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On a construction of majorizing measures on subsets of ℝⁿ with special metrics

Jakub Olejnik (2010)

Studia Mathematica

We consider processes Xₜ with values in L p ( Ω , , P ) 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.

On a relation between norms of the maximal function and the square function of a martingale

Masato Kikuchi (2013)

Colloquium Mathematicae

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 f = ( f ) n , 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 | | M g | | X | | M f | | X , then | | S g | | X C | | S f | | X , where C is a constant independent of f and g.

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