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

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Ukrainian Mathematical Journal Aims and scope

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

$$ \left\| x \right\|C = {\left\| x \right\|_{C\left( {{\mathbb{R}^m}} \right)}}: = \sup \left\{ {\left| {x(t)} \right|:t \in {\mathbb{R}^m}} \right\} $$

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

$$ {H^{j,{\omega_j}}}: = \left\{ {x \in C\left( {{\mathbb{R}^m}} \right):\left\| x \right\|{\omega_j} = \left\| x \right\|{H^{j,{\omega_j}}} = \mathop {{\sup }}\limits_{{t_j} \ne 0} \frac{{\left\| {\Delta {t_j}{e_j}x\left( \cdot \right)} \right\|C}}{{{\omega_j}\left( {\left| {{t_j}} \right|} \right)}} < \infty } \right\}. $$

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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