Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

Authors

  • Feng Liu Shandong University of Science and Technology, College of Mathematics and System Science
  • Qingying Xue Beijing Normal University, School of Mathematical Sciences
  • Kôzô Yabuta Kwansei Gakuin University, Research Center for Mathematics and Data Science

Keywords:

Commutator, maximal commutator, local Hardy-Littlewood maximal function, Sobolev space, boundedness, continuity

Abstract

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.
Section
Articles

Published

2021-12-31

How to Cite

Liu, F., Xue, Q., & Yabuta, K. (2021). Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function. Annales Fennici Mathematici, 47(1), 203–235. https://doi.org/10.54330/afm.113296