Characterization of uniform PL-convexity of complex Banach space by functions with two variables

We give a new characterization ofq-uniform PL-convexity of complex Banach space by using the existence of a kind of functions with two variables and then prove a sharp weak (1, 1)-type inequality for analytic martingales with values in the Banach space. Foundation item: Supported by the National Natural Science Foundation of China Biography: HUANG Hai-jun (1976-), male, Ph D candidate, research interests are in martingales and functional analysis.