Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii
| dc.creator | Defant, Andreas | |
| dc.creator | Frerick, Leonhard | |
| dc.date | 2009-03-19 | |
| dc.date.accessioned | 2026-07-07T12:54:18Z | |
| dc.date.available | 2026-07-07T12:54:18Z | |
| dc.description | In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial $\sum_{|α| = m} a_αz^α$ 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 $\leq C^m$ where $C \geq 1$ 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 $\geq \sqrt{\log n/n}$; 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 \leq p \leq \infty $ of the unit ball of $\ell_n^p ,$ is asymptotically $ \geq (\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. | |
| dc.identifier | https://arxiv.org/abs/0903.3395 | |
| dc.identifier | http://arxiv.org/abs/0903.3395 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/223884 | |
| dc.subject | Functional Analysis | |
| dc.title | Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii | |
| dc.type | text |