The UMD property for Musielak--Orlicz spaces

In this paper we show that Musielak–Orlicz spaces are UMD spaces under the so-called ${\mathrm{\Delta }}_2$ 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.