On Weak Lebesgue ( Marcinkiewicz ) Spaces with Variable Exponents

Abstract

In this memory, we study the so-called Marcinkiewicz spaces ( weak Lebesgue spaces ) with variable exponents which are one of the first generalizations of the Lebesgue spaces. In the framework of Marcinkiewicz spaces we will study, among other topics, embedding results, convergence in measure, interpolation results, and the question of normability of the space. We also show a Fatou type lemma for Marcinkiewicz spaces as well as the completeness of the quasi-norm. The Lyapunov inequality and the H¨older inequality are shown to hold.