On the geometric structure of Lorentz and Marcinkiewicz spaces

We explore the geometric properties of Lorentz and Marcinkiewicz spaces. In the course of this exploration, the increasing rearrangement of a function is defined as a counterpart of the well known notion of the decreasing rearrangement. The properties of the increasing rearrangement are studied, and Hardy-Littlewood type inequalities are proved. The connection between the decreasing and increasing rearrangements is also shown. Both types of rearrangements are then applied to study the convexity and concavity constants in Lorentz and Marcinkiewicz spaces.;The Lorentz function and sequence spaces Lambdap,w and d(w,p), 0 < p < infinity, associated to both increasing and decreasing weights w are considered, and in each case exact q-convexity and q-concavity constants are determined. The corollaries on the constants for both Marcinkiewicz spaces MW and the classical Lorentz function spaces Lp,q are then derived. Considering a special case of increasing weight sequences leads to a discussion of the q-convexity of the resulting Lorentz sequence space. These results ultimately lead to the exact convexity constants for the classical Lorentz sequence spaces l1,q.;Finally, the smooth and extreme points of the unit ball in the Marcinkiewicz spaces MW and M0W are characterized. A function in a unit ball of MW is a smooth point if and only if the norm is attained at exactly one point 0 < a < infinity and some limit conditions are satisfied. In the M0W case, the limit conditions are satisfied automatically. Furthermore, if the decreasing rearrangement of a function in the unit ball of MW is equal to the weight, then it is an extreme point, while if the weight is strictly decreasing, the decreasing rearrangement of any extreme point must be equal to the weight. The unit ball of M0W does not have any extreme points.