Spaces of LORENTZ type called ORLICZ-LORENTZ spaces are studied. There are given necessary and sufficient conditions for the spaces to be order continuous, separable, RB-spaces and to contain isometric or isomorphic copy of 2m or co. Moreover a criterion for strict convexity of these spaces is found.Introduction. Weighted LORENTZ spaces as a generalization of LORENTZ spaces have been studied by I. HALPERIN in [ 6 ] , [6]. This kind of spaces have found frequent application to problems in interpolation theory and weighted-norm inequalities. Here we shall examine a generalization of the spaces consisting in replacing a power function by a YOUNG one. This generalization seems to be natural as ORLICZ spaces being a generalization of classical LEBESGUE spaces. They will be called ORLICZ-LORENTZ spaces. These spaces have already appeared naturally in interpolation theory as an intermediate space with respect to the couple of the LORENTZ space and the space of bounded functions (part 4 in [12]). They have been also studied in [I13 in the context of interpolation and indices. The papers [4], [7] are devoted to the study of spaces generated by a family of measures and a YOUNG function with parameter. I n a particular case of a family of measures they become ORLICZ-LORENTZ spaces. So some basic results come from the papers 143, [7]. But, since the situation considered in those papers is far more general than here, in the first part we state those basic results (completeness, inclusions, condition A * ) with some necessary comments. The second one is devoted to the role of condition A 2 in the structure of ORLICZ-LORENTZ spaces. Many important properties such as e.g. order continuity, separability, the existence of isomorphic or isometric copy of 1" or e, are expressed in term of condition A s . In the third part we examine an isometric structure of the space, finding a criterion for strict convexity in terms of YOUNG and weighted functions. The methods used in the paper both arise from ORLICZ and LORENTZ spaces.Throughout the paper let R, R, and N denote the sets of reals, nonnegative reals and positive integers respectively. We say that 9 : R, -+ R,, is a YOUNG function if is convex, ~( 0 ) = 0 and q(u) > 0 for u > 0. By w : [0, y ) -+ R,, y I_ 60, denote a nonincreasing, locally integrable function, called a weight function. Let A be the family of 12 1 1 n J t E A , : I/(r)l > 0 -> ?n{t E A , : I/(t)l > 0 ) which means that dh(0) > d,(O), for 3*
Some general results on geometry of Banach lattices are given. It is shown among others that uniform rotundity or rotundity coincide to uniform or strict monotonicity, respectively, on order intervals in positive cones of Banach lattices. Several equivalent conditions on uniform and strict monotonicity are also discussed. In particular, it is proved that in Banach function lattices uniform and strict monotonicity may be equivalently defined on orthogonal elements. It is then applied to show that p-convexification E (p) of E is uniformly monotone if and only if E possesses that property. A characterization of local uniform rotundity of Calderón-Lozanovskii spaces is also presented.
Abstract. We study order convexity and concavity of quasi-Banach Lorentz spaces Λ p,w , where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that Λ p,w contains an order isomorphic copy of l p . We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for Λ p,w . We conclude with a characterization of the type and cotype of Λ p,w in the case when Λ p,w is a normable space. 0. Introduction. The purpose of this paper is to characterize order convexity and concavity in quasi-Banach Lorentz spaces Λ p,w , where w is a locally integrable arbitrary weight and 0 < p < ∞. First results on this topic in Lorentz spaces belong to Creekmore [7], who has studied the spaces L p,q with 1 < p, q < ∞, as well as to Carothers [4] and Reisner [24], who considered the Lorentz spaces Λ p,w with a decreasing weight w and p ≥ 1. These spaces have been further investigated by the authors in [14], where convexity and concavity properties as well as the type and cotype of Λ p,w have been characterized by means of several equivalent integral inequalities as well as by indices of w and its integral W (t) = t 0 w. It is well known that Λ p,w are Banach spaces whenever w is decreasing and p ≥ 1 ([18]). The present article is a continuation of [14]. We extend our study to 0 < p < ∞ and an arbitrary weight w such that Λ p,w is a quasi-Banach space. In this general setting, when the weight is not decreasing and 0 < p < ∞ is arbitrary, different methods and techniques must be used.The paper is organized as follows. In the preliminaries we set up notations and we recall the definitions, notions and results which will be used later on. Among other results, we recall that Λ p,w is a quasi-normed space whenever 2000 Mathematics Subject Classification: Primary 46E30, 46B20; Secondary 46B42, 46B25.
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