Geodesic
Difference properties of functions of several variables
Izvestiya. Mathematics , Tome 3 (1969) no. 3, pp. 633-642

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In this article, new identities are established for the so-called total, mixed, and partial differences of functions of several variables. As applications of these identities, expressions are given for mixed derivatives in terms of directional derivatives, as well as inequalities for total and mixed moduli of smoothness of continuous functions of $n$ variables. Moreover, it is shown that in the spaces $L_p$ $(1$, the orders of decrease of the total modulus of smoothness and of the sum of the moduli of smoothness in the coordinate directions only are the same. 
@article{IM2_1969_3_3_a7,
     author = {M. F. Timan},
     title = {Difference properties of functions of several variables},
     journal = {Izvestiya. Mathematics },
     pages = {633--642},
     publisher = {mathdoc},
     volume = {3},
     number = {3},
     year = {1969},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a7/}
}
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M. F. Timan. Difference properties of functions of several variables. Izvestiya. Mathematics , Tome 3 (1969) no. 3, pp. 633-642. http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a7/