Mapping properties of the elliptic maximal function

Abstract

We prove that the elliptic maximal function maps the Sobolev space $W_{4,\eta }({\mathbb{R}}^2)$ into $L^4({\mathbb{R}}^2)$ for all $\eta >1/6$. The main ingredients of the proof are an analysis of the intersection properties of elliptic annuli and a combinatorial method of Kolasa and Wolff.