For domains \(G\subset R^ n\) verifying the flexible \(\lambda\)-horn condition the author gives integral representations for functions \(f: G\to R\) by differences, extending those given in O. V. Besov, V. P. Il’in and S. M. Nikolskij, Integral representations of functions and embedding theorems (Russian) (1975; Zbl 0352.46023). These representations are used to obtain inequalities relating \(L_ p\)-norms of differences of derivatives, implying evaluations for the \(L_ q\)- moduli of continuity of these derivatives and embedding theorems for spaces of type \(B^{\ell}_{p,\theta}(G)\). The results were announced in Dokl. Akad. Nauk SSSR 275, 1036-1041 (1984).