Inequalities in Lp−1Lp−1 for the extended LpLp best approximation operator

The best polynomial approximation operator was recently extended by one of the authors from LpLp to Lp−1Lp−1. In this paper, we study weak and strong inequalities for maximal operators related with the extended best polynomial approximation operator. As an application, we obtain norm convergence of the coefficients of the best polynomial approximation.