Weighted Multidimensional Inequalities for Monotone Functions

Let + := {(x1, . . . , xN ) ; xi 0, i = 1, 2, . . . , N} and + := + . Assume that f : + → + is monotone which means that it is monotone with respect to each variable. We denote f ↓, when f is decreasing (= nonincreasing) and f ↑ when f is increasing (= nondecreasing). Throughout this paper ω, u, v are positive measurable functions defined on + , N 1. A function P on [0,∞) is called a modular function if it is strictly increasing, with the values 0 at 0 and ∞ at ∞. For the definition of an N-function we refer to [7]. We say that a modular function P is weakly convex if 2P (t) P (Mt), for all t > 0 and some constant M > 1. All convex modular functions are obviously weakly convex. The function P1(t) = t, 0 < p < 1 and the function P2(t) = exp( √ t)− 1 are weakly convex, but not convex. See also [6].