On the Bellman function of Nazarov, Treil and Volberg
type: Item
instance of: scholarly article
title: On the Bellman function of Nazarov, Treil and Volberg
zbMATH Open document ID: 1311.42065
author: Rodrigo Bañuelos
author: Adam Osȩkowski
published in: Mathematische Zeitschrift
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.7197
review text: The paper depends on and complements two papers by \textit{F. L. Nazarov} and \textit{S. P. Treil} [St. Petersbg. Math. J. 8, No. 5, 721--824 (1997); translation from Algebra Anal. 8, No. 5, 32--132 (1996; Zbl 0873.42011)] and \textit{F. Nazarov} et al. [J. Am. Math. Soc. 12, No. 4, 909--928 (1999; Zbl 0951.42007)]. It is known that if \(\left\{ h_n\right\}\) is the standard Haar system on \([0,1)\) and \( \left\{a_n\right\}\) and \(\left\{ b_n \right\}\) are two sequences taking values in a Hilbert space \({\mathcal H}\) with \(\|a_n\|\leq \|b_n\|,\) for all \(n,\) then \[ \left\| \sum_{k=0}^n a_k h_k \right\|_p \leq (p^*-1) \left\| \sum_{k=0}^n b_k h_k \right\|_p,\; n=0, 1,2, \dots, \; 1<p<\infty , \tag{\(\star\)} \] where \(p^*=\max\left\{ p, p/(p-1)\right\}\) and the constant \(p^*-1\) cannot be replaced by a smaller constant. In [loc. cit.], Nazarov and Treil showed that Eq. (\(\star\)) can be deduced from the existence of a function \(B_p\) defined on the set \[ D=\left\{ (\zeta, \eta, Z, H) \in {\mathcal H}\times {\mathcal H}\times [0,\infty) \times [0,\infty) : Z\geq \|\zeta\|^p, H\geq \|\eta\|^q \right\} \] satisfying the following two conditions {\parindent=8mm \begin{itemize}\item[(I)] \(0\leq B_p(\zeta, \eta, Z, H)\leq (p^*-1) Z^{1/p} H^{1/q}\) on \(D\). \item[(II)] For any \(a_{\pm}=(\zeta_{\pm}, \eta_{\pm}, Z_{\pm}, H_{\pm})\in D,\) we have \[ B_p \left( \frac{a_- +a_+}{2}\right)-\frac{B_p(a_-) + B_p (a_+)}{2}\geq \left\| \frac{\zeta_+ - \zeta_-}{2}\right\| \left\| \frac{\eta_+ - \eta_-}{2}\right\|. \] \end{itemize}} Finding an explicit formula for \(B_p\) for any \(p\) is challenging, but for \(p=2\) it has been shown that \[ B_2(\zeta, \eta, Z, H)=\sqrt{\left(Z-\|\zeta\|^2\right) \left(H-\|\eta\|^2\right)}. \] There were attempts to find \(B_p\) explicitly for other values of \(p\) but without much success. However, Nazarov and Treil [loc. cit.] and \textit{F. L. Nazarov} et al. [Zbl 0951.42007] managed to construct for each \(1<p< \infty ,\) a funct…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 6357855
zbMATH Keywords: Bellman function
zbMATH Keywords: Haar system
zbMATH Keywords: unconditional basis
zbMATH Keywords: best constant
MaRDI profile type: Publication
OpenAlex ID: W2076903699
arXiv ID: 1401.7197
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DOI: 10.1007/S00209-014-1320-8
description: scientific article; zbMATH DE number 6357855