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, $\mathbb{T} = \left[ {0,2\pi } \right]$, L ^{p, p} _l (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), $i = \overline {1,n}$, 1 ≤ p ≤ ∞, l ∈ ℕⁿ, α ∈ ℕ ⁿ₀ = (ℕ ∪ 〈0〉)ⁿ, D ^α f is the mixed derivative of a function ƒ, 0 < µ_i < 1, $i = \overline {0,n}$, and ∑ ⁿ _i=0 . If G ⁿ = ℝ, then µ₀=1−∑ ⁿ _i=0 (α _i /l _i ), µ_i = α_i/l _i , $i = \overline {1,n}$ if $G^n = \mathbb{T}^n$, then µ₀=1−∑ ⁿ _i=0 (α _i /l _i ) − ∑ ⁿ _i=0 (λ/l _i ), µ_i = α_i/ l _i + λ/l _i , $i = \overline {1,n}$, λ ≥ 0. We prove that, for λ = 0, the equality $K(\mathbb{R}^n ) = K(\mathbb{T}^n )$ is true.