On The Mixed (L(1), L(2))-Littlewood Inequalities And Interpolation

It is well-known that the optimal constant of the bilinear Bohnenblust-Hille inequality (i.e., Littlewood's 4/3 inequality) is obtained by interpolating the bilinear mixed (l(1), l(2))Littlewood inequalities. We remark that this cannot be extended to the 3-linear case and, in the opposite direction, we show that the asymptotic growth of the constants of the m -linear Bohnenblust-Hille inequality is the same of the constants of the mixed (l(2m+2/m+2), l(2))-Littlewood inequality. This means that, contrary to what the previous works seem to suggest, interpolation does not play a crucial role in the search of the exact asymptotic growth of the constants of the Bohnenblust-Hille inequality. In the final section we use mixed Littlewood type inequalities to obtain the optimal cotype constants of certain sequence spaces.