Strong weighted and restricted weak weighted estimates of the square function

2018/04/18 by Paata Ivanisvili, Ivanisvili, P., Pavel Mozolyako +3 · 1 citation
Mathematics · #42B20 #42B35 #47A30 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #F.2.2 #FOS: Mathematics #Mathematical Approximation and Integration #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1804.06869

openalex publication_date 2018/04/18 · openalex created_date 2018/04/24 · openalex updated_date 2026/07/28

Abstract

In this note we give a sharp weighted estimate for square function from L²(w) to L²(w), w∈ A₂. This has been known. But we also give a sharpening of this weighted estimate in the spirit of T1-type testing conditions. Finally we show that for any weight w∈ A^d₂ and any characteristic function of a measurable set ‖S_wχ_E‖_{L^{2, ∞}(w^-1)} ≤ C √[w]_A^d2 ‖χ_E‖_w, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for the restricted weak type is at most √[w]_A^d2.

Citations

• Weighted Weak Type Estimates for Square Functions
• Extremizers and Bellman function for martingale weak type inequality
• On A₂ conjecture and corona decomposition of weights
• Sharp weighted estimates for dyadic shifts and the A₂ conjecture
• A Bellman function counterexample to the A₁ conjecture: the blow-up of the weak norm estimates of weighted singular operators

Cited by

• Weighted Weak Type estimates for non-integral Square Functions

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