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 $1<p<\infty$ \[ \mathbb E \|N_t\|^p \leq χ_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0, \] with the sharp constant ${\chi }_{p,X}$ being the norm of a decoupling-type martingale transform and being within the range \[ \max\Bigl\{\sqrt{β_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} \leq \max\{β_{p, X}^{γ,+}, β_{p, X}^{γ, -}\} \leq χ_{p, X} \leq \min\{β_{p, X}, \hbar_{p,X}\}, \] where ${\beta }_{p,X}$ is the UMD${}_p$ constant of $X$, ${\hslash }_{p,X}$ 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.