# Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Authors: Babenko, V.1; Parfinovych, N.V.1; Pichugov, S.2

Source: Ukrainian Mathematical Journal, Volume 62, Number 3, October 2010 , pp. 343-357(15)

Publisher: Springer

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Abstract:

Let $$C\left( {{\mathbb{R}^m}} \right)$$ be the space of bounded and continuous functions $$x:{\mathbb{R}^m} \to \mathbb{R}$$ equipped with the norm and let e j , j = 1,…,m, be a standard basis in $${\mathbb{R}^m}$$ : Given moduli of continuity  j , j = 1,…, m, denote We obtain new sharp Kolmogorov-type inequalities for the norms $$\left\| {D_\varepsilon^\alpha x} \right\|C$$ of mixed fractional derivatives of functions $$x \in \cap_{j = 1}^m{H^{j,{\omega_j}}}$$ . Some applications of these inequalities are presented.

Document Type: Research Article

Affiliations: 1: Dnepropetrovsk National University, Dnepropetrovsk, Ukraine 2: Dnepropetrovsk State Technical University of Railway Transport, Dnepropetrovsk, Ukraine

Publication date: October 2010

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