scholarly journals Two weighted ineqalities for maximal functions related to Cesàro convergence

2003 ◽  
Vol 74 (1) ◽  
pp. 111-120
Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes
Keyword(s):  
Maximal Operator ◽  
Singular Integrals ◽  
Weak Type ◽  
Maximal Functions ◽  
Cesaro Convergence

AbstractWe characterize the pairs of weights (u, v) for which the maximal operator is of weak and restricted weak type (p, p) with respect to u(x)dx and v(x)dx. As a consequence we obtain analogous results for We apply the results to the study of the Cesàro-α convergence of singular integrals.

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 On the lack of dimension-free estimates in Lp for maximal functions associated to radial measures

2010 ◽  
Vol 140 (3) ◽  
pp. 541-552 ◽  
Author(s):  
Alberto Criado
Keyword(s):  
Gaussian Measure ◽  
Maximal Operator ◽  
Recent Article ◽  
Maximal Functions

In a recent article Aldaz proved that the weak L1 bounds for the centred maximal operator associated to finite radial measures cannot be taken independently with respect to the dimension. We show that the same result holds for the Lp bounds of such measures with decreasing densities, at least for small p near to one. We also give some concrete examples, including the Gaussian measure, where better estimates with respect to the general case are obtained.

scholarly journals Weak type estimates for the Hardy-Littlewood maximal functions

1974 ◽  
Vol 49 (3) ◽  
pp. 217-223 ◽  
Author(s):  
Luis Caffarelli ◽  
Calixto Calderón
Keyword(s):  
Weak Type ◽  
Maximal Functions ◽  
Weak Type Estimates

Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems

2004 ◽  
Vol 11 (3) ◽  
pp. 467-478
Author(s):  
György Gát
Keyword(s):  
Maximal Operator ◽  
Weak Type ◽  
Integrable Function ◽  
Two Dimensional ◽  
Vilenkin Groups ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Marcinkiewicz Means

Abstract We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.

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.

Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators

2020 ◽  
pp. 1-34
Author(s):  
Zhang Chao ◽  
José L. Torrea
Keyword(s):  
Maximal Operator ◽  
Singular Integrals ◽  
Heat Semigroup ◽  
Real Sequence ◽  
Differential Transform ◽  
Intermediate Size ◽  
Heat Semigroups ◽  
Image Position ◽  
Reverse Hölder ◽  
Littlewood Maximal Operator

Abstract In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$ , $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}<N_{2}$ , ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$ , of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$ . It is also shown that the local size of the maximal differential transform operators (with $V=0$ ) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$ , we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.

scholarly journals Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Canqin Tang ◽  
Qing Wu ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Type Inequality ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

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