Mathematics – Functional Analysis
Scientific paper
2012-03-14
Mathematics
Functional Analysis
Scientific paper
The complex multilinear Bohnenblust-Hille inequality states that for every positive integer $m$ there is a constant $C_{m}\geq1$ so that [(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}}%,...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{m}\sup_{z_{1},...,z_{m}\in\mathbb{D}^{N}}|U(z_{1},...,z_{m}%)|] for every positive integer $N$ and every $m$-linear form $U:\ell_{\infty}% ^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$, where $(e_{i})_{i=1}^{N}$ denotes the canonical basis of $\mathbb{C}^{N}$ and $\mathbb{D}^{N}$ represents the open unit polydisk in $\mathbb{C}^{N}$. This result is crucial, for example, in the proof of the famous Bohr's absolute convergence problem on Dirichlet series and have found applications to analytic number theory and in several different frameworks. The problem of estimating constants $C_{m}$ for Bohnenblust--Hille-type inequalities, besides its challenging nature, has far reaching consequences and applications. For instance, by finding important information on the estimates of the constants of the Bohnenblust-Hille inequality for homogeneous polynomials, A. Defant, L. Frerick, J. Ortega-Cerd\'{a}, M. Ouna\"{\i}es and K. Seip, in 2011, were able to find the precise asymptotic behavior of the $n$-dimensional Bohr radius and other highly nontrivial information on Bohr's absolute convergence problem. In this note we improve recent upper estimates for $C_{m}$ and we support the conjecture that the constants obtained here are optimal.
Pellegrino Daniel
Seoane-Sepúlveda Juan B.
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