On the equivalence of weighted inequalities for a class of operators

2011 ◽  
Vol 141 (5) ◽  
pp. 1071-1081 ◽  
Author(s):  
Dah-Chin Luor
Keyword(s):  
Weight Function ◽  
Geometric Mean ◽  
Integral Operators ◽  
Maximal Operators ◽  
Weighted Inequalities

A characterization is obtained on weight function u so that $\smash{T\colon L_{p}^+\mapsto L_{q,u}^+}$ is bounded for 1 < p < ∞ and 0 < q < ∞, where T are integral operators and related maximal operators, and for 0 < p, q < ∞, where T are geometric mean operators and related geometric maximal operators. The equivalence of such weighted inequalities for these operators are established.

On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators

2018 ◽  
Vol 123 (2) ◽  
pp. 273-296
Author(s):  
Dah-Chin Luor
Keyword(s):  
Weight Function ◽  
Geometric Mean ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Maximal Operators ◽  
Averaging Operators ◽  
One Dimensional ◽  
Almost Everywhere ◽  
Carry Over ◽  
Necessary And Sufficient

Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p<q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.

scholarly journals On Some Vector-Valued Inequalities of Gronwall Type

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Dah-Chin Luor
Keyword(s):  
Banach Spaces ◽  
Integral Inequality ◽  
Composition Operators ◽  
Geometric Mean ◽  
Integral Operators ◽  
Maximal Operators ◽  
Vector Valued ◽  
Ordered Banach Spaces

In this paper we established some vector-valued inequalities of Gronwall type in ordered Banach spaces. Our results can be applied to investigate systems of real-valued Gronwall-type inequalities. We also show that the classical Gronwall-Bellman-Bihari integral inequality can be generalized from composition operators to a variety of operators, which include integral operators, maximal operators, geometric mean operators, and geometric maximal operators.

scholarly journals ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN OPERATOR

2008 ◽  
Vol 78 (3) ◽  
pp. 463-475
Author(s):  
DAH-CHIN LUOR
Keyword(s):  
Geometric Mean ◽  
Integral Operators ◽  
Weighted Inequalities ◽  
Averaging Operator ◽  
Work Done

AbstractWe give a characterization of pairs of weights for the validity of weighted inequalities involving certain generalized geometric mean operators generated by some Volterra integral operators, which include the Hardy averaging operator and the Riemann–Liouville integral operators. The estimations of the constants are also discussed. Our results generalize the work done by J. A. Cochran, C.-S. Lee, H. P. Heinig, B. Opic, P. Gurka, and L. Pick.

scholarly journals Entropy bump conditions for fractional maximal and integral operators

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Robert Rahm ◽  
Scott Spencer
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Maximal Operators ◽  
Weighted Inequalities ◽  
Entropy Bounds

AbstractWe investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.

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.

Weighted inequalities for the one-sided geometric maximal operators

2011 ◽  
Vol 284 (11-12) ◽  
pp. 1515-1522 ◽  
Author(s):  
Pedro Ortega Salvador ◽  
Consuelo Ramírez Torreblanca
Keyword(s):  
Maximal Operators ◽  
Weighted Inequalities ◽  
The One

Block Decomposition and Weighted Hausdorff Content

2019 ◽  
Vol 63 (1) ◽  
pp. 141-156
Author(s):  
Hiroki Saito ◽  
Hitoshi Tanaka ◽  
Toshikazu Watanabe
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Maximal Operators ◽  
Block Decomposition ◽  
Fractional Integral Operators

AbstractBlock decomposition of $L^{p}$ spaces with weighted Hausdorff content is established for $0<p\leqslant 1$ and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

scholarly journals Weighted Vector-Valued Inequalities for a Class of Multilinear Singular Integral Operators

2018 ◽  
Vol 61 (2) ◽  
pp. 413-436 ◽  
Author(s):  
Guoen Hu ◽  
Kangwei Li
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Maximal Operators ◽  
Weighted Estimates ◽  
Multilinear Singular Integral ◽  
Multiple Weights ◽  
Vector Valued

AbstractIn this paper, some weighted vector-valued inequalities with multiple weights $A_{\vec P}$ (ℝmn)are established for a class of multilinear singular integral operators. The weighted estimates for the multi(sub)linear maximal operators which control the multilinear singular integral operators are also considered.

scholarly journals Lp estimates for maximal functions along surfaces of revolution on product spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1361-1373 ◽  
Author(s):  
Mohammed Ali ◽  
Musa Reyyashi
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Product Spaces ◽  
Surfaces Of Revolution ◽  
Lp Estimates ◽  
Block Space

Abstract This paper is concerned with establishing Lp estimates for a class of maximal operators associated to surfaces of revolution with kernels in Lq(Sn−1 × Sm−1), q > 1. These estimates are used in extrapolation to obtain the Lp boundedness of the maximal operators and the related singular integral operators when their kernels are in the L(logL)κ(Sn−1 × Sm−1) or in the block space $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$(Sn−1 × Sm−1). Our results substantially improve and extend some known results.

Sign in / Sign up

Export Citation Format

Share Document