Geodesic
Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function
Feng Liu1 ; Qingying Xue2 ; Kôzô Yabuta3
1Shandong University of Science and Technology, College of Mathematics and System Science
2Beijing Normal University, School of Mathematical Sciences
3Kwansei Gakuin University, Research Center for Mathematics and Data Science
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 203-235

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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 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.
DOI : 10.54330/afm.113296
Keywords: Commutator, maximal commutator, local Hardy-Littlewood maximal function, Sobolev space, boundedness, continuity

Feng Liu  1   ; Qingying Xue  2   ; Kôzô Yabuta  3

1 Shandong University of Science and Technology, College of Mathematics and System Science
2 Beijing Normal University, School of Mathematical Sciences
3 Kwansei Gakuin University, Research Center for Mathematics and Data Science 
Feng Liu; Qingying Xue; Kôzô Yabuta. Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 203-235. doi: 10.54330/afm.113296
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