The best $L_p$ approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space $L_p(\mathbb R^m)$, $m=2,3$, $1\le p\le\infty$, of the Laplace operator by linear bounded operators in the class of functions for which the square of the Laplace operator belongs to the space $L_p(\mathbb R^m)$. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of the values of the Laplace operator on functions from this class given with an error. We write an operator whose deviation from the Laplace operator is close to the best.