Asymptotic estimates for unimodular multilinear forms with small norms on sequence spaces

Abstract

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\Vert A\Vert$ over all unimodular $d$-linear (complex or real) forms $A$ on ${\ell }_{p_1}^{n_1}\times \cdots \times {\ell }_{p_d}^{n_d}$, for all $p_1,\dots ,p_d\in [2,\infty ]$ and all positive integers $n_1,\dots ,n_d$, behaves (asymptotically) as $\left(n_1^{1/2}+\cdots +n_d^{1/2}\right){\prod }_{j=1}^d{n}_j^{\frac{1}{2}-\frac{1}{p_j}}$. Applications to the theory of the multilinear Hardy--Littlewood inequality are also presented.

16 pages. To appear in Bulletin of the Brazilian Mathematical Society, New Series

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