Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables - Babenko, V. F.; Churilova, M. S. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
We investigate the correlation between the constants K(ℝ^n) and K(T^n), where K(G^n ): = __i = 1^n \| D_i^l_i f \|_L_p (G^n ) e 0^f L_p,p^l (G^n ) \| D^α f \|_L_p (G^n ) \| f \|_L_p (G^n )^μ _0 _i = 1^n \| D_i^l_i f \|_L_p (G^n )^μ _i is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line, T = [0,2π], L^l_p, p (G^n) is the set of functions ƒ ∈ L_p (G^n) such that the partial derivative D_i^l_i f(x) belongs to L_p (G^n), i = 1,n, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^α f is the mixed derivative of a function ƒ, 0 < µi < 1, i = 0,n, and ∑_i=0^n µ_i = 1. If G^n = ℝ, then µ_0 = 1 − ∑_i=0^n (α_i /l_i), µ_i = α_i/l_i, i = 1,n if G^n = T^n, then µ_0 = 1 − ∑_i=0^n (α_i /l_i) − ∑_i=0^n (λ/l_i), µ_i = α_i/ l_i + λ/l_i , i= 1,n, λ ≥ 0. We prove that, for λ = 0, the equality K(ℝ^n) = K(T^n) is true.