CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

On a singular integral of Christ-Journé type with homogeneous kernel

  • Yong Ding,
    School of Mathematical Sciences, Beijing Normal University, People's Republic of China
  • Xudong Lai,
    Institute for Advanced Study in Mathematics, Harbin Institute of Technology, People's Republic of China
Format:   LaTeX   MathJax   PDF  

Abstract

In this paper, we prove that the following singular integral defined by $$T_{\Omega,a}f(x)=\operatorname{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{|x-y|^d}\cdot m_{x,y}a\cdot f(y)dy$$ is bounded on $L^p(\mathbb{R}^d)$ for $1\lt p\lt \infty$ and is of weak type (1,1), where $\Omega\in L\log^+L(\mathbb{S}^{d-1})$ and $m_{x,y}a=:\int_0^1a(sx+(1-s)y)ds$ with $a\in L^\infty(\mathbb{R}^d)$ satisfying some restricted conditions.
Keywords: Calderón commutator, rough kernel, weak type (1, 1) Calderón commutator, rough kernel, weak type (1, 1)
MSC Classifications: 42B20 show english descriptions Singular and oscillatory integrals (Calderon-Zygmund, etc.) 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.)
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/