On Bernstein type quantitative estimates for Ornstein non-inequalities

2022/06/27 by Kazaniecki, Krystian, Wojciechowski, Michał · 1 citation
#2020: 42A55 #42B37 #46E35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2206.13666

Abstract

For the sequence of multi-indexes \α_i\_i=1^m and β we study the inequality ‖D^β f‖_L₁(\mathbbT^d)≤ K_N ∑_{j= 1}^m ‖D^α_jf‖_L₁(\mathbbT^d), where f is a trigonometric polynomial of degree at most N on d-dimensional torus. Assuming some natural geometric property of the set \α_j\∪\β\ we show that K_N≥ C (ln N)^ϕ, where ϕ<1 depends only on the set \α_j\∪\β\.

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