Bilinear endpoint estimates for Calderón commutator with rough kernel

2017/10/26 by Xudong Lai, Lai, Xudong
Mathematics · #42B20 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1710.09664

openalex publication_date 2017/10/26 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish some bilinear endpoint estimates of Calderón commutator C[∇ A,f](x) with a homogeneous kernel when Ω∈ Llog⁺L(S^d-1). More precisely, we prove that C[∇ A,f] maps L^q(ℝ^d)× L¹(ℝ^d) to L^r,∞(ℝ^d) if q>d which improves previous result essentially. If q=d, we show that Calderón commutator maps L^d,1(ℝ^d)× L¹(ℝ^d) to L^r,∞(ℝ^d) which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint estimate of the Mary Weiss maximal function which may have its own interest in the theory of singular integral.

Related

• An endpoint estimate for the maximal Calderón commutator with rough kernel
• L^p(ℝ^d) boundedness for the Calderón commutator with rough kernel
• Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels
• Sharp bilinear estimates for maximal singular integrals with kernels in weighted L^q spaces
• An endpoint estimate for the commutators of singular integral operators with rough kernels