Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Abstract

Let Mg be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b−a), here η is a non-increasing function such that η (0) = 1 and $$\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ2. It gives an A′Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space LΦ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′Φ(g).