Estimates of the norms of derivatives in the one- and multidimensional cases

We consider multiplicative inequalities of the Kolmogorov type for the norms of the intermediate derivatives of a function through the norms of the function itself and its highest derivative in various Lebesgue spaces for all kinds of one-dimensional areas. Some estimates of the constants in the inequalities in these spaces are established and the asymptotic behavior of such constants is given as the orders of both the highest and intermediate derivatives grow infinitely. In addition, we obtain an estimate for the norms of the mixed derivatives of a function in terms of the norms of the derivatives with respect to each variable separately in different Lebesgue spaces for the case of the multidimensional torus. The results obtained are of independent interest and also can be used in solving various problems of mathematical physics. This in particular applies to problems in which theorems substantially involve embeddings of the corresponding function spaces.