Lower bounds for norms of products of polynomials via Bombieri inequality (Q2841341)

For a multi-index \(\alpha=(\alpha_1,\dots, \alpha_N)\) let \(|\alpha|=\alpha_1+\dots+\alpha_N\) and \(\alpha!=\alpha_1!\dots\alpha_N!.\) For \(z=(z_1,\dots,z_N)\in{\mathbb C}^N\) we let \(z^\alpha=z_1^{\alpha_1}\dots z_N^{\alpha_N}.\) The space of \(m\)-homogeneous polynomials on \({\mathbb C}^N\) will be denoted by \({\mathcal P}(^m {\mathbb C}^N).\) Polynomials \(P\) in \({\mathcal P}(^m {\mathbb C}^N)\) will be written as \(P(z)=\sum_{|\alpha|=m}a_{\alpha}z^{\alpha}.\) The uniform norm and the Bombieri norm of \(P\) are defined by NEWLINE\[NEWLINE \| P\|_{\mathcal P}=\sup_{\|z\|=1}|P(z)|\quad\text{and}\quad [P]_2=\left(\sum_{|\alpha|=m} (\alpha!/m!)|a_{\alpha}|^2\right)^{1/2}, NEWLINE\]NEWLINE respectively.NEWLINENEWLINEFor \(P\in{\mathcal P}(^m {\mathbb C}^N)\) the real sequence \(S_n(P)\) is defined by \(S_n(P):=\sup\{\, [PQ]_2\,:\,[Q]_2=1,\,Q\in{\mathcal P}(^n {\mathbb C}^N)\}\). Relating the norm \([\cdot]_2\) to the \(L^2\)-norm with respect to the Gaussian measure on the Borel sets of \({\mathbb C}^N\), the author proves that NEWLINE\[NEWLINE \limsup_{n\rightarrow\infty}S_n(P)=\| P\|_{\mathcal P}. NEWLINE\]NEWLINE The other main result of the paper is the sharp inequality NEWLINE\[NEWLINE \|P_1\|_{\mathcal P}\cdots\|P_n\|_{\mathcal P}\leq \sqrt{{(k_1+\cdots+k_n)^{(k_1+\cdots+k_n)}}\over{k_1^{k_1}\cdots k_n^{k_n}}}\,\|P_1\cdots P_n\|_{\mathcal P} NEWLINE\]NEWLINE where \(P_i \in {\mathcal P}(^{k_i} H)\,(1\leq i\leq n).\) Here \({\mathcal P}(^{m} H)\) denotes the set of \(m\)-homogeneous polynomials defined on a finite or infinite dimensional Hilbert space \(H.\)NEWLINENEWLINE\textit{J. Arias-de-Reyna} [Linear Algebra Appl. 285, No. 1--3, 107--114 (1998; Zbl 0934.15036)] proved that for any orthonormal system \(z_i \;( 1\leq i\leq n)\) of unit vectors in a complex Hilbert space there is an unit vector \(z\in H\) such that NEWLINE\[NEWLINE|\langle z,z_1\rangle\cdots\langle z,z_n \rangle|\geq n^{-n/2}. NEWLINE\]NEWLINE Moreover \(n^{-n/2}\) is the best constant because for any orthonormal system \(\{z_k\}_{k=1}^n,\) NEWLINE\[NEWLINE \sup_{\|z\|=1} |\langle z,z_1\rangle\cdots\langle z,z_n \rangle| = n^{-n/2}.\eqno{(1)} NEWLINE\]NEWLINE It is proved that if (1) holds then \(\{z_i\}_{k=1}^n\) is an orthonormal system.