On some Banach spaces of martingales associated with function spaces (Q2905094)

Two Banach spaces of martingales \({\mathcal K}(X,{\mathcal F})\) and \(\mathbb{K} (X,{\mathcal F})\) are associated with a Banach function space \(X\) and a filtration \(\mathcal F\). The main result of the paper is a characterization of Banach function spaces \(X\) such that \({\mathcal K}(X,{\mathcal F})=\mathbb{K} (X,{\mathcal F})\) for every filtration \(\mathcal F\). The paper has four sections and two appendices in which some results necessary for the study are given. After preliminaries and setting the problem, in the third section, a characterization of those Banach function spaces \(X\) is given such that the two considered spaces of martingales are equal. For a pair \((\mathcal F,\mathcal G)\) of filtrations \(\mathcal F\) and \(\mathcal G\), an operator \(T_{(\mathcal F,\mathcal G)}\) from the space \({\mathcal M}_u(\mathcal F)\) of real-valued uniformly integrable martingales with respect to \(\mathcal F\) into the space \({\mathcal M}_u(\mathcal G)\) is presented. In the fourth section of the paper, a characterization of the function spaces \(X\) such that the restriction of the operator \(T_{(\mathcal F,\mathcal G)}\) to \({\mathcal K}(X,{\mathcal F})\) is a bounded linear operator from \({\mathcal K}(X,{\mathcal F})\) into \({\mathcal K}(X,{\mathcal G})\) for every pair \((\mathcal F,\mathcal G)\) of filtrations is given.