Estimates for the $n$-widths of compact sets of differentiate functions in spaces with weight functions

We obtain two-sided estimates of the widths defined by A. N. Kolmogorov for the unit spheres of the anisotropic Sobolev-Slobodetskii spaces in $L_q(\mu)$ for an arbitrary measure, guaranteeing the compactness of the corresponding embedding. As shown by examples, these estimates turn out to be exact as far as the order is concerned for measures with a “strong” singularity and, in addition, they allow us to justify the formula of the classical spectral asymptotics under very weak (close to necessary) restrictions of the measure $\mu$.