  The article examines Nikolskii and Besov spaces with norms defined using
$L_p$-averaged mixed moduli of continuity of functions of appropriate orders,
instead of mixed moduli of continuity of known orders for certain mixed
derivative functions. The author builds continuous linear mappings of such
spaces of functions defined in domains of certain type to ordinary Nikolskii
and Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are function
extension operators, thus incurring coincidence of both kinds of spaces in the
said domains. It also significantly broadens the class of Nikolskii and Besov
spaces of mixed smoothness for which the theorems of those kind of extension
have been derived. Under certain conditions, operators of partial
differentiation from the aforementioned function spaces of mixed smoothness to
Lebesgue spaces have been established to be continuous.
