The UMD property for Musielak--Orlicz spaces

Abstract

In this paper we show that Musielak--Orlicz spaces are UMD spaces under the so-called ${\Delta }_2$ condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak--Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that reflexive variable Lebesgue spaces $L^{p(\cdot )}$ are UMD spaces.

minor revision