Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications (Q1948537)

New exact Kolmogorov-type inequalities are derived which estimate the \(L_\infty \)-norm of the Riesz derivative \(D^\alpha f\) of a function \(f\) defined on the \(m\)-dimensional Euclidean space \(\mathbb{R} ^m\;(0<\alpha <2-m/s,\;s>m/2)\) by means of \(\| \Delta f\|_{L_s}\) and \(\| f\|_{L_\infty }\), where \(\Delta \) is the Laplace operator. Moreover, the following problems are studied. (1) The approximation of an unbounded operator \(D^\alpha \) by bounded ones. (2) The optimal recovery of the operator \(D^\alpha \) on elements of the class \(\{ f: \| \Delta f\|_s\leq 1\}\) given with known error. (3) Necessary and sufficient conditions for the existence of a function\(f\) for which the given numbers are upper bounds of the moduli of its derivatives of corresponding orders and \(\Delta f\in L_s(\mathbb{R}^m)\).