Sparse Bounds for Random Discrete Carleson Theorems

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 $Λ\subset [-1/2, 1/2]$ is strictly less than $1- α$. $\sup_{λ\in Λ} \Bigl| \sum_{m\neq 0} X_{\lvert m\rvert } \frac{e( λm )}{ {\rm sgn} (m)S_{ |m| }} f(x- m) \Bigr|.$ 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.

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