Necessary and Sufficient Criteria for a Four-Weight Weak-Type Maximal Inequality in the Orlicz Class

Abstract

Let Φi(i=1,2) be two N-functions, f be a μ-measurable function, and ωi(i=1,2,3,4) be four weight functions. This study presents necessary and sufficient conditions for weight functions (ω1,ω2,ω3,ω4) such that the inequality ∫{x:Mf(x)>λ}Φ1(λω1(x))ω2(x)dμ(x)≤c1∫XΦ2(c1|f(x)|ω3(x))ω4(x)dμ(x) holds, which extends several established results.