Properties of Extrema of Estimates for Middle Derivatives of Odd Order in Sobolev Classes

Abstract

The embedding constants for the Sobolev spaces \(\overset{\circ} {W_{2}^{n}} \)[0; 1] ↪ \(\mathop {W_{\infty }^{k}}\limits^{\circ} \)[0; 1] (\(0 \leqslant k \leqslant n - 1\)) are considered. The properties of the functions \({{A}_{{n,k}}}(x)\) arising in the inequalities \({\text{|}}{{f}^{k}}(x){\text{|}} \leqslant A_{{n,k}}^{{}}(x){\text{||}}f{\text{|}}{{{\text{|}}}_{{\mathop {W_{2}^{n}}\limits^{\circ}[0;1]} }}\) are studied. The extremum points of \({{A}_{{n,k}}}\) are calculated for k = 3, 5 and all admissible n. The global maximum of these functions is found, and the exact embedding constants are calculated.