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.

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.

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.

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.

A REMARK ON THE REGULARITY OF THE DISCRETE MAXIMAL OPERATOR

2016 ◽  
Vol 95 (1) ◽  
pp. 108-120 ◽  
Author(s):  
FENG LIU
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Sharp Bounds ◽  
Regularity Properties

We study the regularity properties of several classes of discrete maximal operators acting on $\text{BV}(\mathbb{Z})$ functions or $\ell ^{1}(\mathbb{Z})$ functions. We establish sharp bounds and continuity for the derivative of these discrete maximal functions, in both the centred and uncentred versions. As an immediate consequence, we obtain sharp bounds and continuity for the discrete fractional maximal operators from $\ell ^{1}(\mathbb{Z})$ to $\text{BV}(\mathbb{Z})$.

Variable exponent Herz type Besov and Triebel–Lizorkin spaces

2018 ◽  
Vol 25 (1) ◽  
pp. 135-148 ◽  
Author(s):  
Jingshi Xu ◽  
Xiaodi Yang
Keyword(s):  
Maximal Operator ◽  
Variable Exponent ◽  
Maximal Functions ◽  
Herz Spaces ◽  
Vector Valued ◽  
Littlewood Maximal Operator

AbstractWe establish the boundedness of the vector-valued Hardy–Littlewood maximal operator in variable exponent Herz spaces, which were introduced by Samko in [33]. We also introduce variable exponent Herz type Besov and Triebel–Lizorkin spaces and give characterizations of these new spaces by maximal functions.

scholarly journals New Herz Type Besov and Triebel-Lizorkin Spaces with Variable Exponents

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Baohua Dong ◽  
Jingshi Xu
Keyword(s):  
Maximal Operator ◽  
Maximal Functions ◽  
Variable Exponents ◽  
Herz Spaces ◽  
Vector Valued ◽  
Littlewood Maximal Operator

The authors establish the boundedness of vector-valued Hardy-Littlewood maximal operator in Herz spaces with variable exponents. Then new Herz type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, characterizations of these new spaces by maximal functions are given.

Regularity properties of Schrödinger equations in vector-valued spaces and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 149-166
Author(s):  
Veli Shakhmurov
Keyword(s):  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

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 Commutators of the One-Sided Hardy-Littlewood Maximal Functions

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Daiqing Zhang
Keyword(s):  
Sobolev Spaces ◽  
Maximal Functions ◽  
Regularity Properties ◽  
Discrete Analogues ◽  
The One ◽  
Derivatives Of

In this paper, the regularity properties of two classes of commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants are investigated. Some new bounds for the derivatives of the above commutators and the boundedness and continuity for the above commutators on the Sobolev spaces will be presented. The corresponding results for the discrete analogues are also considered.

scholarly journals Weighted Estimates for Rough Maximal Functions with Applications to Angular Integrability

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Feng Liu ◽  
Fangfang Xu
Keyword(s):  
Function Spaces ◽  
Maximal Functions ◽  
Rough Kernels ◽  
Weighted Estimates ◽  
Polynomial Curves ◽  
Definition Of ◽  
Vector Valued Function ◽  
Vector Valued

In this note we establish certain weighted estimates for a class of maximal functions with rough kernels along “polynomial curves” on Rn. As applications, we obtain the bounds of the above operators on the mixed radial-angular spaces, on the vector-valued mixed radial-angular spaces, and on the vector-valued function spaces. Particularly, the above bounds are independent of the coefficients of the polynomials in the definition of the operators.

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