### Extension of the best approximation operator in Orlicz spaces and weak-type inequalities.

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In this paper we study a maximal operator $\mathcal{M}f$ related with the best $\varphi $ approximation by constants for a function $f\in {L}_{\text{loc}}^{{\varphi}^{\text{'}}}\left({\mathbb{R}}^{n}\right)$, where we denote by ${\varphi}^{\text{'}}$ derivative function of the ${C}^{1}$ convex function $\varphi $. We get a necessary and sufficient condition which assure strong inequalities of the type ${\int}_{{\mathbb{R}}^{n}}\theta \left(\mathcal{M}\right|f\left|\right)dx\le K{\int}_{{\mathbb{R}}^{n}}\theta \left(\right|f\left|\right)dx$, where $K$ is a constant independent of $f$. Some pointwise and mean convergence results are obtained. In the particular case $\varphi \left(t\right)={t}^{p+1}$ we obtain several equivalent conditions on the functions $\theta $ that assures strong inequalities of this type....

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