Spaces of positive smoothness functions on irregular domains

Weighted spaces of positive smoothness functions defined on irregular regions (of certain classes) of n-dimensional Euclidean space will be constructed. The smoothness of functions is measured by the behavior of the local deviations of the function from the polynomials in the integral metric. Integral representations of such functions will be constructed and embedding theorems connecting these spaces with Sobolev, Lebesgue spaces and with each other will be given.