From the Littlewood-Paley-Stein Inequality to the Burkholder-Gundy Inequality

2021/11/09 by Zhendong Xu, Hao Zhang, Xu, Zhendong +1
Mathematics · #46L99 #47D07 #60G42. Secondary: 46B09 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2111.05164

openalex publication_date 2021/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \T_t\_t>0 be a symmetric diffusion semigroup on a σ-finite measure space (Ω, \mathscrA, μ) and G^T the associated Littlewood-Paley g-function operator: G^T(f)=(∫₀^∞ |t(∂)/(∂ t) T_t(f)|²(dt)/(t))^\frac12. The classical Littlewood-Paley-Stein inequality asserts that for any 10 of L_p(Ω). Recently, Xu proved that L^T _p\lesssim p as p→∞, and raised the problem abut the optimal order of L^T _p as p→∞. We solve Xu's open problem by showing that this upper estimate of L^T _p is in fact optimal. Our argument is based on the construction of a special symmetric diffusion semigroup associated to any given martingale such that its square function G^T(f) for any f∈ L_p(Ω) is pointwise comparable with the martingale square function of f. Our method also extends to the vector-valued and noncommutative setting.

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