Sparse Bounds for Random Discrete Carleson Theorems

TL;DR

This paper establishes sparse bounds and weighted estimates for discrete random variants of the Carleson maximal operator, revealing new insights into their boundedness and behavior in harmonic analysis.

Contribution

It introduces novel sparse bounds and weighted estimates for random discrete Carleson operators, extending understanding of their boundedness properties.

Findings

Almost sure boundedness from ^p to ^p for certain ^p spaces

Sparse bounds imply new weighted ^2 estimates

Boundedness depends on the Minkowski dimension of  subset

Abstract

We study discrete random variants of the Carleson maximal operator. Intriguingly, these questions remain subtle and difficult, even in this setting. Let $\{X_m\}$ be an independent sequence of $\{0,1\}$ random variables with expectations \[ \mathbb E X_m = \sigma_m = m^{-\alpha}, \ 0 < \alpha < 1/2, \] and $ S_m = \sum_{k=1} ^{m} X_k$. Then the maximal operator below almost surely is bounded from $ \ell ^{p}$ to $ \ell ^{p}$, provided the Minkowski dimension of $ \Lambda \subset [-1/2, 1/2]$ is strictly less than $ 1- \alpha $. \[ \sup_{\lambda \in \Lambda } \Bigl| \sum_{m\neq 0} X_{\lvert m\rvert } \frac{e( \lambda m )}{ {\rm sgn} (m)S_{ |m| }} f(x- m) \Bigr|. \] This operator also satisfies a sparse type bound. The form of the sparse bound immediately implies weighted estimates in all $ \ell ^{2}$, which are novel in this setting. Variants and extensions are also considered.