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ON the DUAL of ORLICZ–LORENTZ SPACE 1. Introduction PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 6, Pages 1645{1654 S 0002-9939(02)05997-X Article electronically published on January 25, 2002 ON THE DUAL OF ORLICZ{LORENTZ SPACE H. HUDZIK, A. KAMINSKA,´ AND M. MASTYLO (Communicated by Dale Alspach) Abstract. A description of the K¨othe dual of the Orlicz{Lorentz space Λ';w generated by an Orlicz function ' and a regular weight function w is pre- sented. It is also shown that in the case of separable Orlicz{Lorentz spaces the regularity condition on w is necessary and sufficient for the coincidence of the Banach dual space with the described K¨othe dual space. 1. Introduction Let (Ω,µ):=(Ω; Σ,µ) be a measure space with the complete and σ-finite measure µ,andletL0(µ) denote the space of all µ-equivalence classes of Σ-measurable functions on Ω with the topology of convergence in measure on µ-finite sets. A Banach space (E;k·kE)issaidtobeaBanach function space on (Ω,µ)ifit is a subspace of L0(µ) such that there exists h 2 L0(µ)withh>0a.e.inΩand the assumptions that f 2 L0(µ), g 2 E and jf|≤jgj a.e. in Ω imply f 2 E and 0 kfkE ≤kgkE. If in addition the unit ball BE = ff : kfkE ≤ 1g is closed in L (µ), then we say that E has the Fatou property. A Banach function space defined on (N; 2N,µ) with the counting measure µ is called a Banach sequence space (on N). A Banach function space E on (Ω,µ)issaidtobesymmetric if for every f 2 0 L (µ)andg 2 E with µf = µg,wehavef 2 E and kfkE = kgkE,whereforany 0 h 2 L (µ),µh is the distribution function defined by µh(t)=µ(f! 2 Ω:jh(!)j >tg);t≥ 0: If E is a Banach function space on (Ω,µ), then the K¨othe dual E0 of E is a Banach function space, which can be identified with the space of all functionals possessing an integral representation, that is, Z 0 f 2 0 k k 0 j j 1g E = g L (µ): g E =sup fg dµ < : kfkE ≤1 Ω It is well known that if E has order continuous norm (i.e., kfnkE ! 0 whenever ∗ 0 E 3 fn # 0), then the dual space E can be naturally identified with E ([8]). In this paper we are interested in the description of the K¨othe duals for symmetric Orlicz{Lorentz spaces defined on either nonatomic or purely atomic measure space. Since after minor modifications the proofs presented in the paper work in essentially Received by the editors September 24, 1999 and, in revised form, April 20, 2000. 1991 Mathematics Subject Classification. Primary 46B10, 46E30. The research of the second and third authors was supported by NATO Collaborative Grant CRG 972918. c 2002 American Mathematical Society 1645 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1646 H. HUDZIK, A. KAMINSKA,´ AND M. MASTYLO the same way in both cases, for simplicity, we consider only the case of Orlicz{ Lorentz spaces defined on (I;m), where either I =(0; 1) or I =(0; 1)andm is the Lebesgue measure. We recall that if ' :[0; 1) ! [0; 1)isanOrlicz function (i.e., a convex function which assumes value zero only at zero) and w : I ! (0; 1)isaweight function (i.e.,R nonincreasing and locally integrable with respect to the measure m and such 1 1 1 that 0 wdm= if I =(0; )), then the Orlicz{Lorentz function space Λ';w on (Ω,µ)isthesetofallf 2 L0(µ) such that Z '(λf ∗)wdm<1 Ω for some λ>0, where for any f 2 L0(µ), f ∗ denotes the nonincreasing rearrange- ment of f defined by ∗ f (t)=inffλ>0: µf (λ) ≤ tg for any t>0 (by convention inf ; = 1). N In the case of counting measure on 2 the Orlicz{Lorentz sequence space λ';w on N is defined by n X1 o ∗ λ';w = x = fx(k)g : '(λx (k)) w(k) < 1 for some λ>0 : k=1 Here w =Pfw(k)g is a weight sequence, a nonincreasing sequence of positive reals 1 1 such that k=1 w(k)= . It is easy to check that Λ';w (resp., λ';w) is a symmetric function space (resp., symmetric sequence space) with the Fatou property, equipped with the norm n Z o kfk =inf λ>0: '(f ∗/λ)wdm≤ 1 ; I respectively n X1 o kxk =inf λ>0: '(x∗(k)/λ)w(k) ≤ 1 : k=1 Note that if w ≡ 1(resp.,w(k) = 1 for all k 2 N), then