On maximal inequalities arising in best approximation

Abstract

Let f be a function in an Orlicz space L^Φ and μ(f,la) be the set all the best Φ-approximants to f, given a $sigma-σ−−lattice L. Weak type inequalities are proved for the maximal operator f∗ = supn |fn|, where fn is any selection of functions in µ(f, Ln), and Ln is an increasing sequence of σ-lattices. Strong inequalities are proved in an abstract set up which can be used for an operator as f ∗ .