Maximal operators and decoupling for $\Lambda(p)$ Cantor measures

TL;DR

This paper constructs specific Cantor measures with low Hausdorff dimension on the real line for which a maximal operator is bounded on certain Sobolev spaces, using a decoupling inequality approach.

Contribution

It introduces a new method to establish boundedness of maximal operators for fractal measures with no Fourier decay, including self-similar measures of arbitrarily low dimension.

Findings

Constructed Cantor measures with Hausdorff dimension less than 2/p

Established boundedness of maximal operators on Sobolev spaces for these measures

Used a decoupling inequality similar to prior work by Laba and Wang

Abstract

For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta (\mathbb{R})$ to $L^p(\mathbb{R})$. Maximal theorems for fractal measures on the line were previously obtained by Laba and Pramanik. The result here is weaker in that we are not able to obtain $L^p$ estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension $\alpha>0$, and have no Fourier decay. The proof is based on a decoupling inequality similar to that of Laba and Wang.