Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

In this paper, we study the spaces $B_{pq}^s(G)$ and $L_{pq}^s(G)$ of functions $f$ with positive exponent of smoothness $s > 0$ given on a domain $G\subset\mathbb R^n$. The norms on these spaces are defined via integral norms of the difference of the function $f$ of order $m > s$ treated as a function of the point of the domain and of the difference increment. For an arbitrary domain $G\subset\mathbb R^n$, we characterize these spaces in terms of the local approximations of the function by polynomials of degree $m-1$.