Sharp Weak Type Estimates for Maximal Operators associated to Rare Bases

2022/04/27 by Paul Hagelstein, Hagelstein, Paul, Giorgi Oniani +3 · 1 citation
Computer Science · Mathematics · #42B25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2204.12871

openalex publication_date 2022/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B denote a nonempty translation invariant collection of intervals in ℝⁿ (which we regard as a rare basis), and define the associated geometric maximal operator M_B by M_Bf(x) = sup_{x ∈ R ∈ B} (1)/(|R|)∫_R |f|. We provide a sufficient condition on B so that the estimate |\x ∈ ℝⁿ : M_Bf(x) gt; α\|≤ C_n ∫_ℝⁿ \frac|f|α(1+log⁺\frac|f|α)^n-1 is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including Córdoba, Soria and Zygmund bases.

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