Multivariate Inequalities of Kolmogorov Type and Their Applications

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

“…In the case of functions of two or more variables, very few results of this type are known (see [9][10][11][12][13][14][15]). Many questions in analysis require the consideration of derivatives and antiderivatives of fractional order (see, e.g., [16]).…”

“…1 ; : : : ; ! m /: We assume that inequality (12) is true and select 0 < L 0 Ä M 0 such that M˛D 2 m 1 A˛Z It is obvious that x 2 T m j D1 H j;! j and, furthermore,…”

On the set of extremal functions in certain Kolmogorov-type inequalities

“…are extremal in inequalities (6) and (7). By virtue of Theorems 3 and 4 of the present paper, these functions exhaust the set of extremal functions in inequalities (6) and (7).…”

“…By virtue of Theorems 3 and 4 of the present paper, these functions exhaust the set of extremal functions in inequalities (6) and (7). Since the proof of the results of the present paper is based on the Kolmogorov comparison theorem [1], we give its statement below.…”

On Kolmogorov-type inequalities for fractional derivatives of functions of two variables

“…Note that there are very few exact Kolmogorov-type inequalities known for the functions of two and more variables (see [11][12][13][14][15]). …”