Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm (2212.07804v3)

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(\Omega,\, X)}^p\leq C^p\left( |\mathbb{E} f|\mathcal{F}0|^p{L^p(\Omega,\, X)}+\sum_{n=1}^{\infty} |\Delta_n f|^p_{L^p(\Omega,\, X)}\right),\qquad f\in L^p(\Omega,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}.