Mathematics – Functional Analysis
Scientific paper
2009-03-19
Mathematics
Functional Analysis
Scientific paper
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 $\leq C^m$ 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{\`a}, Ouna\"ies 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.
Defant Andreas
Frerick Leonhard
No associations
LandOfFree
Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-648910