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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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References:
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