Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (6): 1446-1460.

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ρ-Variation for Singular Integral Operators with Variable Kernels

Zhenbing Gong(),Yanping Chen*(),Wenyu Tao()   

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083
  • Received:2020-03-22 Online:2020-12-26 Published:2020-12-29
  • Contact: Yanping Chen;;
  • Supported by:
    the NSFC(11871096);the NSFC(11471033)


In this paper, we will prove that the variation of singular integral operators with rough variable kernels are bounded on $L^{2}({\mathbb S} ^{n})$ if $\Omega\in L^{\infty}({\mathbb R} ^{n})\times L^{q}({\mathbb S}^{n-1})$ for $q>2(n-1)/n$, and $n\geq2$. Moreover, we can also obtain the weighted variational inequalities for singular integral operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

Key words: Variational inequalities, Variable kernel, Singular integral, Ap-weight, Morrey space

CLC Number: 

  • O174.2