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, for all \(1<p<\infty\), \[\mathbb E \|N_t\|^p \leq \chi_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0,\] with the sharp constant \(\chi_{p, X}\) being the norm of a decoupling-type martingale transform and lying in the range \[\begin{aligned} \textstyle\max\Bigl\{\sqrt{\beta_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} &\leq \max\{\beta_{p, X}^{\gamma,+}, \beta_{p, X}^{\gamma, -}\}\\& \leq \chi_{p, X} \leq \min\{\beta_{p, X}, \hbar_{p,X}\},\end{aligned}\] 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.

2010 AMS Mathematics Subject Classification: Primary 60G44, 60H05; Secondary 60B11, 32U05.

Keywords and phrases: strongly orthogonal martingales, weak differential subordination, UMD, sharp estimates, decoupling constant, martingale transform, Hilbert transform, diagonally plurisubharmonic function.