Maximal Inequalities for the Best Approximation Operator and Simonenko Indices

Abstract

In an abstract set up, we get strong type inequalities in $L^{p+1}$ by assuming  weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number $p$ is related to Simonenko indices. We apply the results to get strong inequalities for maximal functions associated to best $\Phi$-approximation operatorsin an Orlicz space $L^{\Phi}$.