The Olovyanishnikov Inequality for Multivariate Functions

Let G be the real line R, space R, the negative half-line R− or the octant R− := {(x1, · · · , xm) ∈ R : x1 ≤ 0, · · · , xm ≤ 0}. Let Lp = Lp(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 Lp = Lp(G), r ∈ N, we will denote the space of functions x : G → R, that have locally absolutely continuous derivative x(r−1) and such that x ∈ Lp(G). For 1 ≤ p, s ≤ ∞ we set Lp,s = L r p,s(G) = L r s(G) ∩ Lp(G). Great amount of work has been done on finding inequalities of the form