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

Abstract

Let γ = (γ1 ,..., γd ) be a vector with positive components and let Dγ be the corresponding mixed derivative (of order γj with respect to the j th 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

$$x \in L_\infty ^{r\gamma } \left( {T^d } \right)$$

for all

$$\left\| {D^{k\gamma } x} \right\|_{L_\infty \left( {T^d } \right)} \leqslant K\left\| x \right\|_{L_\infty \left( {T^d } \right)}^{1 - {k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} \left\| {D^{r\gamma } } \right\|_{L_\infty \left( {T^d } \right)}^{{k \mathord{\left/ {\vphantom {k r}} \right. \kern-\nulldelimiterspace} r}} \left( {1 + \ln ^ + \frac{{\left\| {D^{k\gamma } x} \right\|_{L_\infty \left( {T^d } \right)} }}{{\left\| x \right\|_{L_\infty \left( {T^d } \right)} }}} \right)^\beta $$

Moreover, if $\bar \beta$ is the least possible value of the exponent β in this inequality, then

$$\left( {d - 1} \right)\left( {1 - \frac{k}{r}} \right) \leqslant \bar \beta \left( {d,\gamma ,k,r} \right) \leqslant d - 1.$$