The Bohr radius of the unit ball of
Identifieur interne : 001956 ( Istex/Curation ); précédent : 001955; suivant : 001957The Bohr radius of the unit ball of
Auteurs : Andreas Defant [Allemagne] ; Leonhard Frerick [Allemagne]Source :
- Journal für die reine und angewandte Mathematik (Crelles Journal) [ 0075-4102 ] ; 2011-11.
Abstract
By a classical result due to Aizenberg, Boas and Khavinson the Bohr radius of the unit ball in the Minkowski space , 1 ≦ p ≦ ∞, is up to an absolute constant ≦ (log n/n)1–1/min(p, 2). Our main result shows that this estimate is optimal. For p = ∞, this was recently proved in [Defant, Frerick, Ortega-Cerdà, Ounaies and Seip, Ann. Math. 174: 1–13, 2011] as a consequence of the hypercontractivity of the Bohnenblust–Hille inequality for polynomials. Using substantially different methods from local Banach space theory, we give a proof which covers the full scale 1 ≦ p ≦ ∞.
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DOI: 10.1515/crelle.2011.080
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<front><div type="abstract" xml:lang="en">By a classical result due to Aizenberg, Boas and Khavinson the Bohr radius of the unit ball in the Minkowski space , 1 ≦ p ≦ ∞, is up to an absolute constant ≦ (log n/n)1–1/min(p, 2). Our main result shows that this estimate is optimal. For p = ∞, this was recently proved in [Defant, Frerick, Ortega-Cerdà, Ounaies and Seip, Ann. Math. 174: 1–13, 2011] as a consequence of the hypercontractivity of the Bohnenblust–Hille inequality for polynomials. Using substantially different methods from local Banach space theory, we give a proof which covers the full scale 1 ≦ p ≦ ∞.</div>
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