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 $(n_{1}^{1/2} + \cdots + n_{d}^{1/2}) \prod _{j=1}^{d}n_{j}^{\frac{1}{2} - \frac{1}{p_j}}$. Applications to the theory of the multilinear Hardy–Littlewood inequality are also presented.