On the mixed $(\ell _{1},\ell _{2})$-Littlewood inequalities and interpolation

TL;DR

This paper investigates the role of interpolation in the growth of constants in multilinear inequalities, showing it is not crucial for the asymptotic behavior of the Bohnenblust--Hille inequality and applying results to sequence space cotypes.

Contribution

It demonstrates that interpolation does not determine the asymptotic growth of Bohnenblust--Hille constants and extends the analysis to mixed Littlewood inequalities and sequence space cotypes.

Findings

Interpolation does not influence the asymptotic growth of Bohnenblust--Hille constants.

The growth of m-linear Bohnenblust--Hille constants matches that of certain mixed Littlewood inequalities.

Optimal cotype constants of specific sequence spaces are obtained using mixed Littlewood inequalities.

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.