Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Abstract

Let γ = (γ1,...,γ d ) be a vector with positive components and let Dγ be the corresponding mixed derivative (of order γ j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that $$\mathop {\sup }\limits_{x \in L_\infty ^{r\gamma } \left( {T^d } \right)} \frac{{\left\| {D^{k\gamma } x} \right\|_{L_\infty \left( {T^d } \right)} }}{{\left\| x \right\|_{L_\infty \left( {T^d } \right)}^{1 - {k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} \left\| {D^{r\gamma } } \right\|_{L_\infty \left( {T^d } \right)}^{{k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} }}$$ and $$\left\| {D^{k\gamma } x} \right\|_{L_\infty \left( {T^d } \right)} \leqslant K\left\| x \right\|_{L_\infty \left( {T^d } \right)}^{1 - {k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} \left\| {D^{r\gamma } } \right\|_{L_\infty \left( {T^d } \right)}^{{k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} \left( {1 + \ln ^ + \frac{{\left\| {D^{k\gamma } x} \right\|_{L_\infty \left( {T^d } \right)} }}{{\left\| x \right\|_{L_\infty \left( {T^d } \right)} }}} \right)^\beta $$ for all \(x \in L_\infty ^{r\gamma } \left( {T^d } \right)\) Moreover, if \(\bar \beta \) is the least possible value of the exponent β in this inequality, then $$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$