LlogL results for the maximal operator in variable L^p spaces

We generalize the classical Llog L inequalities of Wiener and Stein for the Hardy-Littlewood maximal operator to variable Lp spaces where the exponent function p(·) approaches 1 in value. We prove a modular inequality with no assumptions on the exponent function, and a strong norm inequality if we assume the exponent function is log-H¨older continuous. As an application of our approach we give another proof of a related endpoint result due to Hasto. Si generalizzano le classiche disuguaglianze LlogL di Wiener e Stein per l’operatore massimale di Hardy e Littlewood al caso degli spazi con esponenti variabili. Si dimostra una disuguaglianza modulare senza ipotesi sull’esponente e una disuguaglianza in norma con ipotesi di continuità tipo log-Holder.