Some constants in Martingale Stein inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:15:12Z http://mathoverflow.net/feeds/question/106255 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106255/some-constants-in-martingale-stein-inequality Some constants in Martingale Stein inequality Yanqi QIU 2012-09-03T16:13:38Z 2012-09-03T16:38:22Z <p>Dear all,</p> <p>the following is a special case of Stein inequalities for martingales.</p> <p>$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of increasing conditional expectations $\mathbb{E}_n$, $n \ge 1$. Then for any $1&lt; p&lt; \infty$, there exists a numerical constant $\gamma_p>0$ such that for any finite sequence of real valued functions $f_1, f_2, \cdots, f_n$ in $L_p(\Omega,\mathbb{P})$, we have $$\| (\sum_{k = 1}^n |\mathbb{E}_kf_k|^2)^{1/2}\|_p \le \gamma_p\| (\sum_{k = 1}^n |f_k|^2)^{1/2}\|_p.$$ ($\gamma_p$ is denoted to be the optimal constant in the above inequality.)</p> <p>$\textbf{Question}:$ Is the following holds? </p> <p>There exist $M> 0$, such that $$\gamma_p \ge 1 + M(p-2),$$ when $0 &lt; p-2 &lt;&lt;1$.</p> <p>$\textbf{Special case}:$ Let $C_q$ be the best constant verifying $$\| f_1^2 + \mathbb{E}(f_2)^2\|_q \le C_q \| f_1^2 + f_2^2\|_q.$$ Does there exist $M > 0$ such that $$C_q \ge 1 + M(q-1)$$ for all $0 &lt; q-1 &lt;&lt; 1$.</p>