Sharp weak-type estimate for maximal operators associated to Cartesian families under an arithmetic condition

2023/04/28 by Anthony Gauvan, Gauvan, Anthony · 1 citation
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2304.14792

openalex publication_date 2023/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Given a set of integers A ⊂ ℤ, we consider the smallest family B_A^n-1 invariant by translation which contains the rectangles R_\boldsymbola = I_a1 × … × I_a_n-1 × I_-(a₁+…+a_n-1) for any \boldsymbola = (a₁,…,a_n-1) ∈ A^n-1 and where I_k = [0,2^k] for k integer. We prove that if the set A contains arbitrary large arithmetic progression then the maximal operator M_B_A^n-1 associated to the family B_A^n-1 is sharply bounded from L¹(1+ log⁺ L¹ )^n-1 to L^1,∞.

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