Adam Osekowski: Kolmogorov's inequalities for martingales and the Haar system, p.181-192


Let $(h_k)_{k\geq 0}$ be the Haar system on $[0,1]$ and let $0<p<1$. We show that for any vectors $a_k$ from a separable Hilbert space $\mathcal{H}$, any $\varepsilon_k\in [-1,1]$, $k=0,\,1,\,2,\,\ldots$ and any Borel subset $A$ of $[0,1]$, we have the sharp inequality

\begin{displaymath}\left\vert\left\vert\sum_{k=0}^n \varepsilon_ka_kh_k\right\ve...
...a_kh_k\right\vert\right\vert _{L^1([0,1])}\vert A\vert^{1/p-1},\end{displaymath}

$n=0,\,1,\,2,\,\ldots$. The above estimate is generalized to the sharp estimate

\begin{displaymath}\vert\vert Y\vert\vert _{L^p(A)}\leq 2\left(\frac{2-p}{2-2p}\...
...ert\vert X\vert\vert _{L^1(\Omega)}\cdot \mathbb{P}(A)^{1/p-1},\end{displaymath}

where $X$, $Y$ stand for $\mathcal{H}$-valued continuous-time martingales such that $Y$ is differentially subordinate to $X$. An application to Riesz system of harmonic functions is indicated.

Key Words: Haar system, martingale, differential subordination, best constant.

2000 Mathematics Subject Classification: Primary: 60G44;
Secondary: 60G42, 31B05, 46E30.

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