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The Bohr radius of the \(n\)-dimensional polydisk is equivalent to \(\sqrt{(\log n) / n}\). (English) Zbl 1331.46037
This paper provides at least four striking results: a surprising and far reaching improvement of an inequality due to Ron Blei, the best known constants of the multilinear Bohnenblust–Hille inequality, the subexponentiality of the complex polynomial Bohnenblust-Hille inequality, and the final solution to the famous Bohr radius problem.
For two positive integers \(n,m\), let \[ \mathcal{M}(m,n) =\big\{\mathbf{i}=(i_{1},\dots ,i_{m});\;i_{1},\dots ,i_{m}\in \{1,\dots ,n\}\big\} \] \[ \mathcal{J}(m,n) =\big\{\mathbf{i}\in \mathcal{M}(m,n);\;i_{1}\leq i_{2}\leq \dots \leq i_{m}\big\}. \] For \(1\leq k\leq m\), let \(\mathcal{P}_{k}(m)\) denote the set of subsets of \( \{1,\dots ,m\}\) with cardinality \(k\). For \(S=\{s_{1},\dots ,s_{k}\}\) in \( \mathcal{P}_{k}(m)\), let \(\hat{S}\) denote its complement in \(\{1,\dots ,m\}\), and \(\mathbf{i}_{S}\) denotes \((i_{s_{1}},\dots ,i_{s_{k}})\in \mathcal{M} (k,n)\).
The first main result of the paper is a beautiful and surprising generalization of Blei’s inequality, which is of independent interest. It reads as follows:
Theorem 1. Let \(m,n\geq 1\) and \(1\leq k\leq m\). Then, for all families \((a_{ \mathbf{i}})_{\mathbf{i}\in \mathcal{M}(m,n)}\) of complex numbers, \[ \left( \sum_{\mathbf{i}\in \mathcal{M}(m,n)}|a_{\mathbf{i}}|^{\frac{2m}{m+1} }\right) ^{\frac{m+1}{2m}}\leq \prod_{S\in \mathcal{P}_{k}(m)}\left( \sum_{ \mathbf{i}_{S}}\left( \sum_{\mathbf{i}_{\hat{S}}}|a_{\mathbf{i}}|^{2}\right) ^{\frac{1}{2}\times \frac{2k}{k+1}}\right) ^{\frac{k+1}{2k}\times \frac{1}{ \binom{m}{k}}}. \] The proof uses a very interesting interpolative approach that relies on the mixed Hölder inequality for mixed sums.
Recall that the multilinear Bohnenblust-Hille inequality, proved in [H. F. Bohnenblust and E. Hille, Annals of Math. (2) 32, 600–622 (1931; JFM 57.0266.05)], asserts that, for any \(m\geq 1\), there exists a constant \(C_{m}\geq 1\) such that, for all \(m\)-linear forms \(L:c_{0}\times \dots \times c_{0}\rightarrow \mathbb{K}\), \[ \left( \sum\limits_{i_{1},\ldots ,i_{m}=1}^{\infty }\left| L(e_{i_{^{1}}},\ldots ,e_{i_{m}})\right| ^{\frac{2m}{m+1}}\right) ^{ \frac{m+1}{2m}}\leq C_{m}\left\| L\right\| . \] Estimates for the constants \(C_{m}\) are crucial for applications, even outside pure mathematics, as can be seen in its applications in quantum information theory. The original estimates of \(C_{m}\) due to Bohnenblust and Hille predicted exponential growth, but the authors show that there are constants \(\kappa _{1},\kappa _{2}>0\) such that \[ C_{m}\leq \kappa _{1}m^{\frac{1-\gamma }{2}}<\kappa _{1}m^{0.212} \] for complex scalars and \[ C_{m}\leq \kappa _{2}m^{\frac{2-\log 2-\gamma }{2}}<\kappa _{2}m^{0.365} \] for real scalars.
The third main result of the paper is a generalization of a striking result due to Defant, Frerick, Ortega-Cerdà, Ounaïes and Seip on the constants of the polynomial Bohnenblust-Hille inequality for complex scalars, published in [A. Defant et al., Ann. Math. (2) 174, No. 1, 485–497 (2011; Zbl 1235.32001)]. The important result of Defant \(et\) \(al.\) shows that the constants of the polynomial version of the Bohnenblust-Hille inequality for complex scalars can be taken having exponential growth, and this result is very important for several different applications. Now, in the paper under review, the authors show that these constants can be taken with subexponential growth. More precisely, denoting by \(B_{m}\) these constants, it is shown that, for any \(\varepsilon >0\), there exists \(\kappa >0\) such that, for any \(m\geq 1\), \[ B_{m}\leq \kappa (1+\varepsilon )^{m}. \] Using this result, the authors finish the paper with the solution of the Bohr radius problem. The Bohr radius \(K_{n}\) of the \(n\)-dimensional polydisk is the largest positive number \(r\) such that all polynomials \(\sum_{\alpha }a_{\alpha }z^{\alpha }\) on \(\mathbb{C}^{n}\) satisfy \[ \sup_{z\in r\mathbb{D}^{n}}\sum_{\alpha }|a_{\alpha }z^{\alpha }|\leq \sup_{z\in \mathbb{D}^{n}}\left| \sum_{\alpha }a_{\alpha }z^{\alpha }\right| . \] The Bohr radius \(K_{1}\) was studied and estimated by H. Bohr, and it was shown independently by M. Riesz, I. Schur and F. Wiener that \(K_{1}=1/3\). Since then, many authors have been trying to find the exact asymptotic behavior of the Bohr radius and despite many very important partial results obtained by several mathematicians, the problem was still open. The solution is then given in this paper in the form of the following simple statement: \[ \lim_{n\rightarrow \infty }\frac{K_{n}}{\sqrt{\frac{\log n}{n}}}=1. \]

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
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