Geodesic
Relations between mixed moduli of smoothness, and embedding theorems for Nikol'skii classes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 204-214

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We prove $(L_p,L_q)$ inequalities for mixed moduli of smoothness of positive orders. As corollaries, we obtain embedding theorems for the Nikol'skii classes. 
@article{TM_2010_269_a16,
     author = {M. K. Potapov and B. V. Simonov and S. Yu. Tikhonov},
     title = {Relations between mixed moduli of smoothness, and embedding theorems for {Nikol'skii} classes},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {204--214},
     publisher = {mathdoc},
     volume = {269},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_269_a16/}
}
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M. K. Potapov; B. V. Simonov; S. Yu. Tikhonov. Relations between mixed moduli of smoothness, and embedding theorems for Nikol'skii classes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 204-214. http://geodesic.mathdoc.fr/item/TM_2010_269_a16/