A note on maximal averages in the plane

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

$$GM f(x) = \sup_{0 < \delta < \frac{1}{2}}\;\; \sup_{x\in R\in\ma... ... \frac{1}{\vert\log \delta \vert\cdot \vert R \vert}\int_R \vert f(y)\vert dy.$$

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.