Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii

In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial ${\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=m}{a}_{\alpha }{z}^{\alpha }$ on ${\text{C}}^n$ the ${\mathrm{\ell }}^{\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a constant $C_m$ (depending only on the degree $m$) times the sup norm of the polynomial on the polydisc ${\mathbb{D}}^n$. We prove that this inequality is hypercontractive in the sense that the optimal constant $C_m$ is $\le C^m$ where $C\ge 1$ is an absolute constant. From this we derive that the Bohr radius $K_n$ of the $n$-dimensional polydisc in ${\mathbb{C}}^n$ is up to an absolute constant $\ge \sqrt{\log n\mathrm{/}n}$; this result was independently and with a differnt proof discovered by Ortega-Cerd{`a}, Ouna"ies and Seip. An alternative approach even allows to prove that the Bohr radius $K_n^p$, $1 \leq p \leq \infty $oftheunitballof$\ell_n^p ,$isasymptotically$ \geq (\log n/n) ^{1-1/ \min (p,2)}$\mathrm{.}Thisshowsthattheupperboundsfor$K_n^p$ given by Boas and Khavinson are optimal.