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

Abstract

We investigate the correlation between the constants K(ℝn) and \(K(\mathbb{T}^n )\), where $$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$ is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line, \(\mathbb{T} = \left[ {0,2\pi } \right]\), Llp, p(Gn) is the set of functions ƒ ∈ Lp(Gn) such that the partial derivative \(D_i^{l_i } f(x)\) belongs to Lp(Gn), \(i = \overline {1,n} \), 1 ≤ p ≤ ∞, l ∈ ℕn, α ∈ ℕ0n = (ℕ ∪ 〈0〉)n, Dαf is the mixed derivative of a function ƒ, 0 < µi < 1, \(i = \overline {0,n} \), and ∑i=0n. If Gn = ℝ, then µ0=1−∑i=0n(αi/li), µi = αi/li, \(i = \overline {1,n} \) if \(G^n = \mathbb{T}^n \), then µ0=1−∑i=0n(αi/li) − ∑i=0n(λ/li), µi = αi/ li + λ/li, \(i = \overline {1,n} \), λ ≥ 0. We prove that, for λ = 0, the equality \(K(\mathbb{R}^n ) = K(\mathbb{T}^n )\) is true.