Maximal Inequalities for a Best Approximation Operator in Orlicz Spaces

In this paper we study a maximal operator \(\mathcal{M}f\) related with the best \(\varphi\) approximation by constants for a function \(f\in L^{\varphi'}_{\text{loc}}(\mathbb{R}^n)\), where we denote by \(\varphi'\) derivative function of the \(C^1\) convex function \(\varphi\). We get a necessary and sufficient condition which assure strong inequalities of the type \(\int_{\mathbb{R}^n} \theta(\mathcal{M}|f|)dx\leq K \int_{\mathbb{R}^n} \theta(|f|) dx\), 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.