Geodesic
Characterizations of the Nikol'skii--Besov and Lizorkin--Triebel Function Spaces of Mixed Smoothness
Informatics and Automation, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 53-65

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The Nikol'skii–Besov and Lizorkin–Triebel function spaces of mixed positive smoothness are characterized in terms of fairly general (including local) averages and Peetre maximal functions. Atomic decompositions for functions from these spaces are also obtained. 
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     author = {D. B. Bazarkhanov},
     title = {Characterizations of the {Nikol'skii--Besov} and {Lizorkin--Triebel} {Function} {Spaces} of {Mixed} {Smoothness}},
     journal = {Informatics and Automation},
     pages = {53--65},
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     volume = {243},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2003_243_a5/}
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D. B. Bazarkhanov. Characterizations of the Nikol'skii--Besov and Lizorkin--Triebel Function Spaces of Mixed Smoothness. Informatics and Automation, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 53-65. http://geodesic.mathdoc.fr/item/TRSPY_2003_243_a5/