Geodesic
Strengthened Ul'yanov's inequalities for partial moduli of smoothness for functions from spaces with various metrics
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2019), pp. 26-38

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Ul'yanov's inequality connecting moduli of continuity in different metrics is well known for functions of one variable. In this paper functions of two variables are considered. Sharp Ul'yanov's inequalities connecting partial moduli of smoothness of positive order are proved in different mixed metrics. 
@article{VMUMM_2019_3_a3,
     author = {M. K. Potapov and B. V. Simonov},
     title = {Strengthened {Ul'yanov's} inequalities for partial moduli of smoothness for functions from spaces with various metrics},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {26--38},
     publisher = {mathdoc},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2019_3_a3/}
}
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M. K. Potapov; B. V. Simonov. Strengthened Ul'yanov's inequalities for partial moduli of smoothness for functions from spaces with various metrics. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2019), pp. 26-38. http://geodesic.mathdoc.fr/item/VMUMM_2019_3_a3/