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.

Publication Title

Glasgow Mathematical Journal