Absolutely summing multilinear operators via interpolation

TL;DR

This paper introduces multiple N-separately summing operators using interpolation, extending recent results and recovering optimal estimates for multilinear Bohnenblust-Hille constants, with implications for complex multilinear forms.

Contribution

It develops a unified interpolative framework for summing multilinear operators, generalizing and improving existing bounds for Bohnenblust-Hille constants.

Findings

Recovered best known estimates for multilinear Bohnenblust-Hille constants.

Established bounds for complex m-linear forms using interpolation techniques.

Unified approach extends previous results in multilinear operator theory.

Abstract

We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sep\'ulveda. More precisely, as a consequence of our main result, for $1\leq t<2$ and $m\in \mathbb{N}$ we prove that $$ \left(\sum_{i_{1},\dots,i_{m}=1}^{\infty}\left\vert U\left(e_{i_{1}},\dots,e_{i_{m}}\right) \right\vert^{\frac{2tm}{2+(m-1)t}}\right)^{\frac{2+(m-1)t}{2tm}} \leq \left[\prod_{j=2}^{m}\Gamma \left(2-\frac{2-t}{jt-2t+2}\right) ^{\frac{t(j-2)+2}{2t-2jt}}\right] \left\Vert U\right\Vert $$ for all complex $m$-linear forms $U:c_{0}\times \cdot \cdot \cdot \times c_{0}\rightarrow \mathbb{C}$.