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

  • V. F. Babenko
  • N. V. Parfinovych Днепропетр. нац. ун-т
  • S. A. Pichugov Днепропетр. нац. ун-т ж.-д. трансп.


Let $C(\mathbb{R}^m)$ be the space of bounded and continuous functions $x: \mathbb{R}^m → \mathbb{R}$ equipped with the norm $∥x∥_C = ∥x∥_{C(\mathbb{R}^m)} := \sup \{ |x(t)|:\; t∈ \mathbb{R}^m\}$ 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 $$H^{j,ω_j} := \left\{x ∈ C(\mathbb{R}^m): ∥x∥_{ω_j} = ∥x∥_{H^{j,ω_j}} = \sup_{t_j≠0} \frac{∥Δtjejx(⋅)∥_C}{ω_j(|t_j|)} < ∞\right\}.$$ We obtain new sharp Kolmogorov-type inequalities for the norms $∥D^{α}_{ε}x∥_C$ of mixed fractional derivatives of functions $x ∈ ∩^{m}_{j=1}H^{j,ω_j}$. Some applications of these inequalities are presented.
How to Cite
Babenko, V. F., N. V. Parfinovych, and S. A. Pichugov. “Sharp Kolmogorov-Type Inequalities for Norms of Fractional Derivatives of Multivariate Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, no. 3, Mar. 2010, pp. 301–314,
Research articles