, Volume 23, Issue 1, pp 125–138 | Cite as

Fractional maximal operators with weighted Hausdorff content

  • Hiroki SaitoEmail author
  • Hitoshi Tanaka
  • Toshikazu Watanabe


Let \(n\ge 2\) be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator \({{\mathfrak {M}}}_{\alpha }\) of order \(\alpha \), \(0\le \alpha <n\), on the weighted Choquet–Lorentz space \(L^{p,q}(H^{d}_{w})\), where the weight w is arbitrary and the underlying measure is the weighted d-dimensional Hausdorff content \(H^{d}_{w}\), \(0<d\le n\). Concerning a dependence of two parameters \(\alpha \) and d, we establish a general form of the Fefferman–Stein type inequalities for \({{\mathfrak {M}}}_{\alpha }\). Our results contain the works of Adams (Publ Mat 42:3–66, 1998) and of Orobitg and Verdera (Bull Lond Math Soc 30(2):145–150, 1998) as the special cases. Our results also imply the Tang result (Georgian Math J 18(3):587–596, 2011), if we assume the weight w is in the Muckenhoupt \(A_{1}\)-class.


Choquet integral Choquet–Lorentz space Fefferman–Stein inequality Maximal operator Weighted Hausdorff content 

Mathematics Subject Classification



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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Science and TechnologyNihon UniversityFunabashiJapan
  2. 2.Research and Support Center on Higher Education for the Hearing and Visually ImpairedNational University Corporation Tsukuba University of TechnologyTsukubaJapan
  3. 3.College of Science and TechnologyNihon UniversityTokyoJapan

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