scholarly journals Estimates for Compositions of Maximal Operators with Singular Integrals

2013 ◽  
Vol 56 (4) ◽  
pp. 801-813 ◽  
Author(s):  
Richard Oberlin
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Operator Norm ◽  
Maximal Operators ◽  
Multiplier Operator ◽  
Norm Estimates ◽  
Variation Norm

Abstract.We prove weak-type (1, 1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator Δ*Ψ where Δ* is Bourgain’s maximal multiplier operator and is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the Lq operator norm when 1 < q < 2. We also consider associated variation-norm estimates.

scholarly journals Maximal operators for the holomorphic Laguerre semigroup

2005 ◽  
Vol 97 (2) ◽  
pp. 235 ◽  
Author(s):  
Emanuela Sasso
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Maximal Operators ◽  
Strong Type ◽  
Half Integer ◽  
Starting Point

For each $p$ in $[1,\infty)$ let $\mathbf{E}_p$ denote the closure of the region of holomorphy of the Laguerre semigroup $\{M^{\alpha}_t:t>0\}$ on $L^p$ with respect to the Laguerre measure $\mu_{\alpha}$. We prove weak type and strong type estimates for the maximal operator $f\mapsto \sup\{|M^{\alpha}_z f|:z\in \mathbf{E}_p\}$. In particular, we give a new proof for the weak type $1$ estimate for the maximal operator associated to $M^{\alpha}_t$. Our starting point is the well-known relationship between the Laguerre semigroup of half-integer parameter and the Ornstein-Uhlenbeck semigroup.

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 $$T^{\#}_\Omega\big(\,f\big)\big(x\big)=\sup_{\varepsilon&gt;0} \left| \int_{\mathbb{R}^{n}\setminus B(0,\varepsilon)}|y|^{-n} \Omega(y/|y|)\,f(x-y) \mathrm{d}y\right|,$$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 $1&lt;p&lt;\infty ,$ 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 Weak-type (1, 1) estimates for parabolic singular integrals

2011 ◽  
Vol 54 (1) ◽  
pp. 221-247 ◽  
Author(s):  
Shuichi Sato
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Parabolic Singular Integrals

AbstractWe prove weak-type (1, 1) estimates for rough parabolic singular integrals on ℝ2 under the L log L condition on their kernels.

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.

A criterion on oscillation and variation for the commutators of singular integral operators

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Weighted Norm ◽  
Norm Estimates ◽  
Variation Operators

AbstractThis paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted

Weak type estimates for maximal operators with a cylindric distance function

2006 ◽  
Vol 253 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Sunggeum Hong ◽  
Paul Taylor ◽  
Chan Woo Yang
Keyword(s):  
Distance Function ◽  
Weak Type ◽  
Maximal Operators ◽  
Weak Type Estimates
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