Integral representations of functions which belong to weighted Sobolev classes in regions, with applications. I

Abstract

In the present work for general weighted classes of functions which are anisotropic with respect to the differentiation indices and with respect to the weight indices we isolate classes of regions for which we have imbedding theorems which are of the same nature as those for a half-space. A definitive result of this type for WD(I) spaces was obtained by Besov and II'in in (I0-13). In our case such regions will also be determined by a horn type of condition, but, in contrast to the unweighted case, the form of the horn for weighted classes of functions dePends in an essential fashion on the point in the region. Our conditions on the region are definitive in a certain sense since without the weight the conditions revert to the conditions for the W~ ~)'' classes. A fundamental tool in our investigation is an integral representation of functions which generalizes a known representation due to Win (see (11)). To obtain our integral identity we use a function average with an averaging parameter that is point-dependent. The first such construction was obtained by Kudryav-