Kolmogorov-type inequalities for mixed derivatives of functions of many variables - Babenko, V. F.; Korneichuk, N. P.; Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2004)
Let $γ = (γ_1 ,..., γ_d )$ be a vector with positive components and let $D^γ$ be the corresponding mixed derivative (of order $γ_j$ with respect to the $j$ th variable). In the case where $d > 1$ and $0 < k < r$ are arbitrary, we prove that $$\sup_{x \in L^{rγ}_{∞}(T^d)D^{rγ}x eq0}\frac{||D^{kγ}x||_{L_{∞}(T^d)}}{||x||^{1-k/r}||D^{rγ}||^{k/r}_{L_{∞}(T^d)}} = ∞$$and $$||D^{kγ}x||_{L_{∞}(T^d)} ≤ K||x||^{1 - k/r}_{L_{∞}(T^d)}||D^{rγ}x||_{L_{∞}(T^d)}^{k/r}\left(1 + \ln^{+}\frac{||D^{rγ}x||_{L_{∞}(T^d)}}{||x||_{L_{∞}(T^d)}}\right)^{β}$$for all $x \in L^{rγ}_{∞}(T^d)$ Moreover, if \(\bar β \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar β \left( {d,γ ,k,r} \right) \leqslant d - 1.$$