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 00) if and only if it is finitely representable in the space L(p,infinity). We also show in particular that the weak Lorentz space L(1,infinity) 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.