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On the real polynomial Bohnenblust-Hille inequality. (English) Zbl 1318.46029
Given an \(m\)-homogeneous polynomial \(P=\sum_{|\alpha|=m}a_\alpha x^\alpha\) on \(\ell_\infty^n\), the authors use \(|P|_{{m+1}\over 2m}\) to denote \(\left( \sum_{|\alpha|=m}|a_\alpha|^{{m+1}\over 2m}\right)^{{2m}\over{m+1}}\) and \({\mathcal P}(^m \ell_\infty^n)\) to denote the space of all \(m\)-homogeneous polynomials on \(\ell_\infty^n\). They study the constants \(D_{{\mathbf K},m}\) and \(D_{{\mathbf K},m}(n)\) defined by \[ D_{{\mathbf K},m}:=\inf\{D>0:|P|_{{m+1}\over 2m}\leq D\left(\sup_{\|x\|_\infty\leq 1} |P(x)|\right), \,P\in {\mathcal P}(^m\ell_\infty^n),\,n\in {\mathbb N}\} \] and \[ D_{{\mathbf K},m}(n):=\inf\{D>0:|P|_{{m+1}\over 2m}\leq D\left(\sup_{\|x\|_\infty\leq 1} |P(x)|\right), \,P\in {\mathcal P}(^m\ell_\infty^n)\} \] for \({\mathbb K}\) equal to the field of reals \({\mathbb R}\) or the field of complex numbers \({\mathbb C}\). They show that \[ D_{{\mathbb R},m}\geq \left({{2\root 4\of 3}\over\sqrt{5}}\right)^m>(1.17)^m \] and that \[ \limsup_{m\to\infty}D_{{\mathbb R},m}^{1/m}(n)\geq \root 8\of {27}=1.5098. \] This contrasts to the complex case for which it is known [F. Bayart et al., Adv. Math. 264, 726–746 (2014; Zbl 1331.46037)] that for every \(\epsilon>0\) there is \(C_\epsilon>0\) such that \(D_{{\mathbb C},m}\leq C_\epsilon(1+\epsilon)^m\) for all positive integers \(m\).

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
47H60 Multilinear and polynomial operators
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