Extension theorems with preservation of local approximation properties of functions in the nonisotropic case

Theorems are proved on the extension of functions from a set of sufficiently general form with preservation of the order of local approximation characteristics of these functions, called the $(\alpha,p)$-modulus of continuity. Extension is realized by a linear operator. As corollaries, descriptions are obtained of traces of functions from spaces $B_p^{\lambda\theta}$ and $BMO$ on compacta of sufficiently general form.