Sparse Bounds for Random Discrete Carleson Theorems

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 \[ \mathbb E X_m = σ_m = m^{-α}, \ 0 < α< 1/2, \] 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 $1-\alpha$. \[ \sup_{λ\in Λ} \Bigl| \sum_{m\neq 0} X_{\lvert m\rvert } \frac{e( λm )}{ {\rm sgn} (m)S_{ |m| }} f(x- m) \Bigr|. \] 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.