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

Abstract

We investigate the correlation between the constants $K(ℝ^n)$ and $K(T^n)$, 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, $T = [0,2π],\; L^l_{p, p} (G^n)$ is the set of functions $ƒ ∈ L_p (G^n)$ such that the partial derivative $D_i^{l_i } f(x)$ belongs to $L_p (G^n), i = \overline {1,n}, 1 ≤ p ≤ ∞, l ∈ ℕ^n, α ∈ ℕ_0^n = (ℕ ∪ 〈0〉)^n, D^{α} f$ is the mixed derivative of a function $ƒ, 0 < µi < 1, i = \overline {0,n},$ and $∑_{i=0}^n µ_i = 1$. If $G^n = ℝ$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i),\; µ_i = α_i/l_i,\; i = \overline {1,n}$ if $G^n = T^n$, then $µ_0 = 1 − ∑_{i=0}^n (α_i /l_i) − ∑_{i=0}^n (λ/l_i),\; µ_i = α_i/ l_i + λ/l_i , i= \overline {1,n},\; λ ≥ 0$. We prove that, for $λ = 0$, the equality $K(ℝ^n) = K(T^n)$ is true.