Geodesic
Equivalent (Quasi)Norms for Certain Function Spaces of Generalized Mixed Smoothness
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 26-39.

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For certain (quasi)Banach function spaces of generalized mixed positive smoothness defined by (mixed) weighted norms of smooth dyadic decompositions of their Fourier transforms, characterizations are obtained in terms of rather general averages, their Peetre maximal functions, atomic and molecular representations, as well as in terms of the so-called $\varphi$-transforms. 
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D. B. Bazarkhanov. Equivalent (Quasi)Norms for Certain Function Spaces of Generalized Mixed Smoothness. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Studies on function theory and differential equations, Tome 248 (2005), pp. 26-39. http://geodesic.mathdoc.fr/item/TM_2005_248_a2/

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