A non-homogeneous local Tb theorem for Littlewood-Paley g_λ^*-function with L^p-testing condition

2015/07/19 by Mingming Cao, Qingying Xue, Cao, Mingming +1
Mathematics · #42B25 #47G10 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #math.CA #msc:42B25 #msc:47G10

paper · pdf · doi:10.48550/arxiv.1507.05291

26 pages

arxiv created 2015/07/19 · openalex publication_date 2015/07/19 · arxiv updated 2015/07/21 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present a local Tb theorem for the non-homogeneous Littlewood-Paley g_λ^*-function with non-convolution type kernels and upper power bound measure μ. We show that, under the assumptions \supp b_Q ⊂ Q, |∫_Q b_Q dμ| \gtrsim μ(Q) and ||b_Q||^p_L^p(μ) \lesssim μ(Q), the norm inequality ‖ g_λ^*(f) ‖_L^p(μ) \lesssim ‖ f ‖_L^p(μ) holds if and only if the following testing condition holds : sup_{Q : cubes in \Rn} (1)/(μ(Q))∫_Q (∫₀^ℓ(Q) ∫_\Rn ((t)/(t+|x-y|))^mλ|θ_t(b_Q)(y,t)|² \fracdμ(y) dtt^m+1)^p/2 dμ(x) < ∞. This is the first time to investigate g_λ^*-function in the simultaneous presence of three attributes : local, non-homogeneous and L^p-testing condition. It is important to note that the testing condition here is L^p type with p ∈ (1,2].

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