ANALYSIS IN THEORY AND APPLICATIONS Anal. Theory Appl., Vol. x, No. x (201x), pp. 1-13 Maximal inequalities for the best approximation op- erator and Simonenko indices Sonia Acinas 1,2, ∗ and Sergio Favier 1,3 1 Instituto de Matem´ atica Aplicada San Luis, IMASL, Universidad Nacional de San Luis and CONICET, Ej´ ercito de los Andes 950, D5700HHW San Luis, Argentina. 2 Departamento de Matem´ atica, Facultad de Ciencias Exactas Naturales, Universidad Nacional de La Pampa, L6300CLB Santa Rosa, La Pampa, Argentina. 3 Departamento de Matem´ atica, Universidad Nacional de San Luis, D5700HHW San Luis, Argentina. 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 inequal- ities for maximal functions associated to best Φ-approximation operators in an Orlicz space L Φ . Key Words: Simonenko indices, Maximal Inequalities, Best Approximation. AMS Subject Classifications: 41A10, 41A50, 41A45. 1 Introduction In this paper we denote by I the set of all non decreasing functions ϕ defined for all real number x > 0, such that ϕ( x) > 0 for all x > 0, ϕ(0+)= 0 and lim x→∞ ϕ( x)= ∞. We say that a non decreasing function ϕ : R + 0 → R + 0 satisfies the Δ 2 condition, symbol- ically ϕ ∈ Δ 2 , if there exists a constant Λ ϕ > 0 such that ϕ(2x) ≤ Λ ϕ ϕ( x) for all x ≥ 0. Now, given ϕ ∈I , we consider Φ( x)=  x 0 ϕ(t)dt. Observe that Φ : [0, ∞) → [0, ∞) is a convex function such that Φ( x)= 0 if and only if x = 0. In the literature, a function Φ satisfying the previous conditions is known as a Young function. In addition, as ϕ ∈I we have that Φ is increasing, Φ(x) x →0 as x →0 and Φ(x) x → ∞ as x → ∞. Thus, according to [6], a function Φ with this property is called an N-function. If ϕ ∈I is a right-continuous function that satisfies the Δ 2 condition, then 1 2 ( ϕ( a)+ ϕ(b)) ≤ ϕ( a + b) ≤ Λ ϕ ( ϕ( a)+ ϕ(b)) ∗ Corresponding author. Email addresses: sonia.acinas@gmail.com (S. Acinas), sfavier@unsl.edu.ar (S. Favier). http://www.global-sci.org/ata/ 1 c 201x Global-Science Press