Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

We investigate the correlation between the constants K ℝn) and K (double-struck Tn), where (Equation Presented) is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line, double-struck T = [0,2pi;], Lp,pl(Gn) is the set of functions f ε Lp(Gn) such that the partial derivative Dili f(x) belongs to Lp(G n), i = 1, n̄, 1 ≤ p ≤ ∞, l ε ℕn, α ε ℕ0n = (ℕ ∪ {0})n, Dα f is the mixed derivative of a function f, 0 < μi < 1, i = 0, n̄, and Σi=0nμi = 1. If Gn = ℝn, then μ0 = 1 - Σi=1 n (αi/li), μi = αi/li, i = 1, n̄; if Gn = double-struck Tn, then μ0 = 1 - Σ i=1n (αi/li) - Σi=1n (λ/li), μi = αi/li + λ/li, i = 1, n̄, λ ≥ 0. We prove that, for λ = 0, the equality K(ℝn) = K(double-struck Tn) is true. © 2006 Springer Science+Business Media, Inc.