Maximal operators and differentiation theorems for sparse sets

Abstract

We study maximal averages associated with singular measures on $\mathbb{R}$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1-\epsilon$ with $0\le \epsilon <1/3$ for which the corresponding maximal operators are bounded on $L^p(\mathbb{R})$ for $p>(1+\epsilon )/(1-\epsilon )$. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems for singular measures in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis; in particular, there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions.