Exact constants in generalized inequalities for intermediate derivatives
Considering the Sobolev space \(W_2^n (\mathbb{R}_+)\) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients, the authors study the problem of exact constants \(A_{n,k}\) in inequalities of Kolmogorov-type [cf. \textit{G. A. Kalyabin}, Funct. Anal. Appl. 38, No. 3, 184--191 (2004); translation from Funkts. Anal. Prilozh. 38, No. 3, 29--38 (2004; Zbl 1079.41009); Funct. Anal. Appl. 38, No. 3, 184--191 (2004); translation from Funkts. Anal. Prilozh. 38, No. 3, 29--38 (2004; Zbl 1079.41009)]. The authors propose here a new method for finding the constants \(A_{n,k}\) in the case certain norms considered in the paper. A symmetric property for \(A_{n,k}\) is proved for the general case. The asymptotic behaviour of \(A_{n,k}\) is also studied.
scientific article; zbMATH DE number 5628201