A Note on the Gurov-Reshetnyak Lemma

Let f be positive function summable on the unit cube I. Denote by O(f,Q) the integral oscillation of the function on the cube Q and by A(f,Q) the integral average of the function f on the cube Q. Trivially, O(f,Q) < 2A(f,Q) for all cubes Q which are subcubes of I. Gurov-Reshetnyak Lemma states that if one replace 2 with sufficiently small number then the function f will be in L^p(I) for some p > 1. In the talk I will discus some improvement and generalizations of this phenomena.