 4.3] in the setting of vector measures. Theorem 3.2. For 1 <q≤∞, it holds that L∞(m),L 1(m)0,q;J=L∞(m),L 1 w(m)0,q;J=L∞,q(log L)−1(m). Proof. We set mΩ:= m(Ω). Since L1(m) ⊆L1 w(m), it follows that L∞(m),L 1(m)0,q;J⊆L∞(m),L 1 w(m)0,q;J. In order to see that L∞(m),L 1 w(m)0,q;J⊆L∞,q(log L)−1(m),(4) we assume first that 1 <q<∞. Namely, take any f∈L∞(m),L 1 w(m)0,q;J. According to inequality (2) in Proposition 3.1, we have that fL∞,q (log L)−1(m)=⎛ ⎝ mΩ 0(1 + |log t|)−1f∗(t) qdt t⎞ ⎠ 1 q ⎛ ⎝ mΩ 0(1 + |log t|)−1K(t−1,f;L∞(m),L 1 w(m)) qdt t⎞ ⎠ 1 q . The change of variables u =mΩ tgives that A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 7 fL∞,q (log L)−1(m)⎛ ⎝ ∞ 1!1+"""log mΩ u"""#−1 Ku mΩ ,f;L∞(m),L 1 w(m)qdu u⎞ ⎠ 1 q ≤max 1,1 mΩ⎛ ⎝ ∞ 1!1+"""log mΩ u"""#−1 K(u, f;L∞(m),L 1 w(m))qdu u⎞ ⎠ 1 q . In addition, 1+|log u|=1+""""log u mΩ ·mΩ""""≤1+""""log u mΩ""""+|log mΩ|≤1+""""log u mΩ""""(1 + |log mΩ|) and so !1+"""log !mΩ u#"""#−1=1+""""log u mΩ""""−1 ≤1+|log mΩ| 1+|log u|.(5) Therefore, (5)and (1) yield fL∞,q (log L)−1(m)max 1,1 mΩ(1 + |log mΩ|)⎛ ⎝ ∞ 1$Ku, f;L∞(m),L 1 w(m) 1+logu%qdu u⎞ ⎠ 1 q =max1,1 mΩ(1 + |log mΩ|)f(L∞(m),L1 w(m))log,q;K≃f(L∞(m),L1 w(m))0,q;J. When q=∞the inclusion (4)can be established by using the same reasoning with menor modifications. Now we will prove that if 1 <q≤∞, L∞,q(log L)−1(m)⊆L∞(m),L 1(m)0,q;J.(6) First we suppose that 1 <q<∞. Define the function W(t):= 1 t(1 + |log t|)q. In order to use Hardy inequality for non-increasing functions (see [1, Theorem 1.7] and [40, Chapter 10]), let us check that ∞ r W(t) tqdt ≤q−1 rq r 0 W(t)dt, r > 0.(7) Since t → t(1 +| log t|)is a non-decreasing function on (0, +∞), we have that ∞ r W(t) tqdt = ∞ r 1 tq+1(1 + |log t|)qdt ≤1 rq−1(1 + |log r|)q−1 ∞ r 1 t2(1 + |log t|)dt ≤1 rq−1(1 + |log r|)q−1 ∞ r 1 t2dt =1 rq(1 + |log r|)q−1. In other words, ∞ r W(t) tqdt ≤1 rq(1 + |log r|)q−1,for every r>0.(8) 8A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 On the other hand, we note that r 0 W(t)dt ≥1 q−1·1 (1 + |log r|)q−1,for each r>0. (9) In fact, if 0 <r≤1, it is clear that r 0 W(t)dt = r 0 1 t(1 + |log t|)qdt =1 q−1·1 (1 + |log r|)q−1, and, for r>1, r 0 W(t)dt = r 0 1 t(1 + |log t|)qdt ≥ 1 0 1 t(1 + |log t|)qdt =1 q−1. Then, from (8)and (9), we have that ∞ r W(t) tqdt ≤1 rq(1 + |log r|)q−1≤q−1 rq r 0 W(t)dt, r > 0. Hardy inequality with W(t) implies that, for every f∈L∞,q(log L)−1(m), ⎛ ⎝ ∞ 0 1 t(1 + |log t|)q⎡ ⎣1 t t 0 f∗(s)ds⎤ ⎦ q dt⎞ ⎠ 1 q ⎛ ⎝ ∞ 0f∗(t) 1+|log t|qdt t⎞ ⎠ 1 q =fL∞,q (log L)−1(m).(10) This in particular yields that the function 1 t t 0 f∗(s)ds is finite a.e. Hence, it can be easily deduced that f∈L1(m)and so f∈L1(m). Thus, for any f∈L∞,q(log L)−1(m), it follows from (1), inequality (3) in Proposition 3.1, and (10)that f(L∞(m),L1(m))0,q;J≃f(L∞(m),L1(m))log,q;K=⎛ ⎝ ∞ 1$Kt, f;L∞(m),L 1(m) 1+logt%qdt t⎞ ⎠ 1 q ⎛ ⎝ ∞ 1 ⎡ ⎣tt−1 0f∗(s)ds 1+logt⎤ ⎦ q dt t⎞ ⎠ 1/q =⎛ ⎝ 1 0 1 u(1 −log u)q⎡ ⎣1 u u 0 f∗(s)ds⎤ ⎦ q du⎞ ⎠ 1 q fL∞,q (log L)−1(m). Then (6)is proved when 1 <q<∞. Finally, we will see that (6)also holds if q=∞. Observe that u 0 (1 + |log s|)ds ≤2u(1 + |log u|),for each u>0.(11) A. Fernández, A. Manzano / J. Math. Anal. Appl. 540 (2024) 128573 9 Given f∈L∞,∞(log L)−1(m), from (11)it follows that fL1(m)= mΩ 0 f∗(t)dt = mΩ 0 f∗(t) 1+|log t|(1 + |log t|)dt ≤2mΩ(1 + |log mΩ|)fL∞,∞(log L)−1(m), and so f∈L1(m) ⊆L1(m). By (1), inequality (3)in Proposition 3.1, and (11), we have that f(L∞(m),L1(m))0,∞;J≃f(L∞(m),L1(m))log,∞;K=sup t>1Kt, f;L∞(m),L 1(m) 1+logt* sup t>1⎧ ⎨ ⎩ tt−1 0f∗(s)ds 1+logt⎫ ⎬ ⎭=sup 0<u<1u 0f∗(s)ds u(1 −log u)* =sup 0<u<1u 0 f∗(s) 1−log s(1 −log s)ds u(1 −log u)* ≤sup 0<u<1u 0(1 −log s)ds u(1 −log u)*fL∞,∞(log L)−1(m) ≤2fL∞,∞(log L)−1(m), and the validity of (6)is proved for q=∞. Remark 3.3. Regarding the inclusion (6)in the proof of Theorem 3.2, we note that applying [2, Theorem 6.4 (6.7)] with λ =1, a =qand α=−1, it follows that 1 0 ⎡ ⎣1 t(1 −log t) t 0 f∗(s)ds⎤ ⎦ q dt t 1 01 1−log tf∗(t)qdt t≤fq L∞,q (log L)−1(m), for any f∈L∞,q(log L)−1(m). As a consequence, it is also possible to deduce that f∈L1(m) ⊆ L1(m). 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