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^{\beta} f|{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} |D^{\alpha_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)^{\phi}, ] where $\phi<1$ depends only on the set ${\alpha_j}\cup{\beta}$.