Jump and Variational Inequalities for Rough Operators

Yong Ding; Guixiang Hong; Honghai Liu

doi:10.1007/s00041-016-9484-8

Jump and Variational Inequalities for Rough Operators

Yong Ding
, Guixiang Hong
, Honghai Liu^*

^*Corresponding author for this work

Beijing Normal University
Wuhan University
Henan Polytechnic University

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families T:={Tε}ε>0 of truncated singular integrals and M:={Mt}t>0 of averaging operators with rough kernels, which are defined respectively by (Formula presented.) and (Formula presented.), where the kernel Ω belongs to Llog+L(Sn-1) or H¹(S^n-1) or G_α(S^n-1) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

Original language	English
Pages (from-to)	679-711
Number of pages	33
Journal	Journal of Fourier Analysis and Applications
Volume	23
Issue number	3
DOIs	https://doi.org/10.1007/s00041-016-9484-8
State	Published - 1 Jun 2017
Externally published	Yes

Keywords

Averaging operators
Jump inequalities
Rough kernels
Singular integrals
Variational inequalities

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10.1007/s00041-016-9484-8

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abstract = "In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families T:=\{Tε\}ε>0 of truncated singular integrals and M:=\{Mt\}t>0 of averaging operators with rough kernels, which are defined respectively by (Formula presented.) and (Formula presented.), where the kernel Ω belongs to Llog+L(Sn-1) or H1(Sn-1) or Gα(Sn-1) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.",

keywords = "Averaging operators, Jump inequalities, Rough kernels, Singular integrals, Variational inequalities",

author = "Yong Ding and Guixiang Hong and Honghai Liu",

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TY - JOUR

T1 - Jump and Variational Inequalities for Rough Operators

AU - Ding, Yong

AU - Hong, Guixiang

AU - Liu, Honghai

PY - 2017/6/1

Y1 - 2017/6/1

N2 - In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families T:={Tε}ε>0 of truncated singular integrals and M:={Mt}t>0 of averaging operators with rough kernels, which are defined respectively by (Formula presented.) and (Formula presented.), where the kernel Ω belongs to Llog+L(Sn-1) or H1(Sn-1) or Gα(Sn-1) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

AB - In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families T:={Tε}ε>0 of truncated singular integrals and M:={Mt}t>0 of averaging operators with rough kernels, which are defined respectively by (Formula presented.) and (Formula presented.), where the kernel Ω belongs to Llog+L(Sn-1) or H1(Sn-1) or Gα(Sn-1) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

KW - Averaging operators

KW - Jump inequalities

KW - Rough kernels

KW - Singular integrals

KW - Variational inequalities

UR - https://www.scopus.com/pages/publications/84974801230

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DO - 10.1007/s00041-016-9484-8

M3 - 文章

AN - SCOPUS:84974801230

SN - 1069-5869

VL - 23

SP - 679

EP - 711

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 3

ER -

Jump and Variational Inequalities for Rough Operators

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