Smooth points in the Lorentz spaces G_{p, w} and applications
by
Maciej Ciesielski
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152

Let L⁰ be a space of all Lebesque measurable real extended functions defined over [0, a), where 0 < a ≤ ∞. The distribution function d_f of a function f ∈ L⁰ is given by d_f(l)=m({x ∈ [0, a):|f(x)| > l}), for all l ≥ 0. For any f ∈ L⁰ its decreasing rearrangement is defined as f^*(t)=inf{s > 0:d_f(s) ≤ t}, t > 0. The maximal function of f^* is f^**(t)=[1/t]∫₀^t f^*(s)ds. Let 1 ≤ p < ∞ and w be a measurable positive weight function. The Lorentz space G_{p, w} is a subspace of L⁰ equipped with the norm

∥f∥Gp, w:= æ è ó õ a 0  f**pw ö ø 1/p   .

We investigate the Gâteaux derivative and the smooth points in the Lorentz spaces G_{p, w}. We find supporting functional for each function from G_{p, w} as well as we characterize the closest point, called best approximant, to each of these functions from a closed convex subset.

Date received: February 26, 2008