Λ';w (resp., λ';w)is the Orlicz function space L' (resp., Orlicz sequence space `'). If '(t)=t,then Λ';w (resp., λ';w) is the Lorentz space Λw (resp., λw). We recall that an Orlicz function ' satisfies the ∆2-condition (' 2 ∆2)ifthereexistsC>0 such that '(2t) ≤ C'(t) for all t>0. We will further say that ' is an N-function whenever limt!0 '(t)=t = 0 and limt!1 '(t)=t = 1. We refer to [5] and [7] to study the basic properties of Orlicz{Lorentz spaces as well to the references included therein. In what followsR by a regular weight we mean a weight function w such that, if we t 2 ≥ 2 denote S(t)= 0 w(s) ds for t I,thenS(2t) KS(t) for any t>0 t (0; 1=2) in the case I =(0; 1) ,whereK>1 is independent of t. In the sequence caseP a weight f g ≥ 2 N n w = w(k) is regular if S(2n) KS(n) for any n ,whereS(n)= k=1 w(k) and K>1 is a constant independent of n. It is well known and easy to show that w is regular iff there exists C>0 such that tw(t) ≤ S(t) ≤ Ctw(t) for all t 2 I in the function case and analogously nw(n) ≤ S(n) ≤ Cnw(n) for all n 2 N in the sequence case. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON THE DUAL OF ORLICZ{LORENTZ SPACE 1647 Recall that if ρ : I ! (0; 1) is a concave function, then the Marcinkiewicz space Mρ is defined by R n t o f ∗(s) ds 2 0 k k 0 1 Mρ = f L (m): f Mρ =sup < ; t2I ρ(t) R t and the Marcinkiewicz space MS with S(t)= 0 w(s) ds is the K¨othe dual of Λw, that is, 0 (Λw) = MS with equality of norms (see [3] or [9]). In what follows we will write f g for nonnegative functions f and g whenever C1f ≤ g ≤ C2f for some Cj > 0;j=1; 2. The problem of the description of dual spaces for Lorentz sequence spaces has been considered in [1] and [2]. It was proved there that the regularity of w is a necessary and sufficient condition, in the case p = 1, and a sufficient condition in the case p>1, in order that the K¨othe dual of classical Lorentz sequence space p d(w; p)=λ';w with '(t)=t for t ≥ 0 to consist exactly of those sequences fx(k)g ∗ 1=p 0 for which fx (k)=w(k) g2`p0 ,where1=p +1=p =1. In [1] Allen also raises the question of whether the similar description of the K¨othe dual space is also true for Lorentz function spaces. Reisner [12] answered this question positively and gave a proof which works in essentially the same way in the sequence and in the function spaces. He also proved that regularity of w is necessary for p>1. Following the ideas from [12] we prove in this paper, under the assumption that ' is an N-function satisfying the ∆2-condition, that the regularity of the weight function w is a necessary and sufficient condition for the dual of the Orlicz{Lorentz ∗ space Λ';w on (I;m) to consist exactly of those functions f for which f =w belongs to the Orlicz function space L'∗ on (I;wdm), where '∗ is the Young conjugate of ', i.e., '∗(t)=supfst − '(s): s ≥ 0g for t ≥ 0. We also obtain some partial results for ' being an arbitrary Orlicz function. A similar result is also true for the symmetric Orlicz{Lorentz sequence space λ';w on N. In the proof of the main result we will use the Lozanovski˘ı theorem on the representation of the K¨othe dual space for the Calder´on-Lozanovski˘ı space (see [11]). Let us recall that if (E0;E1) is any couple of Banach function spaces on (Ω,µ)andU denotes the set of all concave and positively homogeneous functions :[0; 1) × [0; 1) ! [0; 1) such that (s; t) = 0 if and only if s = t =0,then the Calder´on{Lozanovski˘ıspace (E0;E1) generated by the couple (E0;E1)anda 0 function 2U consists of all f 2 L (µ) such that jf|≤λψ(jf0j; jf1j)a.e.forsome λ>0andfj 2 Ej, j =0; 1. The space (E0;E1) is a Banach function space on (Ω,µ) (cf. [10], [11]) equipped with the following equivalent norms: k k f j |≤ j j j j k k ≤ g f =inf λ>0: f λψ( f0 ; f1 )a.e.; fj Ej 1;j=0; 1 and k k1 fk k k k j j j j j j g f =inf f0 E0 + f1 E1 : f = ( f0 ; f1 ) ; satisfying the inequalities k k ≤k k1 ≤ k k f f 2 f License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1648 H.
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