Optimal constants for a mixed Littlewood type inequality

Abstract

For $p\in [2,\infty ]$ a mixed Littlewood-type inequality asserts that there is a constant $C_{(m),p}\ge 1$ such that \[ \left( \sum_{i_{1}=1}^{\infty}\left( \sum_{i_{2},...,i_{m}=1}^{\infty }|T(e_{i_{1}},...,e_{i_{m}})|^{2}\right) ^{\frac{1}{2}\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\leq C_{(m),p}\Vert T\Vert \] for all continuous real-valued $m$-linear forms on ${\ell }_p\times c_0\times \cdots \times c_0$ (when $p=\infty$, ${\ell }_p$ is replaced by $c_0)$. We prove that for $p>2.18006$ the optimal constants $C_{(m),p}$ are ${\left(2^{\frac{1}{2}-\frac{1}{p}}\right)}^{m-1}.$ When $p=\infty ,$ we recover the best constants of the mixed $\left({\ell }_1,{\ell }_2\right)$-Littlewood inequality.

arXiv commentThis new version corresponds to the junction of the previous version with the preprint arXiv:1508.02355. The preprint arXiv:1508.02355 does not exist anymore as a separate preprint