Square Functions for Noncommutative Differentially Subordinate Martingales

Abstract

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:

$$\begin{aligned} & \| (a_n)_{n \geq 1} \|_{L_{1, \infty} (\mathcal{M} \overline{\otimes} \ell_{\infty})} + \| (\sum_{n \geq 1} \varepsilon_{n-1} |b_{n} |^{2} )^{1/2} \|_{1, \infty} \\ & + \| (\sum_{n \geq 1} \varepsilon_{n-1} |c_{n}^{*} |^{2} )^{1/2} \|_{1, \infty} \leq C \| x \|_1\end{aligned}$$

(resp.

$${\| (\sum_{n \geq 1} |z_n|^2 )^{{1}/{2}}\|_{1, \infty}+ \| (\sum_{n \geq 1} |w_n^{*}|^{2} )^{1/2} \|_{1, \infty} \leq C \| x \|_1).}$$

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 ≤ 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 → 1.