Geodesic
Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains
Matematičeskie zametki, Tome 74 (2003) no. 3, pp. 340-349.

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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 $m-1$. 
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O. V. Besov. Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains. Matematičeskie zametki, Tome 74 (2003) no. 3, pp. 340-349. http://geodesic.mathdoc.fr/item/MZM_2003_74_3_a2/

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