Several weak-type weighted inequalities in orlicz martingale classes

The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ(Mf > λ) φ(λ) ≤ C ∫Ω ψ(C|f|)σdμ, ∀ λ > 0ρ(Mf > λ) φ(λ) ≤ C ∫Ω ψ(C|f|)σdμ, ∀ λ > 0or ρ(Mf > λ) ≤ C∫Ω φ(Cλ−1|f|)σdμ, ∀ λ > 0ρ(Mf > λ) ≤ C∫Ω φ(Cλ−1|f|)σdμ, ∀ λ > 0holds for every uniformly integral martingale f = (fn), where M is the Doob's maximal operator, Φ, Ψ are both Φ-functions, and σ are weights.