Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

Abstract

In the present paper we identify those filtered probability spaces $(\Omega ,\mathcal{F},\left({\mathcal{F}}_n\right),\mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $({\mathcal{F}}_n)$ of purely atomic $\sigma$-algebras which determine that the upper ${\ell }^p$ estimates \[ \|f\|_{L^p(Ω,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(Ω,\, X)}+\sum_{n=1}^{\infty} \|Δ_n f\|^p_{L^p(Ω,\, X)}\right),\qquad f\in L^p(Ω,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.

We added: References, additional explanations, improvements. We remove typos and misprints