Sparse Bounds for Random Discrete Carleson Theorems

2016/09/27 by Ben Krause, Michael T. Lacey, Krause, Ben +1
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1609.08701

openalex publication_date 2016/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 lt; αlt; 1/2, and S_m = ∑_k=1 ^m X_k. Then the maximal operator below almost surely is bounded from ℓ ^p to ℓ ^p, provided the Minkowski dimension of Λ⊂ [-1/2, 1/2] is strictly less than 1- α. sup_{λ∈ Λ} | ∑_{m≠ 0} X_{| m|} \frace( λm ) \rm sgn (m)S _|m| f(x- m) |. This operator also satisfies a sparse type bound. The form of the sparse bound immediately implies weighted estimates in all ℓ ², which are novel in this setting. Variants and extensions are also considered.

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