Extending functions from isotropic Nikolskii-Besov spaces and their approximating with derivatives

The article examines isotropic Nikolskii and Besov spaces with norms defined using $L_p$-averaged modulus of continuity of functions of appropriate order, instead of modulus of continuity of known order for fixed-order partial derivative functions. The author builds continuous linear mappings of such spaces of functions defined in domains of $(1,\ldots,1)$-type (in a broad sense) to ordinary isotropic Nikolskii and Besov spaces in $\mathbb R^d$ that are function extension operators, thus incurring coincidence of both kinds of spaces in the said domains. It is established that every bounded domain in $\mathbb R^d$ with a Lipschitzian boundary is a $(1,\ldots,1)$-type domain (in a broad sense). The article also provides weak asymptotics of approximation characteristics related to the problem of reconstruction of functions with their derivatives from function values at a given number of points, the S.B.Stechkin's problem for differential operator, and the problem of width asymptotics for isotropic Nikolskii-Besov classes in those domains.