|Sep 12, 2017
We study various ways to improve integrability for averaging operator along two dimensional parabola. We begin with Stein's analytic interpolation, which was the first approach to this problem, to obtain the strong type at the endpoint. Then we use a real interpolation method for restricted weak type estimate, subsequently some recent progress by combinatorial method due to Christ will be introduced. Finally we discuss multilinear approach which is an alternative method for the strong type estimate.