The boundedness of variation associated with the commutators of approximate identities

Yongming Wen,Xianming Hou

doi:10.1186/s13660-021-02679-8

Abstract

In this paper, we establish L^{p}-boundedness and endpoint estimates for variation associated with the commutators of approximate identities, which are new for variation operators. As corollaries, we obtain the corresponding boundedness results for variation associated with the commutators of heat semigroups and Poisson semigroups.

Highlights

1 Introduction and main results The intension of this paper is to obtain boundedness of variation associated with the commutators of approximate identities
Rochberg, and Weiss [8] first studied the Lp-boundedness of commutators of singular integrals with the symbol b ∈ BMO(Rn)
We study variation associated with the commutators of approximate identities

Summary

Introduction

Introduction and main resultsThe intension of this paper is to obtain boundedness of variation associated with the commutators of approximate identities. A few years later, Liu and Wu [18] obtained the weighted Lp-boundedness for variation operators of commutators of truncated singular integrals with the Calderón–Zygmund kernels. Theorem 1.3 Let φ ∈ S(Rn) satisfy Rn φ(x) dx = 1 and b ∈ BMO(Rn), for ρ > 2 and f ∈ Hb1(Rn), Vρ (( f )b) is bounded from Hb1(Rn) to L1(Rn).

Results

Conclusion