most recent 30 from http://mathoverflow.net 2013-05-23T11:50:30Z http://mathoverflow.net/feeds/question/106525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106525/stein-inequality Yanqi QIU 2012-09-06T15:58:06Z 2012-09-06T15:58:06Z

<p>Dear all, </p> <p>we have by Stein that for any sequence $0 \le r_k \le 1$, and any functions $f_1, \cdots, f_n$ which are holomorphic on a neighbourhood of the unit disk, $$\| (\sum_{k =1}^n |f_k(r_k e^{i 2 \pi \theta})|^2)^{1/2}\|_{L_p([0, 1])} \le c_p\| (\sum_{k =1}^n |f_k(e^{i 2 \pi \theta})|^2)^{1/2}\|_{L_p([0, 1])}. $$</p> <p>Let us assume that $c_p$ is the best constant appear in the above inequality.Of course, $c_2 =1$. </p> <p>$\textbf{Question}$: What is the order $c_p$ when $p \to 2$. Could we have $$c_p \ge 1 + M(p-2)$$ for some fixed $M > 0$ when $p = 2 + \varepsilon$ for $\varepsilon$ very small?</p>