We prove a weak-type (1,1) inequality for square functions of non-commutativemartingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely,the following non-commutative analogue of a classical result of Burkholderholds: there exists an absolute constant $K>0$ such that if $\cal{M}$ is asemi-finite von Neumann algebra and $(\cal{M}_n)^{\infty}_{n=1}$ is anincreasing filtration of von Neumann subalgebras of $\cal{M}$ then for anygiven martingale $x=(x_n)^{\infty}_{n=1}$ that is bounded in $L^2(\cal{M})\capL^1(\cal{M})$, adapted to $(\cal{M}_n)^{\infty}_{n=1}$, there exist two\underline{martingale difference} sequences, $a=(a_n)_{n=1}^\infty$ and$b=(b_n)_{n=1}^\infty$, with $dx_n = a_n + b_n$ for every $n\geq 1$, \[ |(\sum^\infty_{n=1} a_n^*a_n)^{{1}/{2}}|_{2} + | (\sum^\infty_{n=1}b_nb_n^*)^{1/2}|_{2} \leq 2| x |_2, \] and \[ | (\sum^\infty_{n=1}a_n^*a_n)^{{1}/{2}}|_{1,\infty} + | (\sum^\infty_{n=1}b_nb_n^*)^{1/2}|_{1,\infty} \leq K| x |_1. \] As an application, we obtain theoptimal orders of growth for the constants involved in the Pisier-Xunon-commutative analogue of the classical Burkholder-Gundy inequalities.
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机译：我们证明了同时限于$ L ^ 2 $和$ L ^ 1 $的非可交换mart的平方函数的弱类型（1,1）不等式。更准确地说，是Burkholderholds经典结果的以下非交换类似物：存在一个绝对常数$ K> 0 $，使得如果$ \ cal {M} $是半有限冯诺依曼代数和$（\ cal {M } _n）^ {\ infty} _ {n = 1} $是对$ \ cal {M} $的冯·诺伊曼子代数的递增过滤，然后对于任何给定的mar $ x =（x_n）^ {\ infty} _ {n = 1 } $以$ L ^ 2（\ cal {M}）\ capL ^ 1（\ cal {M}）$为界，适用于$（\ cal {M} _n）^ {\ infty} _ {n = 1} $，存在两个\下划线{martingale差异}序列，$ a =（a_n）_ {n = 1} ^ \ infty $和$ b =（b_n）_ {n = 1} ^ \ infty $，其中$ dx_n = a_n + b_n $每$ n \ geq 1 $，\ [|（\ sum ^ \ infty_ {n = 1} a_n ^ * a_n）^ {{{1} / {2}} | _ {2} + | （\ sum ^ \ infty_ {n = 1} b_nb_n ^ *）^ {1/2} | __2 {leq 2 | x | _2，\]和\ [| （\ sum ^ \ infty_ {n = 1} a_n ^ * a_n）^ {{1} / {2}} | __1，\ infty} + | （\ sum ^ \ infty_ {n = 1} b_nb_n ^ *）^ {1/2} | _ {1，\ infty} \ leq K | x | _1。 \]作为一种应用，我们获得了经典Burkholder-Gundy不等式的Pisier-Xunon-可交换类似物所涉及的常数的最优增长阶。
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