{ "id": "2203.11541", "version": "v1", "published": "2022-03-22T08:48:46.000Z", "updated": "2022-03-22T08:48:46.000Z", "title": "$L^p(\\mathbb{R}^d)$ boundedness for the Calderón commutator with rough kernel", "authors": [ "Jiecheng Chen", "Guoen Hu", "Xiangxing Tao" ], "comment": "19pages", "categories": [ "math.CA" ], "abstract": "Let $k\\in\\mathbb{N}$, $\\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have vanishing moment of order $k$, $a$ be a function on $\\mathbb{R}^d$ such that $\\nabla a\\in L^{\\infty}(\\mathbb{R}^d)$, and $T_{\\Omega,\\,a;k}$ be the $d$-dimensional Calder\\'on commutator defined by $$T_{\\Omega,\\,a;k}f(x)={\\rm p.\\,v.}\\int_{\\mathbb{R}^d}\\frac{\\Omega(x-y)}{|x-y|^{d+k}}\\big(a(x)-a(y)\\big)^kf(y){d}y.$$ In this paper, the authors prove that if $$\\sup_{\\zeta\\in S^{d-1}}\\int_{S^{d-1}}|\\Omega(\\theta)|\\log ^{\\beta} \\big(\\frac{1}{|\\theta\\cdot\\zeta|}\\big)d\\theta<\\infty,$$ with $\\beta\\in(1,\\,\\infty]$, then for $\\frac{2\\beta}{2\\beta-1}