Hu, Guoen; Lai, Xudong; Xiangxing Tao; Xue, Qingying
arXiv.org,
04/2024
Paper
In this paper, the authors consider the endpoint estimates for the maximal Calderón commutator defined by $$T_{\Omega,\,a}^*f(x)=\sup_{\epsilon>0}\Big|\int_{|x-y|>\epsilon}\frac{\Omega(x-y)}{|x-y|^{d+1}} \big(a(x)-a(y)\big)f(y)dy\Big|,$$ where \(\Omega\) is homogeneous of degree zero, integrable on \(S^{d-1}\) and has vanishing moment of order one, \(a\) be a function on \(\mathbb{R}^d\) such that \(\nabla a\in L^{\infty}(\mathbb{R}^d)\). The authors prove that if \(\Omega\in L\log L(S^{d-1})\), then \(T^*_{\Omega,\,a}\) satisfies an endpoint estimate of \(L\log\log L\) type.
