On Bernstein type quantitative estimates for Ornstein non-inequalities

Abstract

For the sequence of multi-indexes $\{{\alpha }_i{\}}_{i=1}^m$ and $\beta$ we study the inequality \[ \|D^β f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{α_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{{\alpha }_j\}\cup \{\beta \}$ we show that \[ K_{N}\geq C \left(\ln N\right)^ϕ, \] where $\varphi <1$ depends only on the set $\{{\alpha }_j\}\cup \{\beta \}$.

arXiv commentPresentation improved, typos corrected