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

Abstract

We consider an extension of the best approximation operator from an Orlicz space $L^{\varphi }$ to the space $L^{{\varphi }^{\prime }}$, where ${\varphi }^{\prime }$ denotes the derivative of $\varphi$, and we prove a weak-type inequality in this space. Further, we obtain some strong inequalities for suitable $L^{\psi }$ spaces.