Maximal operators and decoupling for Λ(p) Cantor measures

Abstract

For \(2\leq p<\infty\), \(\alpha'>2/p\), and \(\delta>0\), we construct Cantor-type measures on \(\mathbf{R}\) supported on sets of Hausdorff dimension \(\alpha<\alpha'\) for which the associated maximal operator is bounded from \(L^p_\delta (\mathbf{R})\) to \(L^p(\mathbf{R})\). Maximal theorems for fractal measures on the line were previously obtained by Laba and Pramanik [17]. 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 [18].