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.
Categories: math.PR math.CV math.FA