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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 mixedderivative functions. The author builds continuous linear mappings of suchspaces of functions defined in domains of certain type to ordinary Nikolskiiand Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are functionextension operators, thus incurring coincidence of both kinds of spaces in thesaid domains. It also significantly broadens the class of Nikolskii and Besovspaces of mixed smoothness for which the theorems of those kind of extensionhave been derived. Under certain conditions, operators of partialdifferentiation from the aforementioned function spaces of mixed smoothness toLebesgue spaces have been established to be continuous.

Extending functions from Nikolskii-Besov spaces of mixed smoothness beyond domains of certain kind