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

Abstract

In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial ${\sum }_{|\alpha |=m}{a}_{\alpha }{z}^{\alpha }$ on $\C^n$ the ${\ell }^{\frac{2m}{m+1}}$-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 $\ge \sqrt{\log n/n}$; this result was independently and with a differnt proof discovered by Ortega-Cerd{à}, Ounaïes and Seip. An alternative approach even allows to prove that the Bohr radius $K_n^p$, $1\le p\le \infty$ of the unit ball of ${\ell }_n^p,$ is asymptotically $\ge (\log n/n{)}^{1-1/\min (p,2)}$. This shows that the upper bounds for $K_n^p$ given by Boas and Khavinson are optimal.