Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by $$M_g fleft( x right) = sup frac{{int_a^b {fleft( y right)gleft( y right){text{d}}y} }}{{int_a^b {gleft( y right){text{d}}y} }}$$ , 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 $$uleft( {left{ {x in {text{R}}left| {M_g fleft( x right) >lambda } right.} right}} right) leqslant frac{C}{{Phi left( lambda right)}}int_{text{R}} {Phi left( {left| f right|v} right)} $$ holds for every f in the Orlicz space L_Φ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_Φ(g).