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

Authors:

Institution:

(1) Dnepropetrovsk University, Dnepropetrovsk;(2) Institute for Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk;(3) Dnepropetrovsk Agricultural University, Dnepropetrovsk

Abstract:

Let

= (

₁,...,

_d) be a vector with positive components and let D be the corresponding mixed derivative (of order gamma

_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 beta

in this inequality, then

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

Deceased.Translated from Ukrains rsquo

kyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 579–594, May, 2004.

Keywords:

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