Type and order convexity of Marcinkiewicz and Lorentz spaces and applications
We consider order and type properties of Marcinkiewicz and Lorentz function spaces. We show that if 0 < p < l, 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 L1,∞ do not have type 1, while a non-normable Lorentz space L1,p has type 1. We present also criteria for upper r-estimate and r-convexity of Marcinkiewicz spaces.
Glasgow Mathematical Journal
Kalton, N., & Kamińska, A. (2005). Type and order convexity of Marcinkiewicz and Lorentz spaces and applications. Glasgow Mathematical Journal, 47 (1), 123-137. https://doi.org/10.1017/S0017089504002204