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

Weighted Lorentz norm inequalities for general maximal operators associated with certain families of Borel measures

1998 ◽  
Vol 128 (2) ◽  
pp. 403-424 ◽  
Author(s):  
María Dolores Sarrión Gavilán
Keyword(s):  
Maximal Operator ◽  
Maximal Operators ◽  
Weighted Inequalities ◽  
General Measure ◽  
Norm Inequalities ◽  
Fractional Maximal Operator ◽  
Borel Measures ◽  
The One ◽  
Littlewood Maximal Operator ◽  
Lorentz Norm

Given a certain family ℱ of positive Borel measures and γ ∈ [0, 1), we define a general onesided maximal operatorand we study weighted inequalities inLp,qspaces for these operators. Our results contain, as particular cases, the characterisation of weighted Lorentz norm inequalities for some well-known one-sided maximal operators such as the one-sided Hardy–Littlewood maximal operator associated with a general measure, the one-sided fractional maximal operatorand the maximal operatorassociated with the Cesèro-α averages.

On the regularity of one-sided fractional maximal functions

2018 ◽  
Vol 68 (5) ◽  
pp. 1097-1112 ◽  
Author(s):  
Feng Liu
Keyword(s):  
Bounded Variation ◽  
Maximal Functions ◽  
Continuous Case ◽  
Discrete Case ◽  
Maximal Operators ◽  
Functions Of Bounded Variation ◽  
Sharp Bounds ◽  
Regularity Properties ◽  
Natural Extensions ◽  
The One

Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators $ \mathcal{M}_{\beta}^{+} $ and $ \mathcal{M}_{\beta}^{-} $ map $ W^{1,p}(\mathbb{R}) $ into $ W^{1,q}(\mathbb{R}) $ with 1 <p <∞, 0≤β<1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators $ M_{\beta}^{+} $ and $ M_{\beta}^{-} $ from $ \ell^{1}(\mathbb{Z}) $ to $ {\rm BV}(\mathbb{Z}) $. Here $ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

A+∞ Condition

1993 ◽  
Vol 45 (6) ◽  
pp. 1231-1244 ◽  
Author(s):  
F. J. Martín-Reyes ◽  
L. Pick ◽  
A. De La Torre
Keyword(s):  
Fractional Integrals ◽  
Maximal Operators ◽  
The One

AbstractThe good weights for the one-sided Hardy-Littlewood operators have been characterized by conditions . In this paper we introduce a new condition which is analogous to A∞. We show several characterizations of . For example, we prove that the class of weights is the union of classes. We also give a new characterization of weights. Finally, as an application of condition, we characterize the weights for one-sided fractional integrals and one-sided fractional maximal operators.

The one-sided dyadic Hardy—Littlewood maximal operator

2014 ◽  
Vol 144 (5) ◽  
pp. 991-1006
Author(s):  
María Lorente ◽  
Francisco J. Martín-Reyes
Keyword(s):  
Maximal Operator ◽  
Weighted Inequalities ◽  
The One ◽  
Littlewood Maximal Operator

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

scholarly journals Regularity of one-sided multilinear fractional maximal functions

2018 ◽  
Vol 16 (1) ◽  
pp. 1556-1572 ◽  
Author(s):  
Feng Liu ◽  
Lei Xu
Keyword(s):  
Bounded Variation ◽  
Maximal Functions ◽  
Continuous Case ◽  
Maximal Operators ◽  
Functions Of Bounded Variation ◽  
Regularity Properties ◽  
Natural Extensions ◽  
The One ◽  
Continuous Setting ◽  
Discrete Setting

AbstractIn this paper we introduce and investigate the regularity properties of one-sided multilinear fractional maximal operators, both in continuous case and in discrete case. In the continuous setting, we prove that the one-sided multilinear fractional maximal operators$\mathfrak{M}_\beta^{+}\; \text{and}\, \mathfrak{M}_\beta^{-}$map W1,p1 (ℝ)×· · ·×W1,pm (ℝ) into W1,q(ℝ) with 1 < p1, … , pm < ∞, 1 ≤ q < ∞ and $1/q= \sum_{i=1}^m1/p_i-\beta$, boundedly and continuously. In the discrete setting, we show that the discrete one-sided multilinear fractional maximal operators are bounded and continuous from ℓ1(ℤ)×· · ·×ℓ1(ℤ) to BV(ℤ). Here BV(ℤ) denotes the set of functions of bounded variation defined on ℤ. Our main results represent significant and natural extensions of what was known previously.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 586-603 ◽  
Author(s):  
Feng Liu ◽  
Huoxiong Wu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Maximal Operators ◽  
One Dimensional ◽  
Regularity Properties ◽  
The One ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

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