On Bernstein type quantitative estimates for Ornstein non-inequalities. (English) Zbl 1540.42049
Summary: For the sequence of multi-indices \(\{\alpha_j\}_{j=1}^m\) and \(\beta\), we study the inequality
\[
\|D^\beta f \|_{L_1(\mathbb{T}^d)} \leqslant K_N \sum_{j=1}^m \| D^{\alpha_j} f \|_{L_1(\mathbb{T}^d)},
\]
where \(f\) is a trigonometric polynomial of degree at most \(N\) on the \(d\)-dimensional torus. Assuming some natural geometric property of the set \(\{\alpha_j\} \cup \{\beta\}\), we show that
\[
K_N \geqslant C(\ln N)^\phi,
\]
where \(\phi < 1\) depends only on the set \(\{\alpha_j\} \cup \{\beta\}\).
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 42A55 | Lacunary series of trigonometric and other functions; Riesz products |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42B37 | Harmonic analysis and PDEs |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 60E15 | Inequalities; stochastic orderings |
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