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 o ne dimension, and also those in higher dimension for particular choices of the weights (power weights, etc.). We introduce a new kind of conditions, w hich take into account the more complicated structure of monotone functions in dimension n > 1, and give an example that shows how intervals are not e nough to characterize the boundedness of the operators (contrary to what ha ppens for n = 1). We also give several applications of our techniques.