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

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## Abstract

We investigate the correlation between the constants is the exact constant in a Kolmogorov-type inequality, ℝ is the real straight line, \(\mathbb{T} = \left[ {0,2\pi } \right]\),

*K*(ℝ^{n}) and \(K(\mathbb{T}^n )\), where$$K(G^n ): = \mathop {\sup }\limits_{\mathop {\prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )} \ne 0}\limits^{f \in L_{p,p}^l (G^n )} } \frac{{\left\| {D^\alpha f} \right\|_{L_p (G^n )} }}{{\left\| f \right\|_{L_p (G^n )}^{\mu _0 } \prod _{i = 1}^n \left\| {D_i^{l_i } f} \right\|_{L_p (G^n )}^{\mu _i } }}$$

*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 < µ_{i}< 1, \(i = \overline {0,n} \), and ∑_{ i=0}^{ n }. 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 \(G^n = \mathbb{T}^n \), 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 \(K(\mathbb{R}^n ) = K(\mathbb{T}^n )\) is true.## Keywords

Periodic Function Critical Exponent Unbounded Operator Periodic Case Mixed Derivative
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