# On strongly orthogonal martingales in UMD Banach spaces

## TL;DR

This paper introduces strongly orthogonal martingales in UMD Banach spaces and establishes sharp inequalities for weakly differentially subordinate pairs, linking constants like UMD and Hilbert transform norms.

## Contribution

It defines strongly orthogonal martingales and derives optimal bounds for their subordinate pairs in UMD Banach spaces, connecting various key constants.

## Key findings

- Established inequalities with sharp constants for strongly orthogonal martingales.
- Linked decoupling constants, UMD constants, and Hilbert transform norms.
- Provided bounds for the decoupling-type martingale transform constant.

## 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.