We prove continuity of the Riesz potential operator in optimal couples orf rearrangement invariant function spaces defined in R^n with the Lebesgue measure. An applicationis given to the Hardy-Littlewood maximal operator

We give a decomposition for the dual space of some Banach Function Spaces as the Zygmund space EXP_alpha of the exponential integrable functions, the Marcinkiewicz spaceL^p,infty, and the Grand Lebesgue Space L^p),Theta

Let 1<p<?. Given ??Rn a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces Lp)(?) is given by In this paper we consider the classical norm of the Grand Lebesgue space L^p) obtained considering a generic nonnegative measurable function ?(?). We find necessary and sufficient conditions on ? in order to get a functional equivalent to a Banach function norm, and we determine the "interesting" class Bp of functions ?, with the property that every generalized function norm is equivalent to a function norm built with ??Bp.

We provide regularity properties for the inverse map f^-1 under suitable assumptions on q-distorsion function of f, in bounded domains of R^2.