Resumen de On Doob’s inequality and Burkholder’s inequality in weak spaces

Masato Kikuchi

• Let X be a Banach function space over a nonatomic probability space. We associate with X the weak space {\mathrm {w}}\text {-}{X}, which is a quasi-Banach space defined in a natural way. We give necessary and sufficient conditions for Doob’s inequality in {\mathrm {w}}\text {-}{X} to be valid, and necessary and sufficient conditions for Burkholder’s inequality in {\mathrm {w}}\text {-}{X} to be valid.