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Ukrainian Mathematical Journal

, Volume 58, Issue 5, pp 674–684 | Cite as

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

  • V. F. Babenko
  • M. S. Churilova
Article
  • 22 Downloads

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 l p, p (G n) is the set of functions ƒL 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 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 . If G n = ℝ, then µ0=1−∑ i=0 n i /l i ), µi = αi/l i , \(i = \overline {1,n} \) if \(G^n = \mathbb{T}^n \), then µ0=1−∑ i=0 n i /l i ) − ∑ i=0 n (λ/l i ), µi = αi/ 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. F. Babenko
    • 1
  • M. S. Churilova
    • 1
  1. 1.Dnepropetrovsk National UniversityDnepropetrovsk

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