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

Let γ = (γ1,...,γd) 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 equation presented at =∞ and equation presented at for all x ∈ L ∞rγ (T d). Moreover, if β̄ is the least possible value of the exponent β in this inequality, then (d-1)(1-k/r)≤ β̄(d, γ,k,r) ≤ d - 1. © 2004 Springer Science+Business Media, Inc.