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Abstract

Necessary and sufficient conditions are shown in order that the inequalities of the form $ϱ({M_μ f > λ})Φ(λ) ≤ C ʃ_X Ψ(C|f(x)|) σ(x)dμ$, or $ϱ({M_μ f > λ}) ≤ C ʃ_X Φ(Cλ^{-1}|f(x)|) σ(x)dμ$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_μ$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.