Kolmogorov-Type Inequalities for Norms of Riesz Derivatives of Functions of Several Variables with Laplacian Bounded in $L_\infty$ and Related Problems

Let $L_{\infty,\infty}^\Delta(\mathbb R^m)$ be the space of functions $f\in L_\infty(\mathbb R^m)$ such that $\Delta f\in L_\infty(\mathbb R^m)$. We obtain new sharp Kolmogorov-type inequalities for the $L_\infty$-norms of the Riesz derivatives $D^\alpha f$ of the functions $f\in L_{\infty,\infty}^\Delta(\mathbb R^m)$ and solve the Stechkin problem of approximating an unbounded operator $D^\alpha$ by bounded operators on the class $f\in L_{\infty,\infty}^\Delta(\mathbb R^m)$ such that $\|\Delta f\|_\infty\le 1$, and also the problem of the best recovery of the operator $D^\alpha$ from elements of this class given with error $\delta$.