On strongly orthogonal martingales in UMD Banach spaces

Abstract: In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<\infty$ [ \mathbb E |N_t|^p \leq \chi_{p, X}^p \mathbb E |M_t|^p,\;\;\; t\geq 0, ] with the sharp constant $\chi_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range [ \max\Bigl{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr} \leq \max{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}} \leq \chi_{p, X} \leq \min{\beta_{p, X}, \hbar_{p,X}}, ] where $\beta_{p, X}$ is the UMD$p$ constant of $X$, $\hbar{p, X}$ is the norm of the Hilbert transform on $L^p(\mathbb R; X)$, and $\beta_{p, X}^{\gamma,+}$ and $ \beta_{p, X}^{\gamma, -}$ are the Gaussian decoupling constants.