scholarly journals A weak type estimate for rough singular integrals

2019 ◽  
Vol 35 (5) ◽  
pp. 1583-1602 ◽  
Author(s):  
Andrei Lerner
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Type Estimate

scholarly journals On weighted weak type norm inequalities for one-sided oscillatory singular integrals

2011 ◽  
Vol 207 (2) ◽  
pp. 137-151 ◽  
Author(s):  
Zunwei Fu ◽  
Shanzhen Lu ◽  
Shuichi Sato ◽  
Shaoguang Shi
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Norm Inequalities ◽  
Oscillatory Singular Integrals

Weighted inequalities for some integral operators with rough kernels

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
María Riveros ◽  
Marta Urciuolo
Keyword(s):  
Weak Type ◽  
Integral Operators ◽  
Degree Zero ◽  
Weighted Inequalities ◽  
Rough Kernels ◽  
Type Estimate ◽  
Dini Condition ◽  
Weighted Bmo ◽  
Weak Type Estimates ◽  
Invertible Matrices

AbstractIn this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

scholarly journals Weighted weak type (1,1) estimates for singular integrals and Littlewood–Paley functions

2004 ◽  
Vol 163 (2) ◽  
pp. 119-136 ◽  
Author(s):  
Dashan Fan ◽  
Shuichi Sato
Keyword(s):  
Singular Integrals ◽  
Weak Type

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 Two weighted ineqalities for maximal functions related to Cesàro convergence

2003 ◽  
Vol 74 (1) ◽  
pp. 111-120
Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes
Keyword(s):  
Maximal Operator ◽  
Singular Integrals ◽  
Weak Type ◽  
Maximal Functions ◽  
Cesaro Convergence

AbstractWe characterize the pairs of weights (u, v) for which the maximal operator is of weak and restricted weak type (p, p) with respect to u(x)dx and v(x)dx. As a consequence we obtain analogous results for We apply the results to the study of the Cesàro-α convergence of singular integrals.

scholarly journals On a counterexample related to weighted weak type estimates for singular integrals

2017 ◽  
Vol 145 (7) ◽  
pp. 3005-3012 ◽  
Author(s):  
Marcela Caldarelli ◽  
Andrei K. Lerner ◽  
Sheldy Ombrosi
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Weak Type Estimates

scholarly journals Borderline weak-type estimates for singular integrals and square functions

2015 ◽  
Vol 48 (1) ◽  
pp. 63-73 ◽  
Author(s):  
Carlos Domingo-Salazar ◽  
Michael Lacey ◽  
Guillermo Rey
Keyword(s):  
Singular Integrals ◽  
Weak Type ◽  
Square Functions ◽  
Weak Type Estimates
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