Kolmogorov-type inequalities for the norms of Riesz derivatives of multivariable functions and some applications

Let $L_{\infty,s}^1(\mathbb R^m)$ be the space of functions $f\in L_\infty(\mathbb R^m)$ such that $\partial f/\partial x_i\in L_s(\mathbb R^m)$ for each $i=1,\dots,m$. New sharp Kolmogorov-type inequalities are obtained for the norms of the Riesz derivatives $\|D^\alpha f\|_\infty$ of functions $f\in L_{\infty,s}^1(\mathbb R^m)$. Stechkin's problem on the approximation of unbounded operators $D^\alpha$ by bounded operators on the class of functions $f\in L_{\infty,s}^1(\mathbb R^m)$ such that $\|\nabla f\|_s\le1$, as well as the problem on the optimal reconstruction of the operator $D^\alpha$ on elements of this class given with error $\delta$, is solved.