Weak $\Phi$-inequalities for the Haar system and differentially subordinated martingales

Abstract

For a wide class of Young functions $\Phi:[0,\infty)\to[0,\infty)$, we determine the best constant $C_{\Phi}$ such that the following holds. If $(h_{k})_{k\geq 0}$ is the Haar system on $[0,1]$, then for any vectors $a_{k}$ from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_{k}\in \{-1,1\}$, $k=0, 1, 2,\ldots$, we have \begin{equation*} \left|\left\{x\in [0,1]:\left|∑_{k=0}^{n} ɛ_{k}a_{k}h_{k}(x)\right|≥ 1\right\}\right|≤ C_{Φ} ∫_{0}^{1}Φ\left(\left|∑_{k=0}^{n} a_{k}h_{k}(x)\right|\right)\mathrm{d}x,\quad n=0,1,2,…. \end{equation*} This is generalized to the sharp weak-$\Phi$ inequality \begin{equation*} \mathbf{P}(\sup_{t≥ 0}|Y_{t}|≥ 1)≤ C_{Φ}\sup_{t≥ 0}\mathbf{E} Φ(|X_{t}|), \end{equation*} where $X$, $Y$ stand for $\mathcal{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. These statements complement and generalize the results of Burkholder, Suh, the author and others.