Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions - Babenko, V. F.; Parfinovych, N. V.; Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
Let C(R^m) be the space of bounded and continuous functions x: R^m → R equipped with the norm ∥x∥_C = ∥x∥_C(R^m) := { |x(t)|: t∈ R^m} and let e_j, j = 1,…,m, be a standard basis in R^m. Given moduli of continuity ω_j, j = 1,…, m, denote H^j,ω_j := {x ∈ C(R^m): ∥x∥_ω_j = ∥x∥_H^j,ω_j = _t_j≠0 ∥Δtjejx(⋅)∥_Cω_j(|t_j|) < ∞}. We obtain new sharp Kolmogorov-type inequalities for the norms ∥D^α_εx∥_C of mixed fractional derivatives of functions x ∈ ∩^m_j=1H^j,ω_j. Some applications of these inequalities are presented.