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

Abstract

For $2\le p<\infty$, ${\alpha }^{\prime }>2/p$, and $\delta >0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha <{\alpha }^{\prime }$ for which the associated maximal operator is bounded from $L_{\delta }^p(\mathbb{R})$ 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.

26 pages. Minor corrections, two references added