Maximal Inequalities for a Best Approximation Operator in Orlicz Spaces

Sergio Favier, and Felipe Zó. "Maximal Inequalities for a Best Approximation Operator in Orlicz Spaces." Commentationes Mathematicae 51.1 (2011): null. <http://eudml.org/doc/291873>.

@article{SergioFavier2011,
abstract = {In this paper we study a maximal operator $\mathcal \{M\}f$ related with the best $\varphi $ approximation by constants for a function $f\in L^\{\varphi ^\{\prime \}\}_\{\text\{loc\}\}(\mathbb \{R\}^n)$, where we denote by $\varphi ^\{\prime \}$ derivative function of the $C^1$ convex function $\varphi $. We get a necessary and sufficient condition which assure strong inequalities of the type $\int _\{\mathbb \{R\}^n\} \theta (\mathcal \{M\}|f|)dx\le K \int _\{\mathbb \{R\}^n\} \theta (|f|) dx$, where $K$ is a constant independent of $f$. Some pointwise and mean convergence results are obtained. In the particular case $\varphi (t) = t^\{p+1\}$ we obtain several equivalent conditions on the functions $\theta $ that assures strong inequalities of this type.},
author = {Sergio Favier, Felipe Zó},
journal = {Commentationes Mathematicae},
keywords = {Best $\varphi $-approximations by constants; extended best approximation operator; maximal inequalities},
language = {eng},
number = {1},
pages = {null},
title = {Maximal Inequalities for a Best Approximation Operator in Orlicz Spaces},
url = {http://eudml.org/doc/291873},
volume = {51},
year = {2011},
}

TY - JOUR
AU - Sergio Favier
AU - Felipe Zó
TI - Maximal Inequalities for a Best Approximation Operator in Orlicz Spaces
JO - Commentationes Mathematicae
PY - 2011
VL - 51
IS - 1
SP - null
AB - In this paper we study a maximal operator $\mathcal {M}f$ related with the best $\varphi $ approximation by constants for a function $f\in L^{\varphi ^{\prime }}_{\text{loc}}(\mathbb {R}^n)$, where we denote by $\varphi ^{\prime }$ derivative function of the $C^1$ convex function $\varphi $. We get a necessary and sufficient condition which assure strong inequalities of the type $\int _{\mathbb {R}^n} \theta (\mathcal {M}|f|)dx\le K \int _{\mathbb {R}^n} \theta (|f|) dx$, where $K$ is a constant independent of $f$. Some pointwise and mean convergence results are obtained. In the particular case $\varphi (t) = t^{p+1}$ we obtain several equivalent conditions on the functions $\theta $ that assures strong inequalities of this type.
LA - eng
KW - Best $\varphi $-approximations by constants; extended best approximation operator; maximal inequalities
UR - http://eudml.org/doc/291873
ER -