On a class of processes with multiplicity .
We consider processes Xₜ with values in and “time” index t in a subset A of the unit cube. A natural condition of boundedness of increments is assumed. We give a full characterization of the domains A for which all such processes are a.e. continuous. We use the notion of Talagrand’s majorizing measure as well as geometrical Paszkiewicz-type characteristics of the set A. A majorizing measure is constructed.
Let Ω be a nonatomic probability space, let X be a Banach function space over Ω, and let ℳ be the collection of all martingales on Ω. For , let Mf and Sf denote the maximal function and the square function of f, respectively. We give some necessary and sufficient conditions for X to have the property that if f, g ∈ ℳ and , then , where C is a constant independent of f and g.