Maximal Operators Associated to Sets of Directions of Hausdorff and Minkowski Dimension Zero

Abstract

For any set Ω⊂[0,π∕2) we let R _Ω be the set of rectangles in ℝ ² oriented in one of the directions of Ω. The geometric maximal operator associated to Ω is given by

$$\begin{array}{rcl}{ M}_{\Omega }f(x) =\sup\limits_{x\in R\in {R}_{\Omega }} \frac{1} {\left \vert R\right \vert }{\int \nolimits \nolimits }_{R}\left \vert f(y)\right \vert \;dy\;.& & \\ \end{array}$$

In this chapter we show that if M _Ω is bounded on L ^p(ℝ ²) for 1<p≤∞, then Ω must be countable and of Hausdorff and Minkowski dimension zero. We shall see that the converse does not hold, however, by exhibiting an example of a countable set Ω of Hausdorff and Minkowski dimension zero for which the associated maximal operator M _Ω is unbounded on L ^p(ℝ ²) for 1≤p<∞ . All of these results will be seen to be consequences of a recent theorem of Bateman (Duke Math. J. 147:55–77, 2009) regarding geometric maximal operators and N-lacunary sets.