The UMD Constants for Mixed Normed Lebesgue Spaces

After a short introduction of the unconditional martingale property (UMD property) for Banach spaces, I will give an estimate for the constants appeared in the vector-valued martingales" inequalities for defining this property. This estimate yields in particular a very natural construction of a Banach space (in fact Banach lattice) which is super-reflexive but does not have the UMD property.