Geodesic
Maximal operators and decoupling for Λ(p) Cantor measures
Isabella Łaba1
1 University of British Columbia, Department of Mathematics
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 163-186.

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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].  
Keywords: Maximal operators, Cantor sets, Hausdorff dimension, decoupling

Isabella Łaba 1

1 University of British Columbia, Department of Mathematics 
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Isabella Łaba. Maximal operators and decoupling for Λ(p) Cantor measures. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 163-186. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a9/