Grand Lebesgue spaces with respect to measurable functions

Let 1 < p < infinity. Given Omega subset of R-n a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces L-p) (Omega) is given by vertical bar f vertical bar(Lp)) ((Omega)) = (0<epsilon<p-1)sup epsilon(1/p-epsilon) (1/vertical bar Omega vertical bar integral(Omega) vertical bar f vertical bar(p-epsilon) dx)(1/p-epsilon) In this paper we consider the norm vertical bar f(vertical bar Lp)),delta(Omega) obtained replacing epsilon(1/p-epsilon)by a generic nonnegative measurable function delta(epsilon). We find necessary and sufficient conditions on delta in order to get a functional equivalent to a Banach function norm, and we determine the "interesting" class B-p, of functions delta, with the property that every generalized function norm is equivalent to a function norm built with delta is an element of B-p. We then define the L-p),L-delta (Omega) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality. (C) 2013 Elsevier Ltd. All rights reserved.