Estimates of fractional maximal functions in a quasi-metric space

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Let M_α be the fractional maximal operator in a quasi-metric space X. We will prove that M_α is bounded from the Choquet space L^p (H^η_∞) with respect to the η-Hausdorff capacity H^η_∞ to the Choquet space L^<q, p> (H^δ_∞) of Lorentz type with respect to the δ-Hausdorff capacity for some δ. To prove it, we use the Choquet integrals with respect to Hausdorff capacities and the dyadic balls introduced by E. Sawyer and R.L. Wheeden.

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