scholarly journals $L^p$ bounds for the commutators of singular integrals and maximal singular integrals with rough kernels

2014 ◽  
Vol 367 (3) ◽  
pp. 1585-1608 ◽  
Author(s):  
Yanping Chen ◽  
Yong Ding
Keyword(s):  
Singular Integrals ◽  
Rough Kernels ◽  
Maximal Singular Integrals

scholarly journals A note on maximal singular integrals with rough kernels

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Maximal Operators ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Homogeneous Mapping ◽  
Recent Developments ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

Abstract In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in $H^{1}({\mathrm{S}}^{n-1})$ H 1 ( S n − 1 ) , which complement some recent developments related to rough maximal singular integrals.

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

An Endpoint Estimate for Rough Maximal Singular Integrals

2018 ◽  
Vol 2020 (19) ◽  
pp. 6120-6134
Author(s):  
Petr Honzík
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Weak Type ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral ◽  
Open Question

Abstract We study the rough maximal singular integral $$T^{\#}_\Omega\big(\,f\big)\big(x\big)=\sup_{\varepsilon>0} \left| \int_{\mathbb{R}^{n}\setminus B(0,\varepsilon)}|y|^{-n} \Omega(y/|y|)\,f(x-y) \mathrm{d}y\right|,$$where $\Omega$ is a function in $L^\infty (\mathbb{S}^{n-1})$ with vanishing integral. It is well known that the operator is bounded on $L^p$ for $1<p<\infty ,$ but it is an open question whether it is of the weak type 1-1. We show that $T^{\#}_\Omega$ is bounded from $L(\log \log L)^{2+\varepsilon }$ to $L^{1,\infty }$ locally.

BOUNDEDNESS OF SINGULAR INTEGRALS AND MAXIMAL SINGULAR INTEGRALS ON TRIEBEL–LIZORKIN SPACES

2010 ◽  
Vol 21 (02) ◽  
pp. 157-168 ◽  
Author(s):  
CHUNJIE ZHANG ◽  
JIECHENG CHEN
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Formula ◽  
Maximal Singular Integrals ◽  
Maximal Singular Integral

In this paper, assuming Ω ∈ H1(Sn-1), we prove that the singular integral TΩ and the maximal singular integral [Formula: see text] are all bounded on Triebel–Lizorkin spaces, homogeneous or inhomogeneous.

scholarly journals Boundedness and Continuity of Several Integral Operators with Rough Kernels inWFβSn-1on Triebel-Lizorkin Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Feng Liu
Keyword(s):  
Besov Spaces ◽  
Singular Integrals ◽  
Integral Operators ◽  
Function Class ◽  
Function Spaces ◽  
Maximal Functions ◽  
Rough Kernels ◽  
Systematic Treatment ◽  
Marcinkiewicz Integrals

A systematic treatment is given of singular integrals and Marcinkiewicz integrals associated with surfaces generated by polynomial compound mappings as well as related maximal functions with rough kernels inWFβ(Sn-1), which relates to the Grafakos-Stefanov function class. Certain boundedness and continuity for these operators on Triebel-Lizorkin spaces and Besov spaces are proved by applying some criterions of bounds and continuity for several operators on the above function spaces.

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