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 ω 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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 3, pp. 301–314, March, 2010.
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Babenko, V.F., Parfinovych, N. & Pichugov, S.A. Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions. Ukr Math J 62, 343–357 (2010). https://doi.org/10.1007/s11253-010-0358-y
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DOI: https://doi.org/10.1007/s11253-010-0358-y