scholarly journals Maximal singular integrals on product homogeneous groups

2014 ◽  
Vol 222 (1) ◽  
pp. 41-49
Author(s):  
Yong Ding ◽  
Shuichi Sato
Keyword(s):  
Singular Integrals ◽  
Homogeneous Groups ◽  
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.

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.

scholarly journals Singular integrals with flag kernels on homogeneous groups, I

2012 ◽  
Vol 28 (3) ◽  
pp. 631-722 ◽  
Author(s):  
Alexander Nagel ◽  
Fulvio Ricci ◽  
Elias Stein ◽  
Stephen Wainger
Keyword(s):  
Singular Integrals ◽  
Homogeneous Groups ◽  
Flag Kernels

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.

Two-Weighted Inequalities for Integral Operators in Lorentz Spaces Defined on Homogeneous Groups

1999 ◽  
Vol 6 (1) ◽  
pp. 65-82
Author(s):  
V. Kokilashvili ◽  
A. Meskhi
Keyword(s):  
Singular Integrals ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Hardy Operator ◽  
Weighted Inequalities ◽  
Homogeneous Groups ◽  
Necessary And Sufficient

Abstract The optimal sufficient conditions are found for weights, which guarantee the validity of two-weighted inequalities for singular integrals in the Lorentz spaces defined on homogeneous groups. In some particular case the found conditions are necessary for the corresponding inequalities to be valid. Also, the necessary and sufficient conditions are found for pairs of weights, which provide the validity of two-weighted inequalities for the generalized Hardy operator in the Lorentz spaces defined on homogeneous groups.

Sign in / Sign up

Export Citation Format

Share Document