On the Bellman function of Nazarov, Treil and Volberg
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 function which satisfies condition (II) and a version of (I) in which \(p^*-1\) is replaced by a larger constant. The purpose of the paper under review is to fill this gap and give an explicit formula for \(B_p\) satisfying (I) and (II) for all \(1<p< \infty .\)
scientific article; zbMATH DE number 6357855
scientific article