Maximal operators and decoupling for Λ(p) Cantor measures
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
Affiliations des auteurs :
Isabella Łaba 1
@article{AFM_2021_46_1_a9,
author = {Isabella {\L}aba},
title = {Maximal operators and decoupling for {\ensuremath{\Lambda}(p)} {Cantor} measures},
journal = {Annales Fennici Mathematici},
pages = {163--186},
publisher = {mathdoc},
volume = {46},
number = {1},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a9/}
}
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/