Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2022; 59(6): 1539-1555

Online first article September 6, 2022      Printed November 30, 2022

https://doi.org/10.4134/BKMS.b210874

Copyright © The Korean Mathematical Society.

Boundedness and continuity for variation operators on the Triebel--Lizorkin spaces

Feng Liu, Yongming Wen, Xiao Zhang

Shandong University of Science and Technology; Minnan Normal University; Shandong University of Science and Technology

Abstract

In this paper, we establish the boundedness and continuity for variation operators for $\theta$-type Calder\'{o}n--Zygmund singular integrals and their commutators on the Triebel--Lizorkin spaces. As applications, we obtain the corresponding results for the Hilbert transform, the Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators.

Keywords: Variation operator, commutator, Triebel--Lizorkin space, boundedness and continuity

MSC numbers: Primary 42B20, 42B25

Supported by: The first author was partially supported by the National Natural Science Foundation of China (grant No. 11701333), the second author was partially supported by the Scientific Research Project of the Education Department of Fujian Province (No. JAT200331) and President's Fund of Minnan Normal University (No. KJ2020020).