A note on maximal averages in the plane

Abstract:

Let $\mathcal {B}_{\delta }$ be the class of all $h\times \delta h$ rectangles in the plane with $h > 0$ and $0 < \delta < \frac {1}{2}$. The orientation of the rectangles is arbitrary. Form the maximal operator \[ GM f(x) = \sup _{0 < \delta < \frac {1}{2}}\;\; \sup _{x\in R\in \mathcal {B}_{\delta }}\;\; \frac {1}{|\log \delta |\cdot | R |}\int _R |f(y)| dy. \] Note the logarithmic term in the average. It is shown that $GM$ is a bounded maximal operator in $L^2(\mathbb {R}^2)$. The case of a fixed $\delta$ is due to Córdoba.