On the mixed $({\ell }_1,{\ell }_2)$-Littlewood inequalities and interpolation

Abstract

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 $\left({\ell }_1,{\ell }_2\right)$-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 $\left({\ell }_{\frac{2m+2}{m+2}},{\ell }_2\right)$-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.