Maximal operators and differentiation theorems for sparse sets

TL;DR

This paper constructs singular measures on sparse sets in one dimension and proves boundedness of associated maximal operators on L^p spaces for p > (1+ε)/(1−ε), advancing differentiation theorems in harmonic analysis.

Contribution

It introduces a new class of singular Cantor-type measures with bounded maximal operators on L^p for p > (1+ε)/(1−ε), answering a longstanding question in the field.

Findings

Maximal operators are bounded on L^p for p > (1+ε)/(1−ε).

Differentiation theorems hold for p > (1+ε)/(1−ε).

Global maximal operators are bounded for all p > 1.

Abstract

We study maximal averages associated with singular measures on $\rr$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1 - \epsilon$, $0 \leq \epsilon < {1/3}$ 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 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. Updates: Andreas Seeger has provided an argument to the effect that our global maximal operators are in fact bounded on L^p(R) for all p>1; in particular, it follows that our differentiation theorems are also valid for all p>1. Furthermore, David Preiss has proved that no such differentiation theorems (let alone maximal estimates) can hold with p=1. These arguments are included in the new version. We have also improved the exposition in a number of places.