Type and Order Convexity of Marcinkiewicz and Lorentz Spaces and Applications (original) (raw)

2005, Glasgow Mathematical Journal

We consider order and type properties of Marcinkiewicz and Lorentz function spaces. We show that if 0 < p < 1, a p-normable quasi-Banach space is natural (i.e. embeds into a q-convex quasi-Banach lattice for some q > 0) if and only if it is finitely representable in the space L p,∞. We also show in particular that the weak Lorentz space L 1,∞ do not have type 1, while a non-normable Lorentz space L 1,p has type 1. We present also criteria for upper r-estimate and r-convexity of Marcinkiewicz spaces. 2000 Mathematics Subject Classification. 46A16, 46B03, 46B20, 46E30. 1. Introduction. In this note we study the order convexity and type of Marcinkiewicz and Lorentz function spaces. The space weak L p or L p,∞ is well-known to be p-normable if 0 < p < 1, but is q-convex as a lattice when 0 < q < p (see [4] and [5]). We prove that a p-normable quasi-Banach space X embeds into a p-normable quasi-Banach lattice which is r-convex for some r > 0 (i.e. X is natural) if and only if X is finitely representable in L p,∞ (0, 1). We then consider more general Lorentz and Marcinkiewicz spaces. In [6] it was proved that if a quasi-Banach space (X, •) has type 0 < p < 1, then • is a p-norm, and if X has type p > 1 then X is normable. It was also shown that there exist quasi-Banach spaces that have type 1, but they are not normable. In this note we show that Marcinkiewicz spaces have type 1 if and only if they are 1-convex (that is normable), while the class of Lorentz spaces with type 1 coincides to the class of those spaces satisfying an upper 1-estimate. In consequence, there exist Lorentz spaces with type 1 that are not normable. Let us start with basic definitions and notation. Let ‫,ޒ‬ ‫ޒ‬ + and ‫ގ‬ denote the sets of all real, nonnegative real and natural numbers, respectively. Let r n : [0, 1] → ‫,ޒ‬ n ∈ ‫,ގ‬ be Rademacher functions, that is r n (t) = sign (sin 2 n π t). A quasi-Banach space X has type 0 < p ≤ 2 if there is a constant K > 0 such that, for any choice of finitely many The first author acknowledges support from NSF grant 0244515.