AbstractLet Φ(t) and Ψ(t) be the functions having the following representations Φ(t) = ∫ t0a(s)dsand Ψ(t) = ∫ t0b(s)ds, wherea(s) is a positive continuous function such that ∫ ∞1a(s)/s ds = + ∞ andb(s) is an increasing function such that lims→ ∞b(s) = + ∞. Then the following statements for the Hardy ‐ Littlewood maximal functionM f(x) are equivalent:1(i) there exist positive constantsc1ands0such that1(ii) there exist positi