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.

This paper focuses on two-tier city logistics systems for advanced management of urban freight activities and, in particular, on the first layer of such systems where freight is moved from distribution centers on the outskirts of the city to satellite platforms by urban vehicles, from where it will be distributed to customers by a different fleet of dedicated vehicles. We address the issue of planning the services of this first tier system, that is, select services, their routes and schedules, and determine the itineraries of the customer-demand flows through these facilities and services.