非交换微分从属鞅的平方函数
Square functions for noncommutative differentially subordinate martingales
摘要：
证明了涉及非交换微分从属鞅的不等式。更精确地说，我们证明了：如果$x$是自伴非交换鞅，且$y$弱微分从属于$x$，则$y$允许分解$dy=a +b +c$（分别为. $dy=z +w$），其中$a$、$b$和$c$是适应序列（分别为$z$和$w$是鞅差序列），使得：$$ \Big\| \Big(\sum_{n\geq1} |z_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} |w_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1). $$我们还证明了强型$(p,p)$版本的上述弱型结果$1<p<2$。为了更好地理解非交换微分从属与鞅Hardy空间在$1\leq p<2$时的相互作用，我们还给出了几个新的、独立有趣的带尖常数的鞅不等式.作为我们的方法的副产品，我们获得了新的和建设性的证明的非交换Burkholder-Gundy不等式和非交换Burkholder/Rosenthal不等式$1<p<2$与最佳顺序的常数时，$p \to 1$。
英文摘要：
We prove inequalities involving noncommutative differentially subordinate martingales. More precisely, we prove that if $x$ is a self-adjoint noncommutative martingale and $y$ is weakly differentially subordinate to $x$ then $y$ admits a decomposition $dy=a +b +c$ (resp. $dy=z +w$) where $a$, $b$, and $c$ are adapted sequences (resp. $z$ and $w$ are martingale difference sequences) such that: $$ \Big\| (a_n)_{n\geq 1}\Big\|_{L_{1,\infty}({\mathcal M}\overline{\otimes}\ell_\infty)} +\Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|b_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} \mathcal{E}_{n-1}|c_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1 $$ (resp. $$ \Big\| \Big(\sum_{n\geq1} |z_n|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} + \Big\| \Big(\sum_{n\geq 1} |w_n^*|^2 \Big)^{{1}/{2}}\Big\|_{1, \infty} \leq C\big\| x \big\|_1). $$ We also prove strong-type $(p,p)$ versions of the above weak-type results for $1<p<2$. In order to provide more insights into the interactions between noncommutative differential subordinations and martingale Hardy spaces when $1\leq p<2$, we also provide several martingale inequalities with sharp constants which are new and of independent interest. As a byproduct of our approach, we obtain new and constructive proofs of both the noncommutative Burkholder-Gundy inequalities and the noncommutative Burkholder/Rosenthal inequalities for $1<p<2$ with the optimal order of the constants when $p \to 1$.
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