On the intergrability of the martingale square function

Abstract

Letf=(f ₁,f ₂,..) be a martingale. It is proved that theL ₁ norms of sup _n |f _n | and of$\left( {\sum {\left( {f_n - f_{n - 1} } \right)^2 } } \right)^{\tfrac{1}{2}}$ are equivalent. This result completes results of D. L. Burkholder and R. F. Gundy on operators on martingales.