  Results on asymptotic characteristics of classes of functions with mixed
smoothness are obtained in the paper. Our main interest is in estimating the
Kolmogorov widths of classes with small mixed smoothness. We prove the
corresponding bounds for the unit balls of the trigonometric polynomials with
frequencies from a hyperbolic cross. We demonstrate how our results on the
Kolmogorov widths imply new upper bounds for the optimal sampling recovery in
the $L_2$ norm of functions with small mixed smoothness.
