Sharp Weighted Multidimensional Integral Inequalities for Monotone Functions

We prove sharp weighted inequalities for general integral operators acting on monotone functions of several variables. We extend previous results in one dimension, and also those in higher dimension for particular choices of the weights (power weights, etc.). We introduce a new kind of conditions, which take into account the more complicated structure of monotone functions in dimension n > 1, and give an example that shows how intervals are not enough to characterize the boundedness of the operators (contrary to what happens for n = 1). We also give several applications of our techniques.