Maximal operators and decoupling for Λ(p) Cantor measures

Kirjoittajat

  • Isabella Łaba University of British Columbia, Department of Mathematics

Avainsanat:

Maximal operators, Cantor sets, Hausdorff dimension, decoupling

Abstrakti

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].

 

Osasto
Articles

Julkaistu

2021-06-21

Viittaaminen

Łaba, I. (2021). Maximal operators and decoupling for Λ(p) Cantor measures. Annales Fennici Mathematici, 46(1), 163–186. Noudettu osoitteesta https://afm.journal.fi/article/view/109518