ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 26, 41, 42 Kofanov, Vladimir A., Dnepropetrovsk State University, Ukraine Exact Inequalities of Kolmogorov Type for Periodic Functions of Small Smoothness New exact inequalities of the form

∥x^{(k)}∥_{q} ≤ K ∥x∥_{p}^{α} ∥x^{(r)}∥_{s}^{1−α}  (1)

for periodic functions x with maximal possible exponent α will be presented in the following cases: 1) q, p ∈ [1, ∞], s = ∞; r = 2, k = 1 or r = 3, k = 1, 2 or r = 4, k = 3 or r = 6, k = 4, 5; 2) q, p ∈ [1, ∞], s = 1; r = 2, k = 1;

3) p ∈ [1, ∞), s = 1; r = 3, k = 1, q ≥ 2p or r = 4, k = 1, q ≥ 3p/2 or r = 4, k = 2, q ≥ 3p; 4) s ∈ (1, ∞), q = p = ∞; r = 2, k = 1, or r = 3, k = 2; 5) s ∈ (1, ∞), q = ∞, p = 1; r = 2, k = 1. Besides some exact inequalities of the type (1) for arbitrary r but special k will be presented. These results are joint work with V. F. Babenko and S. A. Pichugov.