Limiting weak-type behaviors for singular integrals with rough Llog L(\mathbbSⁿ) kernels

2021/06/26 by Qin, Moyan, Wu, Huoxiong, Xue, Qingying
#42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.14051

Abstract

Let Ω be a function of homogeneous of degree zero and vanish on the unit sphere \mathbb Sⁿ. In this paper, we investigate the limiting weak-type behavior for singular integral operator T_Ω associated with rough kernel Ω. We show that, if Ω∈ Llog L(\mathbb Sⁿ), then lim_λ→0⁺λ|\x∈ℝⁿ:|T_Ω(f)(x)|>λ\| = n^-1‖Ω‖_L¹(\mathbb Sⁿ)‖f‖_L¹(ℝⁿ), 0≤ f∈ L¹(ℝⁿ). Moreover,(n^-1‖Ω‖_L¹(\mathbbS^n-1) is a lower bound of weak-type norm of T_Ω when Ω∈ Llog L(\mathbbS^n-1). Corresponding results for rough bilinear singular integral operators defined in the form T_Ω(f₁,f₂) = T_Ω1(f₁)⋅ T_Ω2(f₂) have also been established.

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