Sharp one-weight and two-weight bounds for maximal operators

Volume 194 / 2009

Kabe Moen Studia Mathematica 194 (2009), 163-180 MSC: Primary 42B25. DOI: 10.4064/sm194-2-4

Abstract

We investigate the boundedness of the fractional maximal operator with respect to a general basis on weighted Lebesgue spaces. We characterize the boundedness of these operators for one-weight and two-weight inequalities extending the work of Jawerth. A new two-weight testing condition for the fractional maximal operator on a general basis is introduced extending the work of Sawyer for the basis of cubes. We also find the sharp dependence in the two-weight case between the operator norm and the testing condition of Sawyer. Finally, our approach leads to a new proof of Buckley's sharp estimate for the Hardy–Littlewood maximal function.

Authors

  • Kabe MoenDepartment of Mathematics
    University of Kansas
    405 Snow Hall
    1460 Jayhawk Blvd.
    Lawrence, KS 66045-7523, U.S.A.
    e-mail

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