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

2009/03/19 by Defant, Andreas, Frerick, Leonhard
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0903.3395

Abstract

In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial ∑_{|α| = m} a_αz^α on \Cⁿ the ℓ^(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 \mathbbDⁿ. We prove that this inequality is hypercontractive in the sense that the optimal constant C_m is ≤ C^m where C ≥ 1 is an absolute constant. From this we derive that the Bohr radius K_n of the n-dimensional polydisc in ℂⁿ is up to an absolute constant ≥ √(log n/n); this result was independently and with a differnt proof discovered by Ortega-Cerdà, Ounaïes and Seip. An alternative approach even allows to prove that the Bohr radius K_n^p, 1 ≤ p ≤ ∞ of the unit ball of ℓ_n^p , is asymptotically ≥ (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.

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