Best constants in noncommutative Burkholder inequalities

The paper contains the study of noncommutative Burkholder inequalities. We prove that the constant involved is of order $O(p\mathrm{/}\log p)$ when $p\to \mathrm{\infty }$, which is known to be optimal even in the commutative setting. This answers an open problem raised by Randrianantoanina in 2007, and by Junge and Xu in 2008.