Weighted Norm Inequalities for Fractional Maximal Operators: a Bellman Function Approach

We study classical weighted L-p -> L-q inequalities for the fractional maximal operators on R-d, proved originally by Muckenhoupt and Wheeden in the 1970s. We establish a slightly stronger version of this inequality with the use of a novel extension of Bellman function method. More precisely, the estimate is deduced from the existence of a certain special function that enjoys appropriate majorization and concavity. From this result and an explicit version of the "A(p-epsilon) theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Perez, and Torres.