Remarks on the Hardy–Littlewood maximal function

1998 ◽  
Vol 128 (1) ◽  
pp. 1-9 ◽  
Author(s):  
J. M. Aldaz
Keyword(s):  
Maximal Function ◽  
Real Line ◽  
Weak Type ◽  
Best Constant ◽  
The Real ◽  
Littlewood Maximal Function

We answer questions of A. Carbery, M. Trinidad Menárguez and F. Soria by proving, firstly, that for the centred Hardy–Littlewood maximal function on the real line, the best constant C for the weak type (1, 1) inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).

A note on the one-dimensional maximal function

1989 ◽  
Vol 111 (3-4) ◽  
pp. 325-328 ◽  
Author(s):  
Antonio Bernal
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Type Inequality ◽  
Best Constant ◽  
Weak Type Inequality ◽  
One Dimensional ◽  
Littlewood Maximal Function ◽  
The One

SynopsisIn this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

The Weak Type (1, 1) Estimates of Maximal Functions on the Laguerre Hypergroup

2010 ◽  
Vol 53 (3) ◽  
pp. 491-502 ◽  
Author(s):  
Jizheng Huang ◽  
Liu Heping
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Maximal Functions ◽  
Laguerre Hypergroup ◽  
Littlewood Maximal Function

AbstractIn this paper, we discuss various maximal functions on the Laguerre hypergroup K including the heat maximal function, the Poisson maximal function, and the Hardy–Littlewood maximal function which is consistent with the structure of hypergroup of K. We shall establish the weak type (1, 1) estimates for these maximal functions. The Lp estimates for p > 1 follow fromthe interpolation. Some applications are included.

Weighted inequalities for a maximal function on the real line

2001 ◽  
Vol 131 (2) ◽  
pp. 267-277 ◽  
Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes
Keyword(s):  
Maximal Function ◽  
Real Line ◽  
Maximal Operator ◽  
Singular Integrals ◽  
Weak Type ◽  
Weighted Inequalities ◽  
Strong Type ◽  
The Real ◽  
Image Position ◽  
Cesaro Convergence

We consider the maximal operator defined on the real line by which is related to the Cesàro convergence of the singular integrals. We characterize the weights w for which Mα is of weak type, strong type and restricted weak type (p, p) with respect to the measure w(x) dx.

scholarly journals Hardy-Littlewood maximal functions on some solvable Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 78-82 ◽  
Author(s):  
G. Gaudry ◽  
S. Giulini ◽  
A. Hulanicki ◽  
A. M. Mantero
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Weak Type ◽  
Maximal Functions ◽  
Solvable Lie Groups ◽  
Littlewood Maximal Function ◽  
The Family ◽  
Fixed Power ◽  
Simply Connected

AbstractLet N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

scholarly journals Dunkl Translation and Uncentered Maximal Operator on the Real Line

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Chokri Abdelkefi ◽  
Mohamed Sifi
Keyword(s):  
Characteristic Function ◽  
Real Line ◽  
Maximal Operator ◽  
Weak Type ◽  
Dunkl Operator ◽  
The Real

We establish estimates of the Dunkl translation of the characteristic functionχ[−ɛ,ɛ],ɛ>0, and we prove that the uncentered maximal operator associated with the Dunkl operator is of weak type(1,1). As a consequence, we obtain theLp-boundedness of this operator for1<p≤+∞.

Bounds for the best constant in Landau/s inequality on the line

1983 ◽  
Vol 95 (3-4) ◽  
pp. 257-262 ◽  
Author(s):  
Z. M. Franco ◽  
Hans G. Kaper ◽  
Man Kam Kwong ◽  
A. Zettl
Keyword(s):  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Analyse Math ◽  
Best Constant ◽  
The Real ◽  
Explicit Formulae

SynopsisExplicit formulae and numerical values for upper and lower bounds for the best constant in Landau/s inequality on the real line are given. For p > 3, the value of the upper bound is less than the value of the best constant conjectured by Gindler and Goldstein (J. Analyse Math. 28 (1975), 213–238).

The Mehler Maximal Function: a Geometric Proof of the Weak Type 1

2000 ◽  
Vol 61 (3) ◽  
pp. 846-856 ◽  
Author(s):  
Trinidad Menárguez ◽  
Sonsoles Pérez ◽  
Fernando Soria
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Geometric Proof

scholarly journals Fefferman–Stein Inequalities for the Hardy–Littlewood Maximal Function on the Infinite Rooted k-ary Tree

2020 ◽  
Author(s):  
Sheldy Ombrosi ◽  
Israel P Rivera-Ríos ◽  
Martín D Safe
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Type Estimate ◽  
Abstract Theorem ◽  
Littlewood Maximal Function ◽  
Infinite Trees ◽  
Vector Valued

Abstract In this paper, weighted endpoint estimates for the Hardy–Littlewood maximal function on the infinite rooted $k$-ary tree are provided. Motivated by Naor and Tao [ 23], the following Fefferman–Stein estimate $$\begin{align*}& w\left(\left\{ x\in T\,:\,Mf(x)&gt;\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}\: \text{d}x\qquad s&gt;1\end{align*}$$is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if $s=1$. Some examples of nontrivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman–Stein-type estimate and as a consequence some vector-valued extensions are obtained. In the appendix, a weighted counterpart of the abstract theorem of Soria and Tradacete [ 38] on infinite trees is established.

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