Weighted Reverse Weak Type Inequalities for the Ergodic Maximal Function and the Classes L log + L

1985 ◽  
Vol 95 (2) ◽  
pp. 275
Author(s):  
Kenneth F. Andersen ◽  
Wo-Sang Young
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
L Log L

L(log L) spaces and weights for the strong maximal function

1984 ◽  
Vol 44 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Richard J. Bagby ◽  
Douglas S. Kurtz
Keyword(s):  
Maximal Function ◽  
L Log L

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.

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).

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

2020 ◽  
pp. 1-25
Author(s):  
HONG CHUONG DOAN
Keyword(s):  
Heat Kernel ◽  
Maximal Function ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Weak Type ◽  
Adjoint Operator ◽  
Beltrami Operator ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Doubling Condition

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$ . Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

scholarly journals FAILURE OF THE POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP

2015 ◽  
Vol 3 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Free Group ◽  
Weak Type ◽  
Markov Operator ◽  
Finite Measure ◽  
Maximal Inequality ◽  
Ergodic Theorems ◽  
Averaging Operators ◽  
Finite Measure Space ◽  
Image Position ◽  
L Log L

Let $F_{2}$ denote the free group on two generators $a$ and $b$. For any measure-preserving system $(X,{\mathcal{X}},{\it\mu},(T_{g})_{g\in F_{2}})$ on a finite measure space $X=(X,{\mathcal{X}},{\it\mu})$, any $f\in L^{1}(X)$, and any $n\geqslant 1$, define the averaging operators $$\begin{eqnarray}\displaystyle {\mathcal{A}}_{n}f(x):=\frac{1}{4\times 3^{n-1}}\mathop{\sum }_{g\in F_{2}:|g|=n}f(T_{g}^{-1}x), & & \displaystyle \nonumber\end{eqnarray}$$ where $|g|$ denotes the word length of $g$. We give an example of a measure-preserving system $X$ and an $f\in L^{1}(X)$ such that the sequence ${\mathcal{A}}_{n}f(x)$ is unbounded in $n$ for almost every $x$, thus showing that the pointwise and maximal ergodic theorems do not hold in $L^{1}$ for actions of $F_{2}$. This is despite the results of Nevo–Stein and Bufetov, who establish pointwise and maximal ergodic theorems in $L^{p}$ for $p>1$ and for $L\log L$ respectively, as well as an estimate of Naor and the author establishing a weak-type $(1,1)$ maximal inequality for the action on $\ell ^{1}(F_{2})$. Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.

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