APPROXIMATION THEORY: A volume dedicated to Borislav Bojanov (D. K. Dimitrov, G. Nikolov, and R. Uluchev, Eds.) additional information (to be provided by the publisher) The Olovyanishnikov Inequality for Multivariate Functions Vladislav Babenko * , Yuliya Babenko 1. Introduction. Let G be the real line R, space R m , the negative half-line R − or the octant R m − := {(x 1 , ··· ,x m ) ∈ R m : x 1 ≤ 0, ··· ,x m ≤ 0}. Let L p = L p (G), 1 ≤ p ≤∞, be the space of functions x : G → R, integrable in the power p on G (essentially bounded when p = ∞), with usual norm. In the case when G = R or G = R − by L r p = L r p (G), r ∈ N, we will denote the space of functions x : G → R, that have locally absolutely con- tinuous derivative x (r−1) such that x (r) ∈ L p (G). For 1 ≤ p, s ≤∞ set L r p,s = L r p,s (G)= L r s (G) ∩ L p (G). Great amount of work has been done on finding inequalities of the form   x (k)    q ≤ K ·‖x‖ α p   x (r)    β s . (1.1) for functions x ∈ L r p,s (G). Inequalities with the best possible constants (sharp inequalities) are especially interesting, and research of a lot of mathematicians was devoted to obtain such inequalities. The first sharp result was obtained by Landau [9] in 1913 for the case x ∈ L 2 ∞,∞ (R − ), k = 1. One of the first complete results in this area was obtained by Kolmogorov [7] in 1939: for any k ∈ N, and k<r and any function x ∈ L r ∞,∞ (R)   x (k)    ∞ ≤ ‖ϕ r−k ‖ ∞ ‖ϕ r ‖ 1−k/r ∞ ‖x‖ 1−k/r ∞   x (r)    k/r ∞ , (1.2) where ϕ r is the r th periodic integral with zero mean value on the period of the function ϕ 0 (t) = sgn sin x. Such functions are called Euler splines. The Kolmogorov inequality (1.2) becomes an equality for any function of the type ϕ r (λt) for a positive real λ. Since this result inequalities of type (1.1) are often called Kolmogorov type inequalities. * Supported in part by FFIU under project No. 01.07/00241