New Proofs of Two-Weight Norm Inequalities for the Maximal Operator

2000 ◽  
Vol 7 (1) ◽  
pp. 33-42 ◽  
Author(s):  
D. Cruz-Uribe
Keyword(s):  
Maximal Operator ◽  
Sufficient Conditions ◽  
Norm Inequality ◽  
Strong Type ◽  
Norm Inequalities ◽  
Fractional Maximal Operator ◽  
Littlewood Maximal Operator

Abstract We give a new and simpler proof of Sawyer's theorem characterizing the weights governing the two-weight, strong-type norm inequality for the Hardy-Littlewood maximal operator and the fractional maximal operator. As a further application of our techniques, we give new proofs of two sufficient conditions for such weights due to Wheeden and Sawyer.

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.

scholarly journals A note on weighted maximal inequalities

1997 ◽  
Vol 40 (1) ◽  
pp. 193-205
Author(s):  
Qinsheng Lai
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Strong Type ◽  
Maximal Inequalities ◽  
Littlewood Maximal Operator

In this paper, we obtain some characterizations for the weighted weak type (1, q) inequality to hold for the Hardy-Littlewood maximal operator in the case 0<q<1; prove that there is no nontrivial weight satisfying one-weight weak type (p, q) inequalities when 0<p≠q< ∞, and discuss the equivalence between the weak type (p, q) inequality and the strong type (p, q) inequality when p≠q.

scholarly journals On the boundedness of the fractional maximal operator on global Orlicz-Morrey spaces

2021 ◽  
Vol 101 (1) ◽  
pp. 17-24
Author(s):  
N.А. Bokayev ◽  
◽  
А.А. Khairkulova ◽  
Keyword(s):  
Characteristic Function ◽  
Maximal Operator ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Boundedness Condition ◽  
Fractional Maximal Operator

The article deals with the global Orlia-Morrey spaces GMΦ,ϕ,θ(Rn). We find sufficient conditions on pairs of functions (ϕ, η) and (Φ, Ψ), which ensure the boundedness of the fractional maximal operator Mα from GMΦ,ϕ,θ(Rn) in GMΨ,η,θ(Rn). It is proved that under some additional conditions on the function ϕ, the conditions obtained are also necessary. In the proof, the boundedness condition is essentially used, the maximal Hardy-Littlewood functions and the estimate of the norm of the characteristic function in global Orlicz-Morrey spaces are used.

scholarly journals A Weak Type Vector-Valued Inequality for the Modified Hardy–Littlewood Maximal Operator for General Radon Measure on ℝn

2020 ◽  
Vol 8 (1) ◽  
pp. 261-267
Author(s):  
Yoshihiro Sawano
Keyword(s):  
Radon Measure ◽  
Maximal Operator ◽  
Weak Type ◽  
Strong Type ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractThe aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝn. Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

scholarly journals STRONG AND WEAK WEIGHTED NORM INEQUALITIES FOR THE GEOMETRIC FRACTIONAL MAXIMAL OPERATOR

2012 ◽  
Vol 86 (2) ◽  
pp. 205-215
Author(s):  
SORINA BARZA ◽  
CONSTANTIN P. NICULESCU
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Weighted Norm ◽  
Weighted Norm Inequalities ◽  
Lebesgue Spaces ◽  
Norm Inequalities ◽  
Fractional Maximal Operator ◽  
Weighted Lebesgue Spaces

AbstractWe characterise the strong- and weak-type boundedness of the geometric fractional maximal operator between weighted Lebesgue spaces in the case 0<p≤q<∞, generalising and improving some older results.

scholarly journals Two-weight norm inequalities for the rough fractional integrals

2001 ◽  
Vol 25 (8) ◽  
pp. 517-524 ◽  
Author(s):  
Yong Ding ◽  
Chin-Cheng Lin
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Weight Functions ◽  
Norm Inequalities ◽  
Fractional Maximal Operator

The authors give the weighted(Lp,Lq)-boundedness of the rough fractional integral operatorTΩ,αand the fractional maximal operatorMΩ,αwith two different weight functions.

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