A Note on Maximal Averages in the Plane

TL;DR

This paper provides a self-contained proof of maximal operator bounds in the plane for arbitrary directions, extending previous results with sharp $L^2$ estimates using variants of the TT* method.

Contribution

It offers a new, self-contained proof of maximal operator bounds in the plane and extends a theorem of Cordoba with sharp $L^2$ estimates.

Findings

Proved maximal operator bounds for arbitrary directions in $ r^2$.

Extended Cordoba's theorem with sharp $L^2$ estimates.

Utilized variants of the TT* method for the proofs.

Abstract

Using variants of the TT* method we give a self-contained proof of the result of Alfonseca, Soria and Vargas on maximal operators on arbitrary directions in $\rr^2$. We also give a sharp $L^2$ estimate for a maximal function extending a Theorem of Cordoba.