Hypercontractivity of the Bohnenblust-hille Inequality for Polynomials and Multidimensional Bohr Radii

In 1931 Bohnenblust and Hille proved that for each mhomogeneous polynomial P |α|=m aαz α on C the l 2m m+1 -norm of its coefficients is bounded from above by a constant Cm (depending only on the degree m) times the sup norm of the polynomial on the polydisc D . We prove that this inequality is hypercontractive in the sense that the optimal constant Cm is ≤ C m where C ≥ 1 is an absolute constant. ¿From this we derive that the Bohr radius Kn of the n-dimensional polydisc in C is up to an absolute constant ≥ p log n/n; this result was independently and with a differnt proof discovered by Ortega-Cerdà, Ounäıes and Seip in [25]. An alternative approach even allows to prove that the Bohr radius K n, 1 ≤ p ≤ ∞ of the unit ball of l p n , is asymptotically ≥ (log n/n) . This shows that the upper bounds for K n given by Boas and Khavinson from [5] are optimal.